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arXiv:2411.15304 · Physical review. B./Physical review. B

Thermal Hall response of an Abelian chiral spin liquid at finite temperatures

Avijit Maity, Haoyu Guo, Subir Sachdev A, Vikram Tripathi

Published May 14, 2025• 3 citations
arXivHTMLPDF

Abstract

Thermal Hall transport has emerged as a valuable tool for probing the fractionalized excitations in chiral quantum spin liquids. Observing quantized thermal Hall response, expected at temperatures below the spectral gap, has been challenging and controversial. The finite-temperature behavior, especially in the quantum critical regime above the spectral gap, can provide useful signatures of the underlying topological order. In this context, we study the spin-$1/2$ Heisenberg antiferromagnet on a kagome lattice that is believed to be a $\mathrm{U}(1)$ Dirac spin liquid over a wide intermediate energy range. Scalar spin chirality perturbations turn this into a gapped Abelian chiral spin liquid with semionic topological order. Using a recently developed large-$N$ technique [Guo et al., Phys. Rev. B 101, 195126 (2020)], we obtain explicit expressions for the thermal Hall conductivity ${\ensuremath{\kappa}}_{xy}$ at finite temperatures, taking into account both matter and gauge fluctuations. At low temperatures below the spectral gap, the quantized thermal Hall response agrees with that expected from conformal field theory and gravitational anomaly arguments. Our main finding is that in a large temperature window spanning the spectral gap and the Curie temperature scales where quantum critical fluctuations dominate, ${\ensuremath{\kappa}}_{xy}/T$ obeys a power law with logarithmic corrections. Our analysis also provides a route to understanding the thermal Hall response at higher temperatures in the quantum critical regime.

DOI
10.1103/physrevb.111.205119

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In the Kitaev honeycomb model, a weak magnetic field gaps the gapless Majorana cone, driving the system into a gapped non-Abelian phase with Ising topological order. The edge supports a single chiral Majorana mode with chiral central charge c−=1/2c_- = 1/2c−​=1/2, yielding a half-quantized thermal Hall conductance κxy/T=π2kB2/6h\kappa_{xy}/T = \pi^2 k_B^2/6hκxy​/T=π2kB2​/6h. We rigorously investigate whether observing this half-quantized value uniquely identifies the non-Abelian Ising anyon sector, or whether Abelian topological orders or extrinsic mechanisms can produce the same signature. By combining the edge-bulk correspondence with the classification of strictly two-dimensional bosonic topological orders, we show that a chiral central charge c−=1/2c_-=1/2c−​=1/2 (mod 8) is the unique fingerprint of the Ising non-Abelian anyon theory among minimal bosonic topological orders. Abelian bosonic orders necessarily have integer c−c_-c−​, with the minimal nonzero value being c−=1c_-=1c−​=1. Thus, a robustly half-quantized κxy/T\kappa_{xy}/Tκxy​/T uniquely distinguishes the Majorana sector from Abelian alternatives, provided the system is a genuine 2D bosonic spin system free of invertible E8E_8E8​ stacking or phonon contamination. ## 1. Introduction The Kitaev honeycomb model is an exactly solvable model of a quantum spin liquid, characterized by fractionalized excitations known as anyons. In the absence of a magnetic field, the model hosts a gapless Majorana cone. A weak magnetic field in the [111] direction gaps this cone and induces a non-Abelian phase with Ising topological order. This phase supports a single chiral Majorana edge mode, which contributes to a quantized thermal Hall conductance. The central question addressed in this preprint is whether the observation of a half-quantized thermal Hall conductance, corresponding to a chiral central charge c−=1/2c_- = 1/2c−​=1/2, uniquely identifies the non-Abelian Ising anyon sector, or if Abelian topological orders can mimic this signature. ## 2. Background The thermal Hall conductance $\kappa_{x
Motivated by recent transport measurements in high-Tc{T}_{c}Tc​ cuprate superconductors in a magnetic field, we study the thermal Hall conductivity in materials with topological order, focusing on the contribution from neutral spinons. Specifically, different Schwinger boson mean-field Ans\"atze for the Heisenberg antiferromagnet on the square lattice are analyzed. We allow for both Dzyaloshinskii-Moriya interactions, and additional terms associated with scalar spin chiralities that break time-reversal and reflection symmetries, but preserve their product. It is shown that these scalar spin chiralities, which can either arise spontaneously or are induced by the orbital coupling of the magnetic field, can lead to spinon bands with nontrivial Chern numbers and significantly enhanced thermal Hall conductivity. Associated states with zero-temperature magnetic order, which is thermally fluctuating at any T>0T>0T>0, also show a similarly enhanced thermal Hall conductivity.
There has been great interest in identifying a Kitaev quantum spin liquid state in frustrated magnets with bond-dependent interactions. In particular, the experimental report of a half-quantized thermal Hall conductivity in $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$ in the presence of a magnetic field has generated excitement as it could be strong evidence for a field-induced chiral spin liquid. More recent experiments, however, provide a conflicting interpretation advocating for topological magnons in the field-polarized state as the origin of the nonquantized thermal Hall conductivity observed in their experiments. An inherent difficulty in distinguishing between the two scenarios is that the phase transition between a putative two-dimensional spin liquid and the field-polarized state exists only at zero temperature, while the behavior at finite temperature is mostly crossover phenomena. In this work, we provide insights into the finite-temperature crossover behavior between the spin excitation continuum in a quantum spin liquid and topological magnons in the field-polarized state in three different theoretical models with large Kitaev interactions. These models allow for a field-induced phase transition from a spin liquid (or an intermediate field-induced spin liquid) to the field-polarized state in the quantum model. We obtain the dynamical spin structure factor as a function of magnetic field using molecular dynamics simulations and compute thermal Hall conductivity in the field-polarized regime. We demonstrate the gradual evolution of the dynamical spin structure factor exhibiting crossover behavior near magnetic fields where zero-temperature phase transitions occur in the quantum model. We also examine nonlinear effects on topological magnons and the validity of thermal Hall conductivity computed using linear spin wave theory. We discuss the implications of our results for existing and future experiments.