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

Technical Summary: Breakdown of the Wiedemann-Franz Law in an Interacting Quantum Hall Metamaterial

Problem Statement
While Coulomb interactions are known to profoundly affect quantum transport in simple ballistic systems (e.g., inducing the multichannel Kondo effect, Tomonaga-Luttinger liquid physics, and Heat Coulomb Blockade), their impact on scaled-up ballistic structures remains underexplored. Specifically, the behavior of a chain of metallic islands (Coulomb islands) connected by ballistic channels in the presence of strong on-site Coulomb repulsion is not fully understood. A critical open question is whether the Wiedemann-Franz (WF) law, which relates electrical and thermal conductance via the Lorenz ratio $L_0 = \pi^2 k_B^2/3e^2$, holds in such interacting, extended 1D arrays. Previous works have largely probed many-body physics via scattering in quantum point contacts or focused on single-island phenomena, leaving the thermal transport properties of multi-island chains largely theoretical.

Methodology
The authors theoretically model a chain of $M$ identical metallic islands connected to source and drain reservoirs at temperatures $T_S$ and $T_D$. The islands are linked by $N$ chiral ballistic edge channels and possess a small stray capacitance $C$ to the ground. The analysis employs a Langevin approach to describe charge conservation and current fluctuations.

The study focuses on the "Heat Coulomb Blockade" (HCB) regime, defined by the condition $k_B T \ll \hbar G_0/C$, where charge dynamics on the islands are frozen ($dq_k/dt \approx 0$). In this regime, the authors decompose the outgoing current noise from each island into distinct propagation modes:

1. Charge (C) Mode: Carries net charge; spontaneous emission is forbidden in the HCB regime.
2. Split Neutral (SN) Mode: Emits equal and opposite charges in opposite directions, preserving the island's net charge but triggering charge cascades in adjacent islands.
3. Left/Right Neutral (LN/RN) Modes: Carry zero net charge and propagate unidirectionally, behaving similarly to non-interacting diffusion.

The authors derive a system of equations for the voltage fluctuations and power balance, leading to a discrete diffusion equation for the squared temperature profile $T_k^2$. This equation accounts for both local diffusion (via LN/RN modes) and non-local interactions (via SN modes).

Key Contributions and Results

1. Identification of a Neutral Transport Mode: The paper identifies a specific neutral mode of transport (the SN mode) that entwines local diffusion with long-range correlations between island charge states. This mode allows energy transport without net charge transfer, distinct from the standard diffusive behavior.
2. Violation of the Wiedemann-Franz Law: The central result is the demonstration that the WF law is violated in this interacting chain. Unlike the single-island HCB effect where heat transport is suppressed, the chain exhibits a regime where heat is transported more efficiently than electricity.
3. Scaling Laws:
   • Temperature Profile: The temperature profile deviates from the standard diffusive linear profile. It is governed by a characteristic decay length $\Delta = \sqrt{(N-1)(M+1)/2}$. For $N=1$, the chain is thermally decoupled from the reservoirs, and all islands reach a uniform mean temperature. For $N \ge 2$, the profile shows exponential features near the reservoirs and a flat region in the center.
   • Heat Flow: In the long-chain limit ($M \gg \Delta$), the heat flow $\dot{Q}$ scales as $M^{-1/2}$, contrasting with the $M^{-1}$ scaling of standard diffusive transport and the $N=1$ case.
   • Lorenz Ratio: The Lorenz ratio $L$ scales as $\sqrt{M}$ for $N > 1$ and $M \gg N$. Specifically, $L \approx \sqrt{(N-1)(M+1)}/N \cdot L_0$. This indicates that a long chain acts as a better thermal conductor than electrical conductor.
4. Role of Channel Number ($N$): The effect is singular for $N=1$ (where the ratio remains $L_0$) but emerges for $N \ge 2$. As $N$ increases, the diffusive dynamics of the neutral modes dominate, and the Lorenz ratio converges back to $L_0$.

Significance and Claims
The authors claim that this work reveals a new collective behavior in interacting quantum systems where thermal transport is enhanced relative to charge transport due to the interplay of charge and neutral modes. The study generalizes previous work on single islands and two-site chains to multiple edge channels, providing a minimal experimental arrangement (a 1D array without quantum point contact partitioning) to observe many-body thermal signatures.

The paper asserts that these results are experimentally measurable, with predicted heat flows in the 0.01–1 fW range for reasonable parameters ($M \lesssim 30$, $T < 80$ mK). The findings suggest that the WF law is not a universal constraint in the presence of strong interactions and specific topological constraints (chiral channels), offering a new signature of many-body physics in quantum Hall metamaterials. The authors note that their approach applies generally to any system with chiral ballistic channels, not just integer quantum Hall edges.