Thermal transport across a continuous metal-insulator transition

The celebrated Wiedemann-Franz (WF) law is believed to be robust in metals as long as interactions between electrons preserve their fermion-quasiparticle character. We study thermal transport and the fate of the WF law close to a continuous metal-insulator transition (MIT) in the Falicov-Kimball model (FKM) using cluster-dynamical mean-field theory (CDMFT). Surprisingly, as for electrical transport, we find robust and novel quantum critical scaling in thermal transport across the MIT. We unearth the deeper reasons for these novel findings in terms of (i) the specific structure of energy-current correlations for the FKM and (ii) the microscopic electronic processes which facilitate energy transport while simultaneously blocking charge transport close to the MIT. However, within (C)DMFT, we also find that the WF law survives at $T⟶0$ in the incoherent metal right up to the MIT, even in absence of Landau quasiparticles.

Figure 1

$\text{dc}$ resistivity ${\rho }_{dc}\left(T\right)$ (a), thermopower $S_{el}\left(T\right)$ (b), thermal conductivity $K_{el}\left(T\right)$ (c), and Lorenz number $L_0\left(T\right)$ (d) for the FKM as functions of $U/t$. At the Mott QCP (bold red circles) at ${(U/t)}_c=1.8$, both $S_{el}\left(T\right)\text{and}{L}_0\left(T\right)$ attain finite values, cleanly separating metallic and insulating behavior. Concomitantly, ${\rho }_{dc}(T\to 0)$ diverges and $K_{el}\left(T\right)\simeq T^{1+\nu }$ with $\nu \simeq 4/3$ [16].

Figure 2

Mott quantum critical scaling in thermopower $S_{el}(U/t,T)$ across the MIT. $\text{log}[S_{el}\left(T\right)/S_c]$ vs $T$ exhibits almost perfect mirror symmetry around ${(U/t)}_c$ (left panel). Collapse of metallic and insulating curves onto two universal curves upon scaling the $T$ axis by $T_0^{th}$(right panel). This is evidence that Mott quantum critical scaling in electrical transport [16] extends to thermal transport as well.

Figure 3

$T_0^{th}\left(\delta U\right)=c{\left|\delta U\right|}^{{\mu }_{th}}$ with ${\mu }_{th}=1$ (left panel). The beta function varies like $\beta \left(s\right)\simeq \log \left(s\right)$ with $s=S_{el}\left(T\right)/S_c$ close to the MIT and is continuous across $U_c$ (right panel).

Figure 4

Quantum critical scaling in scaled electrical conductivity $T^{-\nu }{\sigma }_{dc}\left(T\right)$ [panel (a)], thermopower $S_{el}\left(T\right)$ [panel (b)], scaled thermal conductivity, $T^{-1-\nu }{K}_{el}\left(T\right)$ [panel (c)], and Lorenz number [panel (d)] when plotted as functions of the “scaling variable” $y(U,T)=|U-U_c|/U_{c}T$, demonstrating clean quantum critical scaling in electrical as well as thermal transport at the Mott QCP.

Figure 5

Thomson coefficient ${\tau }_{th}\left(T\right)$ for FKM as a function of $U/t$.

Figure 6

Results similar to those in Fig. 1, but now using single-site DMFT.

Figure 7

(a) In the left panel, $\text{log}[S_{el}\left(T\right)/S_c]$ vs $T/T_0$ and (b) in right panel, $T_0^{th}\left(\delta U\right)$ vs $\delta U$ within single-site DMFT.

Figure 8

Results similar to those obtained in Fig. 4, but now using single-site DMFT.