Breakdown of the Wiedemann-Franz law in an interacting quantum Hall metamaterial

TL;DR

The paper shows that in a chain of Coulomb islands connected by ballistic quantum Hall edge channels, strong on-site interactions can cause heat transport to exceed the diffusive limit and break the Wiedemann-Franz law. By decomposing edge excitations into one charged mode and multiple neutral modes, and by analyzing Johnson–Nyquist fluctuations within a Langevin framework, the authors derive a temperature-profile diffusion equation for $T_k^2$ with a chain-length–dependent decay length $\Delta=\sqrt{(N-1)(M+1)}/2$. They find that the Lorenz ratio scales as $\mathcal{L}\approx\frac{\sqrt{(N-1)(M+1)}}{N}\mathcal{L}_0$ in the long-chain limit, implying heat can be transported more effectively than charge when $N>1$; in the special case $N=1$, diffusion is suppressed and the chain equilibrates to a mid-temperature, restoring a trivial WF behavior. Overall, the work demonstrates interaction-driven enhancement of heat conduction in mesoscopic quantum metamaterials and offers design principles for heat management using chains of Coulomb islands.

Abstract

Quantum heat transport by electrons in ballistic channels is usually well-described in the Landauer-Büttiker framework, which fails when introducing strong Coulomb interactions. We theoretically show that a chain of small metallic dots where charge cannot accumulate, connected by ballistic channels, conducts heat better than charge. We relate this feature to the competition between heat diffusion by neutral excitations and Coulomb interactions in the chain, which defines a temperature gradient over a finite characteristic length. We show that the Lorenz ratio can be arbitrarily large, scaling as the square root of the chain's length, which suggests new approaches for heat manipulation in mesoscopic systems.

Figures (5)

• Figure 1: a) A chain of $M$ identical metallic nodes connecting two large reservoirs having different temperatures $T_\mathrm{S},\,T_\mathrm{D}$. Each metallic node is galvanically connected to its nearest neighbor via $N$ chiral ballistic edge states, and possesses a capacitance to the ground $C$. b)-e) Sketch representations of the four current noise transmission modes in the HCB regime for $N=2$: b) charged mode (C), where a net charge leaves the island, which modifies the stored charge on the capacitor (charge accumulation). c) "Split neutral" (SN) mode, where net charges with equal but opposite values simultaneously leave the island on each side through each channel, leaving the net island's charge unchanged. d) Left and e) right neutral (LN and RN) modes, where charges with zero sum leave the island on the same side, again conserving the net island's charge.
• Figure 2: Squared temperature profile along the chain, for $M=20$ islands, and source and drain temperatures $T_\mathrm{S}=35$ mK, $T_\mathrm{D}=15$ mK, for three channel numbers $N=1,2,10$. Solid lines are the application of Eq. (\ref{['T_profile_simple']}) derived in the long chain limit $M\gg\Delta$. The dashed line corresponds to a high temperature regime $k_BT\gg\hbar G_0/C$ limit, for which the diffusive profile of Eq. (\ref{['T_sq_noninteracting']}) is recovered.
• Figure 3: Source-drain heat flow as a function of the chain length, for $N=1$ (brown), 2 (magenta) and 10 (red) channels, $T_\mathrm{S}= 35$ mK and $T_\mathrm{D}=15$ mK. Dots represent exact heat flows obtained by Eq. (\ref{['Qdot_local_nonlocal']}). Dashed lines are the approximations obtained in the long chain limit $M\gg \Delta$, i.e. the application of Eq. (\ref{['Qdot_0_large_chain']}). Dotted lines are the diffusive heat flows, obtained in the absence of interactions (see text).
• Figure 4: a) Lorenz ratio as a function of the chain size, for $N=1,\,2,\,10$. Dashed magenta and red lines are the application of Eq. (\ref{['Lorenz_ratio']}) for the corresponding $N$. The black dashed line follows the WF law. b) Lorenz ratio as a function of the number of edge channels $N$, for chain sizes $M=3,\,20,\,100$. Dashed colored lines are the corresponding applications of Eq. (\ref{['Lorenz_ratio']}).
• Figure S1: Equivalent representation for voltage build-up, assuming net thermal current noise on each island behaves as an ideal current source.