Hydrodynamic transport in strongly coupled disordered quantum field theories

TL;DR

The work addresses dc thermoelectric transport in strongly coupled QFTs without long-lived quasiparticles, focusing on disorder that is long-wavelength relative to local thermalization scales.It develops a steady-state hydrodynamic framework that unifies memory-matrix results and holographic computations, and extends to non-perturbative disorder via variational methods, introducing matrices such as $\sigma^{\alpha\beta}_{ij}$, $Γ_{ij}$, and $\Sigma^{\alpha\beta}_{ij}$.Key contributions include exact weak-disorder results matching memory-function predictions, a holographic horizon-fluid interpretation for non-perturbative disorder, and rigorous lower/upper bounds that reveal a coherent-to-incoherent crossover at disorder strength $u$ comparable to the mean charge density $\mathcal{Q}_0$, while ruling out insulating localization in many holographic models.Together, these results generalize resistor-network intuitions to strongly interacting, disordered QFTs and offer a broadly applicable toolkit for predicting transport in quantum-critical, strongly correlated systems beyond quasiparticle pictures.

Abstract

We compute direct current (dc) 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.

Figures (3)

• Figure 1: A qualitative sketch of the coherent-incoherent transition realizable in our framework. $\sigma$ denotes the value of a transport coefficient, such as electrical conductivity, and $u$ denotes the "strength of randomness". The solid black line shows our perturbative analytic computation of $\sigma \sim u^{-2}$ as $u\rightarrow 0$. The dashed red line is the qualitative prediction of mean field models that $\sigma$ saturates at a finite value at strong disorder in a theory with quantum critical transport; in particular, $\sigma\ge \sigma^*$. The gray shaded region corresponds to the region of $\sigma$ allowed by variational bounds on $\sigma$, in generic agreement with mean field models. $u\sim \mathcal{Q}_0$ is the scale of the crossover between a coherent and incoherent metal.
• Figure 2: We employ a separation of 3 length scales in this paper. $\bar{\mu}$, and the local fluid properties such as entropy density $\mathcal{S}$, may vary substantially over the distance scale $\xi$. We require $l \ll \xi$ for a hydrodynamic description to be sensible. We will often put our fluids in a large but finite box of length $L\gg \xi$ as well.
• Figure 3: A qualitative sketch of holography. A finite temperature $T$ and density boundary theory is dual to an emergent gravitational theory in one extra spatial dimension (depicted above). Strong disorder in the boundary theory (depicted in green) backreacts and leads to the formation of a lumpy charged black hole of Hawking temperature $T$. The emergent black hole horizon is curved and is denoted in black. The membrane paradigm suggests that dc transport can be computed in an emergent fluid living on the horizon, which can undergo renormalization relative to the "bare fluid" in the boundary theory.