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Half-Quantized Thermal Hall Conductance as a Unique Fingerprint of the Non-Abelian Ising Anyon Sector in Kitaev Spin Liquids

Rowan Brad Quni-Gudzinas

Published Sep 7, 2026• 0 citations
Publisher

Abstract

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 $c_- = 1/2$, yielding a half-quantized thermal Hall conductance $\kappa_{xy}/T = \pi^2 k_B^2/6h$. 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 $c_-=1/2$ (mod 8) is the unique fingerprint of the Ising non-Abelian anyon theory among minimal bosonic topological orders. Abelian bosonic orders necessarily have integer $c_-$, with the minimal nonzero value being $c_-=1$. Thus, a robustly half-quantized $\kappa_{xy}/T$ uniquely distinguishes the Majorana sector from Abelian alternatives, provided the system is a genuine 2D bosonic spin system free of invertible $E_8$ 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 $c_- = 1/2$, 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

DOI
10.5281/zenodo.22557161

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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 $\kappa_{x
The Kitaev model on a honeycomb lattice with bond-dependent Ising interactions offers an exactly solvable model of a quantum spin liquid (QSL) with fractionalized excitations: gapped Z2{Z}_{2}Z2​ fluxes and gapless linearly dispersing Majorana fermions in the isotropic limit (Kx=Ky=Kz{K}_{x}={K}_{y}={K}_{z}Kx​=Ky​=Kz​). We explore the phase diagram along two axes, an external magnetic field hhh, applied out of plane of the honeycomb, and anisotropic interactions Kz{K}_{z}Kz​ larger than the other two (${K}_{x}={K}_{y}\ensuremath{\equiv}K$). For ${K}_{z}/K\ensuremath{\gg}2$ and h=0h=0h=0, the matter Majorana fermions have the largest gap, and the system is described by a gapped Z2{Z}_{2}Z2​ toric code. One of the central questions we address is whether the fractionalized excitations in the different phases have sharp signatures that can be detected in experiments. We show that while the response to single-spin excitations is broad, the spectral function corresponding to two-spin excitations across a bond has sharp signatures that can be attributed to specific anyons. In the toric code regime, the $\ensuremath{\epsilon}=e\ifmmode\times\else\texttimes\fi{}m$ fermion, formed from the bosonic Ising electric (eee) and magnetic (mmm) charges, disperses along a specific one-dimensional direction that provides a fingerprint of fractionalization. At lower Kz{K}_{z}Kz​ in the center of the Abelian phase, in a regime we dub the primordial fractionalized regime, the field generates a hybridization between the $\ensuremath{\epsilon}$ fermion and the Majorana matter fermion, resulting in a $\ensuremath{\psi}$ fermion which too has a distinct quasi-one-dimensional dispersion. All the other phases in the field-anisotropy plane are naturally obtained from this primordial soup. These highly constrained fractonlike dispersions can be observable by inelastic light and neutron scattering, thereby providing ``smoking gun'' signatures of fractionalization in the QSL phase. Our analysis is based on calculations of susceptibilities, topological entanglement entropy, and excitation dynamics, obtained using exact diagonalization and density matrix renormalization group, and supported by perturbation theory.
We study quantum phase transitions in the honeycomb Kitaev model under a magnetic field, focusing on the topological nature of Majorana fermion excitations. We find a gapless phase between the low-field gapless quantum spin liquid and the high-field gapped forced-ferromagnetic state for the antiferromagnetic Kitaev model in the [001] field by using the Majorana mean-field theory, in conjunction with the exact diagonalization and the spin-wave theory supporting the validity of this approach. The transition between the two gapless phases is driven by a topological change of the Majorana spectrum---line node formation interconnecting two Majorana cones. The peculiar change of the Majorana band topology is rationalized by a sign change of the effective Kitaev coupling by the magnetic field, which does not occur in the ferromagnetic Kitaev case. Upon tilting the magnetic field away from [001], the two gapless phases become gapped and topologically nontrivial, characterized by nonzero Chern numbers with different signs. The sign change of the Chern number leads to a reversal of the thermal edge current in the half-quantized thermal Hall effect.
We microscopically construct an Abelian mutual Chern-Simons lattice gauge theory for magnetic field-tuned topological transitions in the Kitaev model and obtain a complete characterization of the phases, including their quasiparticles. At low fields for both ferro (FM) and antiferromagnetic (AFM) Kitaev interactions, we demonstrate non-Abelian Ising topological order (ITO), and explicitly construct the Majorana anyon as an intrinsic excitation---a twist defect in our Abelian gauge theory. For the AFM case, an Abelian chiral phase appears at intermediate fields with trivial topological order and fermionic bulk excitations. Remarkably, both the ITO phase and the intermediate phase have the same chiral central charge c=1/2,c=1/2,c=1/2, implying no change in the quantized thermal Hall response across the transition. For the FM case, there is a direct transition from ITO to a partially polarized nontopological phase. Our paper completes the proof of Kitaev's original proposal of the low-field ITO, with c=1/2c=1/2c=1/2 going beyond his mean-field arguments by including the crucial effect of the gauge fluctuations, and provides a resolution of the debate surrounding the intermediate field phase in the AFM case.
Majorana edge reconstruction and the ν=5/2ν=5/2ν=5/2 non-Abelian thermal Hall puzzle
Pioneering thermal transport measurements on two-dimensional electron gases in high magnetic fields have demonstrated that the quantized Hall state at filling factor ν=5/2ν=5/2ν=5/2 has a thermal Hall conductance κκκ quantized in half-integer multiples of κ0=π2kB2T/3hκ_0 = {π^2 k_B^2 T}/{3h}κ0​=π2kB2​T/3h. Half-integer κ/κ0κ/κ_0κ/κ0​ is a signature of neutral Majorana edge modes, in turn linked to the presence of non-Abelian anyon excitations in the bulk. However, the experimentally observed value of κκκ corresponds to the 'PH-Pfaffian' state, in tension with numerical studies which instead favor either the Pfaffian or the AntiPfaffian. A variety of mechanisms have been invoked to explain this discrepancy, but have been either ruled out by further experiments or else involve fine-tuning. Building on density-matrix-renormalization group studies of physically realistic edges and analytic calculations of edge structure, we propose an alternative resolution of this puzzle involving an 'edge reconstruction' solely involving the neutral Majorana sector of the theory. Such a Majorana edge reconstruction can "screen'' a Pfaffian or AntiPfaffian bulk, so that transport signatures become indistinguishable from those of the PH-Pfaffian. We argue that this physically natural scenario is consistent with experiment.
In this work we study the phase diagram of the Kekul\'e-Kitaev model. The model is defined on a honeycomb lattice with bond-dependent anisotropic exchange interactions, making it exactly solvable in terms of Majorana representation of spins, in close analogy to the Kitaev model. However, the energy spectrum of Majorana fermions has a multiband structure characterized by Chern numbers 0, $\ifmmode\pm\else\textpm\fi{}1$, and $\ifmmode\pm\else\textpm\fi{}2$. We obtained the phase diagram of the model in the plane of exchange couplings and in the presence of a magnetic field and found chiral topological and trivial spin-liquid ground states. In the absence of magnetic field most of the phase diagram is a trivial gapped phase continuously connected to an Abelian phase, while in the presence of the magnetic field a topological phase arises. Furthermore, motivated by recent thermal measurements on the spin-liquid candidate $\ensuremath{\alpha}\ensuremath{-}{\mathrm{RuCl}}_{3}$, we calculated the thermal Hall conductivity at different regimes of parameters and temperatures and found the latter is quantized over a wide range of temperatures.
The chiral central charge ccc measures the gravitational anomaly of a two-dimensional topological phase. It fixes the heat current carried along the edge. So far, it has been accessed through thermal transport, where phonons complicate the signal, or through wave-function probes that need a pure ground state. Here we show that ccc is a spectroscopic quantity of the edge. A weak modulation of the edge couplings that travel along the edge is absorbed according to the Virasoro algebra. Normalized by the response to a static change of the same couplings, the absorbed weights obey Dn=2n⟨L0⟩+c12(n3−n)D_n=2n\langle L_0\rangle+\frac{c}{12}(n^3-n)Dn​=2n⟨L0​⟩+12c​(n3−n), so that C≡2D2−4D1=c\mathcal{C}\equiv2D_2-4D_1=cC≡2D2​−4D1​=c in any state of the ideal edge theory, thermal or not. The central charge thus appears in transition weights, although it cancels from excitation gaps. The ratio of line weights with and without a vison in the bulk, Λ=4hσ/cΛ=4h_σ/cΛ=4hσ​/c, needs no calibration and fixes the conformal spin hσh_σhσ​ of the non-Abelian Ising anyon. For the Kitaev honeycomb spin liquid we find C=0.49999\mathcal{C}=0.49999C=0.49999 and hσ=0.06250h_σ=0.06250hσ​=0.06250 on 7168 spins. Calibrating with the shift of the measured line itself gives ccc within about 1%1\%1% on 96 spins and within 1%1\%1% under 20%20\%20% random coupling errors. Interacting edge Majoranas in a fixed flux sector, treated with DMRG, keep C\mathcal{C}C within 1.5%1.5\%1.5% while the edge velocity changes by ±13%\pm13\%±13%. An exact simulation of a quantum-processor protocol, including the native Floquet circuit, returns ccc within 3.5%3.5\%3.5% on 96 to 600 spins.