We investigate the thermal Hall conductivity in the Kitaev model with additional interactions under a magnetic field, employing a finite-temperature tensor network method benchmarked by a thermal pure quantum state technique. We find that the thermal Hall conductivity divided by temperature,
κxy/T, significantly overshoots the value of the half-integer quantization and exhibits a pronounced hump while decreasing temperature. Moreover, we show that the field-direction dependence of
κxy/T is consistent with the sign of the Chern number associated with the Majorana fermions across a wide range of magnetic fields. We also demonstrate that the additional off-diagonal interactions, known as the
Γ and
Γ′ terms, considerably affect
κxy/T. In particular, we show that positive
Γ and negative
Γ′ lead to a remarkable enhancement in the intermediate temperature region. From the comparison with the classical counterpart, we reveal that the effects of the
Γ term go beyond the classical picture, indicating significant quantum fluctuation effects, while those of the
Γ′ term are well captured at the classical level. These comprehensive analyses indicate that the enhanced thermal Hall response is consistently explained by dominant contributions from topological Majorana fermions, even within the polarized regime beyond the critical field. Our approach not only establishes a robust theoretical framework for understanding the thermal Hall transport in Kitaev materials such as
α-RuCl
3, but also offers a promising pathway to bridge the gap between theories and experiments across a wide range of strongly correlated materials.
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
κxy/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−=1/2 (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
κxy/T uniquely distinguishes the Majorana sector from Abelian alternatives, provided the system is a genuine 2D bosonic spin system free of invertible
E8 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
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
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.
Recent discovery of the half quantized thermal Hall conductivity in [Formula: see text]-RuCl[Formula: see text], a candidate material for the Kitaev spin liquid, suggests the presence of a highly entangled quantum state in external magnetic fields. This field induced phase appears between the low field zig-zag magnetic order and the high field polarized state. Motivated by this experiment, we study possible field induced quantum phases in theoretical models of the Kitaev magnets, using the two dimensional tensor network approach or infinite tensor product states. We find various quantum ground states in addition to the chiral Kitaev spin liquid occupying a small area in the phase diagram. They form a band of emergent quantum phases in an intermediate window of external magnetic fields, somewhat reminiscent of the experiment. We discuss the implications of these results in view of the experiment and previous theoretical studies.
Motivated by the recent experimental observation of the half-quantized thermal Hall conductivity for the candidate material of the Kitaev spin liquid, $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$ [Y. Kasahara et al., Nature 559, 227 (2018)], we investigate effects of non-Kitaev exchange interactions, which exist in the real material, on the gapped chiral spin-liquid state realized in the case with an applied magnetic field. It is found that off-diagonal exchange interactions enhance significantly the mass gap of Majorana fermions. This result provides a possible explanation for robust quantization of the thermal Hall conductivity observed in the above-mentioned experiment. Furthermore, we demonstrate that if the Kitaev spin-liquid state and the zigzag antiferromagnetic order coexist around the border of these two phases, off-diagonal exchange interactions can induce Fermi surfaces of Majorana fermions, leading to a state similar to the U(1) spin liquid.
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.