Thermal Transport in Chiral Conformal Theories and Hierarchical Quantum Hall States

TL;DR

This work shows that the thermal Hall conductance of chiral edge theories is governed by the gravitational anomaly, with K = (π k_B^2 T/6)(c − c̄). Using annulus partition functions, the authors compute finite-size corrections and compare two candidate edge theories for hierarchical quantum Hall states: Abelian multi-component CFTs and W1+∞ minimal models. Both theories share the same leading conductance, but differ in universal finite-size corrections: Abelian theories have vanishing 1/R corrections, while minimal models exhibit a nonzero 1/R term, providing a potential experimental signature to distinguish edge theories. The results illuminate how modular invariance and anomaly structure shape non-equilibrium edge transport and offer a practical route to probing neutral edge modes in hierarchical quantum Hall states.

Abstract

Chiral conformal field theories are characterized by a ground-state current at finite temperature, that could be observed, e.g. in the edge excitations of the quantum Hall effect. We show that the corresponding thermal conductance is directly proportional to the gravitational anomaly of the conformal theory, upon extending the well-known relation between specific heat and conformal anomaly. The thermal current could signal the elusive neutral edge modes that are expected in the hierarchical Hall states. We then compute the thermal conductance for the Abelian multi-component theory and the W-infinity minimal model, two conformal theories that are good candidates for describing the hierarchical states. Their conductances agree to leading order but differ in the first, universal finite-size correction, that could be used as a selective experimental signature.

Figures (2)

• Figure 1: Annulus geometry with inner and outer radii $R_{1}$ and $R_{2}$, respectively. The arrows indicate the sense of propagation of the edge excitations.
• Figure 2: Neutral and charged modes in the plateaus with $\nu=2/5$ and $\nu =2/3$, having equal and opposite senses of propagation, respectively.