Thermoelectric transport in disordered metals without quasiparticles: the SYK models and holography

TL;DR

This work studies disordered, non-quasiparticle metallic states using complex SYK models with a conserved charge and holographic axion theories, revealing a universal low-temperature thermodynamic structure: a universal entropy function $\mathcal{S}(\mathcal{Q})$ with a nonuniversal ground-state energy $E_0(\mathcal{Q})$. A Schwarzian-plus-phase effective action governs $\mathcal{Q}$ and energy fluctuations, constraining transport and linking the Seebeck coefficient to the derivative of entropy via $\lim_{T\to0} S=\frac{\partial \mathcal{S}}{\partial \mathcal{Q}}=2\pi\mathcal{E}$, a relation that holds in both SYK and holographic models. In SYK, quantum chaos with Lyapunov exponent $\lambda_L=2\pi T$ and butterfly velocity $v_B$ sets the thermal diffusivity $D_2$, with $D_2=v_B^2/(2\pi T)$, while charge diffusion $D_1$ remains model-dependent. The holographic counterpart using AdS$_2$ horizons reproduces the same transport structure and shows how universal linear-in-$T$ thermodynamics arises from the near-horizon geometry, reinforcing a deep connection between these solvable theories of non-quasiparticle metals. The results illuminate how disorder and strong interactions yield diffusive transport with a robust thermodynamic-transport link, with implications for understanding strange metals and quantum critical transport beyond Fermi-liquid paradigms.

Abstract

We compute the thermodynamic properties of the Sachdev-Ye-Kitaev (SYK) models of fermions with a conserved fermion number, $\mathcal{Q}$. We extend a previously proposed Schwarzian effective action to include a phase field, and this describes the low temperature energy and $\mathcal{Q}$ fluctuations. We obtain higher-dimensional generalizations of the SYK models which display disordered metallic states without quasiparticle excitations, and we deduce their thermoelectric transport coefficients. We also examine the corresponding properties of Einstein-Maxwell-scalar theories on black brane geometries which interpolate from either AdS$_4$ or AdS$_5$ to an AdS$_2\times \mathbb{R}^2$ or AdS$_2\times \mathbb{R}^3$ near-horizon geometry. These provide holographic descriptions of non-quasiparticle metallic states without momentum conservation. We find a precise match between low temperature transport and thermodynamics of the SYK and holographic models. In both models the Seebeck transport coefficient is exactly equal to the $\mathcal{Q}$-derivative of the entropy. For the SYK models, quantum chaos, as characterized by the butterfly velocity and the Lyapunov rate, universally determines the thermal diffusivity, but not the charge diffusivity.

Figures (6)

• Figure 1: The density $\mathcal{Q}$ as a function of $\mathcal{E}$ and $\Delta$
• Figure 2: The entropy $\mathcal{S}$ as a function of $\mathcal{E}$ and $\Delta$
• Figure 3: The entropy $\mathcal{S}$ as a function of $\mathcal{Q}$ and $\Delta$
• Figure 4: A chain of coupled SYK sites with complex fermions (in this figure we draw $q=4$ case): each site contains $N\gg 1$ fermions with on-site interactions as in (\ref{['h']}). The coupling between nearest neighbor sites are four fermion interaction with two from each site. In general, one can consider other types of $q$-body interactions ($q=4$ in this caption), e.g. $f^\dagger_{x,i_1} f_{x,i_2} f^\dagger_{x+1,i_3} f_{x+1,i_4}$. Such terms will only change the ratio between $D_1$ and $D_2$, i.e. Eq. (\ref{['Dratio']}) by a non-universal coefficient depends on the details of the model. In particular, if we only have $f^\dagger_{x,i_1} f_{x,i_2} f^\dagger_{x+1,i_3} f_{x+1,i_4}$-type terms to couple the nearest neighbour sites, charge diffusion $D_1$ will vanish due to the local charge conservation.
• Figure 5: The entropy $\mathcal{S}(\mathcal{Q})$ obtained from the exact results GPS01 in Section \ref{['sec:entropy']} (full line), and by the numerical solutions (stars).