A Lucas

We compute direct current thermoelectric transport coefficients in strongly coupled quantum field theories without long lived quasiparticles, at finite temperature and charge density, and disordered on long wavelengths compared to the length scale of local thermalization. Many previous transport computations in strongly coupled systems are interpretable hydrodynamically, despite formally going beyond the hydrodynamic regime. This includes momentum relaxation times previously derived by the memory matrix formalism, and non-perturbative holographic results; in the latter case, this is subject to some important subtleties. Our formalism may extend some memory matrix computations to higher orders in the perturbative disorder strength, as well as give valuable insight into non-perturbative regimes. Strongly coupled metals with quantum critical contributions to transport generically transition between coherent and incoherent metals as disorder strength is increased at fixed temperature, analogous to mean field holographic treatments of disorder. From a condensed matter perspective, our theory generalizes the resistor network approximation, and associated variational techniques, to strongly interacting systems where momentum is long lived.

Article funded by SCOAP^{3}.

One of the most exotic and mysterious systems in condensed matter physics is the strange metal, non-Fermi liquid phase of the high

From a theoretical perspective, a generic strongly interacting quantum field theory (QFT) in more than one spatial dimension has only a few quantities (energy, charge and momentum) that are long lived, and so hydrodynamics may be a sensible description of the low energy physics at finite temperature and density of all of the above systems. Though hydrodynamics is an old theory [

In most ways, hydrodynamics is a far simpler theory to understand (and perform computations in) than quasiparticle based approaches, such as kinetic theory. The difficulty in studying these systems theoretically lies in the fact that hydrodynamics does not completely solve the transport problem: the coefficients in the hydrodynamic equations must be related to Green’s functions in a microscopic model. Nonetheless, if hydrodynamics is valid, it does provide universal constraints on transport, and a transparent physical picture to interpret the results. There are two tractable approaches that can compute the requisite microscopic Green’s functions, without reference to quasiparticles. The first is methods from (perturbative) QFT, combined with the memory matrix approach [

Surprisingly, many of the above transport theories from recent years completely match hydrodynamic predictions, at least superficially, despite being formally beyond the regime of validity of hydrodynamics. We take this as an indication that a thorough understanding of hydrodynamic implications for transport in disordered theories is worthwhile, though we will also carefully describe the regime of validity of the approach. In addition, while almost every citation above aims to address the strange metal phase [

Let us begin with the main quantitative motivation for the present paper, which is the physical interpretation of a large body of recent holographic work on transport in QFTs without translational symmetry.

[

We begin with a generic hydrodynamic framework for zero frequency transport calculations in section

We exactly solve the transport problem in section

We further show in section

We then proceed to study hydrodynamic transport in non-perturbatively disordered QFTs. Though not amenable to analytic techniques, we develop a combination of rigorous variational approaches and heuristic approximations, outlined in section

To date, all models of incoherent metals are holographic massive gravity models [^{1}

This is technically not quite right—there is one (set of) bulk scalar fields in these models which is of the form

^{2}

See [

One might suspect that the fact that (

Our work in section ^{3}

In graphene, for example,

We have written (^{4}

We can maintain an electric current without adding any energy by simply shifting to a moving reference frame.

. In contrast, whenWhen

A pictorial summary of (

A qualitative sketch of the coherent–incoherent transition realizable in our framework.

After we had completed this work, [

Though the main quantitative focus of this work is a set of computational tools to study hydrodynamic transport in relativistic fluids, such as in holography, we also emphasize that the framework we are developing (with suitable generalizations) is sensible for a description of transport in strongly interacting condensed matter systems, without any reference to holography.

A common approximation made in condensed matter is what we will refer to below as the ‘resistor lattice’ approximation, which in physical terms is the statement that the slow, hydrodynamic sector of the theory consists of only a conserved charge. One may then model the emergent hydrodynamics—a simple diffusion equation for charge—as a local resistor network: see e.g. [

However, we will point out in section

We emphasize that the calculations in [

We consider a strongly coupled QFT in

We deform the microscopic Hamiltonian

In order for hydrodynamics to be valid, it is necessary that ^{5}

We employ a separation of 3 length scales in this paper.

In a quantum critical theory of dynamical exponent

More liberally, one could only require that

When (

For holographic theories, we do

Let us begin with some simple calculations to get an intuitive feeling for hydrodynamic transport. We work with first order hydrodynamics, and will justify this later in the section. We will also assume that our theory is isotropic, another assumption which we relax later.

A first simple case to consider is when the only slow dynamics in the system are of charge. As we mentioned previously, in this case the dynamics reduce to the solution of a diffusion equation:

Let us now account for convective transport—this means that momentum is a long lived quantity and must be included in hydrodynamics. If we neglect thermal transport, then we must modify (

Let us now describe the complete linear response theory which includes the response of temperature, chemical potential and velocity to external fields.

About our background fluid we perturb the system with an infinitesimal electric field

Let us define

We write down the gradient expansion of hydrodynamics to first order in derivatives acting on _{i}) fluid motion. In particular,

The momentum conservation equation becomes^{6}

Note that

Equation (

Equation (

Having imposed these boundary conditions, we prove in appendix

As mentioned previously, we have truncated the hydrodynamic gradient expansion at first order. Let us give some sensible, though non-rigorous, justifications for this. The hydrodynamic gradient expansion can be organized as follows:_{i} varies over the length scale _{i} must vary on the length scale _{i} on short length scales compared to _{i}. This general framework readily generalizes to account for higher derivative corrections to hydrodynamics, if one wishes to directly include them, but we will not include them in this paper. Equation (

In the absence of other dynamical sectors of the theory, it is necessary that either _{i} and

Let us also briefly mention the issue of momentum relaxation times. In many holographic mean field models of disorder, the momentum relaxation time can be parametrically fast [

It is also worth stressing that henceforth, when we refer to ‘hydrodynamic’ transport we refer to the transport equations being written in terms of (

In this section, we specialize to the weak disorder limit in which slow momentum relaxation dominates the conductivities. In this limit we can make direct contact with the memory matrix formalism [_{0} translation invariant, and

Let us briefly note our conventions in Fourier space. Fourier transforms are defined as

In [^{ti} is the momentum density, and ^{7}

In contrast, [

Interestingly, our hydrodynamic approach requires the disorder is always long wavelength compared to

Let us begin with the case where the operator _{0}: in position space,

Let us make an ansatz for the solution to the hydrodynamic equations, and show that it is consistent with all conservation laws. Our ansatz is that the only divergent terms in linear response, as

The leading order response of ^{8}

Note that this linear term in perturbations is parametrically large in

Now, let us study the momentum conservation equation, averaged over space, so the derivatives of the stress tensor do not contribute:

Let us now argue that the ansatz (and thus results) we have found are self-consistent. If we do not average the momentum conservation equation over space, then the

In this section, we consider the case where

Again, we make the ansatz that the only response at

Let us briefly discuss some simplisitic limiting cases of (

First, let us begin with the case with

An alternative simple case is thermal transport with

To make contact between (

The retarded Green’s function is

We have worked through two specific examples of deriving (

It is possible to compute the transport coefficients at higher orders in perturbation theory, where the memory matrix formalism has become unwieldy enough that such a calculation has not yet been attempted. Even at next order in perturbation theory, the corrections to the conductivity become quite messy. We discuss the general structure of higher order computations in appendix

Finally, let us compare the results of this subsection with the ‘resistor lattice’ approximation:_{i} and

This is impossible in the weak disorder limit, though our comments appear to have broader validity whenever viscosity cannot be neglected.

Let us now compare with holographic results. Many holographic results, valid in the weak disorder limit, are equivalent to the memory matrix results [

Our discussion of holography is brief—for further details, consult the excellent reviews [

A qualitative sketch of holography. A finite temperature

The precise duality allows us to compute correlation functions in our unknown QFT by solving gravitational equations instead. The basic idea is as follows: correlation functions of the stress–energy tensor _{MN} and a gauge field _{M}, all computed at (classical) tree level. We are using _{i} at the AdS boundary, and then compute the expectation value of the current^{9}

This expectation value is also encoded near the boundary of AdS.

in the field theory in the background perturbed by_{i}.

These computations are often intractable analytically. However, in the simple case of dc transport, many analytic computations of dc transport in holography are performed using the membrane paradigm [

It is remarkable that

Let us begin with the striped models studied in [^{10}

Note that their results simplify, in some special cases, to analytic results derived in [

_{t},

_{tt}and

_{t}.

Let us postulate the following equations of state for the emergent fluid on the horizon:

We also need to make two more assumptions. The first is rather simple—let us suppose (

More recently [

In the case where translational symmetry is broken in multiple directions, there is an important subtlety. It turns out that the local ‘current’ in the emergent horizon fluid is

It was recently shown [

These examples suggest that—while the hydrodynamic framework of this paper is extremely helpful providing some physical intuition to these non-perturbative holographic results—this story is not complete. Importantly, however, much of the variational technology that we develop can be directly applied to holographic models.

We cannot be as rigorous in the strong disorder limit and give closed form expressions for the conductivity matrix. Nonetheless, we will develop simple but powerful variational methods that allow us to get a flavor of transport at strong disorder, by providing lower and upper bounds on the conductivity matrix.

We focus on the discussion of

We present the mathematical formalism in the subsections below—explicit examples of calculations may be found in appendix

Define ‘voltage drops’ _{i}. Since

Let us verify this. Energy is dissipated^{11}

Of course, this would lead to temperature growth at second order in perturbation theory, so that the energy conservation equation (up to external sources) exactly holds at all orders.

locally via the dissipative (Σ and_{i}obeys periodic boundary conditions):

_{i}the outward pointing normal.

Let us begin by discussing the lower bounds on conductivities. These are by far the more important bounds to obtain, because—as we will see—they allow us to rule out insulating behavior in a wide variety of strongly disordered hydrodynamic systems.

We obtain lower bounds on conductivities analogous to how one obtains upper bounds on the resistance of a disordered resistor network, via Thomson’s principle [

Let us propose a trial set of charge and heat currents,

In particular, we can immediately obtain bounds for all diagonal entries of

Obtaining upper bounds on conductivities is in principle more simple, but quite a bit more subtle in practice. Let us write^{12}

For example, this may correspond in a conformal fluid to deforming the equation of state with a non-zero bulk viscosity.

. To compute the conductance, we need to demand (as stated previously) thatWe are going to guess a single valued trial function

And since the integrand in (

Here we present a summary of the calculations performed in appendix

Equation (

Many of the statements which lead to (

In (^{13}

When

A second assumption that went into (

Similar bounds can be found for other transport coefficients. In particular, for bounds on ^{2} with

One of the most important results we find is an exact inequality for an isotropic fluid:

The new approaches advocated in this paper, along with the existing mean-field literature, suggests that the fate of most holographic models at strong disorder—at fixed temperature

We have always referred to these hydrodynamic models as incoherent metals. Holographic ‘insulators’ discussed in the literature typically rely on ^{14}

As the percolation problem is trivial in

In a non-holographic context, it is less clear whether or not our hydrodynamic formalism will be valid in a quantum system undergoing a metal–insulator transition, as the validity of hydrodynamics rests on the disorder being long wavelength. The classical ‘metal–insulator’ transition realized by resistor networks [

Finally, as we are studying a strongly disordered system, it is also worthwhile to think about fluctuations in the transport coefficients between different realizations of the quenched disorder. As in [

As we previously mentioned, many free or weakly interacting quantum systems are described by a ‘localized’ phase where transport is exponentially suppressed at low temperatures [

This is consistent with known results in elastic networks and other random resistor networks. Despite the localization of classical eigenfunctions [

The finite momentum or finite frequency response of the system may be more sensitive to localization. In a simple model of disordered RC circuits, interesting new universal phenomena arise [

There is one other point worth making about localized eigenmodes. In a translationally invariant fluid, long-time tails in hydrodynamic correlation functions in

In this paper, we have explored the consequences of hydrodynamics on the transport coefficients of a strongly coupled QFT, disordered on large length scales. We demonstrated that hydrodynamics can be used to understand the memory function computations of momentum relaxation times, which have previously been derived using an abstract and opaque formalism. It is also straightforward—at least in principle—to compute transport coefficients at higher orders in perturbation theory, whereas memory function formulas only give leading order transport coefficients. Remarkably, we also demonstrated that many non-perturbative holographic dc transport computations can be interpreted entirely by solving a hydrodynamic response problem of a new emergent horizon fluid. Thus, the technology of appendix

The fact that this hydrodynamic framework can be used to interpret such a wide variety of results from memory function or holographic computations is suggestive of the fate of such theories at strong disorder. Shortly after this paper was released, it was proved in [

More generally, we have also demonstrated—without recourse to mean field treatments of disorder, or to holography—a framework which generically gives rise to both a coherent metal at weak disorder, and an incoherent metal at strong disorder. Incoherent metallic physics has been proposed recently to be responsible for some of the exotic thermoelectric properties of the cuprate strange metals [

Hydrodynamics provides a valuable framework for interpreting more specific microscopic calculations. There are many natural extensions of this work: two examples are the study of hydrodynamic transport in disordered superfluids and superconductors, or the study of systems perturbed by further deformations than

I would like to thank Ariel Amir, Richard Davison, Blaise Gouteraux, Sarang Gopalakrishnan, Bertrand Halperin, Sean Hartnoll and Michael Knap for helpful discussions, and especially Subir Sachdev for critical discussions on presenting these ideas in a more transparent way.

In this appendix we prove that the thermoelectric conductivity matrix

To do this, we look for a periodic solution _{i}, up to the constant linear terms in

Let us describe how to extend the weak disorder calculations of section

Let us write _{i} may be written as follows:

Equations (

At higher orders, the _{μ} and

This provides a prediction of our hydrodynamic framework which may be compared with a memory matrix calculation at higher orders in perturbation theory (or another method). Of course, we should stress that in principle, memory matrix calculations can account for corrections beyond the regime of validity of hydrodynamics, though in the limits we identified in section

In the above framework, it does not seem as though there are any natural cancellations between various terms at higher orders in perturbation theory. So this approach becomes rapidly unwieldy for computing transport coefficients past leading order in

A simple set of trial functions is

In cases with weak disorder this bound is not strong enough to be useful, and we can do better by allowing for

It is straightforward to see that the smallest eigenvalue of ^{15}

To compute an eigenvalue, one should first properly ‘re-dimensionalize’

Let us compare to the exact results in the perturbative limit in section

The first inequality here follows from the fact that for any vector _{i}, and two positive definite matrices _{ij} and _{ij}, the following inequality holds:

It is also possible to find viscosity-limited bounds on the resistance matrix which can be smaller than the diffusion-limited bound (

For simplicity, we focus on the bounding of

Let us begin by assuming that

For simplicity, suppose that

Plugging this Φ into (

In a theory with

Now, let us consider the effects of finite viscosity. Henceforth, the discussion will be more qualitative, and we will not be particularly concerned with O(1) prefactors, as it turns out to be quite difficult to write down a good non-perturbative analytic solution to (

Let us see what happens if we simply use (_{x} = 0 or _{y} = 0. It is now straightforward to (qualitatively) see what happens. The first contribution to the conductivity is unchaged from (

Of course, the general framework can certainly account for this possibility, and one can directly plug in our ansatzes into (

Essential to the scaling laws in (

Let us verify this is possible. We wish to find a solution to the highly overdetermined equation

We now look for

Thinking about resistivities turns out to be most convenient for models in _{x} in terms of

In particular, let us carry out this computation explicitly for the holographic striped models with equations of state given in section ^{ij} are

_{2}(As

_{1-x}P

_{x})

_{2}superconductors