Quantum critical theories in a periodic potential: strange metallic thermoelectric and magneto-transport

TL;DR

This work uses holographic AdS$_4$ duals to study DC/AC thermoelectric and magneto-transport in 2D quantum critical theories with a zero-average, spatially modulated chemical potential lattice. It analyzes both 1D and 2D lattices in the charge-neutral regime, revealing that transport is not governed by a single momentum-relaxation rate; 1D exhibits predominantly incoherent electrical transport with dual thermal modes, while 2D displays Effective Medium Theory–like behavior with emergent diffusive channels and magnetotransport that shows approximately linear magnetoresistance at large fields. The findings connect to strange metals and graphene, highlighting dimensionality and strong translational symmetry breaking as key determinants of transport, and suggest EMT-inspired interpretations of horizon dynamics in holography. The work provides comprehensive DC/AC analyses and pole-structure insights crucial for understanding universal quantum-critical transport under strong lattice effects and proposes avenues for exploring finite average chemical potential and broader lattice geometries.

Abstract

We study DC and AC thermoelectric and magneto-transport in 2D quantum critical theories with strong translational symmetry breaking due to a % varying chemical potential lattice with zero average $\barμ=0$. The combination of quantum criticality and the absence of the average natural scale implies that such systems have idiosyncratic signatures that may apply more generally when the variance in the lattice potential far exceeds the average or for strong translational symmetry breaking in general. We model such theories holographically through near-extremal AdS black holes. We find that these systems (a) become \emph{better} conductors. In a 2D lattice, this can be explained by currents flowing around obstacles; (b) exhibit bad-metal electrical transport with Drude-like thermal transport, though it is not Drude, and, notably, (c) display an approximately $B$-linear longitudinal magnetoresistance at large fields, similar to Effective Medium Theory. We comment on how these results may apply when $\barμ\neq 0$.

Figures (11)

• Figure 1: DC quantum critical transport in a 1D charge-neutral lattice, ${\mu(x) = \delta \mu \cos(G x)}$. Note that the thermoelectric conductivity ${\alpha = 0}$ (numerically confirmed, not shown) and thus $\bar{\kappa} = \kappa$. (a) Electrical conductivity $\sigma$ as a function of $T/\delta\mu$ for various $\delta\mu/G$ (left colorbar). At low temperatures, the lattice causes the conductivity to increase. Inset: electrical conductivity as a function of $T/G$. (b) Electrical conductivity $\sigma$ as a function of lattice amplitude for various temperatures $T/G$ (right colorbar). In the low-temperature limit for small $\delta \mu/G$, the scaling is that predicted by Chesler, Lucas and Sachdev chesler_conformal_2014, (Eq. \ref{['eq:sigma_CLS']} black dashed line). Inset: difference from the CFT value $\sigma_{\infty}=1$ on a logarithmic scale, which grows as $(\delta \mu)^2$. (c) Thermal conductivity $\bar{\kappa}$ as a function of $T/\delta\mu$. There is a cross-over from a universal high-$T$ regime to a strong thermal conductor-regime at low $T$. Inset: thermal conductivity $\bar{\kappa}$ as a function of $T/G$. (d) Thermal conductivity $\bar{\kappa}$ as a function of lattice amplitude. The thermal conductivity decreases most rapidly with increased lattice amplitude around ${T/G \approx 0.1}$, which corresponds with the minima seen in panel (c). Upper inset: for weak lattices, the thermal conductivity scales as $1/(\delta \mu)^{2}$. Lower inset: for strong lattices, the thermal conductivity approaches the bound $\kappa/T \geq 4 \pi^2/3$ of grozdanov_incoherent_2016.
• Figure 2: Metric components $Q_{xx}$ and $Q_{yy}$ (offset for clarity; top row) and local charge density $n(z) = F^{tz}$ (bottom row) for a 1D charge neutral lattice, $\mu(x) = \delta \mu \cos G x$ with $\delta\mu/G=2$. The evolution in the $z$ coordinate captures the effect of RG flow. At low temperatures $T=0.1G$ (left), the resulting $x/y$ anisotropy at the horizon gives rise to the asymptotic scaling in Eq. \ref{['eq:sigma_CLS']}. At high temperatures $T=10G$ (right), $\, \mathrm{d} s^2 \approx \, \mathrm{d} s_{\text{AdS}_4}^2$, and the renormalization of the charge density is cut off almost immediately. The horizon charge density entering the Stokes equations, \ref{['eq:stokes']}, thus becomes ${n^H \approx \mu(x)}$ for $T\gg G$.
• Figure 3: AC quantum critical transport in a 1D charge-neutral lattice, $\mu(x) = \delta \mu \cos (G x)$ at three different temperatures, ${T/G=0.1, 1, 10}$ (left to right). (a-c) The AC electrical conductivity $\sigma(\omega)$ exhibits an Umklapp sound peak at $\omega \approx G/\sqrt{2}$ (gray dashed line), which is dominant at high temperatures. At lower temperatures (panel (a)), there is a slight downward shift in the peak frequency. (d-f) The AC thermal conductivity $\bar{\kappa}(\omega)$ is dominated by a pair of poles that moves toward $\omega = G$, a process which happens more rapidly at low $T/G$. Inset (f): Same data on a larger vertical scale for small frequencies. The peak moves closer to the origin, but is not a single Drude peak centered at $\omega=0$.
• Figure 4: Pole structure of the AC response in a 1D charge-neutral lattice, $\mu(x) = \delta\mu \cos G x$ for various $T/G$. In all panels $\delta\mu = 6 G$. Left: logarithm of the complex electrical conductivity $\sigma(\omega)$. The Umklapp sound modes moves slightly toward the origin as $T/G$ is decreased for fix $\delta\mu/G$. There is no Drude pole or Umklapped charge diffusion pole present. Right: logarithm of the complex thermal conductivity $\bar{\kappa}(\omega)$. At all finite $\delta \mu$, transport is governed by a pair of modes that move in an arc toward $\omega = G$ as $T/G$ is decreased. This pair of modes arises from a collision of the Drude pole with the Umklapp charge diffusion pole. At charge neutrality Umklapp sound has negligible weight in the thermal channel and is indeed not visible.
• Figure 5: Pole structure of the AC response in a 1D lattice away from charge neutrality; ${\mu(x) = \bar{\mu} + \delta \mu \cos G x}$. Left/right: logarithm of the complex electrical/thermal conductivity for various $\bar{\mu}/G$, at temperature $T=G$ and with $\delta\mu=2 G$. At large enough finite $\bar{\mu}$, there exists both a Drude pole and a charge diffusion pole along the imaginary axis in both the electrical and thermal sector. These eventually collide, turning into the pair of modes that govern the thermal transport at $\bar{\mu}=0$. In the electrical sector, the residue of these poles vanishes as $\bar{\mu}\to 0$ due to a collision with a pair of zeros (top left panel).