4

Compressible quantum matter

Angle-resolved photoemission spectroscopy (ARPES) data showing the presence of a Fermi surface in an optimally doped cuprate. The challenges of combining scale invariance with structure in momentum space will be discussed in this chapter, as will the fate of Fermi surfaces in strongly interacting non-quasiparticle systems. Image taken from [738] with permission.

4.1 Thermodynamics of compressible matter

The condensed matter systems described in Chapter 2 had the common feature of being at some fixed density of electrons (or other quantum particles) commensurate with an underlying lattice: this was important in realizing a low energy description in terms of a conformal field theory. However, the majority of the experimental realizations of quantum matter without quasiparticle excitations are not at such special densities. Rather, they appear at generic densities in a phase diagram in which the density can be continuously varied.

More precisely stated, the present (and following) section will consider systems with a global U(1) symmetry, with an associated conserved charge density ρ which can be varied smoothly as a function of system parameters even at zero temperature. The simplest realization of such a system is the free electron gas, and (as is well known) many of its properties extend smoothly in the presence of interactions to a state of compressible quantum matter called the ‘Fermi liquid’. The Fermi liquid has long-lived quasiparticle excitations, and almost all of its low temperature properties can be efficiently described using a model of a dilute gas of fermionic quasiparticles. Indeed, the term Fermi liquid is a misnomer: the low energy excitations do not interact strongly as in a canonical classical liquid like water.

Our interest here will be in realizations of compressible quantum matter which do not have quasiparticle excitations. One can imagine reaching such a state by turning up the strength of the interactions between the quasiparticles until they reach a true liquid-like state in which the quasiparticles themselves lose their integrity.

Any compressible state has to be gapless, otherwise the density would not vary as a function of the chemical potential. The non-quasiparticle states of interest to us typically have a scale-invariant structure at low energies, and so it is useful to work through some simple scaling arguments at the outset. As in (2.45), we write for the T dependence of the entropy density

where d is the spatial dimensionality, z determines the relative scaling of space and time and θ is the violation of hyperscaling exponent (discussed in §2.2.2 above). In the arguments that follow only the combination (dθ)/z appears. In the presence of a global conserved charge density ρ, it is useful to introduce a conjugate chemical potential μ, and consider the properties of the system as a function of both μ and T.

Thermodynamics is controlled by a ‘master function’, given by the pressure P(μ, T) in the grand canonical ensemble. Scale invariance implies that, so long as the charge density operator does not acquire an anomalous dimension (which it can, see §4.3.4 below), P is a function of the dimensionless ratio μ/T:

Using the thermodynamic (Gibbs-Duhem) identity

In such theories, the entropy density is proportional to the specific heat, generically defined as c = 𝜖/∂T.

Theories accessible with conventional methods will shortly be seen to have θ = 0 or θ = d − 1. The latter case arises when a Fermi surface contributes to the critical physics. A simple model for theories with more general θ consists of N 1 species of free Dirac fermions with power-law mass distributions [457]. Other examples are non-conformal large N supersymmetric gauge theories [215]. We do not know of any non-large N quantum critical model which has other θ. Indeed, θ < 0 suggests that the low energy theory ‘grows’ in dimensionality – as the effective total number of spacetime dimensions with hyperscaling violation is Deff. = z + dθ – which may require N → ∞ degrees of freedom. The emergence of extra scaling spatial dimensions at low energies underlies the appearance of θ < 0 in some holographic models with known large N field theory duals, e.g. [330; 307].

The first subsection below describes various condensed matter realizations of compressible quantum matter. The following subsections take a holographic approach. The holographic realizations can be obtained by ‘doping’ a strongly-coupled conformal field theory, although they are also well-defined low energy fixed point theories in their own right. We will compare the properties of holographic compressible matter with those of the condensed matter systems.

4.2 Condensed matter systems

We begin with a review of some basic ideas from the Fermi liquid theory [628] of interacting fermions in d dimensions, expanding upon the discussion in §1.1. We consider spin-1/2 fermions c with momentum k and spin α =↑, ↓ and dispersion 𝜖k. Thus the non-interacting fermions are described by the action

This Green’s function has a pole at energy 𝜖k with residue 1. Thus there are quasiparticle excitations with residue Z = 1, with both positive and negative energies, as 𝜖k can have either sign; the Fermi surface is the locus of points where 𝜖k changes sign. The positive energy qausiparticles are electron-like, while those with negative energy are hole-like i.e. they correspond to the absence of an electron. A perturbative analysis [628] describes the effect of interactions on this simple free fermion description. A key result is that the pole survives on the Fermi surface: for small |𝜖k|, the interacting fermion retarded Green’s function takes the form

, indicating that the quasiparticles are well-defined. Although the interactions can renormalize the shape of the Fermi surface, Luttinger’s theorem states that the volume enclosed by the Fermi surface remains the same as in the free particle case:

In principle zeros of the Green’s function can contribute to the Luttinger count [493; 170]. However, generally in gapless metallic systems they do not [28]. One argument is that the Green’s function is complex, and so it requires two conditions to find a zero of the Green’s function in the complex plane. In d spatial dimensions, the locus of zeros is a (d − 2)-dimensional surface, and so makes a negligible contribution to the Luttinger computation. In contrast, the locus of poles is (d − 1)-dimensional: the physical structure of the Feynman-Dyson expansion for fermions ensures that the locus of points where the real part of G−1(k, ω = 0) vanishes coincides with the points where the imaginary part vanishes, as is clear from the structure of (4.8). Alternatively stated, the location of the poles of G in (4.8) is independent of Ginc, while the location of the zeros is not.

A universal description of the low energy properties of a Fermi liquid is obtained by focusing in on each point on the Fermi surface. For each direction . Then we identify wavevectors near the Fermi surface by

Now we should expand in small momenta k. For this, we define the infinite set of fields

where 4.6), expanding in k, and Fourier transforming to real space x, we obtain the low energy theory

direction. In other words, the low energy theory of the Fermi liquid is an infinite set of one-dimensional chiral fermions, one chiral fermion for each point on the Fermi surface. Apart from the free Fermi term in (4.12), Landau’s Fermi liquid theory also allows for contact interactions between chiral fermions along different directions [689; 633]. These are labeled by the Landau parameters, and lead only to shifts in the quasiparticle energies which depend upon the densities of the other quasiparticles. Such shifts are important when computing the response of the Fermi liquid to external density or spin perturbations.

The chiral fermion theory in (4.12) allows us to deduce the values of the exponents characterizing the low temperature thermodynamics of a Fermi liquid. We have

from the linear dispersion and the effective dimensionality dθ = 1 of the chiral fermions.

For d > 1, interactions are irrelevant at a Fermi liquid fixed point. The only exception to this last statement is, with time-reversal invariance (that guarantees that if a momentum iron, or liquid helium) is a Fermi liquid. Nonetheless, it is possible to find routes to destabilize the Fermi liquid to obtain compressible states without quasiparticles, as we detail in the following §4.2.1, §4.2.2 and §4.2.3.

4.2.1 Ising-nematic transition

One route to non-Fermi liquid behavior is to study a quantum critical point associated with the breaking of a global symmetry in the presence of a Fermi surface. This is analogous to our study in §2.1.1, where we considered spin rotation symmetry breaking in the presence of massless Dirac fermions whose energy vanished at isolated points in the Brillouin zone. But here we have a (d − 1)-dimensional surface of massless chiral fermions, and so they can have a more singular influence.

We describe here the field theory for the simplest (and experimentally relevant) example of an Ising order parameter, represented by the real scalar field ϕ, associated with the breaking of a point-group symmetry of the lattice. Other order parameters which carry zero momentum have similar critical theories; the case of non-zero momentum order parameters which break translational symmetry will be considered in §4.2.2.

Unfortunately, in this case the field theory cannot be obtained by simply coupling the field theory of ϕ to the chiral fermion theory in (4.12), and keeping all terms consistent with symmetry. A shortcoming of the effective action (4.12) is that it only includes the dispersion of the fermions transverse to the Fermi surface. Thus, if we discretize the directions 4.12)), we obtain the low energy theory [517]

Figure 4.1
Fermionic excitations at the Ising-nematic critical point. We focus on an extended patch of the Fermi surface, and expand in momenta about the point 4.14). The coordinate y represents the d − 1 dimensions parallel to the Fermi surface.

We have added a co-efficient ζ1 to the temporal gradient term for convenience: we are interested in ζ1 = 1, but the disappearance of quasiparticles in the ultimate theory is captured by the renormalization ζ1 0. Notice the additional second-order gradients in y which were missing from (4.12): the co-efficient κ is proportional to the curvature of the Fermi surface at . There are zero energy fermion excitations when

4.15) defines the position of the Fermi surface, which is then part of the low energy theory including its curvature. Note that (4.14) now includes an extended portion of the Fermi surface; contrast that with (4.12), where the one-dimensional chiral fermion theory for each describes only a single point on the Fermi surface.

The field theory for the Ising-nematic critical point now follows the traditional symmetry-restricted route described also in §2.1.1. We restrict our subsequent attention to the important case of d = 2, because this is the dimensionality in which quasiparticles are destroyed by the critical fluctuations. We combine a field theory like (2.1) for ϕ with a ‘Yukawa’ coupling λ between the fermions and bosons

As usual, the coupling s is a tuning parameter across the quantum critical point, and we are interested here in the quantum state at the critical value s = sc. The theory SIN has been much studied in the literature by a variety of sophisticated field-theoretic methods [517; 577; 588; 165]. For the single fermion patch theory as formulated above, exact results on the non-quasiparticle compressible state are available. In the scaling limit, all the ζi 0, and the fermionic and bosonic excitations are both characterized by a common emergent dynamic critical exponent z. Their Green’s functions do not have any quasiparticle poles (unlike (4.8)), and instead we have the following scaling forms describing the non-quasiparticle excitations:

ψ, ϕ are scaling functions. Note that the curvature of the Fermi surface, κ/vF, does not renormalize, and this follows from a Ward identity obeyed by SIN [577]. For the single-patch theory described so far, the exact values of the exponents in d = 2 are [715]

In the presence of additional symmetries in the problem, the single patch description is often not adequate. Many important cases have inversion symmetry, and then we have to include a pair of antipodal patches of the Fermi surface in the same critical theory [577; 588]. The resulting two-patch theory is more complicated, and exact results are not available. Up to three loop order, we find that η ≠ 0, while the value z = 3/2 is preserved. There is also a potentially strong instability to superconductivity in the two patch theory [579]. A strong superconducting instability is seen in Monte Carlo studies of a closely related quantum critical point [513].

4.2.2 Spin density wave transition

Now we consider the case of an order parameter at non-zero momentum in the presence of a Fermi surface. An experimentally common case is a spin density wave transition, represented by a scalar field φa, with a = 1, 2, 3, very similar to that found in §2.1.1 for the Gross-Neveu model. The order parameter carries momentum K, and so the condensation of φa breaks both spin rotation and translational symmetries. The scalar will couple to fermion bilinears which also carry momentum K. As we are interested only in low energy fermionic excitations near the Fermi surface, we need to look for “hot spots” on the Fermi surface which are separated by momentum K. If no such hot spots are found, then the Fermi surface can be largely ignored in the theory of the transition, as it is described by a conventional Wilson-Fisher theory for the scalar φa. Here we consider the case where the hot spots are present, and then there is a relevant Yukawa coupling between φa and fermion bilinears spanning the hot spots.

We consider the simplest case here of a single pair of hot spots, described by the fermion fields ψ1α and ψ2α representing points on the Fermi surface at wavevectors k1 and k2. The low energy fermion action in (4.14) is now replaced by

Here σa are the Pauli matrices. Notice the similarity of Ssdw to the theory for the onset of antiferromagnetism in the honeycomb lattice in §2.1.1: the main difference is that the massless Dirac fermions (which are gapless only at the isolated point k=0) have been replaced by fermions with dispersion illustrated in Figure 4.2 (which are gapless on two lines in the Brillouin zone).

Figure 4.2
Fermi surfaces of ψ1 and ψ2 fermions in the plane defined by the Fermi velocities . The gapless Fermi surfaces are one-dimensional, and are indicated by the full lines. The lines intersect at the hot spot, which is the filled circle at the origin.

The non-Fermi liquid physics associated with the theory Ssdw has been studied in great detail in the literature. The expansion in powers of λ is strongly coupled, and a variety of large N and dimensionality expansions have been employed [578; 716; 556]. We will not enter into the details of this analysis here, and only mention a few important points. The theory has a strong tendency towards the appearance of d-wave superconductivity, and this has been confirmed in quantum Monte Carlo simulations [85; 680; 523; 522; 239]. The non-Fermi liquid behavior is characterized by a common dynamic critical exponent z > 1 for both the fermions and bosons, but with no violation of hyperscaling θ = 0 [349; 619]. The preservation of hyperscaling indicates that the critical non-Fermi liquid behavior is dominated by the physics in the immediate vicinity of the hot spot. The gapless Fermi lines (responsible for violation of hyperscaling in a Fermi liquid) are subdominant in their non-Fermi liquid contributions.

4.2.3 Emergent gauge fields

Compressible quantum phases with topological order and emergent gauge fields are most directly realized by doping the insulating phases discussed in §2.1.3 by mobile charge carriers. We can dope the resonating valence bond state in Figure 2.4a by removing a density p of electrons. We are then left with a density 1 − p of electrons on the square lattice; this is equivalent to a density of 1 + p holes relative to the fully filled band. However, as long as the emergent gauge field of the parent resonating valence bond state remains deconfined, it is more appropriate to think of this system as having a density p of positively charged carriers [686; 646]. As illustrated in Figure 4.3, these charge carriers can either remain as spinless fermionic holons hq, or bind with the quanta of the neutral spin-1/2 scalars zα in Figure 2.4b to form fermions, dα, with electromagnetic charge + e and spin S = 1/2 [467].

Figure 4.3
Schematic of a component of a state obtained by doping the resonating valence bond state in Figure 2.4a by a density p of holes. The circles represent spinless ‘holons’ hq of electromagnetic charge +e, and emergent gauge charge q = ±1. The rectangular dimers, dα, are bound states of holons and the zα quanta: the dα are neutral under the emergent gauge field, but carry electromagnetic charge +e and spin S = 1/2. Finally, the resonance between all the dimers is captured by the emergent gauge field Aμ.

The final low energy description of this state is left with two kinds of fermions, both of which carry the Maxwell electromagnetic charge (which is a global U(1) symmetry, as is conventional in non-relativistic condensed matter contexts). There are the ‘fractionalized’ fermions, hq, which carry charges ±1 under the emergent gauge field Aμ of (2.14). And we have the ‘cohesive’ fermions, dα, which are neutral under Aμ. The term ‘cohesive’ was introduced in [370] in opposition to ‘fractionalized’; we will see below that holographic descriptions of compressible phases also lead naturally to such catergories of fermionic charge carriers. So we can write down an effective low energy theory for the fermions in the spirit of (4.6):

where Ed(k) and Eh(k) are the dispersions of the cohesive and fractionalized fermions respectively. Note that the fractionalized fermions are minimally coupled to the emergent gauge field.

A modified version of the Luttinger relation (4.9) also applies in the presence of topological order [686; 614]. As the fractionalized fermions can convert into cohesive fermions by binding with the zα, the relation only applies to the sum of the volumes enclosed by all the Fermi surfaces [641; 150]

This is expected to be an exact relationship, satisfied by the fully renormalized excitations in the presence of the gauge excitations, as long as the gauge field does not undergo a confinement transition and preserves topological order. In the confining case, the right hand side of (4.22) would be 1 + p, as we would have to count all the holes, not just those relative to the resonating valence bond state.

Actually, there is an important feature we have glossed over in the discussion so far of (4.22). The hq fermions are minimally coupled to a U(1) gauge field, and this coupling leads to a breakdown of the quasiparticle physics near the Fermi surface. Indeed, the theory of the Fermi surface coupled to a gauge field [577; 588] is nearly identical to that Ising-nematic transition discussed in §4.2.1. The field theory (4.16) applies also to the gauge field problem, with the scalar field ϕ representing the component of the gauge field normal to the Fermi surface (in the Coulomb gauge). Consequently, much of the analysis for the structure of the excitations near the Fermi surface can be adapted from the Ising-nematic analysis summarized in §4.2.1. Despite the absence of quasiparticles, however, the position of the Fermi surface in the Brillouin zone can be defined precisely. Now, rather then the quasiparticle condition Eh(k) = 0 used naively in (4.22), we have the more general condition on the hq fermion Green’s function [678]

4.22) continues to apply to this realization of compressible quantum matter without quasiparticles (similar relations also apply to the examples discussed in §4.2.1 and §4.2.2). As discussed below (4.9), we need not consider the zeros of Gh.

The reader is referred to separate reviews [668; 669] for discussions of the applications of models of Fermi surfaces coupled to emergent gauge fields to the physics of the optimally doped cuprates.

4.3 Charged horizons

A simple way to obtain compressible quantum phases in holography is by adding a chemical potential μ to the strongly interacting holographic scale invariant theories described in §2.2. We are ultimately interested in the universal low energy dynamics of these compressible phases. For such questions it does not matter precisely which high energy starting point we use to obtain the compressible matter. It is convenient, then, to take CFTs (z = 1 and θ = 0) as the UV completion of the IR compressible phase. This is analogous, for instance, to obtaining a Fermi liquid by ‘doping’ the particle-hole symmetric point of graphene.

The chemical potential μ is the source for the charge density operator ρ = Jt in the field theory. As explained around equation (1.40) above, and elaborated in §3.4, the bulk field dual to the charge current Jμ is a bulk Maxwell field Aa. In particular, ρ is dual to At in the bulk. The holographic dictionary implies that μ and ρ are contained in the near boundary (r → 0) expansion of At according to

3.48a) and (3.49a) above – has assumed that near the boundary the Maxwell field is described by the Maxwell action (3.43) in an AdSd+2 background. This is typically the case for the class of doped holographic CFTs we will be considering. Away from the boundary, the Maxwell field will backreact on the geometry. This is the difference with §3.4, where the Maxwell field was treated as a perturbation in order to obtain correlation functions in the zero density theory.

The asymptotic behavior (4.24) implies the presence of a bulk electric field in the radial direction. Near the boundary

Thus the charge density of the field theory is precisely given by the asymptotic electric flux in the bulk. This bulk electric field is subject to Gauss’s law (in the bulk). Therefore it must either originate from charged matter in the interior of the bulk geometry or the electric field lines must emanate from a charged horizon. We will see that these two possibilities respectively correspond to the ‘cohesive’ (a term coined by [370]) and ‘fractionalized’ charge densities discussed briefly in §4.2.3.

We saw in previous sections that much of the technical novelty of holographic quantum matter has to do with the geometrization of dissipation by classical event horizons. Similarly, the most novel – and indeed the simplest – compressible phases arising holographically will be those in which the interior IR geometry is described by a charged horizon. We will start with these cases. These horizons can suffer instabilities leading to the spontaneous emission of charged matter from the black hole. Fermionic charged matter in the bulk will be discussed in §4.6 below whereas bosonic charged matter will be the subject of Chapter 6. The different scenarios are illustrated in Figure 4.4.

Figure 4.4
Fractionalized vs. cohesive charge. Left: The electric flux dual to the field theory charge density emanates from a charged event horizon. The charge is said to be ‘fractionalized’. Right: The electric flux is instead sourced by a condensate of charged matter in the bulk. The charge is said to be ‘cohesive’. Figures taken from [351] with permission.

4.3.1 Einstein-Maxwell theory and AdS2 × d (or, z = ∞)

The simplest holographic theory with the required ingredients to describe compressible states of matter is Einstein-Maxwell-AdS theory. This theory describes the interaction of the metric g with a Maxwell field A via the action

Einstein-Maxwell theory will be seen to describe a compressible phase with quite remarkable low temperature properties. Some of these remarkable properties have proven unpalatable to many, for reasons we shall see, leading to the introduction of the more complicated theories considered in later subsections. Nonetheless, the nontrivial and exotic physics captured by this simplest of holographic theories may yet contain clues about the dynamics of strongly interacting compressible phases of electronic matter.

The charged black hole solutions of (4.26) were studied in some detail shortly after the discovery of holography [129], but gained a new lease of life in the application of holographic methods to quantum matter. Both in early and more recent work, Einstein-Maxwell theory is sometimes studied as a special case of a more general classes of gravitational theories that are dual to specific supersymmetric field theories such as 𝒩 = 4 SYM theory at a nonzero chemical potential, e.g. [163; 162; 700]. This is because 𝒩 = 4 SYM admits three commuting global U(1) symmetries, and Einstein-Maxwell theory describes the case in which the corresponding three chemical potentials are all equal.

The charged black hole solutions to (4.26) are relatively simple and well known. We shall simply quote the black hole metric and Maxwell field. In the following subsection we will describe in detail how to find more general charged black hole solutions, and the present planar Reissner-Nordström-AdS (AdS-RN) geometry is a special case of that discussion. We will restrict to d > 1 for simplicity. The metric is

The spacetime therefore has a horizon at r = r+, and tends to AdSd+2 near the boundary as r → 0. We have grouped various coefficients into

This constant determines the relative strengths of bulk gravitational and electromagnetic interactions. The background Maxwell potential is

Note that At vanishes on the horizon. This boundary condition is necessary for the Wilson loop of the Maxwell field around the Euclidean thermal circle to be regular as rr+, where the thermal circle shrinks to zero (see Figure 2.7 above). Comparing with the near-boundary expansion (4.24), we see that the black hole carries a charge density

Finally, the temperature is found following the procedure described in §2.3.1 to be

This expression is readily inverted to obtain an explicit but slightly messy formula for r+ in terms of T and μ. The limiting behaviors are

From the high temperature behavior in the first line above, it is straightforward to see (from the fact that r+ is becoming small, i.e. close to the boundary at r = 0) that the solution becomes the AdS-Schwarzchild black brane geometry together with an electric field. In contrast, in the second line we see that the horizon remains of nonvanishing size as the temperature is taken to zero. That behavior is very different from the neutral AdS-Schwarzchild solution.

The entropy density is obtained from the horizon area according to (2.45). Using the horizon radius (4.33), in the low temperature limit one obtains

In particular, the entropy density is finite at T = 0. Such an event horizon is called extremal. This is a violation of the third law of thermodynamics; mollifying this feature was an important goal driving the study of the more general Einstein-Maxwell-dilaton models discussed below. See however the final paragraph of this section: the solvable SYK model has lead to some intuition for how the absence of quasiparticles can lead to a proliferation of low energy states. In the remainder of this subsection we will trace the origin of the zero temperature entropy density to the fact that this theory has an emergent IR scaling symmetry with z = ∞. Taking z → ∞ (with θ finite) indeed leads to a zero temperature entropy density. Recall that according to (2.45) or (4.1), s ∼ T(dθ)/z → const. as z → ∞.

In later sections we will see how z = ∞ describes a phase of quantum critical matter with fascinating properties. Because z = ∞ implies that time scales but space does not, momentum is dimensionless under the scaling symmetry. Thus low energy critical excitations are present at all momenta, not just small momenta (as would be the case for any finite z, wherein ω ∼ kz). This fact allows fermions living at a Fermi momentum kF to interact strongly with the critical sector (see §4.5). It also allows momentum non-conserving scattering due to a lattice at wavevector kL to participate strongly in the critical dynamics (see §5.6.4). More generally, z = ∞ criticality is compatible, in principle, with arbitrary spatial inhomogeneity. Both of these features – quantum criticality compatible with a Fermi surface and with momentum relaxation – are desirable and difficult to achieve in other kinematic frameworks.

For completeness, note that with the entropy density (4.34) and charge density (4.31) at hand, the energy density and pressure can be obtained from the general thermodynamic relations (4.4) and (4.5). In the latter equation, relating energy and pressure, one should put z = 1 and θ = 0, as this is a UV relationship that holds for all states in the doped CFT. We will expand more on this point in the last paragraph of §4.3.3 below. While the low temperature scaling of the entropy density is determined purely by IR data (the horizon), this is not the case for the energy density and pressure, that receive contributions from all energy scales.

The near-horizon regime of the black hole spacetime will be seen to control all low energy dissipative processes. It is therefore very important to determine exactly what this geometry is. To do so, let us switch to the new coordinate

as the T = 0 limit of the horizon location. Let us temporarily fix T = 0 exactly. Then ζ → ∞ as we approach the horizon. Switching from the r coordinate to ζ, as ζ → ∞ the geometry (4.27) becomes:

d spacetime [295; 265]. The radius of the emergent AdS2 is

and we will see in §6.2 that this fact has implications for possible instabilities of the near-horizon geometry. Furthermore, we see that the emergent scaling regime in the IR does not involve the spatial coordinates! The scaling symmetry is simply ζλζ, tλt. Following the discussion around (2.16) above, we obtain a dynamic critical exponent z = ∞; such a theory is called locally critical, or more properly, as we will see, semi-locally critical [434]. As we have mentioned, semi-local criticality leads to interesting physics, that we will return to later in this chapter. Figure 4.5 illustrates the emergence of the semi-local criticality from a doped CFT.

Figure 4.5
Emergence of AdS2. A CFT, dually described by Einstein-Maxwell theory in the bulk, is placed at a nonzero chemical potential. This induces a renormalization group flow in the bulk, leading to the emergence of semi-locally critical AdS2 × d in the far IR, near horizon region. Figure adapted with permission from [433].

Finally, let us add a small nonzero temperature T. The location of the horizon shifts to

The near-horizon metric now becomes

Indeed, as one might have guessed, the emergent geometry is d times an AdS2-Schwarzchild black brane. That is to say, the event horizon is contained within the emergent low energy AdS2 geometry. The nonzero temperature near horizon geometry (4.41) will control all of the dissipative physics of the theory at low temperatures and low energies {ω, T} μ. Once T ∼ μ, this emergent scaling geometry disappears, swallowed by the horizon, and we transition back to the AdSd+2 black brane.

The spacetime AdS2 × d is a solution of Einstein-Maxwell theory in its own right. While we have obtained the geometry as the low energy description of a CFT deformed by a chemical potential, it should more properly be considered as a self-contained fixed point theory. The AdS2 × d solution by itself will capture all of the universal low energy physics. Indeed, for instance, a small temperature leads to Schwarzschild-AdS2 × d, which is also a solution on its own. We will see many examples of how the IR fixed point geometry controls the low energy physics below.

The emergent semi-locally critical regime described above is a critical phase. No parameter has been tuned to obtain the quantum criticality. Indeed, within the bulk Einstein-Maxwell theory there is no extra parameter to tune. We simply started with a CFT in the UV and deformed by a chemical potential. In more general bulk theories, with extra fields, there can be relevant deformations of the locally critical IR geometry.

It must be kept in mind that a non-zero entropy in the zero temperature limit (which appears in locally critical theories with z = ∞) does not imply an exponentially large ground state degeneracy. In a system with N degrees of freedom, there are eαN states (for some constant α), and so the typical energy level spacing is of order eαN. In systems with quasiparticles, the spacing of the lowest energies near the ground state is no longer exponentially small in N, and the zero temperature limit of the entropy vanishes. In the present holographic system, the energy level spacing remains of order eαN even near the ground state: this understanding has emerged from studies of the SYK models, which we will describe in §5.11. These models, and their holographic duals, allow a systematic study of 1/N corrections [26; 283; 634; 560; 561; 254], and the locally critical scaling holds down to the smallest values of (total energy)/N.

4.3.2 Einstein-Maxwell-dilaton models

The next simplest set of holographic models are Einstein-Maxwell-dilaton (EMD) theories. The bulk action takes the general form

4.26) has been enhanced to include an additional scalar field Φ, the ‘dilaton’ (the term is used loosely here). In consistent truncations of string theory backgrounds, such scalars are ubiquitous. We gave some examples in §1.8 above. The general class of bulk theories (4.43) is divorced from any particular string theoretic realization, and is instead used to explore the space of possible behaviors of the bulk spacetime. Early papers that studied this class of theories as holographic models of compressible matter include [728; 301; 330; 132; 426]. In the following subsection we will see that these theories lead to a wealth of new emergent critical compressible phases, now with a range of possible values of the exponents z and θ. Before that, we illustrate how one goes about finding the bulk equations of motion from an action such as (4.43).

When looking for rotationally and translationally invariant states, the bulk fields will only have a nontrivial dependence on the radial coordinate, r. Therefore, the bulk equations of motion reduce to ordinary rather than partial differential equations. To find these differential equations we make the following ansatz for the dilaton, electrostatic potential and metric

There is some choice in how the metric components are parametrized. The choice made above (see e.g. [543]) proves convenient as near a black hole horizon at r = r+,

regardless of other fields. In the coordinates in which the metric (4.44) has been written, spatial infinity is at r = 0. If we are working in an asymptotically AdS spacetime, dual to a ‘doped CFT’ as described at the start of this §4.3, then a(0) = b(0) = 1.

On the ansatz (4.44), the Einstein-Maxwell-dilaton equations of motion become

In order, these equations come from the rr and tt components of Einstein’s equation, the tt and ii components of Einstein’s equation, the t component of Maxwell’s equation, and the dilaton equation. In fact, the Maxwell equation above is simply Gauss’ Law. After integrating this equation, from (4.25) we can interpret the integration constant as the charge density ρ of the black hole horizon (and the dual field theory):

4.46a); and then the gauge field p is determined from a and Φ through Gauss’ law. The coefficient b is related to the gauge field through (4.46b) – in the case where the black hole is uncharged, then we find that the computation of b(r) for a black hole geometry reduces to a first order differential equation.

Although (4.46) can be derived by hand, in practice it is helpful to use differential geometry packages to evaluate the Ricci tensor in Einstein’s equations. For convenience, we provide below the Mathematica code necessary to obtain (4.46), employing the RGTC package. The code below will only solve the equations in a specific spacetime dimension d = 2, but is easily changed for any d of interest:

% define the coordinate labels

coords = {r,t,x,y};

% define the metric ansatz

g = DiagonalMatrix[{a[r]*Lˆ2/(rˆ2*b[r]), -a[r]*b[r]*Lˆ2/rˆ2, Lˆ2/rˆ2, Lˆ2/rˆ2}];

% generate RMN, RMNRS etc.

RGtensors[g, coords, {0, 0}];

% define the ansatz for Φ

Phi := phi[r];

% define the 1-form ∇MΦ.

dPhid := covD[Phi];

% define the ansatz for AM; the d label reminds us that the index is down

Ad := {0, p[r], 0, 0};

% define FMN

Fdd := Transpose[covD[Ad]] - covD[Ad];

% define FMN; raise indices

FUU := Raise[Fdd, 1, 2];

% Φ contributions to the right hand side of Einstein’s equations; multiplying tensors together

PhiTdd := 2*Outer[Times,dPhid,dPhid] - (Contract[Outer[Times, dPhid,gUU,dPhid], {1,2},{3,4}] + V[Phi])*gdd/2;

% F2 contributions to the right hand side of Einstein’s equations

F2scalar := Contract[Outer[Times, Fdd, FUU], {1, 3}, {2, 4}];

F2Tdd := Z[Phi]*(Contract[Outer[Times, Fdd, gUU, Fdd], {2, 3}, {4, 6}] - F2scalar*gdd/4);

% Einstein’s equations; the output is a (0, 2) tensor, and vanishes on-shell

EOM1 := Rdd - R*gdd/2 - kˆ2/eˆ2 * F2Tdd - PhiTdd;

% Maxwell’s equations; the output is a (1, 0) tensor

EOM2 := covDiv[Z[r]*FUU, {1, {2}}];

% dilaton equation

EOM3 := 4*covDiv[Raise[dPhid,1],1] - V’[Phi]/Lˆ2 - Z’[Phi] * F2scalar * kˆ2/(2*eˆ2);

Given this code, one finds a messy set of equations. With a little practice one can gain intuition for re-organizing them into more transparent forms such as (4.46).

4.3.3 Critical compressible phases with diverse z and θ

In Einstein-Maxwell theory we explained that the universal low energy and low temperature physics is determined by an emergent near horizon AdS2 × d spacetime. We also noted that this scaling geometry was a solution of the Einstein-Maxwell equations of motion on its own. The same phenomenon occurs for an important class of potentials V and Z in Einstein-Maxwell-dilaton theory (4.43). In this section we describe the IR scaling solutions that arise, leading to critical phases of compressible holographic matter. Following the presentation of [424], we parametrize these solutions in terms of “arbitrary” z and θ, up to some constraints which we note below.

Start at zero temperature. We look for a scaling solution by plugging the ansatz

4.46). This ansatz corresponds to the same form of metric as considered in the hyperscaling-violating solutions of §2.2.2 above, albeit in a different coordinate system. Namely,

2.29) through the coordinate change rr1−θ/d. Thus we see that z is the dynamic exponent and θ the hyperscaling violation exponent. The difference with the discussion in §2.2.2 is the presence of a Maxwell field. From (4.46a) we find that the dilaton

and r0 a constant related to the crossover out of the IR scaling region. The presence of the dimensionful quantity r0 in the solution is nontrivial, as we discuss below. As Φ(r) must be real, we see that

This constraint is also implied by the null energy condition, as we noted in §2.2.2.

One can now use the remainder of the equations of motion to show that

The exact prefactors V0 and Z0 can be found in [424; 543]. Finally, one finds that the scalar potential

4.46b).

Logically speaking, of course, the theory with potentials V and Z is prior to any specific solution. With this in mind, the results in the previous paragraph can be rephrased as follows: If the potentials V and Z in the Einstein-Maxwell-dilaton action behave as (4.52) to leading order as Φ → ∞, then the theory admits the above emergent low energy scaling solution in the far interior as r → ∞ (according to (4.50), Φ → ∞ as r → ∞ ). The critical exponents z and θ are determined by α and β according to (4.53).

If we start with a doped CFT in the UV (r → 0), the above (z, θ) scaling solutions will emerge in the far interior of the spacetime (r → ∞), in just the way that AdS2 × d emerged in Einsten-Maxwell theory. Generically the full interpolating geometry – solving the equations of motion (4.46) at all r – can only be found numerically. Let us stress again, however, that all of the universal low energy physics will be determined by the scaling solution. We will see many examples of this shortly. At T = 0, the only scale in the theory is the charge density ρ induced by doping the CFT. This determines a radial scale

One expects to find that near the boundary, for 0 < r r*, the geometry will be approximately AdS, whereas in the interior, for r* r, the geometry will be of the (z, θ) scaling form. Thus the charge density sets the UV cutoff for the emergent scaling region. The factors of e, κ, L in (4.56) are obtained as follows. If we perform the rescaling p = (eL/κ)4.46).

The T = 0 scaling geometries are generally singular in the far IR, as r → ∞. The metrics typically have naked or null curvature singularities. Furthermore, the blowup of the dilaton field indicates the onset of quantum gravity or string theory effects at large r. For discussions of the singular nature of these solutions from various angles see [132; 426; 416; 688; 215; 73; 344]. As we shall see, the singularities are sufficiently mild that infalling boundary conditions can consistently be imposed at the T = 0 ‘horizons’, and the computations of thermodynamic and linear response quantities in these backgrounds make sense. Nonetheless, the singularities mean that the low energy limit ultimately does not commute with the large N classical limit for these backgrounds. While at large N, then, classical gravity describes scale-invariant compressible phases of matter over a parametrically large range of energy scales, the ’t Hooft limit is not strong enough to uncover new strictly controlled finite density T = 0 fixed points in these cases. An exception seems to be when z = 1 + θ/d, in which case the IR geometry is not singular [688; 519]. These regular geometries saturate the first of the null energy constraints in (2.35) and deserve further attention.

A class of black hole solutions to (4.46) can be found analytically. These will describe the universal low energy physics of the critical phase at temperatures 0 < T κ/(eL) μ. The second inequality corresponds to the temperature below which the bulk event horizon will be well within the IR scaling region – see the discussion below (4.56) above. The solution is found by noting that (4.46b) is satisfied (in the IR scaling region) by

Using (4.45), the horizon radius is related to the temperature by

This is the expected scaling (4.1). In particular, for general values of dθ > 0 and z, the entropy s → 0 as T → 0 and the third law of thermodynamics holds true. Furthermore, at any finite T > 0 the geometric singularities are hidden behind a black hole horizon, and so in this sense the singular IR geometry is ‘good’ [319].

An interesting limit of this class of solutions is to take [376; 38]

This limit retains the interesting phenomenology of z = ∞ fixed points without a zero temperature entropy density (s ∼ Tη → 0). Furthermore, the simplest string theory embeddings of Einstein-Maxwell-dilaton theory turn out to realize this limit [162; 330].

The following sections will characterize the spectrum of fluctuations about these backgrounds in some detail. We can make a quick prior comment. In hyperscaling-violating geometries (θ ≠ 0) a dimensionful scale does not decouple at low energy (e.g. r0 in (4.50) or R in (2.29)). This means that observables computed in these backgrounds are not guaranteed to take a scale-invariant form. For instance, consider a scalar field φ in the bulk (not the dilaton, an additional scalar dual to some scalar operator in the boundary QFT). The quadratic action for the scalar field might take the general form

For a simple mass term, B(Φ) = m2/L2, it is straightforward to show that for θ ≠ 0 these dual operators are gapped, with a large separation Green’s function G(x) emx [215]. However, if B(Φ) eβΦ as Φ ∞, so that on the background scaling solution

, with the dimension Δ given by [535]

In the following sections we will see that interesting operators such as the electric current (whose correlators determine the electrical conductivity) are like this second case: the fluctuations inherit scaling properties from the background. This will allow us to discuss power laws in transport in terms of the exponents {z, θ}.

Finally, a comment about the thermodynamical relation (4.2) in a theory with exponents z and θ. We are considering the case in which the (z, θ)-scaling is obtained as the IR limit of an asymptotically AdS spacetime in the UV (a doped CFT). If we apply the holographic dictionary to obtain the thermodynamic variables, we will find that (4.2) is satisfied, but with z = 1 and θ = 0, independently of the emergent IR scaling. However, (4.2) is not optional for the IR theory — in particular, (4.5) follows from a Ward identity in a Lifshitz theory [417]. The resolution to this apparent tension is that there will be an ‘emergent stress tensor’ 𝒯μν (generally with 𝒯ti𝒯it), distinct from from the UV Tμν, and this will obey the emergent Lifshitz Ward identities: see e.g. [494].

4.3.4 Anomalous scaling of charge density

To fully characterize the scaling backgrounds described in the previous paragraph, and other closely related solutions, a third critical exponent – beyond z and θ – turns out to be necessary [308; 285; 306; 305; 456]. In terms of the bulk solution, while θ determines the anomalous power with which the overall metric scales, the new exponent Φ will determine the anomalous scaling of the bulk Maxwell field, relative to that expected given z and θ. For instance, if we put θ = 0 in the near-horizon scaling of the EMD metric components (4.48) and Maxwell field (4.54) then we find gtt ∼ r−2z as expected, but At ∼ rdz. This expression for the bulk Maxwell field has an extra factor of rd relative to that required by a strict Lifshitz scale invariance. Clearly, then, an additional exponent is necessary to characterize the solution.

A natural expectation is that this extra exponent leads to an anomalous scaling dimension for the charge density operator, so that (see e.g. the discussion in [362])

Here [x] is the momentum dimension of x. Conventional wisdom is that Jμ can pick up no anomalous dimensions [751; 666]; the arguments essentially amount to the fact that the gauge invariant derivative in ordinary QFT is ∇μ − iAμ, implying that A can pick up no anomalous dimensions. However, the argument is a little too fast in suitably general scaling theories (while it is provably true in a CFT using the conformal algebra). A first indication of this fact is that a hyperscaling violation exponent θ already leads to an anomalous scaling dimension for the energy density which is also associated with a conserved current. While θ ≠ 0 also shifts [ρ] from its canonical value in (4.65), it is Φ that is associated with an anomalous dimension for A.

Field theoretic models for Φ ≠ 0 are in short supply, beyond a model involving a large number N of free fields with variable masses and charges [457], related to “unparticle” physics [626]. Models with temperature-dependent charge fractionalization [362] or strong non-locality due to charge screening [525] can avoid the non-renormalization theorems on anomalous dimensions for Jμ.

The most common way that Φ is detected in holography is through the conductivity σ. We will explore the conductivity in some detail in later sections; it is a subtle observable in nonzero density systems. The upshot is that a certain ‘incoherent’ conductivity is expected to scale as [178]

For the EMD models we have just discussed, one finds Φ = z. Interestingly, this corresponds to the charge density operator becoming marginal in the emergent scaling theory [178; 358]. This is the way that the IR fixed point manages to retain a nonzero charge density consistently with scaling properties. Another possibility is that the charge density is irrelevant about the IR fixed point, in this case one can find more general values of Φ [308]. Finally, another possibility for realizing the exponent Φ in holographic models involves the use of probe brane physics [456]. We will discuss probe brane models in §7.1.

Additional models which exhibit this anomalous scaling include Proca bulk actions, in which the ‘Maxwell’ field in the bulk has a mass [308]. However, in this case the bulk Proca field is not dual to a conserved current (or, it is dual to a conserved current in a symmetry broken phase) – recall from (1.40) that broken gauge invariance in the bulk implies that in the boundary theory, μJμ ≠ 0. While the exponent in this case can determine, for instance, the scaling of the conductivity with temperature in a symmetry broken phase, as we will see in §6.4.1, it does not correspond to an anomalous dimension for a conserved current.

At the time of writing, it is still not known the extent to which Φ imprints itself on thermodynamic quantities, and if it does so in a manner consistent with the 3 emergent scaling exponents z, θ and Φ.

4.4 Low energy spectrum of excitations

The compressible states of matter accessible to conventional field theoretic techniques, described in §4.2, all contained a Fermi surface. Such a Fermi surface can be defined even in cases without quasiparticles, as we discussed around (4.23) in §4.2.3. Moreover, there was a Luttinger sum rule on the volumes of all the Fermi surfaces. In systems without quasiparticles, Fermi surfaces are often associated with gauge-charged particles, and we expect this to be the case in holography. The two-point correlators of such particles are not accessible in holographic models, which only allow computation of gauge-invariant obervables. The robustness of Fermi surfaces, embodied by the Luttinger theorem, behoves us to search for the imprint of the Fermi surface in gauge-invariant observables – thermodynamics, transport, entanglement – of the new strongly interacting compressible phases that we have constructed holographically.

In this section we characterize the low energy excitations of fractionalized holographic phases with a charged horizon. We take this opportunity to introduce important holographic techniques for computing spectral densities. In the following §4.5 we will study cohesive phases in which the charge is explicitly carried by fermions. One upshot of these two sections will be that various properties of Fermi surfaces that necessarily come together at weak coupling — such as spectral weight at nonzero momentum and the exponent θ = d − 1 — can be dissociated with strong interactions. Some holographic phases have certain features that are intriguingly suggestive of Fermi surfaces-like dynamics but not others, while other holographic phases seem to have no indications of a Fermi surface at all. It has been argued [636; 261; 677] that bulk quantum corrections will be needed to resurrect the Fermi surface in these cases.

4.4.1 Spectral weight: zero temperature

The low energy spectral weight of an operator 𝒪 as a function of momentum k is characterized by

The division by ω is generally required: recall that the imaginary part of the Green’s function is typically odd under ω → −ω (see e.g. [350]). Hence at nonzero temperature, the limit (4.67) is generally finite and nonzero. Furthermore, the imaginary part of the Green’s function appears divided by ω in the Kramers-Kronig relations and in several of the formulae that will appear below. As noted in (3.33) above, the spectral weight is a direct measurement of the spectrum of excitations in the theory. For free fermions, for example, ρ(k) for the charge density 𝒪 = Jt shows a sharp non-analyticity at 2kF. Two electrons can move charge around the Fermi surface at zero energy cost, but the largest momentum transfer possible is if both electrons move from one side of the Fermi surface to the other. Hence ρ(k) vanishes for k > 2kF.

Let us first show that the extremal (zero temperature) AdS-RN black hole of §4.3.1 is dual to a compressible phase with spectral weight at nonzero momentum. We will focus on a neutral scalar operator of UV dimension Δ, and will comment on other operators later. As in §3.3, to find the retarded Green’s function we need to solve the equations of motion for a massive scalar field in the AdS-RN background. Our previous experience suggests that the low energy spectral weight will be completely fixed, at least the singular properties, by the IR scaling geometry (AdS2 × d). The nonzero momentum spectral weight can then be understood as a consequence of the emergent z = ∞ scaling symmetry, in which momentum is dimensionless, as anticipated in §4.3.1. We will go ahead and assume that we can restrict attention to the IR geometry, justifying this approach at the end of the calculation.

In the near-horizon AdS2 × d geometry (4.37), the equation of motion for a massive scalar field φ at spatial momentum k and frequency ω is

Consider first ω = 0. The two linearly independent solutions to (4.68) are power laws:

This expression reveals a remarkable feature of the emergent AdS2 × d geometry [265]: a continuum of operator dimensions labelled by the momentum k. These decoupled operators at each momentum indicate a large number of degrees of freedom at low energy, as we will see shortly once we have computed the actual Green’s functions. Note that in comparing the behavior (4.69) of the field to the more general expression (2.24) we see that for AdS2 × d geometries, we should take Deff. = 1 in (2.24), i.e. only counting the dimensions involved in scaling by putting d = 0 and z = 1 in (2.25).

At nonzero ω, the solution of (4.68) that satisfies infalling boundary conditions (3.40) is

). Adapting the usual holographic dictionary (3.24) we can then write (the prefactor will not be important)

This is really just the definition of the IR Green’s function. We now show that this quantity can be related to the actual Green’s function that is defined via (3.24) at the UV boundary of the full AdS-RN spacetime, r → 0. The following matching argument is general and important.

The complication with taking the low energy (ω → 0) limit of the full Green’s function is that, in the wave equation for φ in the full geometry, taking ω → 0 does not commute with the very near horizon limit ζ → ∞. This can be seen already in the near horizon wave equation (4.68), due to the ζ2ω2 term. The low frequency Green’s function can be found using a matching argument, as we now explain. The matching argument works provided that

Figure 4.6
The matching argument. When ω μ, then there is a parametrically large region of overlap between the range where the near-horizon solution (4.71) is valid, and the asymptotic range where the frequency may be set to zero. This allows the solution satisfying infalling boundary conditions to be extended to the boundary of the full spacetime.

At these low frequencies, the near horizon condition μζ 1 (cf. the discussion around (4.36) above) is compatible with the condition to drop the frequency dependence in the wave equation, 1 ωζ. The overlap of the two regions is illustrated in Figure 4.6. In the near horizon regime we know the full solution to the wave equation, it is given by (4.71). Now we solve the equation for φ away from the horizon, i.e. from 1 ωζ in the near horizon region, all the way out to the boundary of the full AdS-RN geometry at r → 0. In this range we can safely set ω = 0 to leading order in the limit (4.73). Let φ(k, r) be the solution to the scalar equation of motion in this range which asymptotes to the form (4.69) at the boundary of the near-horizon region, parametrized by the two constants A1, 2. This solution further extends to the UV of the full spacetime as r → 0, where it can be expanded as (1.28):

1.42). Because the wave equation for the scalar is linear and real, the coefficients B(0) and B(1) are linear combinations of A1, 2 with real coefficients. Therefore the full Green’s function can be written

). In the final step we used the definition (4.72) of the IR Green’s function.

Taking the imaginary part of (4.75), and noting that νk > 0 in (4.72) so that

Thus, up to a frequency-independent constant, the low frequency spectral weight of the system is completely determined by the near horizon Green’s function and is independent of the form of the spacetime away from the horizon. This simplification is physically intuitive and very helpful in practice – for many purposes it will not be necessary to explicitly solve the wave equation in the full geometry. Versions of the matching argument above have appeared in many papers, including [329; 407; 301; 243; 265; 377; 235]. In fact, such matching in black hole Green’s functions has an older history, predating AdS/CFT and stretching back into the astrophysical literature.

A similar computation to the above goes through for the retarded Green’s function of the charge and energy currents, Jμ and Tμν. The calculation is a little more complicated because there are multiple coupled fields in the bulk that are excited (different components of the metric and Maxwell field). The equations can be turned into decoupled second order equations analogous to (4.68) using ‘gauge-invariant variables’. This is similar to what we did in equation (3.45) above, but with more variables. For d = 2 (i.e. for a 2+1 dimensional dual field theory) extremal AdS-RN these equations were found and solved in [243; 242; 241], building on [489]. The low energy Green’s functions all take the form (4.76), but now the exponents have a more complicated k dependence

The exponents turn out to be the same in the transverse and longitudinal channels in this case, but note that there are still two different exponents. The most singular will dominate generic correlators [235]. In fact, if all that is required are the exponents νk — that, we have seen, control low energy dissipation — then these can be found without using gauge-invariant variables. It suffices to make a power law ansatz for the near-horizon fields with ω = 0, to find the solutions analogous to (4.69) above.

A generalization of the result (4.77) for the exponents controlling the retarded Green’s functions of charge and energy currents can be found for the EMD theories with z → ∞ in the limit described in (4.61) above. One finds [376; 38]

Here there are two transverse exponents and three longitudinal exponents. Recall that η was defined in (4.61). The momentum scale k0 depends on details of how the IR geometry crosses over to the full spacetime. In particular, the fact that the Green’s function takes the form (4.76) shows that the charge and energy currents are ‘nice’ operators whose fluctuations inherit scaling properties in these z = ∞ hyperscaling violating backgrounds, in the sense discussed around equation (4.62) above.

The power-law form of the spectral weight (4.76) in all the cases above shows that z = ∞ scaling geometries (including those with additional hyperscaling violation) admit nonzero momentum excitations at all energy scales. This can be contrasted with the behavior we will find shortly for z < ∞, in which the nonzero momentum spectral weight is exponentially suppressed as ω → 0. A power law spectral weight is the sort of behavior one might expect for a ‘critical’ Fermi surface – there is no longer a sharp Fermi momentum, or ‘2kF’ singularity, but instead the low energy spectral weight spreads over all of momentum space, with a continuously weakening power law at large momentum. This behavior is perhaps illustrated most clearly by the T > 0 result. We will derive the corresponding nonzero temperature result rigorously below, but meanwhile, the reader should not be surprised to know that the result for ρ(k) in (4.67) is

It seems quite remarkable that these completely non-quasiparticle z = ∞ compressible phases exhibit behavior reminiscent of Pauli exclusion/ Fermi surface physics, in which low energy excitations are pushed out to nonzero momenta. The expression (4.79) diverges as T → 0 if 2νk < 1, as can occur for some of the cases discussed above [38].

Given formulae such as (4.70), (4.77) and (4.78) for an exponent νk appearing in the spectral weight (4.76), one can ask if there is any preferred momentum k that could play a role analogous to a Fermi momentum. A characteristic feature of the above formulae is the presence of branch cuts. The (complex!) momenta k of the branch points are special, in that they are the singular points of the Green’s function in the k-plane. In particular, the imaginary part of k determines the exponential decay of correlators with distance [434; 105]. This is the essence of semi-local criticality as defined by [434]: there is an infinite correlation length in time, but finite correlation in space. In this regard one should note that strictly local criticality, in which there is no k dependence at all in the low energy spectral weight, is fine-tuned. Once k is dimensionless with z = ∞ scaling, the dimensions of operators can acquire k dependence and generically they will.

In contrast to the preceding discussion, we have already noted around equation (2.28) above that in scaling geometries with z < ∞ the nonzero momentum spectral weight is exponentially suppressed as ω → 0, like , which is completely real for ω < k. In this case, the low energy spectral weight trivially vanishes at nonzero momenta.

There are several indications that the inclusion of effects beyond classical gravity may reveal more conventional Fermi-surface like behavior. Classical but string-theoretic α corrections were shown to lead to non-analyticities in momentum space correlators in [636]. Faulkner and Iqbal [261] computed quantum corrections on AdS3, and found that inclusion of monopoles in the bulk gauge field led to 2kF Friedel oscillations in the boundary quantum liquid in one spatial dimension. Bulk monopole contributions were argued in [677] to be necessary for signatures of the Fermi surface to appear for two-dimensional quantum liquids. And, we will see in §7.1 below that probe branes, that include the resummation of a class of α effects, lead to slightly more conventional – albeit still strongly interacting – analogues of Fermi-surface like dynamics. The inclusion of α physics retains the benefits of the large N ’t Hooft limit, and is relatively underexplored.

4.4.2 Spectral weight: nonzero temperature

In this section we obtain a rather general and powerful expression (4.89) for the nonzero temperature spectral weight (4.67). It is given purely in terms of the behavior of the fluctuating field at ω = 0 and on the horizon itself. An early use of these kind of formulae is [128]. We will follow the elegant Wronskian derivation of [536].

Consider the equation of motion for a neutral scalar (with action (4.62)) in the background (4.44):

Recall that a and b are functions of r. The crucial observation is that this equation only depends on ω2, not ω. Hence, to linear order in ω, we may write

where φ0(k, r) is the regular solution with ω = 0 and φI(k, r) will also be determined from the ω = 0 equation, as we describe shortly. The zero of b(r) at the horizon means that we cannot ignore ω when solving (4.80). This is the noncommutativity of limits that we have encountered previously. However, upon taking r → r+, infalling boundary conditions (3.28) amount to

Comparing (4.83) with (4.81), we only need to find a solution to the ω = 0 equation which has this precise logarithmic singularity near the horizon to capture the 𝒪(ω) response.

Suppose that we know the solution φ0 to the wave equation (4.80) at ω = 0 that is regular on the horizon, and that we have normalized it so that near the asymptotic boundary

i.e. it is sourced by unity. We can construct the singular solution as an integral (we have conveniently normalized it):

This is a standard result for ordinary differential equations that is obtained using the Wronskian. Expanding near the boundary r = 0 we see that the second solution is normalizable:

Near r = r+, however, this solution is singular. The singular behavior comes from the factor of b(s) in the integral expression (4.85):

4.81), the near horizon behavior (4.83) and the logarithmic singular behavior (4.87) we can conclude that the solution to linear order in ω must be

4.84)) and φ1 (in (4.86)). Using (4.88) together with the basic holographic dictionary, we obtain the spectral weight

Two very useful aspects of this expression bear repeating. Firstly, it gives the low energy spectral weight in terms of data directly at ω = 0; it is not necessary to perform nonzero ω computations and then take the limit. Secondly, it gives the answer in terms of data on the horizon r = r+. Sometimes, it is possible to determine this data without having to solve the equations of motion at all radii explicitly. For these reasons, (4.89) will play a key role in the theory of holographic metallic transport developed in Chapter 5.

To illustrate the use of (4.89), we can return to the example of the spectral weight of an operator in the AdS-RN background. Now we turn on a small temperature T μ. The matching argument has essentially the same structure as the one we used previously around (4.73). We set ω = 0 and use the small temperature as the parameter that guarantees the existence of two parametrically overlapping regions. The near horizon region is again μζ 1. The near horizon geometry is a black hole in AdS2 × 2, given in (4.41) above. To simplify the argument even further, we will use the scaling symmetry to avoid solving the wave equation explicitly in this black hole geometry (it can be done in terms of hypergeometric functions). If we solve the wave equation with ω = 0 in the (T > 0) near horizon background, the solution that is regular at the horizon must take the form

We have used the fact that the wave equation is only a function of ζ/ζ+ when there is no time dependence. Recall that ζ+ ∝ 1/T in (4.41). Regularity at the horizon requires that f(k, 1) is a constant, that without loss of generality we have set to 1. The solution (4.90) can be expanded for small ζ (the boundary of the near horizon region) to give the zero temperature answer (4.69)

Importantly, the αi(k) do not depend on temperature (i.e. ζ+).

At small temperatures T μ, the near horizon solution has an overlapping regime of validity with a ‘far’ solution, valid for 1 ζ/ζ+ ∼ ζT, all the way out to the asymptotic boundary of the full spacetime. This is entirely analogous to the situation depicted in Figure 4.6, replacing ωT. In the far regime, we can safely set T = 0 in the wave equation, similarly to how we set ω = 0 away from the horizon in our discussion around (4.73). But this means that upon integrating out (4.91) to the boundary, no additional factors of temperature will appear. Furthermore, at the boundary we have imposed in (4.84) that φ0 tend to a temperature independent constant. Given that the equation is linear, this will only happen (generically) if the temperature dependence drops out to leading order at low temperatures in (4.91). This requires 4.89), the low temperature spectral weight is immediately found to be

4.89), we used the fact (from (4.33)) that at low temperatures r+ 1, independent of temperature. The result above is that anticipated in (4.79), confirming that there is a power law spectral weight at nonzero momentum in these locally critical theories. It should be clear in the derivation above that almost no details about the full spacetime geometry are needed. The only step where the equation of motion was necessary was in determining the exponents νk in the IR scaling region.

The matching arguments are more complicated in theories with finite z and θ. The starting point is the wave equation (4.80). There are now two dimensionless combinations of parameters that can be made, kT1/z and krd/(dθ) (or alternatively r/r+). The spectral weight can be determined by simple arguments in the limits k T1/z and k T1/z.

When k T1/z we can put k = 0 in the equation of motion. The dimensionless combination of k and r, region — i.e. putting ω = T = 0 — is a power law,

4.69) above. The scaling dimension Δ′ is defined in (4.64). Using the same argument as previously, the spectral weight is then found from (4.89) to be [535]

This is a power law spectral weight, but only for ‘thermally populated’ small momenta.

The zero temperature limit is instead obtained by considering k T1/z. In this limit, to find the matching solution we need to solve the equation of motion (4.80) with k rd/(dθ) 1. This approximation now holds from the horizon at r = r+, all the way out to the matching region at r+ r. As in [267; 376; 469], the large k solution can be found using WKB. The WKB solution is

A short WKB connection argument is necessary to convince oneself that it is this mode – decaying towards the horizon – that corresponds to the regular solution at the horizon. A growing mode would lead to a nonsensical exponentially large spectral weight below. The solution has been normalized so that it carries no temperature dependence into the matching region (r r+). As previously, this is necessary to ensure that the asymptotic UV behavior of the field has no temperature dependence, as required by the boundary condition (4.84). Evaluating (4.95) on the horizon – one does not need to perform the integral, it is sufficient to rescale rr+4.89), we find the spectral weight

Here C is a temperature-independent constant, depending on the UV scale (typically charge density ρ) which supports the emergent scaling geometry. The scaling with k is a little different to the zero temperature suppression quoted in (2.28) above. We see that for any finite z and θ, as T → 0 all nonzero momentum spectral weight is exponentially suppressed. In terms of spectral weight, then, there is no indication of Fermi surface-like physics in Einstein-Maxwell-dilaton models with z < ∞.

4.4.3 Logarithmic violation of the area law of entanglement

In compressible phases with a Fermi surface, the nonlocality (in space) of the many low-lying degrees of freedom at the Fermi surface leads to a logarithmic violation of the ‘area law’ of the entanglement entropy of a region. For free fermions the entanglement entropy of a spatial region of size R has, in addition to the usual area law, a contribution [768; 298]

This result can intuitively be understood by thinking of the low energy excitations near the Fermi surface as a collection of chiral fermions forming CFT2s at each point on the Fermi surface [720]. Hence, the logarithm is related to (1.65), and the area prefactor counts the number of these CFT2s.

We saw in §4.2 above that Fermi surfaces lead to the thermodynamic hyperscaling violation exponent θ = d − 1. It was suggested in [424] that logarithmic area law violation may be a generic property of compressible phases with θ = d − 1. In particular, building on [607], they showed that this is the case in holographic models. We will outline this result.

It is convenient to write the hyperscaling violating metric as (2.29) in the calculation below. Using a holographic calculation similar to that presented in §1.7, one finds that the entanglement entropy of a sphere of radius R is given by

𝜖)d−1. To evaluate the integral (4.98) one first needs to find the surface ρ(r) that minimizes the above functional. This can be done in an expansion about r = 0 (the boundary of the IR region). This is where one expects any singular contribution to originate from. It turns out that ρ is constant to leading order in this limit [424], and so the scaling of the integral is determined the measure dr/rdθ. If dθ > 1, the integral in the hyperscaling violating region is dominated by small r, leading to a further (finite) contribution to the area law. However, if θ = d − 1, then the integral scales as dr/r. This is just like the CFT2 case that we previously studied, which had a logarithmic area law violation. So in this case (schematically):

4.99) has R*ρ−1/d, with no other dimensionful scales appearing, and with a prefactor that is independent of the shape of the entangling surface. These results make nontrivial use of Gauss’s law in the bulk [424]. If θ > d− 1, then the integral (4.98) is dominated by the IR, and we find power law violations of the area law. This suggests imposing the condition

4.1) not to diverge as T → 0. It is also independent of the null energy conditions in (2.35) above.

The fact that backgrounds with θ = d − 1 share both the thermodynamics and entanglement entropy of Fermi surfaces is suggestive of a common physics at work. String theoretic constructions of such backgrounds may shed further light on the microscopic origin of the logarithmic entanglement in holographic models [598]. As we have seen above, however, if z is finite these backgrounds will not have any spectral weight at nonzero momentum (at least in the simple operators considered above). Conversely, the z = ∞ geometries discussed above — that do have finite momentum spectral weight — do not have logarithmic violations of the entanglement entropy area law [719].

The upshot is that EMD backgrounds can have important Fermi-surface-like physics if z = ∞ or if θ = d − 1. However, each case only shares some features with weakly interacting Fermi liquids. When neither of these two relations hold, as can certainly be the case in microscopic models, then there is little evidence for Fermi-surface like dynamics of the charged horizon.

Finally, we can note a tantalizing connection to the condensed matter systems discussed in §4.2 above. Putting θ = d − 1, and furthermore restricting to the unique regular geometries with z = 1 + θ/d, discussed at the end of §4.3.3, leads to z = 1 + (d − 1)/d. For the two dimensional case d = 2 this leads to

4.18) for the single patch Ising-nematic theory.

4.5 Fermions in the bulk I: ‘Classical’ physics

The most direct signature of a Fermi surface is a zero in the inverse fermionic Green’s function; this is expected to be present even in cases where fermionic quasiparticles are absent. We have stated previously that the microscopic electron operators are not readily accessible from the standpoint of emergent universal low energy physics, and certainly do not help to organize that physics. The low energy effective description, however, will typically also contain gauge-invariant fermionic operators. In this section we describe the correlation functions of such fermionic operators in holographic compressible phases. These computations are ‘classical’ in the sense that they involve solving the classical Dirac equation in various backgrounds. The pioneering works in this endeavor were [516; 530; 161; 265]. The Fermi surfaces found below in such computations are analogs of the dα fermions in §4.2.3 [423; 674].

4.5.1 The holographic dictionary

A few words are necessary about the holographic dictionary for fermions. A simple quadratic action for bulk fermion fields is the Dirac action with minimal coupling to a background metric and Maxwell potential:

Here {ΓM, ΓN} = 2δMN are d + 2 dimensional (bulk) Gamma matrices, and ω is the spin connection, necessary to describe fermions on curved spaces. A further important type of bulk fermions are spin-3/2 Rarita-Schwinger fields.

Unlike the bosonic fields we have studied so far, the Dirac action is manifestly first order, and so we must revisit the basic holographic dictionary [589; 430]. As the computations are similar to those for e.g. scalar fields, we will skip most details and outline the necessary changes. Crudely speaking, we think of writing down a second order equation for half of the components of the bulk spinor. It is this half of the bulk spinor which is dual to a fermionic operator in the boundary theory. If the bulk spacetime dimension d + 2 is even, this is consistent with the dual operator also being a Dirac spinor. This is because a Dirac spinor in d + 1 odd dimensions has half the number of components of a Dirac spinor in d + 2 even dimensions. However, if the bulk spacetime dimension is odd, then we are left with “half” of a Dirac spinor in the boundary theory – namely, a Weyl spinor. We can now be more precise – see [589; 430] for more details on the following.

We will assume that the bulk metric depends only on r and is diagonal. This is the case for all the spacetimes we have considered so far. In this case, it is convenient to make the change of variable [265]

4.102) becomes

. Near the asymptotically AdS boundary, this equation reads

Here Γrψ± = ±ψ± – this is the “splitting” of the bulk spinor in half, as advertized. One can further show that the dimension Δ of the fermionic boundary operator is

, an alternate quantization is also possible, analogous to that discussed in §1.6.3 above for scalars. In this quantization the dimension Δ = (d + 1)/2 − mL.

4.5.2 Fermions in semi-locally critical (z = ∞) backgrounds

An important understanding that emerged from holographic studies of fermionic correlators is that semi-locally critical degrees of freedom (as discussed in §4.3.1 and §4.4.1 above) with z = ∞ can efficiently scatter low energy fermions at nonzero momentum kkF. This leads to broad non-Fermi liquid fermionic self-energies at low energies and temperatures, ω, T μ. This was first understood in detail from the AdS-RN background, as we now describe [265; 263; 433]. The same analysis applies to fermions in the more general z = ∞ geometries described around equation (4.61) above [328; 200].

The matching computation for the zero temperature Green’s functions proceeds exactly as for the bulk scalar fields studied in §4.4.1. Thus the result for the low energy Green’s function takes the form

As previously for the scalar fields, these exponents control the momentum-dependent scaling dimensions of operators in the semi-locally critical IR theory. The dimension is again given by νk + 1/2. Note that if qe is large enough, νk can become imaginary. This indicates an instability of the black hole towards pair production of fermions, which we will return to and resolve in §4.6.

In a slight generalization of (4.75) we have allowed the functions in (4.108) to have regular frequency dependence (i.e. positive integer powers of ω starting at ω0). This comes from perturbatively re-inserting the frequency dependence in the wave equation in the far region. Recall that away from the horizon, the ω → 0 limit is analytic. Unlike the exponents in (4.109), the functions are not universal IR data, but depend upon the entire bulk geometry. The functions 1 and 3 are real, following the arguments given in §4.4.1.

Fermi surfaces occur if the denominator of (4.108) vanishes at ω = 0 for some k = kF. That is, if 3(0, kF) = 0. This may or not occur, depending on the full background. Typically the Fermi momenta must be found by solving the equations of motion numerically (see [265] for systematic numerical results in AdS-RN). Occasionally the full ω = 0 wave equation can be solved analytically, as in e.g. [345; 328]. Once kF is found, the Green’s function can be expanded around the Fermi sufrace. Denoting k = kkF to be the momentum transverse to the Fermi surface (as the theory is isotropic), one finds

depend on the full spacetime, not just the IR scaling geometry.

Equation (4.110) is indeed the two point function of a fermionic particle, with a Fermi surface at momentum kF, and with self-energy 4.8):

1. If 2νkF > 1, then as ω → 0, (4.110) may be approximated as GR = Z(ωvFk)−1. The fermion propagator genuinely resembles that of the Fermi liquid. In this case, there is a long-lived quasiparticle, but its decay occurs via interactions with a locally critical bath and is not governed by the standard Fermi liquid phenomenology.

2. When 2νkF = 1, then (4.110) becomes

It was found in 1989 that such a Green’s function explains multiple experimentally anomalous properties of the cuprate strange metal [740], and is called the ‘marginal Fermi liquid’. This possibility is fine tuned in the AdS-RN black hole, because the charge and mass need to be just right for 2νkF = 1, but it is interesting to see these connections with a much older model.

3. If 2νkF < 1, then as ω → 0, (4.110) is , and does not look like a long-lived quasiparticle.

Plots of νkF in the AdS-RN background as a function of the mass and charge of the fermion operator can be found in [265]. It was also found in [265] that whenever Fermi surfaces appear in AdS-RN, then as k → 0 (i.e. at small momenta, not close to kF), νk becomes imaginary. This indicates an instability of the low energy AdS2 × d geometry in all of these cases, as we will discuss below.

The following sections will consider the physical interpretation of these Green’s functions, as well as their implications for the charged horizon background. Further work on the Fermi surfaces themselves includes the addition of bulk dipole couplings between the spinor and the background Maxwell field [245; 244; 317] — such couplings are common in consistent truncations, e.g. [61; 60; 531; 198; 199] — as well as an understanding of the physics of spin-orbit coupling [23; 395]. Sum rules for holographic fermionic Green’s functions have been discussed in [337]. Holographic fermions in periodic potentials were discussed in [532]. For massless fermions, exchanging the choice of quantization – as described at the end of §4.5.1 – inverts the fermion Green’s function [265]. With a bulk dipole coupling, this can lead to zeros in the Green’s functions that are ‘dual’ to Fermi surface poles [27; 739].

4.5.3 Semi-holography: One fermion decaying into a large N bath

The Green’s function (4.110) can be understood from what was termed a ‘semi-holographic’ perspective in [267]. This perspective leads to an intuitive picture that will furthermore clarify the nature of the large N physics going into (4.110).

The semi-holographic picture is motivated as follows. The Fermi surface pole in (4.110) is nothing other than a quasinormal mode of the Dirac equation in the charged black hole background; recall the discussion in §3.5 above. As kkF, the frequency of the mode goes to zero. At ω = 0 exactly, it is a zero energy normal mode (with no imaginary part) in the background. The radial profile of this mode is peaked at some radius set by the crossover from the IR to the UV region [267], consistent with the fact that kF comes from solving the Dirac equation in the ‘far’ region, away from the horizon. This mode is therefore an example of the phenomenon described in §1.6.4 above: it is a low energy excitation that is not geometrized. That is, it is not captured (at ω = 0) by the near-horizon region of the geometry. The reason this can occur is that the Green’s function (4.110) is describing a single fermion at a Fermi surface whereas geometrized gravitational dynamics necessarily describes – as we have seen above – a large N number of degrees of freedom.

At nonzero ω, the decay of this mode is determined by the rate at which it falls through the black hole horizon. The near horizon region describes the low energy dynamics of a strongly interacting large N quantum critical phase. To describe the decay, one can write down an action in which a ‘free fermion’ ψ is coupled to a fermion χ that is part of a strongly interacting large N sector:

The strongly interacting sector Sstrong[χ, …] above is the near horizon region of the AdS-RN geometry. In particular, χ has a dual field in this near horizon spacetime. At large N, the only fermion correlation function of the strongly interacting sector which is not suppressed by a power of N is the two point function χχ – in the bulk this manifests itself in the fact that the action for the field dual to χ is quadratic up to 1/N corrections. Hence, employing the bare Green’s functions

4.72), the full Green’s function for ψ may be computed exactly as a geometric sum of diagrams:

4.110) up to the normalization coefficient, and upon ‘zooming in’ near a Fermi surface where 𝜖kvF(kkF).

The argument above can be turned into a precise derivation using the renormalization group flow of double trace operators at large N [266; 383]. This semi-holographic perspective gives a field-theoretic understanding of why we were able to compute the Green’s function (4.110) with only a small amout of information coming from the full black hole geometry (such as the value of kF). The non-trivial input comes from (i) the large N suppression of higher order correlation functions, (ii) the existence of a single fermion normal mode at ω = 0 and (iii) the semi-locally critical form of the Green’s functions in the large N critical sector. Any model with these features, holographic or not, will lead to (4.110).

The semi-holographic perspective also applies when the IR critical sector has z < ∞. In these cases, the imaginary part of the IR Green’s function at low temperatures and energies, but nonzero kkF, takes the same exponentially suppressed form as we have discussed previously for bosons in (2.28) and (4.96). See e.g. [267; 426; 360; 434; 336; 480]. The decay into the horizon is very ineffective for these bulk fermion normal modes, a geometrization of the fact that there are no low energy excitations at nonzero momenta when z < ∞.

The semi-holographic derivation emphasizes the separation between the single bulk fermion zero mode and the remaining large N low energy degrees of freedom that are described by the near horizon dynamics. Consistent with this picture, we will see in §4.6.1 below that the holographic Fermi surface comes with an order one amount of charge carried outside of the black hole horizon, by the bulk fermion field. This can be contrasted with the large N amount of charge behind the horizon, suggesting that the Fermi surface is only capturing the dynamics of a very small part of the system [351; 424; 432]. An alternative interpretation starts by noting that the bulk fermion is dual to a gauge-invariant composite fermionic operator in the boundary [330; 198; 152]. One can roughly think of the composite operator as the product of a colored (gauge-charged) boson with a colored fermion. In a generic state the colored bosons can be condensed, this then allows the composite operator to have an overlap with a large N number of colored ‘gaugino’ fermions. The Fermi surfaces detected holographically would then count the large N amount of charge carried by these gaugino Fermi surfaces.

Independently of the interpretation of the bulk Fermi surface, the presence of a small amount of charge in a Fermi surface outside of the horizon is responsible for important bulk quantum mechanical effects. These effects are subleading in large N. This means firstly that the properties of the Fermi surface do not imprint themselves onto leading order in N observables. For instance, the electrical conductivity at leading order in large N will be determined by fluctuations of the bulk Maxwell field (as we have discussed above) and will not notice the existence or not of a Fermi surface. This last perspective emphasizes how in strongly coupled dynamics one must be prepared to disassociate physics (such as Fermi surfaces and conductivities) that is closely connected at weak coupling. Of course, in a microscopically grounded holographic dual of a given boundary theory, the single theory fixes the background and the equation for the Maxwell field and for the Dirac field, and therefore the conductivity and the Fermi surfaces arise from a common underlying structure.

Secondly, it turns out that many of the quantum mechanical effects associated with Fermi surfaces, as is well known in condensed matter, are singular at low energies. These include quantum oscillations, Cooper pairing and singular contributions to observables such as conductivities. The presence of quantum mechanical effects in the bulk that can dominate observables at low energies amounts to a breakdown of the large N expansion. In the following §4.6 we will describe some of these effects, including the quantum mechanical instability of the near horizon region in the presence of fermionic operators with imaginary scaling dimension.

4.6 Fermions in the bulk II: Quantum effects

4.6.1 Luttinger’s theorem in holography

Towards the end of the previous section we noted that the presence of Fermi surface poles in Green’s functions is suggestive of the presence of charge in the bulk. Because the bulk fermions obey Pauli exclusion, they cannot carry the charge by condensing into a macroscopically occupied ground state. Instead they will fill up a Fermi surface in the bulk, outside of the black hole event horizon. A derivation of this fact necessarily requires quantum mechanics. In this section we will sketch the result that

Here ρ is, as always, the charge density of the boundary QFT, ρhor. is the electric flux coming through the charged horizon and the sum is over the (d-dimensional) momentum-space volumes 4.115) was first proven in a WKB limit in the bulk in [360; 434], away from this limit in [676] and in complete generality (allowing for horizons in the bulk) in [432].

The expression (4.115) is strongly reminiscent of the modified Luttinger relation (4.22), in which the charge density is made up of a sum of gauge-neutral ‘cohesive’ Fermi surfaces and gauge-charged ‘fractionalized’ Fermi surfaces [423; 351; 361; 424]. At large N, ‘gauge-charged’ operators are more rigorously thought of as those built up as very long traces of the fundamental fields (and hence with a large scaling dimension). For instance, the deconfinement transition can be profitably discussed in this language [17]. Fractionalization in this context is an analogue of such large N deconfinement in which the charge is carried by long rather than short operators. The large degeneracy of long operators at large N leads to this charge being hidden behind an event horizon. Equation (4.115) does not say whether the charge behind the horizon is carried by Fermi surfaces or not. The reader is referred to our discussion in §4.4 that bulk quantum corrections could lead to signatures of a Fermi surface [636; 261; 677] from the charge behind the horizon. The work of [17] has not yet been generalized to a discussion of possible large N fractionalization in weakly interacting theories in the ’t Hooft limit. An attempt to find a holographic order parameter for charge fractionalization can be found in [370].

To prove (4.115), consider the free bulk Dirac fermion (4.102), of charge q, coupled to e.g. EMD theory. We can sketch the derivation in [676; 432], and for simplicity work at zero temperature. The fermion can be integrated out, leading to the effective bulk action

where Vd is the spatial volume of the boundary theory, 𝜖l(k) are the eigenvalues of the bulk Dirac equation, with eigenspinors χl(k, r), subject to appropriate boundary conditions [676; 432]. The effective action (4.117) shows that the free energy due to fermions is the energy of all the occupied sites in a bulk Fermi sea. 𝜖l(k) is implicitly a function of the background EMD fields. As advertised, equation (4.117) shows the Pauli exclusion principle at work. The bulk geometry contains fermionic matter placed into its lowest energy states.

The equations of motion for the metric and Maxwell fields originating from the effective action (4.117) are nonlocal. This makes the effects of the backreaction of the Fermi surfaces on the spacetime difficult to study. A backreacting and gravitating Fermi surface is nothing other than a star. For this reason, the backreacted solutions were called ‘electron stars’ in [377], by analogy to astrophysical neutron stars. The asymptotically Anti-de Sitter boundary conditions provide an additional gravitational potential well that allows a charged star to exist, despite the repulsion between the individual charged fermions. Indeed, taking a cue from astrophysics, the backreacted equations are most easily solved in a WKB approximation for the fermions. This is a generalization of the Oppenheimer-Volkoff-Tolman approximation [610; 734] for neutron stars to include charge and a negative cosmological constant. Neutron stars themselves have also been considered in Anti-de Sitter spacetime [184]. In a condensed matter context, the WKB approximation for many electrons interacting with an electromagnetic field is called the Thomas-Fermi approximation [731; 268]. For discussion on the connection between WKB fermion wavefunctions and the effective ‘fluid’ descriptions of these solutions, see e.g. [662; 54]. WKB electron star solutions are studied in §4.6.3 below.

Away from the WKB limit, the backreaction problem is very challenging. Important progress has been made in [25; 24]. One result of these papers in that the WKB approximation works well somewhat beyond its naive regime of applicability. Without solving the full backreaction equations, however, we proceed to show how the result (4.115) can be obtained exactly from one of the bulk Maxwell equations.

Assuming the usual ansatz A = p(r)dt for the Maxwell field, one obtains [676; 432]

4.117). Integrating over r from the AdS boundary to the horizon gives

where we have used the AdS/CFT dictionary at the boundary, orthonormality of χl, defined ρhor as a (normalized) radial electric flux across the horizon, and defined VFS as the total volume of the bulk Fermi surfaces. In this way we establish (4.115), the holographic analogue of the Luttinger relation (4.22) in the presence of both cohesive and fractionalized charge carriers. qVFS counts the cohesive charge which can detected by a gauge-neutral fermionic operator (for instance, via Fermi surface singularities). ρhor counts the fractionalized charge which is not directly detected by such gauge-neutral operators, but remains hidden behind the horizon. We note in passing that (4.119) has a nonzero temperature generalization [676; 432]. Prior to the fermions considered in this subsection, the solutions we have considered have ρ = ρhor and all the charge was fractionalized. Cohesive charge can also be carried by condensed bosons, as we will see in Chapter 6 below.

The semi-holographic description of §4.5.3 makes clear that while cohesive fermions obey a conventional Luttinger relation, their decay is not that of a Fermi liquid. Rather, the fermions decay into a large N critical sector. An ordinary Landau Fermi liquid is obtained holographically if the critical sector is removed, by considering a gapped geometry of the kind discussed in §2.4. In these spacetimes there is no horizon for the bulk fermions to fall into. An explicit toy model of this situation, using a ‘hard wall’ in AdS, was considered in [676]. As anticipated from §2.4, at a classical level the imaginary part of the fermion self-energy vanishes. One expects quantum bulk computations to follow standard many-body treatments, leading to Im(GR(ω → 0)) ∼ ω2. That is because, in the absence of a horizon, one is simply studying interacting fermions in a fancy (i.e. curved spacetime) box. There are no geometrized low energy degree of freedom and much of the interesting ingredients of holography are gone. The situation is analogous to confinement in QCD. The strongly interacting dynamics leads to a mass gap, and the remaining ‘semi-holographic’ degrees of freedom (hadrons and mesons in the case of QCD) interact weakly at low energies.

4.6.2 1/N corrections

We now will explore three consequences of the bulk fermion matter whose existence has been revealed by the holographic Luttinger theorem (4.115). These will be IR singular quantum mechanical effects in the bulk. The study of these effects (quantum oscillations, Cooper pairing and contributions to the conductivity) will follow very closely the conventional treatment in standard weakly interacting fermion systems. In this sense, bulk fermion matter is fairly conventional. The two important differences are that (i) we must learn to do the computations in a curved spacetime and (ii) the presence of an event horizon means that the tree level fermion propagators already have significant imaginary self-energy terms, as shown in §4.5.2 above. As explained in §4.5.3, this latter effect is a consequence/artifact of having a large N critical bath into which the fermions can decay.

The computations outlined below follow the logic described in §3.5.2. The reader should read that section before continuing. The Fermi surface pole in the fermion Green’s function corresponds to a quasinormal mode with complex frequency approaching zero. As we explained in §3.5.2, the non-analytic one-loop effects of such poles can be extracted zooming in on the portion of the propagator controlled by this pole, as in (3.92) or (3.95) above.

4.6.2.1 Quantum oscillations

We begin with quantum oscillations. This is a characteristic phenomenon of a Fermi surface in a magnetic field: the magnetic susceptibility (the second derivative of the free energy with respect to the magnetic field) oscillates like ℏ = c = e = kB = 1. These oscillations are resonances that occur when the cyclotron orbit of the fermion (in momentum space) coincides with a cross section of the Fermi surface, so that there are many low energy excitations that can participate in the motion. The existence of the oscillations is very robust and independent of the form of the self-energy of the fermions, so long as the fermion Green’s function has a pole along a Fermi surface. More details of the following computation can be found in [193; 194; 357].

The amplitude of the quantum (or de Haas-van Alphen) oscillations is a function of temperature. This temperature dependence is strongly sensitive to the self-energy of the fermions. In a Fermi liquid, the temperature dependence of the amplitude takes a celebrated form called the Lifshitz-Kosevich formula [524]. In particular, at large temperatures compared to the magnetic field, the amplitude decays exponentially as, schematically, . The bulk fermions in a holographic semi-locally critical background will lead to quantum oscillations. We now outline the computation of A(T) in this case, restricting attention to the simplest case of d = 2 space dimensions.

The quantity to be computed is the determinant of the Dirac operator from the bulk fermion action (4.102). The logarithm of this determinant is the one loop contribution of the fermions to the free energy. Using methods similar to those leading to (3.92) above, a general formula can be derived for the ‘oscillatory’ part of this fermion contribution to the free energy [357] (see also [749] for a more diagrammatic derivation – this is not a holographic formula per se)

The fermionic Matsubara frequencies are 4.121) is the Landau level. To leading order in a small magnetic field, the effects of the magnetic field are captured by starting with the zero magnetic field holographic Green’s function and replacing the momentum dependence by k2 → 2ℓB.

To evaluate (4.120) for semi-locally critical fermions, it is important to use the full nonzero temperature IR Green’s function for the fermions. For the case of AdS-RN, this can be found by solving the Dirac equation in the AdS2-Schwarzschild near-horizon geometry (4.41). The upshot is that in the expression (4.110) for the fermion retarded Green’s function, the frequency dependence of the self-energy is generalized according to [263]

ℝ) symmetry of the AdS2 IR theory [263]. This is an additional symmetry that the theory enjoys and is logically distinct from the fact that z = ∞.

Inserting the IR Green’s function (4.122) into the full Green’s function (4.110), the oscillatory part of the free energy (4.120) can be evaluated. The full expression is a little complicated, see [357], but a distinctive behavior emerges in the limit of large temperatures compared to the magnetic field, where the magnetic susceptibility

Here c is a constant set by the Fermi momentum and chemical potential, and ν is the exponent νk evaluated at the Fermi momentum k = kF. In this non-Fermi liquid case where the amplitude of the oscillations is distinct from the Lifshitz-Kosevich form and depends upon the dimension ν of the fermionic operator in the low energy critical theory.

The oscillatory susceptibility in (4.123) is a quantum effect in the bulk and is therefore suppressed by factors of N compared to the leading classical contribution, that does not oscillate. It is notable that none of the holographic models of compressible matter discussed so far exhibit quantum oscillations in the leading order in N magnetic susceptibility, despite some backgrounds manifesting Fermi-surface like physics as discussed in §4.4. The only known, somewhat artificial, way to make the oscillations contribute at leading order is in the semiclassical electron star discussed in §4.6.3 below.

4.6.2.2 Cooper pairing

Fermi surfaces are well known to be unstable at low temperatures towards condensation of Cooper pairs. This condensation spontaneously breaks the U(1) symmetry. Holographic Fermi surfaces can undergo Cooper pair condensation following the usual BCS mechanism adapted to curved spacetime. More details of the following computation can be found in [345].

Cooper pairing requires an attractive force between the fermions. In [345] the following bulk contact interaction was considered

Here Γ5 = iΓ0Γ1Γ2Γ3, MF determines the mass scale of the interaction and Ψc is the charge conjugate fermion (see [345]). The action (4.124) can be considered a toy model for exchange interactions between the fermions mediated by other bulk fields. The interaction has been chosen to decouple nicely in the ‘Cooper channel’. This decoupling is achieved by introducing a charged Hubbard-Strotonovich field Δ so that the interaction is written

The next step is to integrate out the fermions to obtain the one-loop effective action for Δ. An instability arises when the effective mass squared for Δ becomes negative. Restricting to configurations of Δ = Δ(r) that only depend on the radial direction, one finds the quadratic effective Euclidean action to be

Here V2 is the volume of the boundary spatial directions and T is the temperature (inverse period of the Euclidean time circle). After some standard manipulations with fermionic Green’s functions and with Gamma matrices and, crucially, using (3.95) to zoom into the singular part of the Green’s function, one obtains [345],

4.110), at low energies and with momenta close to the Fermi surface, with the self energy generalized to the nonzero temperature expression (4.122). The Green’s functions describe the fermion loop sourced by two insertions of Δ (as specified in (4.126) above). The momentum integral has also been restricted to the contribution close to the Fermi surface. The hyperbolic tangent arises from standard manipulations in which a sum over fermionic Matsubara frequencies is expressed as a contour integral. Finally, ψ0(r) is the radial profile of the normalizable fermion zero mode at ω = T = 0 and k = kF. These last terms arise from the numerator of (3.95).

The IR singular contribution to the integrals in (4.127) comes from the frequency range T Ω μ and is readily evaluated. If

This is essentially the standard BCS logarithmic IR divergence. As T → 0, the logarithm leads to a negative mass squared term in the effective potential (4.126) and condensation of Δ. The instability onsets below the critical temperature at which the two terms in (4.126) are equal. The critical temperature is given, very schematically, by 4.124). From the perspective of the dual compressible phase of matter, this implies that the critical temperature is determined by the magnitude of a certain four point correlation function of the composite fermion operator Ψ dual to ψ in the bulk.

Recall that interesting vertex corrections or long range attractive forces at this leading order in N level.

The condensation of Δ discussed above only impacts an order one part (that in bulk Fermi surfaces) of a large N amount of charge (mostly behind the horizon). Most large N observables, therefore, will not notice that the U(1) has been spontaneously broken. In contrast, in Chapter 6 below we will describe a distinct, fully holographic and leading order in large N mechanism for holographic superconductivity.

4.6.2.3 Corrections to the conductivity

The charge carried by the bulk Fermi surface will, of course, move when an electric field is applied and will hence contribute to the conductivity. The conductivity is obtained holographically as described around (3.50) above, by solving Maxwell’s equations in the bulk. More details for the following computation can be found in [262; 263; 264]. The fermion contribution is a quantum correction to this process, whereby the fermions ‘screen’ the bulk Maxwell field. That is, there is a fermion one loop correction to the Maxwell field propagator in the bulk. This one loop process is evaluated using standard methods, together with (3.95) to zoom into the singular part of the Green’s function, and the result takes the form [262; 264]

4.110), at low energies and with momenta close to the Fermi surface, with the self energy generalized to the nonzero temperature expression (4.122), and Λ(k) is a smooth function of k that describes (i) the vertex coupling the fermions to the Maxwell field, (ii) the ‘bulk to boundary’ propagator of the Maxwell field (and metric) and (iii) integrals over the radial profile ψ0(r) of the normalizable fermion zero mode at ω = T = 0 and k = kF. In general the function Λ is frequency and temperature dependent, but these dependences drop out at low energies and temperatures and close to the Fermi surface.

The IR singular contribution of the integrals in (4.129) is readily extracted. The low temperature dc conductivity is found to be [262; 264]

where, as previously, ν = νkF is the exponent at the Fermi momentum. The resistivity is linear in temperature when μ with [262; 264]

4.130) and (4.131) will be strongly suppressed compared to the leading order in large N conductivities that will be the main topic of Chapter 5 below. The computation of the fermion conductivity outlined above avoids many of the subtleties typically associated with computing conductivities in quantum critical theories. This is because of the semi-holographic coupling to a large N critical bath. The fermions acquire nontrivial self-energies at leading order in N due to dissipation into the black hole, but there are no vertex corrections at this leading order. Furthermore, because the momentum of the order one ‘probe’ fermions is not conserved at large N – it is lost to the quantum critical bath – one does not need to worry about the momentum-conservation delta functions that will be discussed at depth in Chapter 5.

4.6.3 Endpoint of the near-horizon instability in the fluid approximation

We explained in §4.6.1 above that the backreaction of bulk fermions on the geometry is most easily captured in the WKB approximation. The individual wavefunctions of the Dirac field become sufficiently localized in space that they are insensitive to variations in the metric, Maxwell or other fields. We noted that this amounts to a conflation of the Thomas-Fermi approximation in condensed matter physics with the Oppenheimer-Volkoff-Tolman approximation of astrophysics. Namely, the fermions are treated as a charged, gravitating ideal fluid. This limit was first studied in [368] and further developed in [377]. In this subsection we overview the physics of the resulting semiclassical ‘electron stars’.

In this subsection we will restrict attention to d = 2 space dimensions. This follows most of the literature on this topic. The wavefunctions of fermions in a background coming from the Einstein-Maxwell-dilaton action (4.43) enter a WKB regime when (e.g. [360])

In this limit the mass of the fermion appears in Dirac equation in the combination (κm)/(qe). This ratio quantifies the relative strength of gravitational and electric interactions between fermions. For the ratio to be order one, the dimension of the dual QFT operator Δ (mL)2 (qeL)22 1. This large operator dimension is also the condition for the quantum wavelength of the bulk fermions to be small compared to the background AdS curvature scale. This latter condition however is not strictly implied by the limit (4.132). Further discussion of the conditions for the WKB limit to hold can be found in [184; 377; 54; 24]. The WKB limit is not necessarily natural from the point of the view of the dual compressible phase of matter. Its function instead is to obtain a tractable bulk description of the fermion backreaction and some intuition for the resulting physics.

In §4.6.1 we saw that every bulk Fermi surface comes with an associated bulk charge carried by fermions. For typical parameter values (order one charges and scaling dimensions) we have noted that the amount of charge carried by such bulk Fermi surfaces is subleading compared to charge behind the black hole horizon. In the large N limit, then, the gravitational and electric effects of the backreaction of this charge is negligible. In addition, because these Fermi surfaces are semi-holographic and localized away from the event horizon, their backreaction does not alter the low energy quantum critical dynamics.

The more dramatic effect of backreaction arises instead from the near-horizon instability of modes with imaginary scaling dimension under the IR critical scaling. From (4.109) we see that such an instability occurs in the AdS-RN background (over some range of low momenta) whenever

d [629]. A fruitful way to think of (4.133) is as a competition between electromagnetic screening and gravitational anti-screening. Suppose a fermion anti-fermion pair is spontaneously produced in the near-horizon region. The electromagnetic interaction will act to screen the charge of the black hole: the fermion with an opposite sign charge to the horizon will be attracted towards the horizon, while the fermion with the same sign charge will be pushed away. This effect tends to discharge the black hole and populates the bulk outside of the black hole with charged fermions. The gravitational interaction, in contrast, famously anti-screens or clumps. Both of the charges will be gravitationally attracted towards the horizon and the pair production will have no net effect. (4.133) therefore expresses the fact that the horizon will discharge if the charge of the fermions is large relative to their mass, so that the electromagnetic screening effect wins out. This process is illustrated in Figure 4.7.

Figure 4.7
Pair production in the near horizon region leads to a discharging of the black hole if electromagnetic screening overcomes gravitational clumping. Figure taken with permission from [351].

The fate of the fermionic charge in the bulk can be determined, in the WKB approximation, by solving the Einstein-Maxwell-(dilaton) equations of motion coupled to a charged, gravitating fluid. At zero temperature and with no rotation, such a fluid can be described by a generalization [377] of the Schutz action [681]

and Pf is the pressure of the fermions:

. To include the effects of nonzero temperature and rotation in the fluid, see [681; 377; 351].

From the expression (4.136) for the pressure, it is clear that a fluid of fermions will be present in the geometry wherever f > m. In the semiclassical limit, this is where the local chemical potential is large enough to overcome the rest mass energy and populate a local Fermi surface. Near the asymptotically AdS boundary, since gtt ∞ and At → μ, then μf 0. Hence, at any nonzero mass m, the bulk fermion fluid can only appear towards the IR, if at all. In the case in which the fluid forms, it is therefore present beyond some radius rs at which f(rs) = m. This is the boundary of the electron star.

The most important question for universal low energy physics is the fate of the geometry in the far interior, as r → ∞. One can note that if there is an emergent Lifshitz scaling invariance (cf. (2.16)) so that gtt ∼ r−2z and At ∼ rz, then the local chemical potential (4.135) remains constant as r → ∞. If this constant is large enough compared to the mass, then this emergent scaling will be consistent with the presence of a fluid all the way to r → ∞. Precisely this scenario is realized in Einstein-Maxwell theory, whenever the criterion (4.133) for instability is satisfied [368; 377]. A fluid extends from ∞ < r < rs, all the charge is carried by fluid, and the far interior is given by a Lifshitz geometry:

While the full solution is found numerically, the emergent scaling solution in the interior can be found analytically. The value of z depends on the mass and charge of the fermion, it can take any value from 1 < z < ∞. The solution is illustrated in Figure 4.8.

Figure 4.8
The electron star. A charged, gravitating fluid is present in the spacetime for ∞ < r < rs. All the charge is carried by the cohesive fermion fluid. Figure taken with permission from [351].

The IR fixed point solution (4.137) is somewhat remarkable as a gravitational background. Typically a spatially uniform and static matter density is impossible in a theory of gravity, as the energy density causes the universe itself to contract. The solution (4.137) gets around this in an intrinsically relativistic way: a scale-invariant gravitational redshift allows the gravitational and electric interactions to consistently and stably balance themselves.

The discussion so far has been at T = 0. At sufficiently high temperatures, one expects the star to collapse to a black hole. That is because as the temperature is increased at fixed chemical potential, the mass of the black hole increases relative to its charge and hence gravitational clumping is increasingly favored over electric screening. Eventually, pair production is no longer possible and the high temperature black hole is stable. This scenario has been studied quantitatively in [367; 644]. The main points are as follows. At any T > 0, the local chemical potential μf 0 near the nonzero temperature horizon. This follows from the definition of μf and the fact that near the horizon At ∼ gtt ∼ rrh. This behavior of At at a horizon was noted below (4.30) above. Thus, even at very low temperatures, the fermion fluid is pushed away from the horizon, and exists only in an intermediate region in the bulk . The range of radii over which this ‘electron cloud’ exists narrows as the temperature is increased, until the fluid disappears altogether in a third order phase transition at a critical temperature Tc. This is, then, a temperature-driven fractionalization phase transition: At T = 0 all charge is cohesive, at nonzero temperature some amount of the charge is fractionalized, and at T > Tc, all of the charge is fractionalized.

Fractionalization transitions can also occur at zero temperature as parameters in the theory are varied. Quantum fractionalization transitions were studied in [361] in a class of Einstein-Maxwell-dilaton theories coupled to a charged fluid. The novel feature compared to the Einstein-Maxwell case is that in cases where Z(Φ) ∞ in the IR, then some of the charge remains on the horizon (and hence fractionalized, in the dual theory). The EMD fields look similar to the theories described in §4.3.3 in this case. In particular, note from (4.48) and (4.54) that

as r → ∞ in the IR. This behavior – a non-constant μf in the scaling regime – is a direct consequence of the anomalous scaling dimension of the charge density, as discussed in the first paragraph of §4.3.4. According to (4.138), if a fermion fluid forms at all in the bulk, it can only be over an intermediate range of radii. Depending on the details of the model, μf may be so small that no Fermi surfaces are populated, and all the charge is fractionalized. The quantum phase transitions between fractionalized, partially fractionalized and cohesive compressible phases can be first order or continuous in these models.

As the bulk fermionic fluid describes cohesive boundary charge, then, according to the holographic Luttinger theorem (4.115), there must be poles in fermionic Green’s functions at nonzero momenta. The Dirac equation was solved in the electron star backgrounds in [360; 434; 160]. In the WKB limit (4.132) a large number of closely spaced Fermi surfaces are present. These essentially correspond to the distinct bulk Fermi surfaces at each radius of the star. The electron star is therefore a distinctly holographic phase of matter, in which the distribution of Fermi surfaces captures the radial distribution of charge in the bulk. While the many Fermi surfaces should presumably be treated as a large N ‘artifact’, the corresponding smearing of the fermionic spectral weight over a range of momenta does have some interesting phenomenological consequences. For instance, the closely spaced Fermi surfaces lead to cancellations in the quantum oscillations of §4.6.2 so that only a single ‘extremal’ oscillation survives [359].

Observables such as the charge density spectral weight and electrical conductivities are computed in the same way as in other holographic backgrounds. Namely, by solving the bulk Maxwell equations in the presence of the charged fluid. These quantities turn out to be controlled by the properties of the IR scaling solutions such as (4.137), much as in the fractionalized cases [377; 306]. The full hydrodynamic behavior of these geometries remains to be investigated.

4.7 Magnetic fields

Magnetic fields are an important probe of compressible phases. The boundary QFT has a global U(1) symmetry, and we can put this theory in a background (non-dynamical) magnetic field. Per the essential holographic dictionary applied to conserved currents in §1.5.5 above, the source

The gauge symmetry AA + dλ ensures that the leading behavior near the boundary is a constant in r. The boundary magnetic field is then

To determine the effects of such a magnetic field, we need to solve the bulk equations of motion subject to this boundary condition. We will start by considering the case of d = 2 spatial boundary dimensions, so that is the only component of the boundary magnetic field. Later we will also consider the case of d = 3, in which a magnetic field necessarily breaks isotropy by pointing in a specific direction.

4.7.1 d = 2: Hall transport and duality

We will limit ourselves to constant magnetic fields so that we can write A(0) = Bxdy. In d = 2 dimensions we can then search for a bulk solution of the same form (4.44) as previously, except that now we allow the bulk Maxwell field to take the more general form

4.24) and (4.25) above, whereas the near boundary behavior of B(r) controls the boundary magnetic field. Note a certain asymmetry: the asymptotic electric flux in the bulk is an expectation value in the dual QFT (charge density) whereas the asymptotic magnetic flux in the bulk is a source in the QFT (background magnetic field).

An especially simple solution can be found in (d = 2) Einstein-Maxwell theory (4.26). This is the AdS-RN solution. The Maxwell potential takes the form

4.30) above. The charge density ρ is still given by (4.31). The metric still takes the form (4.27), but now with the redshift given by

Thus we see that the radial dependence of the metric is virtually unchanged from the purely electric case. Similarly to the purely electric case, the T = 0 limit of this solution has an AdS2 × 2 near horizon geometry, very similar to (4.37) above, and associated zero temperature entropy. Details of the thermodynamics of this solution can be found in [363; 364]. In particular, the magnetic susceptibility χ = 2P/∂B2 remains a smooth negative function at T = 0, showing no quantum oscillations as a function of the magnetic field and qualitatively rather similar to the magnetic susceptibility for free bosons [193]. Recall that we discussed possible subleading in 1/N quantum oscillations in §4.6.2 above.

A magnetic field introduces qualitatively new features into transport. In particular, because the magnetic field breaks time reversal invariance, Hall conductivities are now allowed. Hall conductivities are zero in a time reversal invariant and isotropic state because then the retarded Green’s function

⟩ = 0. We will proceed to discuss aspects of the electrical conductivities following from the dyonic AdS-RN solution of Einstein-Maxwell theory above. We follow the presentation of [353].

With time reversal invariance broken, there is a matrix of electrical conductivities

We have assumed isotropy. This information is usefully packaged into a complex conductivity. If we define

Note that in frequency space, σxx(ω) and σxy(ω) are already complex functions. The rate of work done by an electric field in the presence of Hall conductivities is

This result is obtained from manipulations similar to those leading to (3.33) above.

Transport in compressible phases will be the topic of Chapter 5. We will be able to discuss aspects of transport in magnetic fields prior to our more general exposition below. This is because it turns out that a magnetic field avoids the complications related to conservation of momentum that will be a central issue later. Technically this is due to fact while the magnetic field itself is uniform, the vector potential (such as A(0) = Bxdy) necessarily breaks translation invariance. As a consequence, there is a Lorentz force on the resulting system which violates momentum conservation.

As in §3.4 above, the conductivities σij(ω) are obtained by perturbing the bulk Maxwell field about the dyonic black hole background. Because of the background charge density and magnetic field, the perturbations of the Maxwell field couple to metric perturbations. At k = 0, a self-consistent set of modes to perturb are δAx, δAy, δgtx, δgty. Let us define the bulk electric and magnetic fields to be

then the linearized Einstein-Maxwell equations about the dyonic background can be written as [353]

This transformation will be important shortly.

As usual, we must solve the above equations of motion subject to infalling boundary conditions at the horizon. The conductivities (4.147) are then extracted from the boundary behavior according to

4.153) implies that

4.147) is inverted. This is of course a generalization of the particle-vortex duality (3.86) to the case with magnetic field, charge density and Hall conductivities. The factors of 2π appear because under the duality the electromagnetic coupling is inverted as 2π/e2e2/2π. We will see this duality from a more hydrodynamic perspective in §5.7 below.

In fact, the duality map (4.155) is part of a larger SL(2, ) group of dualities. See [767] and references therein. Electromagnetic duality is the S generator of the group. The T generator is understood by adding a topological theta term to the bulk Einstein-Maxwell theory (4.26) in d = 2:

Here θ is a constant. This term has no effect on the bulk dynamics. Its only effect is that under a shift θθ + 2π, the Hall conductivity shifts so that

4.156) makes to the on-shell bulk action, which determines the current expectation value according to (1.33).

To evaluate the conductivities, one must solve the equations of motion (4.152a) and (4.152b). The ω = 0 limit is especially simple; b+ = −e2ρ/Be+ and hence from (4.154), as first found in [363],

This is a general result that will be rederived in §5.7 from relativistic hydrodynamics.

A second limit in which the equations of motion can be solved analytically is at small frequencies, with ρ2 ∼ B2 ∼ ω also taken to be small. The small frequency expansion is implemented similarly to in §3.4.3 above, by first factoring out the non-analytic frequency dependence of the infalling boundary condition at the horizon,

4.154) that

It is simple to check that the expression (4.160) is consistent with the duality (4.155). This collective cyclotron mode can also be obtained from hydrodynamics, as discussed in §5.7 below. We will elaborate more on its physics in that later section and will give the separate expressions for σxx and σxy. The cyclotron mode is just the longest lived of many quasinormal modes that extend into the lower half complex frequency plane. In fact, the analytic structure of the conductivity obtained in this case is similar to that shown in Figure 3.5 above. Now the magnetic field and charge density are the parameters that deform away from the self-dual point. Poles and zeros alternate and are exchanged under electromagnetic duality.

For a discussion of how the physics described above generalizes to Einstein-Maxwell-dilaton theories, see [300; 527].

4.7.2 d = 3: Chern-Simons term and quantum phase transition

The above results have been for d = 2. The case of d = 3 also contains rich physics. The role of charge density ρ and magnetic field B are quite different in this case, and a quantum phase transition can occur as a function of the dimensionless ratio B/ρ2/3. This system has been studied in a series of papers including [204; 205; 206; 207]. These papers and others are summarized in [208]. We will note a few salient results.

Firstly, in 4 + 1 bulk dimensions (corresponding to d = 3) there is a natural Chern-Simons term that often appears in Einstein-Maxwell theory. This is

This interaction term is not a total derivative, and alters the Einstein-Maxwell equations. It leads to an FF term on the right hand side of Maxwell’s equations. Thus the electromagnetic field itself now carries charge. This fact leads to instabilities in this theory as will be discussed in Chapter 6 below. It also leads to the possibility of cohesive phases within Einstein-Maxwell-Chern-Simons theory, with all charge being carried by the bulk electromagnetic fields. The coupling (4.162) is also closely connected to an anomaly in the global symmetry dual to the bulk Maxwell field [764]. Such relativistic anomalies are not unheard of in condensed matter systems, as we will see in §5.7.1.

We will not write down the form of the bulk fields. These are still homogeneous – only depending on a radial coordinate r – but no longer isotropic, because the magnetic field singles out a direction. This leads to a greater number of functions of r (metric components etc.) that must be solved for. Let us describe the T = 0 solutions for different values of B/ρ2/3. The asymptotic near boundary geometry is fixed to be AdS5. The most important question for low energy observables, as we have seen, is the near horizon geometry. At B = 0 the near horizon geometry is just the AdS2 × 3 of the five dimensional planar AdS-RN solution. However, at ρ = 0 the near horizon geometry is AdS3 × 2 [204]. This asymmetry arises because in the electrically charged case the bulk Maxwell field is in the t, r directions (that define the AdS2) while in the magnetic case the bulk Maxwell field is in the x, y directions, that define the 2. Given these different IR geometries, it is clear that a phase transition will need to occur at intermediate values of B/ρ2/3. Indeed this is the case [206; 207], and the phase diagram is shown in Figure 4.9 below.

Figure 4.9
Phase diagram of Einstein-Maxwell-Chern-Simons theory as a function of temperature and magnetic field, at fixed nonzero charge density. The Chern-Simons coupling k > 3/4 (for 3/4 > k > 1/2 the temperature scaling of s in the critical region is different). Figure taken with permission from [208]. This phase diagram ignores certain spatial modulation instabilities that will be discussed in Chapter 6.

This phase diagram is best understood when the Chern-Simon coupling in (4.162) is sufficiently large, k > 1/2. In this case the T = 0 near-horizon geometry is known at all magnetic fields above the critical field [207]. It is given by

Now, for αo > 0, the second term in brackets in (4.163) is subleading in the far interior as r → ∞. In this case the near horizon geometry reduces to AdS3 × 2 written in null coordinates, as in the pure magnetic case. There is no electric flux through the horizon. The quantum critical point, however, is characterized by a value of the magnetic field at which αo = 0 in the near-horizon geometry (4.163). In this case the second term in the brackets must be kept and the scaling symmetry of the IR geometry is changed to

The critical IR geometry is of the Schrödinger form mentioned briefly in §2.2.3 above, and may be the most microscopically well-grounded and simple of the known realizations of this geometry in string theory backgrounds. The IR geometry determines the temperature scaling in the quantum critical regime [207; 208]. Note that the quantum critical point separates two phases that are in themselves critical (i.e. gapless).

The phase transition in Figure 4.9 is ‘metamagnetic’, that is to say, no symmetries are broken across the transition. Instead, this transition is a fractionalization quantum phase transition, analogous to those discussed for charged fermions towards the end of §4.6.3 above. Below the critical magnetic field, some of the asymptotic electric field emanates from behind the horizon, corresponding to fractionalized charge, and the rest is sourced by the bulk electromagnetic field via the Chern-Simons interaction (4.162), corresponding to cohesive charge. Above the critical magnetic field, all of the charge is cohesive and there is no electric flux through the horizon.

The low temperature thermodynamics of these solutions admits a physical interpretation [208]. At large magnetic fields, the behavior s ∼ T describes the excitations of a chiral CFT that propagate in the direction of the magnetic field. At small magnetic fields there remains a zero temperature ground state entropy density. While a compelling connection has yet to be made, it is worth noting that a ground state entropy density does in fact arise for free relativistic fermions in a magnetic field: the ‘zero point’ energy of the lowest Landau level is precisely cancelled by the Zeeman splitting energy gain of the spin up electrons. Therefore, the entire lowest Landau level is at zero energy.

Exercises

4.1.  Scalars in a hyperscaling violating geometry. In this exercise, we will study the two-point functions of scalars in a hyperscaling violating geometry (2.29) at T = 0.

a) Consider a conventional massive scalar in the bulk, with action (1.24). Focus on a hyperscaling violating geometry with d = 2θ, z = 1. Show that the two point function of the dual operator 𝒪 is [215]

Plot this function of k, and comment on the physical interpretation. Discuss qualitatively the position space form of this Green’s function.

b) Now, consider a scalar φ with action (4.62) and with B(Φ) depending exponentially on the background dilaton, as described below (4.62). Compute G𝒪𝒪(k) for the dual operator 𝒪 for general d, z and θ.

c) In many top-down constructions, a dilatonic potential V(Φ) ∝ eβΦ arises from integrating over extra dimensions with volume ∝ eβΦ. Suppose that a probe scalar φ were present in the higher dimensional theory. Which of the two actions for φ studied in this problem is more likely to arise in the dimensionally reduced theory? Comment on your answer, especially in relation to the physical differences between the correlation functions resulting from the different actions.

4.2.  IR Green’s functions from SL(2, ) invariance. We have seen that semi-local criticality leads to the zero frequency spectral weight (4.92), as well as the zero temperature spectral weight (4.76). In this exercise you will see how the full IR Green’s function ℝ) symmetry in the semi-locally critical geometry [263]. This occurs in Einstein-Maxwell theory.

a) For T μ, the temperature is part of the IR geometry. In Einstein-Maxwell theory the IR geometry is then the Schwarzschild-AdS2 × d solution (4.41). Write down the wave equation for a massive neutral scalar field in this geometry, generalizing (4.68) to include the emblackening factor f2(ζ) in (4.41).

b) Solve the wave equation above in terms of hypergeometric functions. Impose infalling boundary conditions at the horizon and then expand towards to the boundary of AdS2, as in §4.4.1 to obtain the IR Green’s function

4.70). The overall constant is unimportant, as in any case it is remormalized in going between the UV and IR Green’s functions as in (4.76). Check that the limits ω/T 1 and ω/T 1 reproduce the previous results in the text. This formula is the neutral, boson version of (4.122) quoted in the main text. The charged, fermion result is obtained analogously.

c) The result (4.167) is obtained from symmetry as follows. First, show that upon making the coordinate transformation from t, ζ to τ, σ:

4.41) becomes simply the AdS2 metric in (4.37) with coordinates τ, σ.

d) At the boundary, σ = ζ = 0. The transformation becomes

Now, Fourier transform the T = 0 Green’s function ω2ν to position space, apply the transformation (4.170), and Fourier transform back to frequency space. Does your result agree with (4.167)?

4.3.  Viscosity. This exercise gives an elegant derivation of the holographic result (3.60) for the shear viscosity, following [372].

a) Find the equation of motion for a ‘shear’ metric perturbation in linear response, in an arbitrary EMD background of the type studied in §4.3.2.

b) Generalize the methods of §4.4.2 to δgxy, and derive (3.60).

4.4.  Nucleation of an electron star. In this exercise you will find the critical temperature at which the electron star discussed in §4.6.3 appears in the spacetime as the temperature is lowered in a charged black hole background. You will also show that this leads to a third order phase transition. This exercise follows [367].

a) Start with the Reissner-Nordström-AdS background (4.27) in d = 2. Recall that the charged fluid will be present in the geometry wherever the local chemical potential in (4.135) satisfies m < qμf(r). Show that at high temperatures, there is no charged fluid anywhere in the spacetime.

b) As the temperature is lowered there will eventually be a critical temperature Tc at which the fluid starts to nucleate, at one radius rc in the spacetime. This extremal radius satisfies

Show that these conditions can also be understood as determining the radius at which a charged point particle can remain stationary, with gravitational and electromagnetic forces balancing.

c) By solving the conditions (4.171) make a plot of the dimensionless ratios rc/r+ and γTc as a function of the ratio m/q of the fermions.

d) At temperatures slightly below Tc, the fluid spreads out to form a cloud of width Δr = r2r1. Using the fact that at the endpoints r1, 2 of the cloud are defined by f(r1, 2) = m, show that Δr ∝ (1 − T/Tc)1/2 to leading order. The change ΔΩ in the free energy due to the appearance of the cloud then follows from the (Euclidean version of the) action (4.134):

Here V2 is the volume in the boundary spatial directions. Using the definition of the pressure in (4.136), show that this leads to the scaling ΔΩ ∝ (1 −T/Tc)3 and hence to a third order phase transistion.