Conductivity of weakly disordered strange metals: from conformal to hyperscaling-violating regimes

TL;DR

This work develops a semi-analytic holographic construction that interpolates between UV AdS and IR hyperscaling-violating geometries to model strange metals at finite temperature and weak disorder. Using two bulk U(1) gauge fields and an Einstein-Maxwell-dilaton action, the authors generate a one-parameter family of finite-$T$ geometries that capture the crossover from conformal to HV behavior, and they compute DC conductivity including disorder via a horizon-based formula. They validate scaling predictions for resistivity across regimes, analyze sample-to-sample fluctuations, and uncover logarithmic corrections in certain holographic limits, while also examining the onset of superconductivity driven by a charged scalar in these backgrounds. The results reveal a consistent crossover scale set by $\mathcal{Q}^{1/d}$ and illuminate how momentum relaxation and disorder influence transport in non-Fermi liquid holographic metals, with implications for both theory and potential condensed-matter applications.

Abstract

We present a semi-analytic method for constructing holographic black holes that interpolate from anti-de Sitter space to hyperscaling-violating geometries. These are holographic duals of conformal field theories in the presence of an applied chemical potential, $μ$, at a non-zero temperature, $T$, and allow us to describe the crossover from `strange metal' physics at $T \ll μ$, to conformal physics at $T \gg μ$. Our holographic technique adds an extra gauge field and exploits structure of the Einstein-Maxwell system to manifestly find 1-parameter families of solutions of the Einstein-matter system in terms of a small family of functions, obeying a nested set of differential equations. Using these interpolating geometries, we re-consider holographically some recent questions of interest about hyperscaling-violating field theories. Our focus is a more detailed holographic computation of the conductivity of strange metals, weakly perturbed by disorder coupled to scalar operators, including both the average conductivity as well as sample-to-sample fluctuations. Our findings are consistent with previous scaling arguments, though we point out logarithmic corrections in some special (holographic) cases. We also discuss the nature of superconducting instabilities in hyperscaling-violating geometries with appropriate choices of scalar couplings.

Figures (6)

• Figure 1: A sketch of $\rho_{\mathrm{dc}}$ in $d=2$, showing $\rho_{\mathrm{dc}}(T)$, as well as the portion of the geometry capturing the processes dominating the resistivity.
• Figure 2: We show $\hat{\rho}_{\mathrm{dc}}$ vs. $T$ for the $d=2$ geometries with (a) $z=3$, $\theta=1$, (b) $z=2$, $\theta=-2$. In both cases $\Delta_{\mathrm{UV}}=2$ and $\Delta_{\mathrm{IR}}=3$. The solid lines are fits to analytical predictions (up to overall coefficients), which are computed using Eq. (\ref{['rhosketch']}). Error in numerical methods is substantially smaller than the data markers. Note that $\Delta_{\mathrm{UV}}$ is "marginal" from the Harris criterion; though at very large $T \gg T_{\mathrm{pc}}$ the geometry may deviate from AdS logarithmically due to the presence of disorder, such an effect is well beyond our regime of validity. This choice of $\Delta_{\mathrm{UV}}$ appeared in patel as well.
• Figure 3: We show $\hat{\rho}_{\mathrm{dc}}$ vs. $T$ for the $d=3$ geometry with $\eta=3$. We took $\Delta_{\mathrm{UV}} = 12/5$ and $\Delta_{\mathrm{IR}} = 3z/4$. Circular data points denote data where we have done a full numerical computation of the strength of the disordered scalar hair. For square data points, we have used matched asymptotic expansions (see e.g. sera) to extend the numerical methods over an additional 8 orders of magnitude in $T$. Note that $\rho_{\mathrm{dc}}$ has a minimum for $T\hat{\mathcal{Q}}^{-1/d}\sim 10^{-5}$. Error in numerical methods is substantially smaller than the data markers.
• Figure 4: We show the values of $Q$ and $T$ for which there is a likely instability to superconductivity for our previous $d=z=-\theta=2$ geometry. The boundary between red and white defines where a marginal mode of the charged scalar exists.
• Figure 5: The 9 functions of our construction corresponding to various matter fields. We are taking the "bare" (no scaling with $q$) functions for $b_2$, $\tilde{p}^\prime$, and $\tilde{Z}$. Red lines correspond to the geometry with $d=2$, $z=3$, $\theta=1$; blue to $d=2$, $z=2$, $\theta=-2$; olive to $d=3$, $z=\infty$, $\eta=3$.