Connections to experiments

The rate of change of the resistivity with temperature shows similarities across several families of metals with T-linear resistivity, possibly suggestive of a bound at work. In this chapter we consider several instances in which holography has motivated different ways of organizing experimental data. Image taken from [114] with permission.

8.1 Probing non-quasiparticle physics

As we noted in the introduction, §1.1, modern quantum materials provide many examples of ‘strange metal’ states without quasiparticle excitations. Traditional field theoretic condensed matter studies of such states employ a variety of expansion methods which extrapolate to strong-coupling from a weak-coupling starting point with quasiparticles. The advantage of the holographic methods described here is that they directly yield a solvable framework for metallic states without quasiparticle excitations, and this has many consequences for a complete description of the observable properties of such states.

Before turning to specific materials below, the most important ‘practical’ output of holography is that it suggests a shifted emphasis for what kind of observables reveal the nature of non-quasiparticle dynamics. This is helpful independently of the system of interest.

8.1.1 Parametrizing hydrodynamics

Holographic examples explicitly demonstrate that conducting quantum fluids exist in strongly interacting systems, and that one of their defining properties is rapid local equilbration in a time of order /(kBT). Classical hydrodynamics is the effective theory of such fluids beyond this time scale. Indeed, solutions of holographic models point the way to direct hydrodynamic analyses of transport, without the need to extrapolate from a quasiparticle framework; as we noted in §5.6.3, the latter can often lead to misleading results.

A very basic distinction between quasiparticles and collective hydrodynamic transport is captured by the Lorenz ratio L ≡ κ/(). Quasiparticles transport both charge and heat and hence tie electronic and thermal conductivities together into the Wiedemann-Franz (WF) law (5.2), as long as interactions can be neglected. Measured violations of the WF law have long been considered a tell-tale sign of non-quasiparticle or at the very least exotic physics, e.g. [643; 213; 713; 727]. Hydrodynamic transport in particular, however, offers clean ways of thinking about violations of the WF law with distinctive signatures. In a hydrodynamic regime charge and heat are independent hydrodynamic variables. In an incoherent hydrodynamic metal they are essentially decoupled into distinct diffusive modes, see §5.8 above and [352]. In hydrodynamic metals with a long-lived momentum they are related by the relative efficiency with which sound propagation drags heat and charge as in (5.127); see also [554]. A further effect in this regime (with large enough charge density) is a dramatic distinction between open and closed circuit thermal conductivity, κ and , discussed towards the end of §5.4.3.

Recent experimental violations of the WF law have been usefully interpreted in terms of both coherent [159; 303] and incoherent [515] hydrodynamics. In a hydrodynamic regime the Lorenz ratio is also an interesting observable at higher temperatures. At higher temperatures one must either find a compelling way to subtract out the phonon contribution to the thermal conductivity [515], look at the Hall Lorenz ratio [777], to which the neutral phonons do not directly couple, or alternatively understand the phonons themselves as an intrinsic part of the metallic system [775].

A major open question from this standpoint is whether the many families of strange and bad metals with T-linear resistivity extending to high temperatures are in a non-quasiparticle hydrodynamic regime. If so, one would like to know whether the appropriate hydrodynamic framework is purely diffusive [352] or based around a further long-lived mode that could be momentum [182] or a Goldstone-related excitation such as a phase-disordered density wave [189]. These long-lived modes directly couple to the currents and so their lifetime should be directly visible in the optical conductivity.

Direct evidence for hydrodynamic flow can be found in an unconventional sensitivity to the geometry of the current flow. Recent pioneering experiments have initiated progress in this direction [67; 582; 303], as we discuss in more detail in §8.2.1 below. It will be very exciting if these and similar direct probes of hydrodynamic flow can be extended to other materials.

8.1.2 Parametrizing low energy spectral weight

The low energy spectral weight as a function of energy and momentum is a basic characteristic of any system. Conventional metallic phases of matter contain a Fermi surface of weakly interacting quasiparticles, and this strongly determines the structure of the low energy spectral weight. Without quasiparticles the universally defined spectral weight to consider is that of the conserved charges and currents, such as . Direct, high resolution information about this observable would give invaluable insights into strange metal regimes.

A phenomenologically interesting possibility that emerges in some of the simplest holographic models is that of a z = ∞, semi-locally critical sector as in §4.3.1. Such a sector retains some features of a Fermi surface without reference to single particle concepts. As described in §4.4.1 and §4.4.2 this leads to zero temperature spectral weight of the form ω2ν(k) and a low temperature dependence of T2ν(k). The k dependence of the exponent is generically present once z = ∞ and momentum is dimensionless. Requiring strictly local criticality with no k dependence whatsoever – as has occasionally been considered in the condensed matter literature – amounts to a fine tuning. In clean systems, singular k dependence linked to the underlying Fermi surface is generally expected, but such effects are suppressed in holographic analyses in the leading large N limit [261; 636; 677]. Remarkably, a recent experimental measurement of in the strange metal regime of a cuprate material has found an unexpectedly weak k dependence [581].

While the spectral weights of currents and charges are the most universally defined, any operator in a locally critical regime can be expected to show similar scaling. Thus we discussed the impact of semi-local criticality on fermionic operators in §4.5.2 and on ‘Cooper pair’ operators in §6.3.1. The case of semi-locally critical fermions has motivated analysis of ARPES data with a continuously varying exponent [653], as well as a way to fit the form of measured unconventional quantum oscillations [726]. Quantum oscillations – discussed briefly in §4.6.2 – are especially interesting here. The Lifshitz-Kosevich formula for the amplitude of quantum oscillations with temperature is a direct measurement of Fermi-Dirac quasiparticles, see e.g. [682]. If non-quasiparticle physics underlies at least some strange metals, strong deviations from Lifshitz-Kosevich must arise. It is challenging to measure quantum oscillations in quantum critical regimes because the effective mass becomes large, as seen in e.g. [648].

Direct, unambiguous evidence for semi-local criticality in any observable would be exciting as it would (i) signal that the simplest holographic model captures the correct low temperature kinematics and (ii) would allow the rich phenomenology of z = ∞ fixed points to be realized. The latter includes the strong effects of umklapp scattering on dc transport, as in (5.136), strong scattering of fermions by the critical sector, as in (4.110), tendency towards density wave instabilities, as in §6.5.2 and the ability to incorporate strong spatial inhomogeneities into the critical dynamics, mentioned in e.g. §5.10.2.

8.1.3 Parametrizing quantum criticality

Scaling arguments give a powerful way to organize observables in the absence of quasiparticle ‘building blocks’. It is well known that quantum critical points or phases lead to dynamics that is characterized by a dynamic critical exponent z as well as by the scaling dimensions of operators in the critical theory [673].

Compressible phases of matter offer a challenge to scaling theory because there is an additional scale, the charge density, which plays a key role. Even in a weakly coupled Fermi liquid the appropriate scaling theory for the low energy physics is nontrivial for this reason, as we discussed in §4.2. In the strongly coupled compressible phases described holographically in §4.3.3 and §4.3.4 two additional exponents played a central role: the hyperscaling violation exponent θ and the anomalous dimension for the charge density Φ. These exponents determine the temperature dependence of thermodynamic and transport obervables.

Experimental determination of the exponents θ and especially Φ in strange metals is of great interest. It was emphasized in [362] that the Lorenz ratio, already discussed in §8.1.1, is a direct probe of the exponent Φ in a scaling theory. If the Lorenz ratio has a nontrivial scaling with temperature in a quantum critical regime, then Φ must be nonzero. It has also been proposed to measure Φ through the nonlocal charge reponse that it induces [525].

Attempts to fit observed scaling of quantities as a function of temperature and magnetic field in the cuprates with the three exponents {z, θ, Φ} have been only partially successful [362; 472]. One challenge is that transport observables are potentially sensitive to irrelevant operators that break translation invariance, as we described in e.g. §5.6.1 above. Various assumptions that need to be made for a scaling theory to get off the ground are outlined in [362]. Finally, some of the observed scalings are better established than others. More systematic measurements to high temperatures of the Hall Lorenz ratio in the strange metal regime across the cuprate family, so far limited to [777; 572; 573; 571], with contradictory results, would be especially desirable.

Holographic studies have also lead to new parametrizations of observables in zero density critical systems described by CFTs. At high frequencies and short time scales, we learned that the operator product expansion controls deviations from criticality in response functions (§3.2.2 and §3.4.5). At intermediate time scales holography motivates searching for universal behavior in the leading order non-hydrodynamic decay to equilibrium, which is controlled by quasinormal modes [72]. Going beyond linear response, we have seen the similar imprints of critical operator dimensions in the behavior of quantum many-body systems undergoing a quench (§7.3). We hope that some of this novel far-from-equilibrium dynamics can be observed in quantum critical cold atomic gases [776; 253], and/or in the strange metals discussed above.

8.1.4 Ordered phases and insulators

It is an experimental fact that a large number of strange metallic regimes are unstable to ordering at low temperatures. As we noted in §6.1, from a weakly-interacting single-particle perspective one expects quantum critical bosonic fluctuations to both enhance and inhibit ordering. The quantum critical modes can provide a strong ‘superglue’ for pairing, but also strongly reduce the density of states available for pairing at the Fermi surface. In Chapter 6 we described an alternate non-quasiparticle description of ordering in a critical theory. The instability is ubiquitous in holographic models and occurs as a symmetry-breaking operator acquires a complex scaling exponent. This mechanism has several distinctive signatures including BKT-like exponential scalings, as discussed around (6.50), and unconventional Cooper pair fluctuations above Tc, described by (6.8). We noted in §6.1 that a weakly interacting cousin of this physics seems to arise when the RG flow of a ‘Yukawa’ coupling between fermions and a quantum critical boson couples to a nontrivial flow for the BCS coupling [647].

Many strange metals also arise in close proximity to localized phases. In §5.9.1 we saw that holographic models realized a novel scenario in which insulating behavior arises from relevant translational symmetry breaking operators in a z = ∞ scaling theory (see §8.1.2). This is an intrinsically non single-particle mechanism for localization. The temperature dependence of transport quantities in this scenario will be controlled by the scaling dimension of the relevant operator breaking translation invariance as in e.g. [236].

Both ordered and localized holographic phases often exhibit a soft, power law gap in e.g. the optical conductivity. This is due to the topologically ordered nature of holographic states, with deconfined gauge fields. We will discuss the possible relevance of topological order in the cuprates in §8.2.2 below. Such soft gaps are also characteristic of quantum spin liquid candidates [249; 627].

8.1.5 Fundamental bounds on transport

Quasiparticles lead to an infinite number of long-lived operators, δnk, and hence transport can be studied in terms of the scattering that these many variables experience. Without quasiparticles we have fewer moving parts to work with. It is natural, then, to turn to possible results that can apply to a large number of systems, following from basic principles of quantum mechanics and statistical mechanics (two handles on the system that we certainly always have!).

Fundamental bounds on dephasing times were discussed in §1.1, and transport bounds were discussed in §5.8. These were motivated from several angles. Firstly it connects naturally to bounds that apply to quasiparticle systems and that can be derived from the uncertainty relations of single-particle quantum mechanics. Secondly, holographic models that are at infinite coupling could potentially have led to infinite scattering and hence vanishing transport coefficients. Yet, instead, appropriate dimensionless transport data in holography is typically ‘order one’ in some suitable sense (see e.g. §3.4.2). Thirdly, conducting states in holography were found to survive even with arbitrarily strong disorder in §5.9 and §5.10: in particular, in some holographic models [313; 314] we saw that conductivities can be exactly bounded. Hydrodynamics can also be used to lower bound the conductivity in the presence of arbitrarily strong but long-wavelength disorder [542]. Fourthly, a recent bound (1.2) on chaotic timescales has been proven. This bound may plausibly have consequences for transport.

Examining non-quasiparticle transport through the lens of potential universal bounds has proved useful in several recent experimental studies [114; 380; 775; 548; 303]. Conductivity and/or diffusion bounds seem natural in holographic models, and it is possible there is a deep connection to such experiments. In this direction, a bound relating diffusive transport, the local equilibration rate and a certain ‘lightcone’ velocity has recently been established [346].

8.2 Experimental realizations of strange metals

We now turn to some of the best studied families of experimentally realized strange metals where the ideas we have discussed in this review are (in our view) most likely to be relevant.

8.2.1 Graphene

As we noted in §2.1, graphene is described at low energies by electronic excitations with a massless Dirac spectrum in 2+1 spacetime dimensions with the 1/r repulsive Coulomb interaction. It was argued in [592; 282; 590] that breakdown of screening near the Dirac point should lead to strange metal behavior which can be described by relativistic hydrodynamics in the presence of weak disorder. The needed transport results, with the σQ transport coefficient in (5.31), were first worked out in [364], initially using holographic inspiration, but also by direct hydrodynamic arguments; here we can extract the results from §5.6.2.

Crossno et al. [159] measured the thermal and electric conductivities of graphene for a range of temperatures and densities near the charge neutrality point. We focus here on the Wiedemann-Franz ratio, which from e.g. (5.127) is

We have re-instated factors of the effective speed of light vF and fundamental charge e. We have also used the relativistic relations (5.75) for αQ and have long-wavelength “puddle” character, and the whole fluid was described by inhomogeneous hydrodynamics [538]. Other work has examined the influence of long relaxation times between the electron and hole subsystems [276; 687].

Another test of the hydrodynamic nature of electron flow in graphene appeared in the experiments of Bandurin et al. [67]. In a finite geometry with boundaries, they observed a ‘negative nonlocal electrical resistance’ inconsistent with Ohmic diffusive electron flow, while consistent with viscous hydrodynamic flow. Such an effect requires the electron-electron interaction time to be shorter than the electron-impurity scattering time. This is a more direct probe of hydrodynamic coefficients such as viscosity, compared to global transport measurements [159]. Unfortunately, both of these experiments rely on electrical measurements. If we study relativistic hydrodynamics near charge neutrality carefully (see §5.4.1), we observe that the charge sector decouples from the energy-momentum sector. Hence, we expect the viscous ‘backflow’ signature of [67] to vanish near charge neutrality. Indeed, the signal of [67] was strongest in the Fermi liquid regime of graphene, when μ/kBT ∼ 10. So while these explorations of viscous electronic dynamics are in many instances inspired by the holographic developments we have discussed, conventional kinetic theory (and its hydrodynamic limit) is the correct approach to quantitative modeling of these systems. The quantitative modeling of the ballistic-to-hydrodynamic crossover in Fermi liquids, as in [188], may be the simplest way to quantitatively connect theories of hydrodynamic electron flow to experiment.

An important open question is how to directly measure the viscosity of the charge neutral plasma in graphene in experiment. This will allow us to test the theoretical prediction [591] that this plasma is very strongly interacting, with η/s comparable to the ‘universal’ holographic result (3.60): η/s = /4πkB. [538] suggested that η/s ∼ 10/kB, but the transport data of [159] is not specific enough to reliably measure this number. Unfortunately, the presence of disorder makes a direct measurement of η rather delicate. One proposal has been the detection of electronic sound waves [534], but this may not be feasible.

8.2.2 Cuprates

The hole-doped cuprate high temperature superconductors provide a prominent strange metal near optimal doping [679]. It is likely that this strange metal exhibits the T > 0 physics of a T = 0 critical point, or phase, near optimal doping. A great deal of theoretical and experimental work has tried to deduce the theory of this critical point by examining the nature of the adjacent phases at lower and higher hole densities. Much has been learnt about symmetry breaking in the under-doped ‘pseudogap’ regime: charge and spin density wave orders, Ising-nematic order, time-reversal and/or inversion symmetry breaking are present in a complex phase diagram as a function of density and T [470]. But it appears that none of these order parameters can explain the gap-like features observed in most spectrocopic probes over a wide range range of temperature in the underdoped regime. An attractive possibility, especially given the recent observation of a pseudogap metal at low T [56], is that the fundamental characteristic of the pseudogap metal is the presence of topological order (see §4.2.3). Then the various conventional symmetry-breaking order parameters are proposed to be incidental features of the topologically-ordered state. The optimal doping criticality controlling the strange metal is associated with the loss of topological order into a conventional Fermi liquid state at high doping. The critical theory of such a transition invariably involves a Fermi surface coupled to emergent gauge fields, and such theories were briefly noted in §4.2.3; a specific candidate for such a ‘deconfined’ theory is in [669], and this candidate includes spectator conventional orders that vanish at the critical point. From the perspective of the present article, the transport properties of such critical theories with emergent gauge fields are likely to be described well by holographic models.

In the electrical transport properties of the strange metal, a prominent deviation from quasiparticle physics is in the frequency dependence of the optical conductivity [565], σ ∼ ωα with α ≈ 2/3. Remarkably, the exponent α = 2/3 has been argued to be a robust property of Fermi surfaces coupled to emergent gauge fields [478; 240].

The dc conductivity (i.e. at zero frequency, ω = 0) is characterized by a well-known linear-in-T resistivity, which is a strong indication of the absence of quasiparticles. A direct signature of the absence of quasiparticles appears in the Hall resistivity measurements [143] which cannot be fit to a quasiparticle model [40]. Hydrodynamic and holographic models can provide a natural fit to the data [102] (see §5.7), although the connection of these models to the microscopic theory has not been established.

Recent experiments have focused on the thermal diffusivity of the strange metal [775]. They provide striking evidence of strong electron-phonon coupling, with the both the electrons and phonons excitations exhibiting a scattering time of order /(kBT). For the future, it would be of great interest to extend the graphene experiments discussed in §8.2.1 to the cuprates: those could provide crucial information on the nature of hydrodynamic flow in the electron and phonon subsystems.

8.2.3 Pnictides

Unlike the hole-doped cuprates, the pnictides typically have a spin density wave quantum critical point near the optimal doping for superconductivity. There also appears to be an interesting interplay between the spin density wave order, and Ising-nematic order which breaks tetragonal crystalline symmetry down to orthorhombic. Moreover, strange metal behavior is also dominant near optimal doping [692], suggesting a direct connection between spin density wave or Ising-nematic criticality and the non-quasiparticle transport. Electrical transport for spin density wave criticality, in the presence of weak disorder coupling to the order parameter, leads to linear-in-T resistivity [621], as we noted in §5.6.3: this is a possible explanation for the strange metal behavior. However, it remains to be seen whether such models can reproduce the remarkable scaling with B/T in measurements of the magnetoresistance [380]. The pnictides furthermore exhibit resistivities above the MIR bound, suggestive of non-quasiparticle physics.

8.2.4 Heavy fermions

The rare-earth intermetallic compounds provide realizations of Kondo lattice models, where localized spin moments residing on the rare-earth sites interact with itinerant electrons from the other elements. These provide numerous examples of quantum criticality and strange metal behavior [693; 149]. Although the Kondo lattice models look quite distinct from the single-band Hubbard models applied to the cuprates, the same fundamental issues on the nature of the quantum phases and the phase transitions apply to both models. In particular, both models support the same class of metallic states with topological order [645; 646].

The phase transitions in the Kondo lattice model are traditionally interpreted to be of two types [693]: (i) transitions with the onset of spin density wave order, which are described by the Landau-Ginzburg-Wilson framework of §4.2.2; and (ii) Kondo-breakdown transitions, in which the local moments decouple from the conduction electrons. Using a more general language, the second class of transitions are more precisely viewed as phase transitions involving the onset of topological order, in which the size of the Fermi surface can change [686]. Theories of the second class of transition therefore involve emergent gauge fields, as in §4.2.3, and are likely to be closely related to theories of the cuprate strange metal. A remarkable example of a possible topological phase transition was studied in β-YbAlB4 [735].