5

Metallic transport without quasiparticles

Data depicting an order of magnitude violation of the Wiedemann-Franz law observed in charge neutral graphene. This law, relating thermal and electrical transport, is respected by all conventional metals. This breakdown is evidence for hydrodynamic electron flow, discussed in this chapter, and is one of the simplest examples of metallic transport without quasiparticles. Image taken from [159] with permission.

5.1 Metallic transport with quasiparticles

We begin this section with a historical perspective on metallic transport in metals with quasiparticle excitations. Essentially all of the condensed matter literature on quantum transport in metals is built on a theory of the scattering of quasiparticles formulated using the quantum Boltzmann equation, and its Baym-Kadanoff-Keldysh extensions [451; 452].

In the simplest and most common cases, the scattering of quasiparticles is dominated by elastic scattering off impurities in the crystal. We can then define an impurity mean-free path, qp:imp, and a corresponding impurity scattering time τqp:imp, related by qp:imp = vFτqp:imp, where vF is the Fermi velocity (the cumbersome notation is needed to distinguish other impurity-related times and lengths we define in subsequent subsections). It is assumed that the elastic impurity scattering time is much shorter than the quasiparticle lifetime from inelastic processes, including those due to electron-phonon (τqp:phonon) and electron-electron (τqp:ee) interactions

Such a theory of quasiparticle transport has a number of well-known characteristics, amply verified by experimental observations:

1. The low temperature resistivity ρ = ρ0 + AT2, where ρ0 is the impurity-induced ‘residual’ resistivity at zero temperature, and A arises from electron-electron interactions which relax momentum, often including umklapp processes.

2. The ratio of the electrical, σ, and thermal, κ, conductivities obeys the Wiedemann-Franz law

3. Identifying a quasiparticle requires a length scale larger than the typical wavelength of a quasiparticle excitation . The inequality, kFqp:imp > 1, then leads to the proposal of the Mott-Ioffe-Regel lower bound on the conductivity of a metal [334]

We will return to this bound in §5.8.

5.2 The momentum bottleneck

Our interest here is in metals without quasiparticles where, as we will describe below, all three of the quasiparticle transport characteristics highlighted above do not apply. But it is instructive to first discuss their breakdown in a situation which arises already within the quasiparticle framework.

Consider a clean metal where τqp:imp is very long, and the dominant scattering of electronic quasiparticles is off the phonons [778]. Indeed, just such a situation was considered in the classic work by Bloch [108; 109] in 1929, in which he derived ‘Bloch’s law’, stating that at low T the resistivity of metals from electron-phonon scattering varies with temperature as Td+2. However, in 1930, Peierls wrote insightful papers [624; 625] pointing out a crucial error in Bloch’s argument. In the electron-phonon scattering event, the momentum carried by the quasiparticles is deposited into the phonons, and the total momentum of the electron+phonon system is conserved (umklapp events freeze out at low T). So after multiple scattering events, the combined electron+phonon system will reach a state of maximum entropy consistent with the conservation of total momentum. In a generic state of quantum matter at non-zero density, the two-point correlator between the momentum and current is non-zero (although, such a correlator vanishes in CFTs), and consequently such an equilibrated state has a non-zero electrical current, proportional to the initial total momentum. So the electrical current does not decay to zero, and the resistivity vanishes. This is the ‘phonon drag’ effect, in which the mobile quasiparticles drag the phonons along with them, and the combined system then flows without decay of the electrical current to zero.

In practice, however, it is found that Bloch’s law works well in most metals, and Peierls’ phonon drag has turned out to be difficult to observe. This is due to a combination of two factors: (i) the electron-phonon coupling is weak, and so it takes a long time for the electron-phonon system to reach local thermal equilibrium; (ii) impurities are invariably present in experiments, and they can absorb the momentum deposited into the phonons before the electron-phonon system has equilibrated. We need metals with exceptional purity, and with appreciable electron-phonon (or electron-electron) interactions, to observe the long-lived flow of electrical current in a thermally equilibrated electronic system with momentum conservation. Only recently have such experiments become possible [188; 144; 67; 582].

We turn, finally, to metals without quasiparticle excitations. We can imagine reaching such a state by turning up the strength of the electron-electron interactions in a quasiparticle system, and so reducing the value of τqp:ee. In a moderately clean sample, (5.1) will be violated before the quasiparticles become ill-defined. Consequently, the ‘drag’ mechanism applies forcefully to metals without quasiparticles, and we should describe transport in terms of a quantum fluid which has locally thermally equilibrated. A number of well-known computations of non-Fermi liquid transport [514; 429; 428; 190] use a Boltzmann-Baym-Kadanoff-Keldysh framework to describe scattering of charged excitations around a Fermi surface by their strong coupling to a neutral bosonic excitation, and then relate this scattering rate to the transport co-efficients. However, these computations ignore the fact that the bosonic excitations will rapidly equilibrate with the charged fermionic excitations in a short time of order τeq 1/T, as we discussed in §1.1, and so yield incorrect results for the transport coeffecients for the models studied [568].

The holographic perspective on strange metals has the advantage of naturally building in the momentum bottleneck. Such a formulation of transport makes no direct reference to the Fermi surface, or quasiparticles of any kind, and instead describes a thermally equilibrated fluid of fermions and bosons which obeys all the important global conservation laws. There are important connections between the holographic approach to such liquids, and more traditional hydrodynamic and memory function approaches. We will review these connections below, and show that such methods lead to a unified theory of metals without quasiparticles with mutual consistency in overlapping regimes of validity. We also note that hydrodynamic perspectives on electronic transport have been suggested for ultra-clean metals with well-defined quasiparticles [338; 339; 708; 43]. More recently, the conventional theory of transport was unified with a hydrodynamic theory of quasiparticle transport through a microscopic solution of the Boltzmann equations [541].

A further possibility afforded by non-quasiparticle transport is that momentum may also be strongly degraded. This occurs if the effects of disorder or lattice umklapp scattering are large. With quasiparticles, this readily results in localization. Without quasiparticles, one seems more likely to enter an incoherent metallic regime instead (see §5.8). Incoherent metals are again locally equilibrated and hence hydrodynamic, but there is no (approximate) sound mode and hence the only collective motion of charge is diffusive. Incoherent metals can have large resistivities, in contrast to ‘momentum drag’ transport in which the conductivity is large. They are thus a natural framework for describing ‘bad metals’ that violate the Mott-Ioffe-Regel bound (5.3), as discussed in [352]. Incoherent metals appear readily in holographic theories, and will be discussed in §5.9 and §5.10; explicit quantum matter realizations of incoherent metals are harder to find, and are present in the SYK models discussed in §5.11.

Figure 5.1 summarizes how the various regimes discussed in the previous paragraphs are related to each other.

Figure 5.1
A ‘phase space’ of metals. In this section we will discuss the coherent and incoherent non-quasiparticle metals that appear on the left column. Figure developed in discussion with Aharon Kapitulnik.

In the first half of the remaining section, we will detail the major developments in the theory of transport in strongly interacting metals with weak momentum relaxation. The latter part of this section consists of holographic approaches to transport that are non-perturbative in the strength of momentum relaxation. We postpone our review of non-holographic condensed matter models to §5.6.3 and §5.11, until after we have had a chance to lay out the formalisms which are relevant for the discussion.

5.3 Thermoelectric conductivity matrix

Electric and thermal transport generally couple together in charged quantum matter. (We emphasize that in this section the label ’charged’ merely implies a non-zero density away from a particle-hole symmetric point; it does not imply the presence of long-range Coulomb interactions.) Hence, we will want to compute not just the electrical conductivity σ, as in Chapter 3, but a more general matrix of thermoelectric conductivities. We will need to consider the transport coefficients

The heat current naturally couples to a temperature gradient,

as we will demonstrate below, following [350]. The classic discussion can be found in [550]. The matrix of thermoelectric conductivities defined above is guaranteed to be symmetric, assuming time-reversal symmetry – this is called Onsager reciprocity.

We want to impose a uniform electric field δEi, and a uniform temperature gradient δζi. For simplicity, we suppose that the field theory lives on flat space. An electric field is imposed through an external gauge field

Imposing a temperature gradient is more subtle. A clean way to do this is to think about Euclidean time in the rescaled coordinate

This Euclidean coordinate has periodicity

Imposing a constant temperature gradient δζi therefore leads to

. We now make a coordinate change

Thus the coordinate change (5.11) leads to the following set of perturbations describing the temperature gradient:

Let us now collect our results and rescale back to the original time coordinate t. If we perturb the metric and gauge field, using (5.7) and (5.13) the action of the field theory is deformed to

5.4). In holography, we have also learned how to encode a thermal gradient by perturbing gti and Ai at the boundary. For simplicity assuming that the boundary is asymptotically AdS:

5.14) implies that the thermoelectric conductivities (5.4) are given by retarded Green’s functions of the currents Ji and Qi. Recall that these Green’s functions relate sources and expectation values, as in e.g. (1.38) above. The appearance of time derivatives (factors of ω) in (5.14) means that some care is required in the manipulation of the Green’s functions. The answer is:

The susceptibilities χAB are equal to the nonzero temperature Euclidean Green’s function at Matsubara frequency ωn = 0. The susceptibilities in the above expressions are often zero by gauge invariance at T > 0 (because they would generate mass terms if the theory is coupled to dynamical photons or gravitons). We will return to these quantities later. The derivation of (5.16b) and (5.16d) from (5.14) can be found in [550]. In [550] the answer is given in terms of Kubo functions. We will relate these to the quantities appearing in (5.16b) and (5.16d) later.

There is a zoo of thermoelectric coefficients used in the condensed matter literature. All such coefficients are related to the matrices defined in (5.4), and are defined by changing the “boundary conditions” (instead of measuring Ji and Qi given Ei and ζi, we choose to fix Ji and ζi, for example). Let us note two famous ones for convenience. The thermal conductivity at vanishing electric current is:

Observables with Ji = 0 arise naturally in experimental situations that typically work with open circuit boundary conditions. Charge accumulates at one end of the sample and the resulting electric field precisely cancels out any net electric current.

5.4 Hydrodynamic transport (with momentum)

Before discussing holographic transport, it is important to understand what features of holography are genuinely novel, and which features are already expected on general field theoretic grounds. In fact, we will see that the transport problem is tightly constrained by hydrodynamics.

Hydrodynamics is the effective theory describing the relaxation of an interacting classical or quantum system towards thermal equilibrium. The key assumption of hydrodynamics is that the field theory has locally reached thermal equilibrium. In thermal equilibrium, we assign to a QFT with conserved charge and energy-momentum a chemical potential μ, and a temperature T and four-velocity uμ, defined such that the local density matrix ρ is “approximately”

with Pμ the total energy-momentum in a volume of linear size lth, where lth is the “thermalization length scale”, and Q the charge. More precisely, (5.19) is true so long as we only ask for the expectation value of products of local operators. The equations of hydrodynamics describe the dissipative dynamics under which a theory with long wavelength inhomogeneity in μ, T and uμ slowly relaxes to global equilibrium (or as close to it as boundary conditions allow).

Hydrodynamics characterizes the dynamics perturbatively in powers of lth, where ξ is the scale over which there is spatial variation. We will focus, to start with, on the case of relativistic dynamics, and so after trivial multiplication by the (effective) speed of light c the same considerations apply to time scales: the local thermalization time is fast compared to the scale of hydrodynamic phenomena. The hydrodynamic expansion is achieved by expanding Tμν and Jμ as functions of T, μ and uμ, and their derivatives. We will perform this task explicitly in the next subsection. But beforehand let us emphasize that the hydrodynamic limit is parametrically opposite to the standard limit of quantum field theory computations – ξ/lth crudely counts the “number of collisions” (in a quasiparticle framework), which apparently must be large for the hydrodynamic limit to be sensible. Hence, recovery of hydrodynamics from quasiparticle approaches such as kinetic theory requires some conceptual ‘care’, although it is well-known how to do so [53; 52]. In kinetic theory, dissipative coefficients such as diffusion and viscosity are non-perturbative in the coupling constant λ, typically scaling as 12 as λ → 0.

5.4.1 Relativistic hydrodynamics near quantum criticality

We start by considering relativistic hydrodynamics. This is constrained by additional symmetries, and is directly applicable to a CFT deformed by temperature and chemical potential. As we have discussed, this is a natural framework for holography and for compressible matter more generally.

The equations of motion of hydrodynamics read

where Tμν is the expectation value of the local stress tensor, Jμ is the charge current, and Fμν is an external electromagnetic field tensor. Our goal is to construct Tμν and Jμ for a relativistic theory, following [364].

The expansion in lth will be an expansion in derivatives μ. So we begin by constructing Tμν and Jμ at zeroth order in derivatives. The most general possible answer is

We were only able to use ημν and uμ to construct these tensors, since Fμν = μAννAμ is first order in derivatives. In the rest frame of the fluid, we readily interpret 𝜖 as the energy density, P as the pressure, and ρ as the charge density.

There is a further conservation law at this order in derivatives:

and s is the entropy density. We can derive (5.22) by using that at leading order in derivatives, taking the divergence of (5.21a),

Further employing (5.20b), (5.21b), along with the thermodynamic identity (4.4), we obtain (5.22). (5.22) asserts that entropy is conserved at leading order, and is the second law of thermodynamics for a non-dissipative system. At higher orders in hydrodynamics, we will require that a certain entropy current satisfy

so that the fluid is dissipative, and entropy increases over time locally.

The positivity constraints coming from (5.26) are important. However, the need to construct an entropy current is a weakness of the conventional formulation of hydrodynamics, as identifying the correct entropy current can be subtle. While we will follow the conventional description [505], the reader may be interested in more modern treatments that dispense with (5.26) as an additional postulate [442; 69; 340; 158].

The next step is to construct the first order derivative corrections to Tμν and Jμ. A priori, we have to add terms containing all possible combinations of μμ, μT and μuν. However, we immediately run into an ambiguity with how to define the fluid variables in an inhomogeneous (out of equilibrium) background [496]. This is the issue of picking a “fluid frame”: a simultaneous re-definition of uμ, T and μ at first order in derivatives is allowed. We will work in the Landau frame, which chooses

are the first order in derivative terms in the conserved currents.

Now, we use the equations of motion to construct the most general 5.22), but now include first order corrections. We find

The entropy current at first order in derivatives is the quantity appearing on the left hand side of the above equation,

This is just the heat current, up to a factor of T: Qμ = Tsμ. With this identification of sμ, the most general form of 5.29),

η and ζ are the shear and bulk viscosity of the fluid. σQ is a “quantum critical conductivity” which plays a very important role in transport [364], as we will see. In the zero density limit, σQ reduces to the “relativistic” conductivity described in Chapter 3. The equations (5.31) are called the first order constitutive relations. The zeroth order constitutive relations were (5.21). Combining the constitutive relations with the conservation laws (5.20) leads to a set of dynamical equations for the conserved densities 𝜖, ρ, uμ.

As we have emphasized previously, the black hole geometries employed in holography are dual to field theories with consistent thermodynamics. Therefore, it must be that when these black holes are perturbed on very long wavelengths, hydrodynamic behavior arises. In fact, there is a mapping between solutions of hydrodynamics and those of charged black branes, perturbed on very long length scales. We will not discuss this “fluid-gravity correspondence” further. For original papers see [92; 68; 256], and for a review see [420].

5.4.2 Sound waves

With the hydrodynamic equations set up, the next step is to solve the equations and find the hydrodynamic modes. We got a taste of this in §3.4.3, where we found a diffusive mode carrying charge fluctuations in an overall neutral system. What, however, are the hydrodynamic modes of a charged fluid? The most important feature of charged hydrodynamics in a medium with a conserved momentum is the presence of sound waves that carry charge. We will see that these modes have a huge effect on charge transport.

To find the collective modes of linear transport, we linearize the hydrodynamic equations about a fluid at rest:

Following [496], we have used δ𝜖 and δρ as hydrodynamic variables instead of δμ and δT. Assuming that δvi, δ𝜖 and δρ have x and t dependence given by ei(kxωt), it is straightforward to use the conservation laws (5.20) and the constitutive relations (5.21) and (5.31) to obtain a set of algebraic equations. The condition that these equations admit non-vanishing solutions defines the hydrodynamic normal modes.

The hydrodynamic modes are easily found. The modes decompose into transverse and longitudinal sectors. The perpendicular components of the velocity field obey a diffusion equation with dispersion relation

That is, the shear viscosity controls transverse momentum diffusion. The longitudinal modes are as follows: there is a diffusive mode with with dispersion relation ω = −iDk2, with diffusivity

This is the generalization of the diffusive mode derived in §3.4.3. This ‘incoherent’ mode does not carry pressure or momentum and is discussed in some detail in [178]. Finally, there is a sound mode

and attenuation constant

The sound mode carries momentum, heat and charge.

In holographic models, the sound modes can be found explicitly by perturbing the background black hole. The general computational strategy is identical to that in §3.4.3 above, in which we found the diffusive mode of a neutral black hole. The sound modes are also low lying quasinormal modes of the black hole. The technical difficulty is that there are now more bulk fields that are coupled and it is more difficult to find gauge-invariant variables analogous to (3.45) that can lead to (ideally decoupled) second order differential equations. It is often useful to fix the gauge δgrM = 0 and δAr = 0. To find the diffusive momentum mode (5.35), one must then solve for the bulk metric perturbations δgty and δgxy. For the remaining longitudinal modes one must couple δAt, δAx, δgii, δgxx, δgtx, δgtt, and any further scalar modes such as a dilaton.

Once the fluctuation equations have been obtained, the procedure is the same as in §3.4.3. That is, one has to integrate the equations from the horizon to the boundary and demand the absence of a source at the boundary. This integration can be done perturbatively in ω, k → 0, with ω/k fixed. The analysis was done for neutral AdS-Schwarzschild black holes in [639; 387; 500] and for charged AdS-RN black holes in [241; 180]. Modes are found at precisely the frequencies (5.37) predicted by hydroynamics. Decoupled gauge-invariant equations of motion for both longitudinal and transverse channels have also been found for some Einstein-Maxwell-dilaton theories in [38], although the sound modes have not been studied.

Hydrodynamics requires ω, k T in order for the derivative expansion to hold. Perhaps surprisingly, holographic sound modes were found to exist even at T = 0 in the AdS-RN black hole [181], and obey the expected dispersion relations. While reminiscent of ‘zero sound’ modes in Fermi liquids, the raison d’être of these gapless collective modes is not clear at the time of writing. A linearly dispersing collective mode at zero temperature has also been found in an EMD model without a ground state entropy, although still with z = ∞ [179]. It would be interesting to see if the existence of these modes could be directly tied to the presence of low energy, nonzero momentum spectral weight when z = ∞. The two properties (spectral weight and zero sound modes) are tied together in weakly interacting Fermi surfaces. We shall come across similar modes in §7.1.3 below on probe brane models.

5.4.3 Transport coefficients

We are now almost ready to use hydrodynamics to compute the transport coefficients {σ, α, α, κ} defined in (5.4). However, there is an immediate problem which arises: at nonzero ρ and s, the transport coefficients are all divergent as ω → 0 in any fluid with translation invariance! We shall prove this result in due course. To get a feel for where it comes from, consider the following easy way to generate nonzero currents Ji and Qi: start with a fluid at rest, uμ = (1, 0, …), and perform a small ‘Galilean’ boost of velocity vx. Going to a moving frame leads to electric and heat currents Jx = ρvx and Qx = sTvx, respectively. But there is no temperature gradient and no electric field. Evidently, all of the coefficients in (5.4) are divergent in any direction with translation invariance. This is not inconsistent with our results for σ from Chapter 3, because for those systems ρ = 0. While it might appear from this argument that boost rather than translation invariance is the symmetry at work here, we will see shortly that this is not the case. Boost invariance gives extra structure: it fixes the coefficients of the delta functions in equations (5.43) – (5.46) below.

Nonetheless, let us now compute σ(ω) at ω > 0 from hydrodynamics; for simplicity, we assume the theory is isotropic. We shall do this first by directly solving the hydrodynamic equations of motion in the presence of a uniform electric field and temperature gradient. We will freely use the conservation equations (5.20) and constitutive relations (5.21) and (5.31). Consider first a uniform electric field E, so Fxt = −E, which is infinitesimally small. This will induce a perturbatively small velocity field v, and hence a perturbatively small momentum current Ttx = (𝜖 + P)v. Integrating the Lorentz force (5.20a) over space gives

since Jt = −ρ up to spatial derivatives. Solving (5.40) for v and using the leading order constitutive relation

In fact, exactly at ω = 0, analyticity properties of σ(ω) – in particular the fact σ(ω) is a retarded Green’s function (5.16b) and hence analytic in the upper half plane and subject to the Kramers-Kronig relations [350] – demand that there be a δ function:

We will see in later sections more ways to understand the emergence of this δ function. Using the constitutive relation for Qx, we also obtain

κ we need an expression for the force due to a thermal gradient. A thermal ‘drive’ is equivalent to a background metric and Maxwell field according to (5.13). The hydrodynamic equation of motion (5.20a) is placed in a nontrivial background metric by making the derivatives covariant and contracting indices with the background metric. In a background given by flat spacetime perturbed by (5.13) one obtains

5.40) above. Following the same steps as previously leads to the thermal conductivity

A very systematic way to obtain transport coefficients from hydrodynamic equations of motion was developed by Kadanoff and Martin [450]. The result of that analysis is as follows [450; 496]. Write the hydrodynamic equations of motion in the matrix form

where, for example, ϕA = {δρ, δs, δT0i} are the fluctuating hydrodynamic variables and λA = {δμ, δT, δvi} are fluctuations of the corresponding sources. The fluctuations of sources and hydrodynamic variables are related by the static susceptibilities

These are Green’s functions for the conserved densities. They clearly have poles on the hydrodynamic modes where (5.47) is satisfied. The Green’s functions for the currents JA = {Ji, Qi, Tij} – needed to compute the matrix of conductivities using formulae such as (5.16b) and (5.16d) – is obtained using the conservation equations, which hold as operator relations. Thus, for instance, 5.46).

The results (5.43), (5.44) and (5.46) obey the following relations that can be obtained as relativistic Ward identities [353; 388]

These relations are valid beyond hydrodynamics. In a relativistic theory, it is therefore sufficient to compute the electrical conductivity σ to obtain the entire matrix of thermoelectric conductivities. This is why there is only one independent dissipative coefficient σQ.

Given that (5.43), (5.44) and (5.46) follow from general considerations, they must of course be true in holographic models. This can be verified explicitly from bulk computations [353]. Furthermore, in holographic models the coefficient σQ will acquire a particular form. In Einstein-Maxwell-dilaton theories one obtains [353; 437; 128; 178]

4.43) evaluated on the horizon. We will discuss the temperature scaling of σQ shortly.

In experiments, we have explained that it is often convenient to measure κ, defined in (κ, there is no divergence in κ as ω → 0. One finds

5.17) has a simple physical explanation [554], as we now explain. κ is the thermal conductivity with open circuit boundary conditions. With such boundary conditions, as we noted, electric current does not flow. However, in a charged system, momentum necessarily transports charge (the sound mode carries charge). Therefore with no electric current, there can be no momentum and the sound wave is not excited. Thus there is no divergence. The association between the sound mode and the divergent conductivities will be seen very explicitly below. The fact that κ is finite but σ diverges in a fully hydrodynamic and nonzero density theory leads to a strong violation of the Wiedemann-Franz law (5.2) because the Lorenz ratio L ≡ κ/() → 0.

A further quantity that remains finite as ω → 0 is the conductivity of the ‘incoherent current’ [178]

This is the current corresponding to the incoherent combination of charges. Recall that this combination does not couple to the velocity and hence diffuses, with diffusivity (5.36). This mode that is decoupled from momentum behaves in much the same way as charge in a neutral relativistic system, described in Chapter 3. In fact, using (5.53) together with (5.43), (5.44) and (5.46), it is clear that the incoherent conductivity is nothing other than

As noted in the discussion around equation (4.66) above, this corresponds to an anomalous dimension for the charge density operator with Φ = z, making the operator marginal in the IR. The incoherent conductivity is sensitive to the IR physics precisely because it has decoupled from the ‘dragging’ effect of the sound mode.

5.4.4 Drude weights and conserved quantities

Delta functions in the conductivity, such as those appearing in (5.43), (5.44) and (5.46), always arise when a conserved operator overlaps with the current operator. In this section we will prove this fact. Let us focus on the electrical conductivity for concreteness. The conductivity can be decomposed into a ‘Drude weight’ part and a regular part

𝒟. From the expression for the conductivity in terms of Green’s functions (5.16b) it is clear that

5.57) to conserved quantities. This argument is originally due to Mazur [574] and Suzuki [718]. We follow the presentation in [335]. Both of the two terms in (5.57) admit a spectral representation. While they are rather similar, χJxJx is a zero frequency Euclidean Green’s function and GJxJxR(0) a zero frequency retarded Green’s function. Zero energy excitations contribute to the former and not the latter, so that

Here Z is the partition function and V the volume. The expression (5.58) is furthermore the time-averaged correlation function, so that

This can again be seen by using the spectral representation of the correlator. Formula (5.59) is intuitive. The delta function in the conductivity is due to current that does not relax.

Suppose that there are conserved Hermitian operators QA in the system. Without loss of generality we can take these to be orthogonal with respect to an inner product on the space of operators:

5.16d) – is not obvious but can be shown, see [275] and also §5.6 below. Using this inner product we can write

). Inserting this decomposition into (5.59) one obtains

5.58) of the time averaged correlator which is manifestly positive. This is the advertized result: conserved charges that overlap with the current operator lead to delta functions in the conductivity.

In the hydrodynamic results (5.43), (5.44) and (5.46) the inequality (5.62) is saturated by the overlap of the conserved momentum with the electric and thermal current operators. That is

These relations encode the intuitive facts that a net density causes overlap between the current and momentum, while a net entropy causes an overlap between heat and momentum.

In this chapter the long-lived operator will always be the momentum. Another important case is when the long-lived operator is the supercurrent [174]. Some aspects of superfluid transport will be considered in Chapter 6.

5.4.5 General linearized hydrodynamics

Let us briefly mention the extension of the (linearized) hydrodynamics derived above to models which are not Lorentz invariant. A powerful approach to this more general problem is the memory matrix formalism described in §5.6 below. However, a hydrodynamic perspective is also instructive. We must again choose a frame to fix the ambiguity in the definition of μ, T and vi out of equilibrium. A convenient choice, which can be thought of as a generalization of the Landau frame conditions (5.27), is as follows. Require that the thermodynamic expressions for the momentum density 𝔤i (while 𝔤i = Tti in a relativistic theory, we prefer to use manifestly non-relativistic notation here) and energy and charge densities 𝜖 and ρ are obeyed to all orders in the derivative expansion. Thus in particular for the momentum density at linear order

5.116) below. It is analogous to the “mass density”. Finally recall that in a non-relativistic theory Tti ≠ Tit.

In the non-relativistic notation, the hydrodynamic equations become

and Ji. This is done following the same logic as previously for the relativistic case. The spatial components of the stress tensor are found to be identical to before:

with E the external electric field, and αQ, , even when neglecting dissipative effects. The charge current is now

Note that the non-dissipative terms in these constitutive relations (e.g. Ji = ρvi) are no longer fixed by Lorentz invariance, but by the absence of entropy production. Let us see how this works by computing the entropy production to leading order in derivatives. Because entropy production occurs at quadratic order away from equilibrium, it is necessary here to keep a vid𝔤i term in the first law of thermodynamics for d𝜖. Everywhere else in our discussion we have dropped velocity contributions to thermodynamic relations, as they are quadratic in fluctuations about the equilibrium state (which we are taking to have vi = 𝔤i = 0) and are hence irrelevant for linear response. Thus we have

5.65). The second line collects an overall total derivative term involving the entropy current. The third line uses the constitutive relations (5.66), (5.67) and (5.68) to leading order (recall 5.31).

The transport coefficients are found by solving the hydrodynamic equations exactly as we did previously, the results are now

One can compute α by either applying a temperature gradient and computing Ji, or applying an electric field and computing the heat current. Demanding Onsager reciprocity (that these two computations of α yield the same result) gives

𝔤 is (see e.g. [440; 127])

In addition to the two viscosities, there is a single dissipative coefficient .

In the case where Lorentz invariance is restored, we have

5.33), further corrections are required at the nonlinear level to restore full Lorentz invariance. We do not have a systematic understanding of how this arises. Indeed, to the best of our knowledge, there has not been a systematic development of hydrodynamics for theories which are neither Galilean nor Lorentz invariant.

5.5 Weak momentum relaxation I: Inhomogeneous hydrodynamics

To obtain finite transport coefficients we must break translation invariance. In this section we describe how this can be done, in a certain limit, within hydrodynamics. The simplest hydrodynamic approach to translation symmetry breaking is a “mean-field” approximation, where momentum is relaxed at some constant (small) rate τ. The spatial components of the hydrodynamic ‘Newton’s law’ (5.20a) are modified to [364]:

𝔤i is the momentum density, as in the previous subsection. Following the previous section, we may derive the electrical conductivity

5.76), we may simply set 5.77) is now finite. We can directly see that this is due to the fact that momentum is no longer conserved.

In the quasiparticle limit, ρ → ne and → nm, with n the number density of quasiparticles of charge e and mass m. Then (5.77) is simply the classical Drude formula. Thus a generalized Drude formula (5.77) is valid for any model with a clean (momentum-relaxation dominated) hydrodynamic limit of transport. Note that in a quasiparticle metal away from this clean hydrodynamic limit (i.e. most conventional metals), one should strictly not use the Drude formula but rather a Boltzmann equation that allows for the fact that momentum is not a privileged observable, but rather decays at the same long timescale as generic quasiparticle excitations δnk [778].

To leading order in perturbation theory in 5.77)). Thus, in this framework, σQ cannot reliably be computed in the clean theory without taking into account such ‘spectral weight transfer’ from the coherent to the incoherent part of the conductivity. Holography gives a particularly convenient way to account for these corrections, as we will see.

To go further, we need a more ‘microscopic’ approach in order to derive an expression for τ. A physically transparent way to account for the presence of τ is to directly solve the hydrodynamic equations described in previous sections, but to explicitly break translational symmetry in the background fluid with long wavelength, hydrodynamic modes. These modes could be disordered or could form a long wavelength lattice. We will refer to the position dependence of the sources as ‘disorder’ for convenience. A particularly simple way this can be done is to source the fluid with charged disorder. Although the resulting inhomogeneous hydrodynamic equations must in general be solved numerically, this approach is non-perturbative in the amplitude of (long-wavelength) disorder [537; 538]. In the remainder of this section we consider a simpler limit in which the disorder is weak and can be treated perturbatively. In this limit of weak, long wavelength disorder we will obtain an explicit formula for τ from hydrodynamics alone. Similar ideas, of self-consistently describing momentum relaxation within hydrodynamics, have been developed in [43; 63; 182]. The discussion below follows [537; 538].

Suppose that we have a translationally invariant fluid at nonzero temperature and density. The translation-invariant Hamiltonian H0 is perturbed by a time-independent, spatially-varying source which linearly couples to a scalar operator O:

𝒪(h2). This assumption can be relaxed [538], leading to more complicated expressions for τ that hide the simple physics at work. There is the following Ward identity (see [475] for a field theoretic derivation)

⟩ the thermal expectation value of O; for simplicity henceforth we drop the angled brackets. To solve the hydrodynamic equations of motion, we must also account for the dependence of pressure on h. The pressure is microscopically P = W/Vd, with W the generator of correlation functions in the QFT. By definition ∂W/∂h(k) = O(k). Therefore

This allows us to find an exact, non-dissipative solution of the hydrodynamic equations with μ and T uniform, uμ = (1, 0, …), and h = h0. The Ward identity (5.79), in the absence of an electric field, together with (5.80) implies that the stress tensor takes the required ideal fluid form Tij = ij, since

Now, we linearly perturb this background with an external force. For simplicity, we apply an electric field Ei. A thermal gradient works similarly. The argument below is not long, but is somewhat subtle. We are performing a strict linear response calculation in E. Only after taking this limit, we then impose the perturbative limit that the strength of the background disorder h0 0. We also turn on a small frequency ω. The interesting frequency scales will be .

We make an ansatz that the only two fields which pick up corrections due to the external electric field are a constant shift to the velocity, 5.79) reads

In the above formula ρ, vj are uniform, that is to say, evaluated at zeroth order in a derivative expansion. Inhomogeneities are induced at higher order in h0, and these are the terms that are dropped in (5.82). At linear order in perturbations, averaging (5.82) over space leads to

The objective now is to solve this equation to obtain a relationship between the electric field Ei and the induced velocity vi and hence obtain the conductivity.

In order to evaluate (5.83), it turns out to be easier to evaluate δO than δh. Happily, these can be related by a static susceptibility, or retarded Green’s function, in Fourier space:

5.57) the susceptibility and zero frequency Green’s functions are equal. To evaluate the integral in (5.83) we need to obtain the δO that is induced due to a flow vi (that is itself induced by the electric field). Because δO is a scalar but vi is a vector, δO can only be sourced in the presence of an inhomogeneities that result in nonzero gradients. Therefore we expect δO ∼ viih0. To obtain the precise relation it is easiest to work in the fluid rest frame. In this case the inhomogeneous source (5.78) is

where we have only kept terms to linear order in our perturbative expansion in E, as v ∼ E. Hence from (5.83), we obtain

5.84) on the background, along with analyticity properties of Green’s functions. Assuming that the source h0 is isotropic, then we may replace kikj (k2/d)δij above. This leads to the explicit formula for τ:

along with the constitutive relation Jiρvi at this order in perturbation theory, we obtain the Drude formula

which is what we obtained from hydrodynamics, but without the σQ contribution, which is generically subleading at this order in perturbation theory.

5.6 Weak momentum relaxation II: The memory matrix formalism

Let us now re-derive the results of the previous section, and more, using the memory matrix formalism. This will require a couple of pages of rather formal manipulations and definitions, but the payback will be worth the effort (the pragmatic reader can skip to the results in the subsections below). Specifically, the memory matrix extends the results of the previous section beyond the hydrodynamic limit. That is, there will be no assumption that the disorder is slowly varying. For instance, the memory matrix can describe generalized umklapp scattering that occurs at microscopic distances. The memory matrix is the toolkit of choice for describing non-quasiparticle physics in which a small number of slowly decaying variables dominate the dynamics. Our formal development will follow [275].

Define the following correlation function between two Hermitian operators A and B:

○] is the Liouvillian operator. What is the usefulness of this object? We take a time derivative and perform some basic manipulations:

where in the last step we have used time translation invariance. The essential point of 𝒞AB(t) is this fact that it represents the integral of a commutator. The commutator of course plays a central role in the definition of the retarded Green’s function, so that now we have that

In the last step we assume that the operators A and B decay at late times so that the corresponding Drude weight vanishes in (5.57). The result (5.94) is somewhat counterintuitive in that it relates a fixed time (t = 0) quantity to an integral over all time (ω = 0).

We may also do the ‘Laplace transform’ integral:

In the above expressions z is a complex frequency in the upper half complex plane, so that the integral above converges. Results at real frequencies are obtained by setting z = ω + i0+. Note that we are using the same symbol for 𝒞 and its Laplace transform. We will keep the argument explicit in order to distinguish them.

Suppose that we pick A = B = Jx, and that we have a metal with finite electrical conductivity (so that 5.16b), so that

We will leave the + i0+ implicit in expressions below. Analogous identities hold for the other thermoelectric conductivities in (5.16b) and (5.16d). The identification of the conductivity with the correlation function 𝒞 is the starting point for the memory matrix method. We now proceed to perform manipulations on 𝒞.

As we have seen in previous sections, the conductivity and hence 𝒞AB is parametrically large when translation symmetry breaking is weak. The goal of the memory matrix formalism is to re-organize the computation of 𝒞AB so that we can instead compute parametrically small quantities, which can then be analyzed easily perturbatively. In order to do this, we will introduce a further abstraction: an inner product on an “operator Hilbert space”

which will soon come in handy. We can now again perform the Laplace transform, this time writing the answer as

5.100) in order to cleanly extract out the perturbatively small “denominator” to the conductivity 𝒞AB. We pick a selective set of “long-lived” modes A, B. These will be the hydrodynamic degrees of freedom in a nearly clean fluid. More generally they can be any set of operators we wish, although they must include any operators whose correlation functions we wish to compute. The strategy will be to treat separately the time dependence of these operators and the remaining “fast decaying” operators. To this end we introduce the projection operator (that acts on the space of operators)

with χ the reduced susceptibility matrix, only including the long-lived modes. It is simple to check that 𝔮2 = 𝔮, and hence 𝔮 projects out the slow degrees of freedom. We strongly emphasize that we are free to choose any set of operators to call the “long-lived” modes, at this point, so long as we only compute correlation functions between this set of privileged operators. Only at the end of the calculation will it be clear why to choose the modes which are, in fact, long-lived to be privileged.

Our goal is now to separate zL in (5.100) into the contributions from fast and slow modes. Note the exact operator identities

and

we find that

5.106) more transparent, introduce the following two matrices on the space of long-lived operators. The eponymous memory matrix is

Note that M is a symmetric matrix, while N is antisymmetric. Further, N = 0 in a time-reversal symmetric theory if the operators A and B transform identically under time reversal, as will often be the case. More generally note that N is a constant whereas M depends on z. The N matrix is clearly directly related to the first term in brackets in (5.106). To see the appearance of the memory matrix itself from the second term in (5.106), note that

𝔮 in this chain of equations. Then, (5.106) becomes

with σAB the generalized conductivity between operators A and B, as we noted in (5.97) above. We have analytically continued to real frequencies. Of particular relevance to us will be the thermoelectric conductivities (cf. (5.16b) and (5.16d) above):

5.111) for the conductivity has a lot of physical information which will be elaborated upon in the following subsections. The basic idea is that the susceptibilities χAC, χDB determine the overlap of the current operators with the long-lived modes, as in equation (5.62) above. This overlap can be thought of as a fast process. Then the extended Drude-peak structure of the rest of the formula will describe on how the long-lived operators themselves decay. Both M and N are proportional to time derivatives of the long-lived operators. Therefore, if these time derivatives are small, M and N will also be small, and the Drude-like peak will be sharp. In this way the formula (5.111), with no assumptions whatsoever at this point, has split the computation of the conductivity into a thermodynamic piece (the susceptibilities) and a piece that only depends on the dynamics of long-lived operators. The formula will be useful when a small number of parametrically long-lived operators exist.

5.6.1 The Drude conductivities

The results of the previous section can now be used to re-derive the results (5.88) and (5.90) for the conductivities. As advertised previously, the memory matrix derivation does not require momentum relaxation to be a long wavelength process. We again consider a clean and isotropic fluid, weakly perturbed by an inhomogeneous field coupling to a scalar operator O, as in (5.78). If this breaking of translation invariance is small enough, then the longest-lived mode that will have substantial overlap with the (vector) current Jx is the momentum Px. Hence, we will allow the long-lived modes in the memory matrix formalism to be Jx and Px. These are both time-reversal odd and so, assuming a time reversal invariant state, we may set the matrix N = 0. The next key observation is that as ω → 0, M has a zero eigenvalue associated with MPP in the clean limit. This follows from the definition of MPP because in a clean metal 5.78) breaking translation invariance

5.113) at hand we can evaluate the memory matrix. The basic idea is that the inhomogeneous coupling will lift the zero eigenvalue of the clean theory and replace it by a small nonzero eigenvalue, which we can calculate. This eigenvalue will control momentum relaxation and hence the conductivities. At first nontrivial order we will see that (i) the computation can be reduced to evaluating correlators in the clean theory and (ii) the projection operators that appear in the definition of the memory matrix can be set to unity. Both of these facts offer substantial simplifications that will allow the momentum relaxation rate to be evaluated explicitly in later sections. The derivation below follows [358], that built on [364; 354].

The operator in the clean theory. The following manipulations aim to exploit this fact. Firstly write

5.114) is 𝒪(h2). We must first explain why the second term after the third equality is subleading. The sum in that term is over A, B ∈{J, P}. That term includes the expression 𝒪(h3) or higher.

Using the expression (5.114) for MPP in the definition of 𝒞(z) in (5.100), as well as the connection to the retarded Green’s function in (5.96), we can obtain the following important result at leading order in h [354; 358]:

The Green’s functions in the above formula may be evaluated in the clean theory. The final step in the above uses the explicit expression (5.113) for the operator is even in ω to focus on the imaginary part as ω → 0.

Two aspects of (5.115) are worth highlighting. This quantity will shortly become the width of the Drude peak in the frequency dependent conductivity, controlling the current and momentum decay rate. Firstly, the expression (5.115) is fairly intuitive. It can be thought of as a non-quasiparticle version of Fermi’s Golden rule. The rate of decay of the momentum is determined by the number of low energy excitations (measured by the spectral density, the imaginary part of the Green’s function) that overlap with the operator . Secondly, because the decay rate is given in terms of a low energy spectral density, it is purely a property of the low energy critical theory. It can be calculated within the low energy effective theory. In particular, in order for perturbation theory in h to be valid, it is sufficient that the inhomogeneous coupling be small in the effective theory. If the coupling is irrelevant at low energies, perturbation theory can be used at low temperatures even if the inhomogeneities are large at short distance and high energy scales.

Similar arguments to the above show that MJP ∼ h2. One factor of h comes from 5.111) for the conductivity.

To evaluate the conductivity we also need the susceptibilities. At leading order in weak momentum relaxation these may also be evaluated in the clean theory. Note, however, that the susceptibility is then the susceptibility computed within the low energy theory. This can have a nontrivial relationship in general with the microscopic susceptibility if disorder is important at high energies. In the clean theory, the susceptibility χJxPx is the charge density ρ. To see this, recall that the velocity is thermodynamically conjugate to momentum, the same way that μ is conjugate to ρ. Thus χJP ≡ Jx/vx = ρ. We noted in our discussion of zeroth order hydrodynamics around (5.68) above that the relation Jx = ρvx is determined by thermodynamics and does not require any kind of boost invariance. We define further χPxPx . In a relativistic fluid, = 𝜖 + P, the enthalpy. In a Galilean-invariant fluid, is the mass density. At leading order in h, as noted in the previous paragraph, we may neglect MJP and MJJ when inverting M − iωχ. (5.111) becomes

5.115), we obtain

5.90) for the Drude peak. The formula for τimp is the same as we previously obtained in (5.88), but in now in significantly more general circumstances.

5.6.2 The incoherent conductivities

It is also possible to account for the ‘incoherent’ contributions σQ to the conductivities (5.77) from the memory function formalism. This requires some care about which quantities are small or large in perturbation theory, and teaches us a limit in which the issues of spectral weight transfer discussed in the paragraphs below (5.77) can be avoided, so that both terms in (5.77) can be trusted. This was first discussed in [544]. Here we present a cleaner derivation, following [174].

As a warm up, consider first the case with ρ = 0. Here we expect that σ = σQ at low frequencies. In this limit χJP = 0 and hence the memory matrix formalism immediately gives us

In fact, this expression is always exactly true, as we are free to consider the set of slow operators to contain only J. Setting ω = 0 we obtain

Next, let us turn on a nonzero charge density ρ. The conductivity is

Recall that in the perturbative limit MPP ∼ MJP ∼ h2. Let us zoom in on frequencies that saturate the Drude peak, so that ω ∼ h2. In order for an ‘incoherent’ frequency-independent term to contribute at leading order as h → 0, on equal footing with the Drude term, it is necessary for the ‘incoherent susceptibility’ to be large. We define the incoherent susceptibility as [174]

that we encountered previously in equation (5.53) above in the relativistic case. Now suppose that . In this scaling limit, the conductivity becomes

5.77). In the final line we obtained an expression for the incoherent conductivity

There is in fact some freedom with how the scaling limit is taken and how this formula is written. We could equally well have taken χJJ 1/h and obtained precisely the result (5.119) above for σQ, with no reference to incoherent currents. The expression (5.124) is natural given that σQ in hydrodynamics is the dc conductivity of the incoherent current, according to (5.54), and therefore (5.124) is the natural nonzero density generalization of (5.119).

The crucial point, however, about the above derivation is that we see that to perturbatively recover σQ in an exact formalism, we have to treat σQ as anomalously large in perturbation theory. In general this will not be the case. We will observe corrections to (5.77) at the order of σQ in §5.9.2 below.

To account for the incoherent thermal conductivities αQ and

The net σ, α and

Here χQP = Ts gives the entropy density in terms of microscopic Green’s functions. As we noted previously with χJP = ρ, this is fixed by the absence of entropy production at zeroth order the hydrodynamic derivative expansion. We have only included leading order contributions to the memory matrix M.

Accounting for slow momentum relaxation as before, we recover the thermoelectric conductivities from σ = χ(M − iωχ)−1χ, and keeping only the leading order contributions in the scaling limit as h → 0:

5.70), upon setting .

Note that MJJ changes (but σQ does not) depending on whether or not Q is included as an index in the memory matrix, due to the change in the projection operator 𝔮. In general, computing χJJ, MJJ etc. will require a microscopic computation more complicated than the perturbative computation of τimp, which is readily expressed in terms of microscopic Green’s functions as in (5.115).

Finally, it is worth noting that in a relativistic theory, Q = PμJ holds as an operator equation. In this case, Q should not be a separate index in the memory matrix, and instead one computes σJQ = σJPμσJJ, e.g. Similarly, in a Galilean invariant fluid, the momentum P = mJ, and hence J should be removed as an independent index in the memory matrix. Of course, an inhomogeneous coupling breaks both Lorentz and Galilean invariance, so these simplifications are likely only possible at leading order in a small h expansion.

5.6.3 Transport in field-theoretic condensed matter models

With the general form of the thermoelectric transport response functions at hand in (5.70) via hydrodynamics, and also in (5.127) via memory matrices, we now return to the condensed matter models of compressible quantum matter in §4.2.1-4.2.3, and discuss the frequency and temperature dependence of their transport properties.

First, we have the ‘diffusive’ or ‘incoherent’ components, σQ, αQ, and κQ. These are not expected to experience special constraints from total momentum conservation, and so should be finite scaling functions of ω/T. Applying a naive scaling dimensional analysis to these terms, we can expect that they obey scaling forms similar to those discussed in Chapter 3:

with different scaling functions ϒ for each of the transport coefficients. As discussed at the beginning of Chapter 4, thermodynamic quantities related to the free energy density typically violate hyperscaling. This violation amounts to a replacement d → dθ in scaling forms. So we might similarly expect that (5.128) is replaced more generally by

This issue has been investigated in recent works in the models of non-Fermi liquids described in §4.2 for the case of σQ, and it has been found that (5.129) is indeed obeyed. The Ising-nematic transition of §4.2.1 was investigated in [240], and σQ obeyed (5.129) in d = 2 with θ = 1. Similarly, for the spin density wave transition of §4.2.2, the expected value, θ = 0, appeared also in the hot spot contributions of σQ in d = 2 [349; 619]. (We should note that for the spin density wave case, there are important contributions to the conductivity for ω > T from ‘lukewarm’ regions of the Fermi surface away from the hot spots [349; 148; 147]; we do not discuss these here, but these can be more singular than the hot-spot contributions.) These computations do not show evidence for the separate anomalous dimension for the charge density discussed in §4.3.4.

Next, let us consider the “momentum drag” terms in (5.70) and (5.127). A key characteristic of the theories of non-Fermi liquids in §4.2.1 and §4.2.2 is that they can all be formulated in a manner in which the singular low-energy processes (responsible for the breakdown of quasiparticles) obey a Lagrangian which has continuous translational symmetry. Consequently, the resistance of such theories is strictly zero, contrary to computations in the literature [514; 429; 428; 190]. Let us emphasize, again, that here it is the emergent translation invariance of the low energy critical theory that matters. Following the discussion in §5.2, we need additional perturbations which relax momentum, and these can be inserted in (5.115) and then (5.127) to compute the transport coefficients.

The thermodynamic parameters ρ and are invariably dominated by non-critical ‘background’ contributions of the compressible state. So we can take them to be non-singular T- and ω-independent constants. The singular behavior of the entropy density, s, was discussed already at the beginning of Chapter 4. So it only remains to discuss the singular behavior of τimp.

The memory matrix result for τimp in (5.115) was expressed in terms of the spectral density of the operator O coupling to a space-dependent source, h(x), in (5.78). If we assume that average of |h(k)|2 is k-independent (this is the case for a Gaussian random source which is uncorrelated at different spatial points), and the scaling dimension dim[O] = Δ in the theory without the random source, then a scaling analysis applied to (5.115) yields [364; 354; 366; 535]

at small disorder.

The expression (5.115) has been evaluated for specific models of disorder in the non-Fermi liquids discussed above. For the Ising-nematic critical point, [366] argued that the most relevant disorder was a ‘random field’ term that coupled linearly to the Ising-nematic order parameter ϕ in §4.2.1. However, it was found that the order parameter spectral weight, i.e. Im Gϕϕ, violated the expected scaling form because of an emergent gauge-invariance in the underlying critical theory. Consequently, (5.130) is not obeyed in this specific model. It was found that, in d = 2, 1/τimp ∼ T−1/2 (up to logarithms) at low T. The fact that τimp becomes large at low temperatures in this model is a reflection of the fact that disorder is strongly relevant at the critical point. The computation is only self-consistent, then, if the disorder is tuned to be very small, so that the clean fixed point can still control the physics down to low temperatures. Later parts of this section and §7.2 will consider the physics of strongly disordered fixed points, leading to incoherent transport. One can also consider a higher temperature regime in the pure Ising-nematic fixed point, where the Landau damping of the order parameter fluctuations is negligible and so they obey z = 1 scaling [670; 271]: here, a linear-in-T resistivity was found [366; 543] via 1/τimp ∼ T.

For the spin-density-wave transition of §4.2.2, we have to consider an additional subtlety before evaluating (5.115). The critical theory (4.19) is expressed in terms of fermions at hot spots on the Fermi surface which couple strongly to the spin-density-wave order parameter. However, fermions away from the hot spots can also carry charge and energy and hence contribute to the transport co-efficients. In [621], it was assumed that higher-order corrections to the critical theory [349] are strong enough to rapidly equilibrate momentum around the entire Fermi surface. With this assumption, the slowest process is the impurity-induced momentum relaxation, and this is dominated by the contributions of the hot spot fermions, and can be computed [621] from (5.115). Here, the disorder was assumed to be associated with local variations in the position of the critical point, and so coupled linearly to (φa)2. In this case, the scaling law (5.130) was found to apply, and yields 1/τimp ∼ T (up to logarithms), implying a resistivity linear in T.

Finally, we consider the scaling of the shear viscosity η. The viscosity is well-defined in these critical theories with an emergent translation invariance. Using the same scaling arguments leading to (5.128) and (4.1), or from the holographic result in (3.60), we expect that

However, explicit computations [620] on non-Fermi liquid models of a Fermi surface coupled to Ising-nematic or gauge fluctuations lead to a rather different conclusion. Although strong interactions can destroy quasiparticles near the Fermi surface, they do leave the Fermi surface intact, as we discussed in §4.2.1. So far, we have found that this residual Fermi surface leads to the same scaling of transport properties as might be expected in a critical and isotropic quantum fluid in dθ spatial dimensions. But for the shear viscosity, the anisotropic structure of momentum space near each point on the Fermi surface turns out to have important consequences, among which is the breakdown of (5.131): computations on the non-Fermi liquid model show that the ratio η/s diverges [620]

5.132) does not emerge from holography using classical gravity. It depends strongly on the Fermi surface structure of the field theory. See the discussion in §4.4 concerning Fermi surface physics in holographic strange metals.

5.6.4 Transport in holographic compressible phases

Let us now go through a few examples of the holographic application of the memory matrix formula (5.115) for the momentum relaxation rate. We have really done the hard work in §4.4.2 by computing the spectral weight of scalar operators in various critical holographic phases. The ease with which the results below are obtained — in particular, without solving any partial differential equations in the bulk spacetime, corresponding to the inhomogeneous sources — reflects two powerful features of the approach. Firstly, the memory matrix has isolated the universal low energy data, the low energy spectral weight in (5.115), that determines the momentum relaxation rate. Secondly, in §4.4.2 we showed how the low temperature scaling of the low energy spectral weight could be computed from near-horizon data. Combining these two facts, the momentum relaxation rate and hence the transport can be obtained purely from the geometrical properties of the emergent critical phase in the far IR of the spacetime.

Throughout this section we are assuming that the effects of translation symmetry breaking are weak in the IR, so that perturbation theory is possible. Holographically speaking, this means that inhomogeneities induced by the boundary source (5.78) decay towards the interior of the spacetime. If, instead, the inhomogeneities are relevant and grow towards the interior, then a new, intrinsically inhomogeneous IR emerges. This will lead to incoherent hydrodynamic transport, without momentum, that is the subject of §5.8–5.10 below. For generic situations without extra symmetries (to be discussed later), there is no way around solving bulk PDEs in those cases. The cases of relevant and irrelevant disorder are illustrated in Figure 5.2.

Figure 5.2
Relevant and irrelevant inhomogeneities. In the left plot the inhomogeneities grow towards the interior of the spacetime whereas in the right plot they decay towards the interior. In the irrelevant case one case use perturbation theory in the interior, almost homogeneous, geometry to describe momentum relaxation. In the relevant case one must find the new inhomogeneous interior spacetime, typically by solving PDEs.

There are two important distinctions in the holographic models: firstly, whether momentum relaxation occurs due to a periodic lattice or a random potential, and secondly, whether the dynamic critical exponent z is finite or infinite. Let us discuss these in turn. Note first that the memory matrix expression (5.115) for the momentum relaxation rate depends only on the k-dependent spectral weight ρ(k) in (4.67), so that we can write

μ. This formula will thus determine the temperature dependence of τimp from the temperature dependence of ρ(k).

Consider first the case of a regular, periodic lattice. For simplicity, focus on the periodicity in one direction and take the lattice potential to be a single cosine mode with lattice wavevector kL. These are not crucial assumptions for the physics we are about to describe. Then in (5.133) we have

The essential point here is that the momentum relaxation rate is given in terms of the low energy spectral density at a nonzero wavevector. The lattice can only relax momentum efficiently if there exist low energy degrees of freedom that the lattice can scatter into. This again brings out the analogy to Fermi’s Golden rule.

If z < ∞, then we saw in (4.96) above that low energy, nonzero wavevector spectral weight is exponentially suppressed. This reflects the intuitive fact that if z < ∞ then low energy excitations scale towards the origin of momentum space. A spatially periodic lattice is highly irrelevant with such scaling. It follows that momentum relaxation is very inefficient and momentum is exponentially long-lived

5.117), was found numerically in the holographic computations of [409], for the normal component of the conductivity in the superconducting phase. It reflects the emergence (up to logarithmic corrections) of a z = 1, AdS4 geometry in the far interior of the spacetime, and the corresponding inefficient scattering by a periodic lattice. Exponentially large conductivities can occur in conventional metals in a ‘phonon drag’ regime, where the bottleneck for momentum relaxation is phonon umklapp scattering. Evidence for this effect was presented in [398]. The kinematic cause is the same: phonons do not have low energy spectral weight at nonzero wavevector and hence cannot scatter efficiently off a lattice.

As was emphasized in §4.4.1 and §4.4.2, a key aspect of z = ∞ scaling is the existence of low energy, nonzero momentum spectral weight. In particular, at low temperatures we can use (4.92) in the formula (5.133) for the relaxation rate to obtain [358]

d, with z = ∞. The above expression (5.136) leads to a power law in temperature dc resistivity via (5.117), with the power determined by the scaling dimension νkL. This power law can be seen explicitly in full blown numerical results [411]. Perturbation theory is valid so long as the lattice is irrelevant and the resistivity goes to zero at T = 0, which requires . In later sections we will discuss the relevant case, leading to insulators, as well as the case of νk imaginary, leading to instabilities.

The discussion in §4.4.1 and §4.4.2 emphasized that the existence of low energy spectral weight was a property that z = ∞ theories shared with Fermi liquids. Similarly, the result (5.136) parallels the T2 power law resistivity found from umklapp scattering by a lattice in Fermi liquids. Umklapp scattering is an interaction in which charge density is moved around the Fermi surface (at zero energy cost) with a net change in momentum by kL. A unified description of umklapp, holographic and conventional, is developed in [358].

The simplest model of weak disorder is Gaussian short range disorder. In this model the inhomogeneous source h(x) of (5.78) is drawn randomly from the space of functions with average and variance

We will discuss the physics of random disorder further in §7.2 below. For the moment we can simply use (5.137) in the formula (5.133) for the momentum relaxation rate, where it amounts to

Unlike the case of a periodic lattice, the random disorder contains modes of all wavelengths, and this is why an integral over k survives. This means that disorder can scatter long wavelength modes and can efficiently relax momentum in theories with z < ∞.

For the Einstein-Maxwell-dilaton models with z < ∞, we saw in (4.94) and (4.96) above that the spectral weight is exponentially suppressed for T1/z k, and a constant power of the temperature for k T1/z. Thus from (5.138) we obtain the resistivity [364; 354; 535]

5.130) quoted above, as it should be. The disorder leads to a resistivity determined by the scaling dimension of the disordered operator. The perturbative computation is controlled so long as the resistivity goes to zero as T → 0. We will discuss relevance and irrelevance of disorder in §7.2.

Finally, we can consider disorder in holographic models with z = ∞. The spectral weight is again (4.92) and hence the momentum relaxation rate and resistivity [358; 38; 543]

Assuming that νk grows at large k, as it typically does, at low temperatures this integral is dominated by small k. At low T we can write the integrand as a exponential, which then has saddle point

He we used the fact that generically k/dk ∼ k as k → 0. See for instance (4.77) above for the AdS-RN case (in some interesting cases there are cancellations such that dνk/dk ∼ k3, as in for instance [182], in such cases ). Thus the resistivity

4.77) and hence the resistivity is purely given by the ‘universal’ logarithmic correction. Note that the result (5.142) only depends on the dimension of the disordered operator as k → 0 in the semi-locally critical IR theory.

5.6.5 From holography to memory matrices

The memory matrix results for the Drude conductivities, (5.115) and (5.117), follow from general principles and apply to all systems. In the previous section we have used these results to obtain the conductivity in holographic models. It is also instructive to see how the memory matrix formulae themselves can be directly obtained holographically, and how the holographic computation of σ(ω) proceeds in a momentum-relaxing geometry. We will present the derivation in this section, following [536]. The derivation will be perturbative in the disorder strength, which is required to be small at all energy scales.

Consider a homogeneous background EMD geometry, of the kind discussed in §4.3.2. This background is now perturbed by an additional scalar field φ, that we can take to have action (4.62). We choose φ to be dual to an operator of dimension Δ > (d + 1)/2. The asymptotic boundary conditions as r → 0 corresponding to the inhomogeneous source (5.78) are

where φ0 solves the wave equation of motion for φ in the homogeneous bulk background, is regular at the black hole horizon and satisfies φ0(k, r → 0) = rd+1−ΔLd/2 + .

Our goal now is to compute the conductivity in the homogeneous background perturbed by the scalar field (5.144). We will find that the conductivity σ(ω) contains a Drude peak so long as h is perturbatively small, and we will recover (5.115). The Drude peak is a perturbative limit of a double expansion in small ω and small h. From (5.117), σdc ∼ τ ∼ h−2; evidently we must treat h2 and ω as small parameters on the same footing. Now, the backreaction of the inhomogeneous field (5.144) onto the background means that the original EMD fields are altered at order h2. However, these perturbative corrections to the background cannot lead to a singular conductivity as h, ω → 0. We will see that the leading order effect of inhomogeneity – smearing out the δ function in σ(ω) – is achieved by the inhomogeneous scalar (5.144) directly. We may thus treat the EMD fields as uniform, and only consider the nonzero momentum response in the φ field [107]. We further need only compute the holographic linear response problem to linear order in ω [536].

We first identify the perturbations which will be excited by an infinitesimal uniform electric field. Since the electric field is uniform, the lowest order processes in h occur when a linear perturbation δφ(k) scatters off of the background φ(−k). Hence the perturbations of interest will be δφ(k), which furthermore couples to spatially uniform spin 1 perturbations under the spatial SO(d) group: δAx and Mathematica, along with a differential geometry/general relativity package, to perform many tedious steps in such computations):

4.44), Z appears in the EMD action (4.43) and ρ is defined in (4.47).

Terms ∼ ω2 in the above equations are beyond the order in perturbation theory to which we work, so we may drop them. Now (5.145) collapses to a set of 3 differential equations:

(specifically, we assume for simplicity that the modes h(k) are chosen so that σij = σδij; the theory is microscopically disordered but macroscopically isotropic), and defined the new field

We now proceed to directly construct (at leading order) the bulk response to the applied electric field. Counting derivatives in (5.146) shows there are 4 linearly independent modes. With the benefit of hindsight, none are singular in h, and so we may look for these modes assuming h = 0. The first such mode – and the only one containing δ𝒫x, is:

𝒫x = 0. There is a “diffeomorphism mode” mode:

and a third mode (explicitly given in [536]) which is the “Wronskian” partner to this Galilean boost mode; it is logarithmically divergent at the horizon. (5.146a) near r = r+ allows us to fix the constant term in δAx in (5.149). Recall that p is the background gauge field in (4.44).

What is the bulk response if we impose the boundary condition that δAx(ω, r = 0) = 1 and infalling boundary conditions at the horizon? We do not source a heat current, and so 5.148).

Next consider a nonzero but small ω. We are after the 𝒪(ω) correction to the response of the previous paragraph. At nonzero frequency we must impose infalling boundary conditions (3.28) at the horizon. Recalling the behavior (4.45) of the function b near the horizon, we find that equation (5.146c) requires the near horizon behavior

To get the factor of ρ in (5.150), we use the ω = 0 result from (5.148) that δ𝒫x = ρ/Ld δAx = ρ/Ld. The last step here uses the boundary condition δAx = 1. Note that in the scaling limit ω ∼ h2, then 𝒪(ω0) in perturbation theory. At this order in perturbation theory we can set the ω in the exponent to zero, effectively removing the oscillating part for most purposes.

The lifetime defined in (5.151) is in fact precisely that obtained from the memory matrix in (5.115). To see this, recall the formula derived in (4.89) above, in which the low energy spectral weight is given in terms of the value of the field on the horizon, which is precisely what appears in (5.151).

To obtain the Drude peak itself, with the lifetime (5.151), we turn to the perturbation δAx. Regularity at the horizon requires that the only mode sourced – upon including the metric perturbation (5.150) at order ω/h2 – is the Galilean boost mode (5.149). Adding this term to the ω = 0 solution of δAx = 1, we obtain

We have neglected the oscillatory part (r+r)−iω/(4πT) that is unimportant at small ω, except very close to the horizon. The Drude form (5.117) for the conductivity now follows from (5.152) using basic entries of the holographic dictionary: The conductivity is given by (3.50), with the current given by (3.49b), with z = 1, and using (3.49a) to relate the subleading behavior in p to ρ (recall p(0) = μ).

The following comment on the above result is useful. We saw that the bulk response is dominated by a ‘Galilean boost’. This is the holographic analogue of the derivation of the Drude conductivities from hydrodynamics in (5.88). In the holographic argumentation, however, we did not need to assume that the inhomogeneities were slowly varying. We will see other instances below of how holographic models often admit a ‘hydrodynamic’ description beyond its naive regime of validity.

5.7 Magnetotransport

This section will discuss transport in a magnetic field as well as disorder. We will restrict to d = 2; in higher dimensions, adding a magnetic field breaks isotropy. As previously, we start with a discussion of the predictions of hydrodynamics. We will generalize both the hydrodynamic discussion of §5.5 as well as the memory matrix approach of §5.6.

The setup is similar to before: a fluid with translational symmetry broken weakly by an inhomogeneous field h(x) coupled to a scalar operator 𝒪, and now additionally in a uniform magnetic field B. In the case in which the inhomogeneities are slowly varying, the entire discussion can be made within hydrodynamics, as in §5.5 above. The ‘momentum conservation’ equation now reads

with Fμν the (externally imposed) gauge flux; in our case, we will have Fxy = B, as well as Fit = −E. We assume that 5.82) for the fluctuations now becomes

The treatment of δh is identical to before – B-dependent corrections to τ are subleading, just as ω-dependent corrections were. Thus upon spatial averaging we obtain

5.88). Solving for the velocities in terms of the electric field, and using Ji = ρvi, the longitudinal and Hall conductivities are obtained as

5.153) and in the definition of the conductivities, instead of only including the latter term. One finds [364]:

Thermoelectric conductivities have also been obtained in [364]. In the above formulas we introduced the hydrodynamic cyclotron frequency and decay rate

4.161). We will discuss the physics of this cyclotron mode below.

The agreement with the holographic results implies that the hydrodynamic expression (5.157) will exhibit a particle-vortex duality, dual to the Maxwell duality in the bulk. The transformation (4.155) of the conductivities is manifested as follows. Under the transformations

where ρ = σ−1 is the resistivity matrix.

It is instructive to derive (5.156) from the memory matrix formalism. To do so, we must employ the full expression (5.111) for the conductivity, including the matrix N which breaks time reversal symmetry. We will outline the computations contained in [544; 174]. For simplicity we will focus on the case τimp = ∞, that is, with no inhomogeneities. Momentum is relaxed by the magnetic field. The operator equation for momentum relaxation is now

The space of ‘slow operators’ will now be {Px, Py, Jx, Jy}.

To evaluate the memory matrix expression for the conductivity, firstly we need the susceptibilities. These are symmetric in the operators and hence, using isotropy, the nonzero components are

5.161) and (5.108) and the N matrix is antisymmetric. The nonzero components of N are evaluated using (5.108) to be

5.107). These are simplified by the fact that the projection operator 𝔮 in the memory matrix projects out Ji, and these same Ji appear in the expression (5.161) for . Thus one finds

5.111), one recovers precisely the conductivities (5.157) in the scaling limit ω ∼ B. In order for the incoherent conductivity σQ to contribute at leading order in this limit one must furthermore take . See the discussion in §5.6.2 above. The incoherent conductivity is now given by

5.124) above. As discussed in that section, one can express σQ in terms of incoherent currents, but the results are not distinguishable at leading order in this scaling limit. In the present setup there is also an incoherent Hall conductivity , but this is expected to be subleading in powers of B in the scaling limit and hence we have not included it.

One of the interesting features of (5.157) is a violation of Kohn’s theorem in the clean limit τimp = ∞. Suppose we take a translationally invariant fluid with σQ = 0. Then we see that the conductivities given by (5.156) diverge at ±ω = ωc. This is called the cyclotron resonance, and is associated with exciting a collective mode which uniformly rotates the momentum of the fluid. Kohn’s theorem states that this cyclotron resonance frequency is robust against electron-electron interactions in a Galilean-invariant fluid [490]. In a quantum critical fluid with σQ ≠ 0, but still τimp = ∞, we see that there is no longer any divergence, as the cyclotron resonance is damped by the particle-hole “friction”. The loophole in Kohn’s theorem is that we have both electrons and holes, and that a generic quantum critical system does not conserve the number of electrons (due to electron-hole creation/annihilation processes). It is not Galilean invariant.

Another interesting feature of (5.157) is the possibility for anomalous scaling in the Hall angle, defined as

The present discussion is at ω = 0. If σQ = 0, then

The last relation is concerned with the temperature scaling, which comes from τimp. Mechanisms that evade the connection (5.167) have been of interest since it has been known for some time that (5.167) is violated in the cuprates [143]. A nonzero σQ in the formulae (5.157) provides such a mechanism [102]: suppose that σxx(B = 0) ≈ σQ (recall from (5.77) that σxx is a sum of two terms). Using (5.157), the scaling tan θH ∼ τimp is unchanged. It follows that if τimp and σQ have unrelated T-scaling, then the scaling tan θH ∼ σxx need not hold. This conclusion requires σQ to dominate the dc conductivity in (5.77). If, instead, τimp is very large, perhaps in an ultra-clean sample, then tan θH ∼ σxx is recovered.

As we will later discuss below (5.188), at higher orders in perturbation theory in 1imp, corrections appear to the hydrodynamic theory of [364], and to the equations (5.157). In particular, if the exact holographic mean-field results are qualitatively different to the theory of [364] even at ω = 0 [37; 104; 476].

5.7.1 Weyl semimetals: Anomalies and magnetotransport

Let us also briefly discuss transport in models of 3+1 dimensional metals with approximate axial anomalies: Weyl semimetals. At the level of band theory, there are many predictions and experimental realizations [551; 770; 533] of metallic systems with ‘Weyl points’ in the Brillouin zone where the low energy degrees of freedom exhibit a chiral (axial) anomaly. This axial anomaly is an emergent IR effect: (i) high energy processes allow for scattering between fermions located near distinct Weyl points, and (ii) there is a theorem which states that the total axial anomaly of lattice fermions must vanish [603]. Temporarily putting aside this theorem, at the hydrodynamic level, the axial anomaly causes the breakdown of charge conservation [702; 601]:

5.168) is proportional to . In a background magnetic field, the transport problem for charge is ill-posed with such a violated conservation law, so clearly the microscopic processes which restore the conservation of global charge will play an important role.

A hydrodynamic approach to thermoelectric transport in such anomalous theories was developed in [539], by coupling together chiral relativistic fluids, allowing for the exchange of energy and charge between the fluids. The key result is that, if the magnetic field is oriented in the z-direction, there is an additional contribution to the conductivity σzz as a consequence of the axial anomaly:

where Γanom is a coefficient related to the rate at which the chiral fluids exchange charge and energy. It can be defined using the memory matrix formalism, and is small in the limit where the ‘Weyl’ behavior of the semimetal is pronounced, and hence Δσzz will be large. The B-dependence of this signal, only present parallel to , is called negative magnetoresistance (NMR). NMR in σzz was predicted earlier using kinetic approaches in [698], and hydrodynamic/holographic methods in [507]. Experimental signatures for electrical NMR were first definitively observed in [769; 419], along with many additional experiments since.

It was further noted in [539] that there is NMR in α and 5.168), and anomalous violation of energy-momentum conservation as well.) The link between NMR and the axial-gravitational anomaly is independent of the strong coupling limit, and can be observed in the weakly coupled limit accessible in experiment. A recent thermal transport measurement has confirmed this prediction [304].

There has been recent development of holographic approaches to studying the strongly coupled analogues of Weyl semimetals [507; 445; 509; 508]. The analogue of Γanom is added by coupling the axial gauge field to a certain background scalar [445]. When Γanom = 0, one might expect (5.169) to hold even beyond the hydrodynamic limit as the consequence of a new ‘Ward identity’, analogous to the case of a conserved momentum (5.42). A holographic calculation seems to confirm this conjecture [714]. In the future, it would be interesting to understand the interplay of NMR with strong disorder. It would also be interesting to study thermal transport holographically. However, following the lines of [539] would require a model with two separate ‘energy-momentum tensors’, only one of which is conserved. This may require the addition of a massive spin-2 field in the bulk, which is a delicate matter.

5.8 Hydrodynamic transport (without momentum)

So far, we have discussed hydrodynamic transport assuming that (up to weak disorder) energy, momentum and charge are all good conserved quantities. If translational symmetry is very strongly broken – for example, the amplitude of short wavelength disorder is large – then there is no reason to expect that the dynamics associated with momentum relaxation are slower than other microscopic processes. In this limit, the only conserved quantities are charge and energy [352]. (If inelastic phonon scattering is important, one should consider the combined electron-phonon system.) Therefore, the hydrodynamic description should only include fluctuations of the chemical potential μ and temperature T. We will call such a system an “incoherent metal”.

In the absence of external sources the two hydrodynamic equations of motion are conservation of charge and energy

There is no longer a conserved momentum. This furthermore means that velocity is no longer a hydrodynamic variable, and hence the constitutive relations for the charge and heat currents, to first order in the derivative expansion, are simply

As previously the heat current Q = JEμJ. It is immediate from the above expression that σ0, α0 and κ0 are nothing other than the thermoelectric conductivities. Recall that the matrix of thermoelectric susceptibilities is

5.170) with the constitutive relations (5.171) and susceptibilities (5.172) one obtains coupled diffusion equations for entropy and charge

5.170) imply that within linear response: . The above equations can also be expressed as the coupled diffusion of energy and charge [352]. The diffusion equations reveal the generalized Einstein relation according to which the matrix of conductivities is equal to the matrix of diffusivities times the matrix of susceptibilities.

Thus incoherent metals are characterized by two independent diffusive modes, the eigenvectors of the coupled diffusion equations above. If thermoelectric effects are weak (as is the case in e.g. conventional metals), then heat and charge diffusion are directly decoupled. Charge diffusion alone controls the electrical conductivity. The loss of Lorentz or Galliliean invariance means that we can proceed no further on symmetries alone – unlike the hydrodynamics described previously. There are few non-trivial constraints beyond the second law of thermodynamics, which enforces that the matrix of conducitivities is positive definite.

Until recently, most understanding of hydrodynamic transport in this incoherent limit has been phenomenological and non-rigorous. There is a large literature on ‘phases’ of transport in disordered media. Various methods for tackling such problems have been developed, including effective medium theory (EMT) [506] and analogies with resistor networks [483]. For simple inhomogeneous metals, EMT can work quite well. A simple model of the metal-insulator transition beyond EMT consists of a resistor network with a random fraction p of resistors deleted (so the resistance R = ∞ for that link); the classical percolation transition of these deleted links then becomes the metal-insulator transition [483]. Some of these techniques generalize to diffusive transport controlled by (5.173), and indeed to hydrodynamic transport with momentum [537].

Ultimately to use the approaches mentioned in the previous paragraph, we need knowledge of the coefficients in (5.171) or (5.173), and there has been recent progress in this regard. In the absence of useful symmetries, a compelling approach is to ask whether any of these coefficients are subject to universal bounds following from general principles of quantum mechanics and statistical physics. We will briefly discuss some (still speculative) ideas in this direction. The following sections below will discuss specific, calculable models of incoherent metals that are obtained in holography, and in the SYK models in §5.11.

A simple argument leads to the so-called Mott-Ioffe-Regel bound in quasiparticle theories of transport [334], in the presence of a sharp Fermi surface (in momentum space, so that excitations have a well defined momentum). There are several ways to formulate this bound but the one that is most relevant to our discussion is the following. A well-defined quasiparticle must have a mean free path lmfp = vFτ larger than its wavelength kF/vF, we obtain from the Drude formula

This bound is violated in many strange metals [334], and if these systems lack quasiparticles this is not surprising. Nonetheless, these compounds do seem to saturate the bound [114]

ℏ/(kBT). Such a lifetime bound is of course in the same spirit [673; 772] as the bound (1.1). It is desirable, however, to formulate a potential bound directly in the language of incoherent transport, namely, diffusion.

A precedent for the reformulation of uncertainty principle bounds as diffusivity bounds already exists. An argument identical to that of (5.175) can be applied to the ratio of shear viscosity to entropy density [495]. In a CFT at nonzero temperature, this ratio is precisely the transverse diffusivity of momentum [701], so that the famous conjectured viscosity bound of [495] can be written

We will be agnostic about the numerical prefactor in this and (most) other bounds [153]. Similarly, the bound in (5.175) can be reformulated, using the Einstein relation σ = and assuming decoupled heat and charge diffusion for simplicity, as a bound on charge diffusion [352]

under plausible assumptions about the charge susceptibility χ. Such a bound could provide a simple explanation for the ubiquitous scaling σ ∼ 1/T observed in incoherent metals [114].

Various aspects of the proposed bound (5.177) are unsatisfactory. In particular, vF is a weakly coupled notion, whereas the objective of formulating diffusivity bounds is to move away from weak coupling. Relatedly, clearly the diffusivity goes to zero at low temperatures in an insulator. Indeed, even the constant T = 0 diffusivity of free electrons in a disordered potential in three dimensions violates (5.177) if vF is interpreted too literally. An interesting recent proposal is that the velocity that appears in diffusivity bounds such as (5.177) is the ‘butterfly velocity’ vB [100; 101]. This is appealing because the butterfly velocity can be defined in any system, with or without quasiparticles.

In weakly coupled models the butterfly velocity is close to the Fermi velocity [723; 22]. In certain strongly coupled models, in contrast, it is found to be temperature dependent [316; 622; 175]. Recent results have furthermore suggested that the diffusive process most directly related to the butterfly velocity is in fact thermal diffusion [622; 103; 175; 57], with the potential bound

The butterfly velocity characterizes the spread of entanglement and chaos in quantum systems [691; 657], and such phase randomization is connected to energy fluctuations (and hence thermal diffusivity) because the Schrödinger equation maps the time derivative of the phase to the energy. In contrast, many different physical effects, unrelated to scrambling, can contribute to the charge diffusivity. The rate of growth of quantum chaos has recently been bounded [559], as discussed in §1.1. A bound in the spirit of (5.178) has recently been used to understand anomalous aspects of thermal transport in a cuprate [775]. In §5.11.2, we will present solvable models for which the equality (5.231) is consistent with (5.178).

Much of the evidence for a relation of the form (5.177) or (5.178) between D and vB comes from studying homogeneous momentum-relaxing models, discussed in §5.9 below. However, upon generalizing to inhomogeneous models, strong violations of (5.178) have been observed [315]. In a holographic setting, inhomogeneity affects Dρ and 5.177) with vFvB [547].

Although we have framed the discussion here around diffusion bounds, we will discuss specific holographic models in §5.10 where electrical and thermal conductivity (but not diffusion) can be rigorously bounded from below. The correct formulation of possible universal, model-independent bounds on transport (of energy, charge and momentum) remains a fascinating question at the time of writing. Proofs of appropriately formulated transport bounds may be within reach. A recent result has upper bounded the diffusivity in terms of the local equilibration time and a ‘lightcone’ velocity [346].

5.9 Strong momentum relaxation I: ‘Mean-field’ methods

The simplest way to model an incoherent metal holographically is through a ‘mean-field’ approach, where translational symmetry has been broken, but the spacetime geometry is homogeneous. Although this might seem contradictory, it is not. The essential ‘trick’ is to use a source that breaks translations but preserves some combination of translations with a different (often internal) symmetry. We will see that this leads to a homogeneous energy-momentum tensor and hence a homogeneous spacetime. These are not realistic models of translation symmetry breaking, but are often the simplest to work with: the bulk geometry, and response functions, are computed using ODEs rather than PDEs. Some results obtained have been found to be qualitatively similar to those following from more generic breaking of translation invariance.

Several versions of ‘mean-field’ translation symmetry breaking have been developed. In holographic ‘Q-lattices’ [223], the background is invariant under a combination of translation combined with rotation under an internal U(1) symmetry. With ‘helical lattices’ [236], the preserved symmetry is geometric; translation in a specific spatial direction (in 3 boundary spatial dimensions) is combined with a rotation in the plane perpendicular to the translation. This symmetry leads to a homogeneous but anisotropic spacetime and is an instance of the Bianchi classification of spacetimes with homogeneous boundary sources [425]. In ‘linear axion’ models [42], translations are combined with an internal shift symmetry.

We will give some explicit computations in linear axion models [42], which are perhaps the simplest of all. Furthermore, as elaborated in [730; 58; 21; 20], this broad class of models – where translations are broken by a linear source that preserves a combination of translations and shift symmetry – is precisely the Stückelburg formulation of a bulk ‘massive gravity’ theory dual to broken translation invariance. Momentum relaxation through massive gravity was initially formulated on its own terms without gauge invariance [743; 173]. Results obtained with Q-lattices and helical lattices are qualitatively similar to those we shall derive for linear axion models. We will gives references below. Let us emphasize again that there is no mystery concerning how these models are relaxing momentum. They have explicit non-translationally invariant sources (periodic or linear in a spatial coordinate).

We will illustrate the physics of mean-field models with the following action [42]:

Note that the number of scalar fields χ is equal to the number of spatial dimensions. We will be considering solutions in which the EMD fields depend only on the radial direction (as in all the backgrounds we have considered thus far), while the scalar fields

The parameter m quantifies the strength of translation symmetry breaking. The χ fields are often called axions, as they have a shift symmetry, just like an axion in particle physics. In a homogeneous background for the other EMD fields, it is easy to see that (5.180) solves the χ equation of motion. Now, note that the dilaton and graviton equations of motion depend only on derivatives of χI, and that the derivatives of χI are constants, and thus homogeneous in the spatial directions. Hence, the assumption that the EMD fields are homogeneous is justified.

In mean-field models, an explicit formula for σdc in terms of horizon data can be obtained. There are a variety of methods to do this. Let us short-cut to the ‘membrane paradigm’ technique developed in [106]. This is closely related to the method we used in §3.4.2 for the simpler case of a translationally invariant and zero density theory. For the present EMD-axion theory, we have essentially already computed the equations of motion in §5.6.5: in particular, (5.146) is valid for any homogeneous geometry, and if we rewrite (5.146c) in position space, and use the axion profile (5.180), we obtain

Now, we can carry through the same calculation as previously, but taking ω → 0 while not taking a perturbative limit m → 0. (5.146b) implies that

with C0 a constant. We can evaluate C0 both at the boundary and at the horizon. Using the near-boundary asymptotics of δAx, Z, b and we can conclude that as r → 0 only the δAx term contributes. In this way the constant is obtained in terms of the dc conductivity as

and similarly for other fields. (5.146a) implies that near the horizon ρδAx = Ldδ𝒫x, and combining this fact with (5.181), evaluating the constant in (5.183) on the horizon and setting it equal to (5.184) we obtain

where Z+ and Y+ are the values of Z(Φ) and Y(Φ) on the horizon, respectively. An expression analagous to (5.186) was found in very similar models by [224; 305].

Many models lead to results similar to (5.186), where the conductivity is a sum of two terms. The split is rather suggestive. While the formula (5.186) is nonperturbative in the disorder strength, there is an obvious comparison to make with the hydrodynamic prediction (5.77): the zeroth order term in ρ analogous to the incoherent ‘σQ’ – compare also the result (3.58) – and the quadratic term in ρ the analogue of the Drude term. We will see shortly that this comparison is not quite right. Let us point out three further important points:

  1. As m → 0, wherein translational symmetry is weakly broken, we recover the perturbative memory matrix regime [107; 535].
  2. As the translation symmetry breaking parameter m → ∞, with other quantities held fixed, the conductivity is not driven to zero but rather saturates at the value set by the first term in (5.186). This is rather interesting from a condensed matter perspective, as it is suggestive of the phenomenological ‘parallel resistor’ formula used to model resistivity saturation, see e.g. [334]. We will see that this is a generic prediction of many holographic models in d = 2.
  3. σdc is a sum of two contributions which can have different low temperature T-scalings. While the temperature dependence of Z+, Y+ and s can be expected to follow the quantum critical dimensional analysis developed in Chapter 4, ρ and m are temperature independent, and hence the dimensions to be soaked up by powers of temperature are different.

The last of the above points indicates a limitation of the mean-field approach. Because the background geometry is homogeneous despite breaking translation invariance, thermodynamic variables can have conventional critical behavior even though there is a scale m in the problem. In more generic models of strongly inhomogeneous fixed points, the nature of any emergent scaling regime will have to be strongly tied up with the nature of translation symmetry breaking. Disordered fixed points will be discussed in §7.2. Another interesting possibility is that z = ∞ scaling is a natural home for inhomogeneous fixed points with strong momentum relaxation, because space is not involved in the critical scaling [374].

One should also take into account the possibility that as m → ∞ a quantum phase transition to a different (e.g. gapped) phase might occur before the momentum relaxation rate becomes too large, cf. [12]. In our opinion this possibility is relatively underexplored. Recently it was noted that some of these linear axion models are unstable at large m [122]: this instability is signalled by the energy density becoming negative. As 𝜖 < 0 is a common feature of many of the simplest models [177], this may have important consequences for the existence of some of the simplest holographic incoherent metals.

5.9.1 Metal-insulator transitions

The ability to perform controlled calculations with very strong momentum relaxation makes the mean-field models a natural framework for metal-insulator transitions. We should first describe insulators. In an insulator the resistivity diverges rather than going to zero as T → 0. Holographic models allow for gapless as well as gapped insulators, as we will discuss in turn. Gapless holographic insulators were constructed in [236; 224; 305; 234; 59; 309]. The essential part of the computation is to find near-horizon scaling solutions to the equations of motion in the presence of fields (such as the linear axion) that break translation invariance. The geometry itself is homogeneous and so still has the form (4.44). Nonetheless, solving the equations of motion can be tricky, even if they are only ODEs. We shall not discuss the explicit backgrounds here – the reader is referred to the papers for details. With the solutions at hand, the temperature dependence of the resistivity is obtained from formulae such as (5.186). The background is an insulator if both of the terms in (5.186) go to zero as T → 0. Typically in these cases, the first term – that does not depend explicitly on the charge density – is the larger one at low temperatures.

The holographic insulators obtained in this way are quite different from the more familiar classes of insulators in condensed matter physics: band insulators, Anderson insulators and Mott insulators [212]. Band and Anderson insulators make essential reference to single-particle properties whereas a Mott insulator requires commensurability leading to an emergent particle-hole symmetry. Instead the holographic insulators described in this section are best understood as the consequence of relevant (i.e. strong in the IR) generalized umklapp scattering from a periodic potential. That is to say, the scattering from the potential is simply too strong to allow the current to flow as T → 0. This physics allows for anisotropic systems that are insulating in one direction but conductive in another [236].

In a conventional Fermi liquid metal, umklapp processes are irrelevant [358] and lead to the well known T2 resistivity mentioned in §5.1 and §5.6.4 above. In one spatial dimension, it is known that umklapp can become relevant in a Luttinger liquid [250]. In holographic models, the most natural starting point for relevant umklapp scattering (i.e. for an operator in the theory that has a periodic space dependence) is a z = ∞ compressible phase. We have already noted around equation (5.136) that z = ∞ fixed points are especially susceptible to spatially periodic deformations. In equation (5.136) it is explicitly seen that if the exponent of the latticized operator then the resistivity diverges at low temperatures. This divergence indicates that at sufficiently low temperatures the lattice will backreact strongly on the metric. In general, the field ϕ describing the amplitude of the lattice will grow like

as r → 0 in the IR of the z = ∞ geometry. This is the behavior we have already seen in e.g. (4.69). The growth of the field shows that eventually backreaction will be important. In the cases studied in [236; 224; 305; 234], this backreaction induces an RG flow to a new, insulating interior geometry.

If the dimension νkL of the lattice deformation in the low energy compressible phase can be continuously tuned at T = 0 from being irrelevant to relevant – for instance by varying the charge density at fixed lattice wavevector – then a metal-insulator transition results [236]. This appears to be an infinite order quantum phase transition, similar to the quantum BKT transitions that we discuss in §6.6.

Once the background is understood, the temperature and frequency dependent conductivities are computed using the same holographic methods we have described previously. In Figure 5.3 we show the low temperature optical conductivity in the metallic and insulating phases of the model described in [236]. These plots show the canonical spectral weight transfer expected through a metal-insulator transition. The spectral weight in the Drude peak is transferred to ‘interband’ energy scales of order the chemical potential.

Figure 5.3
Spectral weight transfer: Optical conductivity in the metallic (left) and insulating (right) phases of the model from [236]. Curves with lower (higher) σdc are at higher temperature in the metallic (insulating) phase. Figure adapted with permission.

As can be seen in Figure 5.3, the insulating phase only has a ‘soft’ power law gap in the spectral weight. For the particular model shown, in fact, σ ∼ ω4/3. Relatedly, the resistivity ρdc ∼ T−4/3. The soft gap is due to the persistence of a black hole horizon in the solution with a scaling near-horizon geometry. There exist quantum critical excitations, but they are not able to conduct efficiently due to strong interactions with the lattice.

A gapped insulator is instead obtained by starting with a geometry that is gapped to charged excitations (cf. §2.4 above), and then breaking translation invariance in a manner that does not destroy the charge gap [132; 65; 482]. The role of strong interactions here is not to have strong momentum relaxation but instead to gap out the charge sector in a situation where there is no translational zero mode.

At zero charge density, the second term in (5.186) is zero. In this case a vanishing conductivity at T = 0 can also be obtained if Z+ = 0 on the zero temperature horizon [576]. In such zero density systems, it is not necessary to break translation invariance.

5.9.2 AC transport

The low frequency conductivity σ(ω) can also be computed in mean-field models. In particular, by including next-to-leading order effects in the translational symmetry breaking parameter m2, taken to be small, one can see the relationship between the two terms in the exact result (5.186) and the two terms in the dc limit of the hydrodynamic formula (5.77). For a simple linear axion model of the form (5.179) in d = 2, the conductivity at small m and ω was obtained in [176; 99] as:

and σQ is the clean incoherent conductivity given in (5.51) above. The temperature dependent quantity λ is given in [176; 99].

To leading order at small m, and using (5.189), the weight of the Drude peak in (5.188) is seen to agree with the hydrodynamic result (5.77). However, in the dc limit ω → 0, the subleading terms in the spectral weight of the Drude peak contribute at the same m0 order as the clean incoherent contribution. In fact, we see that there is a cancellation so that in addition to σ(ω), one sees that the hydrodynamic results §5.4.5 and §5.5 do not have enough parameters to reproduce the full answers at order m0 [176; 99].

At higher orders in perturbation theory, [177] has demonstrated the loss of a Drude peak in once m ∼ T in this same axion model, in agreement with the qualitative predictions of incoherent transport in [352].

5.9.3 Thermoelectric conductivities

So far, we have focused on computing the electrical conductivity σ. Computations of α and can be a bit more involved, because fluctuations of the bulk metric participate, but let us sketch out the procedure [35; 225; 36]. We follow a similar approach to that used in §3.4.2; again, we assume d = 2 for simplicity, and use the axion model from before. This time, the bulk perturbations take the form

along with time-independent perturbations δχx and δgrx. The time-dependent parts of the perturbations above are fixed by employing the tx component of the linearized Einstein’s equations and the x-component of the Maxwell equations, and showing the t-linear piece only vanishes when the above choices are made. Asymptotic analysis near r = 0 confirms that these perturbations encode the electric field E and thermal “drive” ζ.

The linearized Maxwell equations can be shown to imply that rJ = 0, where

The asymptotic behavior as r → 0 implies that J is nothing more than the expectation value of the electric current in the dual field theory. Next, the t-independent part of the tx-Einstein equation implies that rQ = 0, where

and again boundary asymptotics imply Q is the boundary heat current. The strategy is now to evaluate the constants J and Q at the horizon and in this way relate the currents to the electric field and thermal drive and hence obtain the conductivities.

Demanding regularity near the horizon, one finds that

5.192) drops out because At = 0 on the horizon. Furthermore, solving the linearized rx-Einstein equation, along with the background equations of motion, pertubatively near the horizon enforces

5.193) and using the definitions of and

Using the expression for J, we can independently compute σ, as given in (5.186), and confirm that Onsager reciprocity holds by an independent computation of α.

When m → 0, we recover the Drude predictions for thermoelectric conductivities as derived from hydrodynamics. As m gets larger, we see deviations from the hydrodynamic predictions – there is no σQ dependence in α or 5.46). This is the same issue that we discovered in §5.9.2 – namely, the transport theory of §5.4.5 and §5.5 is not correct beyond leading order in perturbation theory in m.

5.10 Strong momentum relaxation II: Exact methods

5.10.1 Analytic methods

Recent remarkable results have in a certain sense ‘solved’ the problem of computing the dc thermoelectric conductivities in holographic models with arbitrary inhomogeneous sources. These results go well beyond the mean-field models considered thus far, and amount to a substantial generalization of the mean-field expressions (5.186) and (5.195). Specifically, it was shown in [226; 70] that the dc transport problem can be reduced to the solution of some ‘hydrodynamic’ PDEs on the black hole horizon. The two key facts here are (i) even if the boundary sources themselves vary over microscopic distance scales and with large amplitudes, there exists an effective hydrodynamic description that computes the dc conductivities and (ii) this hydrodynamic description depends only on the far IR geometry of the spacetime, the black hole event horizon. In these results the old black hole membrane paradigm discussed in §1.2 has finally found its holographic home.

The derivation of the dc conductivities in [226; 70] is essentially a significantly souped up version of the membrane paradigm calculation of thermoelectric conductivities that we just reviewed in the previous section for the axion model, and so we simply state the results. We assume that the black hole horizon is at constant temperature T > 0 and is connected, with the topology of a torus (see [70] for more general results). The answer is simplest if we employ Gaussian normal coordinates near the horizon [575], in which case the background EMD fields have near-horizon expansion (switching to radial coordinate

Here γij is the induced metric on the black hole horizon. The horizon data itself is three functions of x: {γij(x), ϕ(x), p(x)}. One then writes down the following linear partial differential equations for variables {vi(x), δμ(x), δΘ(x)} that depend on these background fields as well as two constant sources Ei and ζi (note that with indices raised by γij, Ei and ζi will often not be constant):

With solutions to the horizon fluid equations (5.197) at hand, one can show that the spatially averaged charge and heat currents in the boundary theory are given by

A linear expression for {vi(x), δμ(x), δΘ(x)} in terms of the sources Ei and ζi is obtained by solving the fluid equations. Therefore, with such a solution, the formulae (5.199) give expressions for J and Q that are linear in E and ζ, and hence it is straightforward to read off the thermoelectric conductivities (5.4).

The ‘horizon fluid’ equations (5.197) look like the equations describing transport in a relativistic fluid on a curved space, with inhomogeneous background fields such as the chemical potential [537], but with strange equations of state. In particular, the thermodynamic Maxwell relations are not obeyed for any sensible equation of state.1 In a simple axion model, [99; 98] has perturbatively shown that these equations of state can be understood from a hydrodynamic perspective, by choosing a curious fluid frame where the fluid velocity is proportional to the heat current (cf. (5.199b) above). This seems to be a convenient frame in holography. One can gain further intuition by sprinkling factors of 5.198) – then sh follows from the Bekenstein-Hawking formula, and ρh is the local charge density on the horizon [537].

In general, the horizon fluid equations are a dramatic simplification from a numerical point of view – given a complicated black hole geometry, one can solve a lower dimensional PDE to compute the dc transport coefficients. Remarkably, there are some powerful statements that can also be made analytically. For example, Onsager reciprocity immediately follows from the hydrodynamic form of these equations [537].

One can analytically solve the transport equations if translational symmetry is only broken in a single direction. In fact this was noted earlier [227; 649]. To understand why this is so, consider the following simple limit. If we have a charge-neutral system, then the hydrodynamic equations reduce to a simple Laplace-like equation, which governs steady-state diffusion of charge. Discrete diffusion equations are resistor networks (one thinks of 5.197). One finds using this approach that the inverse conductivity matrix is expressed as a single integral over local quantities on the horizon [70], just as the power dissipated in the resistor network is the sum of local Joule heating:

where we have assumed that translation invariance is broken in the x-direction, and assumed that γxj = 0 for xj, for simplicity. The combinations of conductivities appearing in (5.200) are the components of the inverse of the thermoelectric conductivity matrix σ from (5.4). This determines Joule heating through (σ−1)ABJAJB, with JA being the electric and thermal current operators. It is straightforward to obtain an expression for σ, 5.200), but the inverse matrix is more physically transparent.

While the horizon fluid equations cannot be solved analytically when translation invariance is broken in more than one direction, they do allow for the proof of non-trivial lower bounds on conductivities for bulk theories dual to d = 2 metals. For simplicity, we assume that the metal is isotropic. The intuition behind this approach again follows from resistor network tricks [537]. That is, given any arbitrary conserved flows of heat and charge in the horizon fluid, the one which dissipates the minimal power solves the equations of motion. Hence, the power dissipated on other configurations gives us lower bounds on the conductivities. The calculations are a bit technically involved, so let us state without proof the results, which hold for any isotropic theory with a connected black hole horizon. The first is a bound on the electrical conductivity in the Einstein-Maxwell system [313]:

This bound generalizes in a nautral way to any EMD theory [97]. In that more general setting the thermal conductivity κ can also be bounded [314]:

We have written the coupling in the Einstein-Hilbert action as 2κ2 = 16πGN to avoid the symbol κ meaning two different things in the same equation. For theories with bounded potentials (which can generally occur for models with θ ≥ 0 in the clean limit) we hence find a strictly finite thermal conductivity at finite T (the factor of T is trivially necessary by dimensional analysis).

Both of these bounds are saturated in some of the simplest mean-field models, implying that these mean-field approaches to breaking translational symmetry are – at least in part – quantitatively sensible. In particular, in a simple Einstein-Maxwell model with massive gravity [106] or axions [42], one finds the dc conductivity (5.186) with Z+ = Y+ = 1, which we readily see saturates (5.201) either when m → ∞ or ρ → 0. In this same model, the thermal conductivity is given by [225]

where the inequality follows from bounding the entropy density in this particular model by ss(T = 0) = π(2m2 + μ2)/3. This again agrees with (5.202) upon using that V = −6 for the Einstein-Maxwell system.

The bounds on σ and κ above are powerful non-localization theorems in specific strongly interacting metals. However much the strength of disorder is increased, the conductivity is never driven to zero. These results amount to a no-go theorem on the possibility of realizing a many-body localized state in certain phases of certain holographic models. A many-body localized state is non-ergodic, with the absence of thermalization and transport coefficients vanishing at finite energy density [597]. This is in contrast with (5.201) and (5.202) – hence we conclude that at least some holographic models with connected black hole horizons do not realize such many-body localization. We will discuss disconnected horizons shortly, in §5.10.2. Another possibility to keep in mind are possible first order phase transitions to geometries with no horizon. We have already noted in §5.9.1 that insulators are obtained from inhomogeneous geometries with a charge gap.

A bound away from zero on the electrical conductivity such as (5.201) is impossible even with horizons in the gapless (mean-field) insulating geometries discussed in §5.9.1 above. In an insulator the conductivity goes to zero at zero temperature. Those models are either anisotropic or have additional fields that effectively drive Z+ to zero in the horizon formula (5.186) for the dc conductivity (in addition to translation symmetry breaking being strong enough that the second term in (5.186) is absent at T = 0).

The essence of the challenge of bad metals (that violate the MIR bound (5.3)) is not to show that insulators cannot exist, which they clearly do, but rather to demonstrate that it is possible for strongly interacting non-quasiparticle systems to remain metallic even in the face of strong disorder. From this perspective, a bound such as (5.201) on the conductivity in a specific system (Einstein-Maxwell theory) is an exciting development.

It has recently been shown that the dc magnetotransport problem in EMD models reduces once again to solving fluid-like equations on a black hole horizon [229]. Interestingly, particle-vortex duality in holography is robust to disorder [313; 229], and the bound (5.201) can partially be understood as a consequence of this duality [313].

5.10.2 Numerical methods

To employ the horizon fluid equations (5.197) in a specific model one needs, of course, to first construct an inhomogeneous black hole background. In essentially all cases, this must be done numerically – a fact that has spurred several recent developments in numerical general relativity [211]. Furthermore, to compute conductivities at finite frequency, we need to linearize around the inhomogeneous black hole background. Both of these procedures are computationally intensive. We will briefly discuss some classes of inhomogeneous backgrounds that have been studied numerically.

Breaking translation invariance in a single direction, x, minimizes the number of inhomogeneous dimensions and hence offers a significant saving in computational cost while still capturing important momentum-relaxing physics. For this reason, this case has been widely studied. A minimal way to do this is allow an inhomogeneous chemical potential of the form [410; 529]

This can also be done with scalar fields [411]. These potentials often go by the name “holographic lattices”. From a microscopic perspective, this is a little different to an actual ionic lattice in a traditional condensed matter model: there is no relation between the lattice spacing and the charge density, a point we will return to later. However, as we have repeatedly stressed above, the real objective is to understand the universal low energy physics, and from this point of the view one should consider (5.204) as simply a way to source a spatially periodic structure. Furthermore, given that cases with an irrelevant lattice can be understood from perturbation theory as in §5.6.4 above, the most important objective of numerical studies is to identify new intrinsically inhomogeneous IR geometries that are beyond perturbation theory.

As described in §5.6.4 above, a periodic source such as (5.204) is not expected to have strong effects in z < ∞ geometries. Each lattice mode decays exponentially towards the interior of the spacetime. Indeed, when μ0 = 0, numerical study shows that even a strong periodic deformation of an AdS spacetime simply flows back to a rescaled AdS in the far IR [138]. Even when μ0 ≠ 0, in z = ∞ spacetimes (as would arise in Einstein-Maxwell theory), the amplitude of the lattice scales as (5.187) – that behavior is not specific to homogeneous lattices. In Einstein-Maxwell theory, this scaling corresponds to a decay towards the interior of the spacetime because for all k one has 2 IR geometry prior to deformation by a lattice) [358]. Numerical work supports this conclusion [227], though other numerical results suggest that nonlinear effects might lead to the survival of inhomogeneities as T → 0 [374] (this evidence is most compelling with a large amplitude lattice). The T → 0 limit is challenging to precisely address numerically with present day methods, so it is worth looking for new numerical or analytical techniques to conclusively settle this issue.

Beyond Einstein-Maxwell theory (in more general EMD models) the exponents νk can be such that the lattice grows towards the interior of a z = ∞ IR spacetime. This is what is going on in the paper [649], who have shown a metal-insulator transition as a function of the wavelength of a periodic potential source for the dilaton (at fixed charge density). The insulating phase has (presumably) an inhomogeneous low temperature horizon. This transistion is entirely analogous to the original metal-insulator transition of [236], discussed in §5.9.1 above, and confirms the persistence of that physics in a more generic (inhomogeneous) setting.

Beyond universal IR physics, the numerically constructed solutions allow one to compute the ac conductivities at all frequencies. This is a probe of higher energy physics. An example of the kind of results one obtains in d = 2 is shown in Figure 5.4. This is for relatively weak momentum relaxation, so that a Drude peak (5.90) controls the low frequency conductivity. An additional structure is seen at intermediate frequencies ω ∼ kL, associated with resonances with the modulating potential [410; 138]. At high frequencies ω, which are much larger than all other scales in the problem, one finds that σ(ω) 1 (in units with the coupling e = 1), in agreement with (3.72).

Figure 5.4
ac conductivity in an inhomogeneous background. Illustrative plot of the real and imaginary parts of σ(ω) in a holographic model deformed by the periodic chemical potential (5.204). The dashed lines show the corresponding conductivities without the lattice deformation. The lattice plot shows a Drude peak at small frequencies, a resonance due to scattering off the lattice with ω ∼ kL and the asymptotic behavior σ → 1. Figure taken from [410] with permission.

In addition to lattices, a natural class of inhomogeneous sources to consider are random potentials. These describe random disorder. As we have discussed in §5.6.4, random disorder contains inhomogeneities at all wavelengths and is therefore capable of having a strong effect on the IR physics even for z < ∞. We will discuss the IR geometry of disordered black holes in §7.2.

A final class of inhomogeneities that has revealed some rather interesting physics are pointlike defects [405; 413]. A sufficiently strong charged defect can lead to a deformation of the horizon even in the zero temperature limit. Furthermore, it was found that as the temperature is lowered the defect can cause part of the horizon to split off, leading to a localized ‘hovering’ black hole above the deep IR black brane horizon. This suggests a novel kind of ‘topological order’ associated with the defect. The hovering black holes are challenging to find numerically, but can be understood heuristically as the statement that some black hole spacetimes admit static, charged geodesics. It is natural to imagine placing a point particle of charge q and mass m at such a geodesic, and ‘growing it’ into a tiny black hole. The existence of such hovering defect solutions suggests the possible existence of disordered or lattice black holes with many such ‘hovering’ horizons. In this ‘point mass’ approximation, geometries with many hovering black holes have been proposed [45]. Such geometries with a density of disconnected horizons may avoid the ‘no-go theorems’ on many-body localization that we have mentioned above [313; 314].

5.11 SYK models

A class of random fermion Sachdev-Ye-Kitaev (SYK) models [671; 484] have emerged as important solvable models of incoherent metal states that are stable at T = 0. An explicit derivation of a holographic gravitational dual of the long time energy and number fluctuations can be established, with properties closely related to the AdS2 horizons of charged black holes discussed in §4.3.1. A class of higher-dimensional SYK models [316] exhibit diffusion of energy and number density at T = 0 in a metallic state without quasipartices: thus, they provide an explicit quantum matter realization of the holographic models with strong momentum relaxation that were examined in §5.8, §5.9, and §5.10, and of the holographic disordered physics discussed in §7.2. Finally, as we noted in §1.1, these models saturate the bound in (1.2) for the Lyapunov time to quantum chaos [484; 560].

The simplest SYK model of direct relevance to strange metals has the nonlocal Hamiltonian [667]

A local generalization will be considered in §5.11.2, enabling the realization of incoherent transport with strong momentum relaxation. Here, the ci are fermion operators on sites i = 1…N, which obey the standard fermion anti-commutation relations

The properties described below hold for all 0 < ρ < 1. The couplings Jij; kℓ are independent random variables with Jij; kℓ = 0 and |Jij; kℓ|2 = J2.

At this point, the reader is likely concerned about the random nature of the SYK model, and its role in the context of holographic duality. However, a key property is that the disorder largely self-averages in the limit of large N: in particular, the on-site Green’s function on any site of a single sample is the same, and equal to the disorder-averaged value. The holographic dual will describe these disorder-averaged Green’s functions, and so will not have any explicitly random couplings.

We now describe the critical strange metal obtained in the N → ∞ limit. The simplest way to generate the large N saddle point equations is to simply perform a Feynman graph expansion in the Jijkℓ, followed by a graph-by-graph average. Then we obtain the following equations for the on-site fermion Green’s function [671]

here ω is a frequency and τ is imaginary time. There is a remarkably rich structure in the solution of these equations, and we will present a few highlights here. It is not difficult to establish that any solution of these equations must be gapless [671]: briefly, if there were a gap, the first equation in (5.208a) would imply that G and Σ must have the same gap, while the second equation implies that the gap in Σ is 3 times the gap in G, which is a contradiction. The low frequency structure of the Green’s function can be determined analytically [671]

for some complex number A, where z is a frequency in the upper-half of the complex plane. The result for the local Green’s function in (5.209) indicates a divergence in the local density of states at the Fermi level, and the absence of quasiparticles in a compressible quantum state with a continuously variable density ρ.

We now itemize some further properties of HSYK.

1. The imaginary time Green’s function at T = 0 can be written as

5.209). It is clear that is a measure of the particle-hole asymmetry in the fermion density of states: for > 0 ( < 0), the density of states for inserting a particle (hole) is larger. The value of can be analytically related to the value of ρ via an argument similar to that required to prove the Luttinger theorem [293].

2. It is also possible to solve (5.208a) for the Green’s function at small non-zero temperatures T J. It was found that G = gs with [615]

This form resembles the local Green’s function of a conformal field theory.

3. The entropy, S, of the large N state was computed in [293], and it was found that this entropy did not vanish in the low temperature limit S(T → 0) = Ns0. An expression was obtained for s0 as a function of the density ρ. Note that a non-zero s0 does not imply an exponentially large ground state degeneracy; rather it is a consequence of the exponentially small level spacing of the many-body spectrum, generically found in states with finite energy density, extending all the way down to the ground state [283].

After the appearance of black hole models of strange metals, it was realized in [674] that all three properties of the SYK model listed above match precisely with those of the charged black holes we examined in §4.3.1 and §4.5.2: specifically gs(τ) in (5.211) is the Fourier transform of the IR Green’s functions of AdS2 horizons [259] examined in §4.5.2 (specifically, the nonzero temperature Green’s function of equation (4.122)), while the entropy Ns0 of [293] matches the non-zero Bekenstein-Hawking entropy of the AdS2 horizon in (4.34). More recently [667], it was noted that the match between the Green’s functions in (5.211) and (4.122) implied that the particle-hole asymmetry parameter, , was equal to the near-horizon electric field of the black hole, which can be determined from (4.30). Furthermore, it was found that [667] both the SYK model, and the black hole solution in §4.3.1 obeyed the relationship [683; 667]

4.31). This result is further evidence for the identification of the entropy of the SYK model [293] with the Bekenstein-Hawking entropy of the black hole, and of the close connection between SYK models and AdS2 [674; 667].

5.11.1 Fluctuations

Recent work has made the holographic connection between SYK models and AdS2 horizons quite explicit by deriving a common effective action for T > 0 energy and density fluctuations in both models. Furthermore, this work has highlighted the crucial role of many-body quantum chaos, and established [484; 560] that the SYK models saturate the lower bound for the Lyapunov time in (1.2). Black holes also saturate this chaos bound [691], this further strengthens the connection between SYK and quantum gravity on AdS2. We will not present the chaos computation here, but will outline the derivation of the effective action for energy and density fluctuations starting from the SYK side.

The basic structure of the effective action can be largely deduced by a careful considerations of symmetries. While solving the equations for the Green’s function and the self energy, (5.208a), we found that, at ω, T J, the iω + μ term in the inverse Green’s function could be ignored in determining the low energy structure of the solution. The μ cancels with the leading term in Σ(z) in (5.209), while the iω is less important than the 5.208a) are invariant under [484; 667]

where f(σ) and g(σ) are arbitrary functions representing the reparameterizations and U(1) gauge transformations respectively. Furthermore, it can be shown that not just the saddle-point equations, but also the entire action functional for the bilocal Green’s function [650; 293; 667; 444] is invariant under (5.213) at low energies.

The second key aspect is that the large set of (approximate) symmetries in (5.213) are spontaneously broken by the saddle point solution in (5.210) and (5.211); the saddle point is only invariant under a small subgroup of these transformations. For the reparameterizations, we can only choose the SL(2, ) subgroup [484]

5.210) is invariant under (5.213). Similarly, we can only choose the global U(1) transformation for g(τ) = e−iϕ. (At T > 0, (5.211) implies that there is subtle intertwining of the SL(2, ) and U(1) symmetries, for which we refer the reader to [175].)

The reader should now note the remarkable match between the low energy symmetries of the SYK model, and those of the Einstein-Maxwell theory of AdS2 horizons: this is ultimately the reason for the mapping between these models. Like the SYK model, the Einstein-Maxwell theory has a reparameterization and U(1) gauge invariance, and SL(2, ) is the isometry group of AdS2.

We now present the effective action for the analog of the Nambu-Goldstone modes associated with the breaking of the approximate reparameterization and U(1) gauge symmetries to SL(2, ) and a global U(1) respectively. To this end, we focus only a small portion of the fluctuations of the bilocal G by writing

5.211). Now we need an effective action for f(τ) and ϕ(τ) which obeys the crucial constraint that the action vanishes exactly for all f(τ) and ϕ(τ) which leave (5.215) invariant, i.e. the G(τ1, τ2) on the l.h.s. equals gs in (5.211); for such f(τ) and ϕ(τ) there is no change in the Green’s function, and hence the action must be zero. The action is obtained by a careful examination of this constraint, and also by explicit computation from the SYK model [560; 175]; after writing

The curly brackets in the last term represent a Schwarzian

5.214). The couplings in (5.217) are fully determined by thermodynamics: in terms of (5.172), the matrix of thermodynamic susceptibilities is

where c is the specific heat at fixed chemical potential, χ = (∂ρ/∂μ)T is the compressibility, and ζ = (∂ρ/∂T)μ.

We can obtain correlators of the heat and density fluctuations in the SYK model by the identifications

where E is the energy operator. We define

5.217) to quadratic order in ϕ and 𝜖 to obtain the Gaussian action

where ωn is a Matsubara frequency. Note the restrictions on n = 0, ±1 frequencies in (5.222), which are needed to eliminate the zero modes associated with SL(2, ) and U(1) gauge invariances. Computing correlators under (5.222) for the observables in (5.220) we obtain results consistent with the fluctuation-dissipation theorem and the susceptibilities in (5.219): indeed, this computation is a derivation of (5.219).

Finally, a key observation is that the action (5.217) can also be obtained from a gravitational theory of a black hole. This has been established so far for the case without density fluctuations represented by ϕ(τ), although we do expect that the gravitational correspondence is more general. Integrating out the bulk modes from a gravitational theory in AdS2 leads to an effective action for the boundary fluctuations of gravity, represented by f(τ), which coincides with the Schwarzian term in (5.217): we refer the reader to [561; 441; 254; 164] for details. This result confirms the intimate connection between SYK and holographic theories of strange metals.

5.11.2 Higher dimensional models

Gu et al. have defined a set of higher-dimensional SYK models [316] which turn out to match the holographic transport results in §5.9.3, in the case of a ‘maximally incoherent’ Einstein gravity model with axions. For the complex SYK model in (5.205), the one-dimensional Hamiltonian on sites, x, can be written as

5.205) on each site

Fig 5.5:

Figure 5.5
A chain of coupled SYK sites with complex fermions: each site contains N 1 fermions with on-site interactions as in (5.205). The coupling between nearest neighbor sites are four fermion interaction with two from each site. Figure adapted from [316] with permission.

The coupling constants {Jjklm, x} and with zero mean and the variances:

This model can be analyzed along the lines of §5.11.1. The diagonal single fermion Green’s functions retain the form in (5.209) of the original SYK model: in the context of the higher-dimensional models, the spatial independence of the Green’s function implies that they obey a ‘locally critical’ z = ∞ scaling. Along with this z = ∞ scaling, the non-vanishing T → 0 limit of the entropy also remains unchanged in the higher dimensional case. However, there is an important difference for the effective action for the density and heat fluctuations in (5.222); for the higher dimensional case, this becomes an action for diffusive density and heat modes

𝜖 are functions of Matsubara frequency ωn and wavevector q. The action (5.227) contains two temperature-independent diffusion constants D1, 2 which control the transport of heat and number density; their values are determined by the off-site couplings J′ in the Hamiltonian.

We can now proceed as for (5.222), and compute correlators ϕ and 𝜖, and hence the full thermoelectric conductivity matrix in (5.171). We obtain [175]

We now compare the transport coefficients in (5.228), (5.229), and the thermodynamic susceptibilities in (5.219), with the holographic results in §5.9.3, and find an excellent match. Equation (5.228) shows that the SYK model realizes the incoherent transport of §5.8; the three transport coefficients σ, α and κ are determined by just two diffusivities, D1, 2, and the thermodynamic susceptibilities. In the SYK case, this further leads to the following relationship between the thermoelectric and electric conductivities [175]

5.228) and (5.212). It can be verified that (5.230) is also obeyed exactly by the black holes with AdS2 horizons and mean-field disorder, where α is of the form discussed in §5.9.3. The match between the higher-dimensional SYK and holographic models also extends to the facts that they obey z = 2 scaling for the heat and number diffusivities (as implied by their T independence), while obeying z = ∞ scaling for the fermion Green’s functions along with non-zero entropies in the T → 0 limit.

Recent work has computed properties associated with many-body quantum chaos, which were discussed briefly in §1.1 and §5.8. It was found that the diffusivity, D2, and the butterfly velocity, vB, are related by

in both the SYK models and momentum-dissipating holographic theories with AdS2 horizons [316; 103; 175]. This equality is consistent with the bound (5.178), as we noted in §5.8. However, generalizing these higher-dimensional SYK models by allowing [315].

Exercises

5.1.  Fluid-gravity correspondence. In §5.4.1, we mentioned that the dynamical Einstein equations for asymptotically AdS black holes reproduce hydrodynamics in the dual theory. This exercise will give a flavor for why this is true. Consider the geometry

where Pμν = ημν + uμuν, uμ(x)uμ(x) = −1, and f(r, x) = 1 − (4πT(x)r/(d + 1))d+1. Here xμ = (v, xi), where xi correspond to spatial coordinates and v is an ‘infalling’ time coordinate. If T(x) and uμ(x) were xμ-independent, this would be the AdS-Schwarzchild black hole.

a) Show that if T and uμ are slowly-varying functions of the length scale L, and TL 1, then (5.232) solves all components of Einstein’s equations, up to corrections of 𝒪((TL)−1).

b) Furthermore, if , show that Eμr = 0 is exactly satisfied at first order in derivatives only when μTμν = 0, where Tμν is the stress tensor of the conformal plasma – determined by T(x) and υμ(X) – at leading order in the derivative expansion.

5.2.  Sound modes from black holes. In this exercise you will show that sound modes arise as hydrodynamic quasinormal modes of planar black holes, as discussed in §5.4.2. This exercise follows the logic in [500] and the method is similar to that used in §3.4.3 to find the diffusion mode. You will find the sound mode in pure gravity.

a) The background is the AdS-Schwarzschild black hole given in (2.37) with z = 1 and θ = 0. The emblackening factor f(r) is (2.42) and the temperature is (2.43). This exercise will be in dimension d = 2. The sound mode is in the longitudinal channel. Therefore, if the wave is travelling in the x direction, it is sufficient to consider a perturbation to the background of the form

A choice of gauge allows the δgμr components to be set to zero. Add this perturbation to the background and linearize the Einstein equations (1.22) to obtain coupled differential equations for {Htt, Hxx, Hyy, Htx}. You should do this and the rest of this exercise using e.g. Mathematica.

b) Show that the combination of variables

obeys a single decoupled second order differential equation.

c) Look for a solution to the decoupled equation of the form

. Plug the ansatz above into the equation, expand in small 𝜖 and obtain equations for s1(r) and s2(r).

d) Verify or show that the solution that is regular at the horizon is

e) Impose limr→0[r2 (s1(r) + s2(r))] = 0, implying the absence of a source at the boundary. The formal expansion parameter has been set to 𝜖 = 1. Thereby obtain a dispersion relation ω(k) for a quasinormal mode. Expand at low wavevectors k. Does your result agree with (5.37)? Hint: Remember to set ρ = μ = 0, and use (3.60) and (4.4).

5.3.  Bianchi spacetimes. Consider the Einstein-Proca action (2.17).

a) Show that the following ansatz solves the equations of motion:

for suitable choices of the constants α, β, k, A0, G0, Λ and mA [425].

b) Show that the following ansatz solves the equations of motion:

for suitable choices of the constants α, β, γ, A0, Λ and mA [425].

c) In each case, determine which components of the thermal conductivity ij are infinite, and which components are finite.

5.4.  Membrane paradigm for an inhomogeneous black hole. Consider a static, inhomogeneous solution to the EMD system (4.43), but with vanishing gauge field A = 0. In this exercise, we will show how to generalize the membrane paradigm to compute the conductivity. It will be convenient for our purposes to work in a coordinate system where an asymptotically AdS boundary is located at r = ∞, and a black hole horizon of temperature T is located at r = 0. In this problem, you may assume for simplicity that the metric as r → ∞ is ds2 = L2[r−2dr2 + r2dxμdxμ].

a) Explain how regularity enforces (in a reasonable gauge) the following conditions on the near-horizon metric:

conditions A(r → ∞) = −Eitdxi. Show that regularity as r → 0 enforces

is independent of bulk radius r, and that it equals the spatially averaged current in the boundary theory. Here Vd is the volume of the spatial directions in the boundary theory.

d) Show that α obeys a constraint equation on the horizon:

By evaluating Ji near r = 0, show that the conductivity is given by the algorithm described in §5.10.1 for an uncharged black hole. Note that α defined above is equivalent to − δμ in (5.197b).

5.5.  Conductivity bounds. In this exercise we will derive (5.201), following [313]. We define the functional

𝒱i = 0.

a) Suppose that vi, δΘ and δμ solve (5.197) with ϕ = 0 and Z = 1 (these are the ‘horizon fluid’ equations for the Einstein-Maxwell system) for a given Ei and ζi. Let

5.199), show that

. Show that

𝒫 is minimized on the solution to the equations of motion. This variational principle for the inverse conductivity matrix was developed in [537]. Hint: 𝒫 is positive-definite. Plug in for the explicit form of 5.197).

c) Consider an arbitrary solution to the Einstein-Maxwell-AdS4 theory. Let us assume that the geometry is disordered and in the thermodynamic limit, such that σij = σδij. Explain why, without loss of generality, we may fix the metric on the horizon to be ds2 = eω(dx2 + dy2). Then, using the ansatz .

d) Can one generalize (5.201) to other dimensions d? Why or why not?

5.6.  Random matrix models. Consider a simpler version of HSYK in (5.205) in which the fermions have random hopping, but are non-interacting:

. This model is also exactly solvable in the large N limit, but has a quasiparticle decomposition of the excited states. We will see this here by computing the effects of interactions in perturbation theory.

a) Show by a diagrammatic expansion that the fermion self energy is given in the large N limit by

5.208a) for the SYK model. Solve the Dyson equation to obtain the single-particle Green’s function

The imaginary part of this result yields the famous ‘semicircular’ density of states. Note that there is no singularity at ω = 0, and the density of states at the Fermi level is finite. Contrast this with (5.209), which shows a divergence at ω = 0 for the SYK model.

b) Use perturbation theory to compute the effect of interactions in the combined Hamiltonian H2 + HSYK as T → 0 in the regime J t. Compute the lifetime of a quasipartice, τα, in an exact eigenstate of H2, ψα(i). Use Fermi’s Golden rule, and the fact that the wavefunctions are uncorrelated with the Jij; kℓ to obtain

where f(E) is the Fermi function, and ρ0 is the density of states (computed above) at the Fermi level. The T2 dependence is the characteristic signature of Fermi liquid behavior and the presence of quasiparticles.

Note

1 Thermodynamic Maxwell relations require the entropy sh and charge ρh to be obtained as derivatives of the pressure with respect to temperature and chemical potential. This is only possible in (5.198) if the pressure is a separable function of μ and T.