arXiv:2608.29792

Thermal Hall Signatures of Distinct Schwinger-Boson Flux Sectors on the Honeycomb Lattice

Daiki Sasamoto

Published Aug 30, 2026• 0 citations

Abstract

Thermal transport offers a bulk probe of charge-neutral excitations in frustrated magnets. It remains challenging, however, to determine whether such transport can distinguish different parton mean-field structures of the same spin system. This question is particularly relevant on the honeycomb lattice, where distinct $0$-flux, $π$-flux, and chiral $π/2$-flux sectors have been proposed. In this paper, we compare self-consistent saddle-point solutions in these three sectors for a frustrated $J_1-J_2$ Heisenberg model with a next-nearest-neighbor Dzyaloshinskii-Moriya interaction and a perpendicular magnetic field. Using finite-temperature Schwinger-boson mean-field theory and the Kubo formula for bosonic Bogoliubov-de Gennes systems, we evaluate the intrinsic spinon thermal Hall conductivity. At a common parameter point where all three branches remain gapped, the $0$-flux response is positive and the $π$-flux response is negative for $D>0$ and $h>0$ under the sign conventions used in this paper, whereas the $π/2$-flux response changes sign with increasing temperature. We further show that a projective symmetry combining a sixfold rotation with time reversal forces the zero-field response of a fixed chiral $π/2$-flux domain to vanish. Within the mean-field regime examined in this paper, the sign and temperature dependence of $κ_{xy}/T$ therefore provide a flux-sensitive transport signature.

DOI: 10.48550/arxiv.2608.29792

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arXiv:2608.29792

Thermal Hall Signatures of Distinct Schwinger-Boson Flux Sectors on the Honeycomb Lattice

Daiki Sasamoto

Published Aug 30, 2026• 0 citations

arXiv HTML PDF

Abstract

0

-flux,

π

-flux, and chiral

π/2

-flux sectors have been proposed. In this paper, we compare self-consistent saddle-point solutions in these three sectors for a frustrated

J_1-J_2

Heisenberg model with a next-nearest-neighbor Dzyaloshinskii-Moriya interaction and a perpendicular magnetic field. Using finite-temperature Schwinger-boson mean-field theory and the Kubo formula for bosonic Bogoliubov-de Gennes systems, we evaluate the intrinsic spinon thermal Hall conductivity. At a common parameter point where all three branches remain gapped, the

0

-flux response is positive and the

π

-flux response is negative for

D>0

and

h>0

under the sign conventions used in this paper, whereas the

π/2

-flux response changes sign with increasing temperature. We further show that a projective symmetry combining a sixfold rotation with time reversal forces the zero-field response of a fixed chiral

π/2

-flux domain to vanish. Within the mean-field regime examined in this paper, the sign and temperature dependence of

κ_{xy}/T

therefore provide a flux-sensitive transport signature.

DOI: 10.48550/arxiv.2608.29792

Related papers

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Rohit Mukherjee, Ritajit Kundu, Avinash Singh +1 more

We theoretically investigate, within Schwinger-boson mean-field theory, the transition from a gapped ${\mathbb{Z}}_{2}$ quantum spin liquid, in a ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg spin- $\frac{1}{2}$ system in a honeycomb lattice, to a chiral ${\mathbb{Z}}_{2}$ spin-liquid phase under the presence of time-reversal symmetry-breaking scalar chiral interaction (with amplitude ${J}_{\ensuremath{\chi}}$). We numerically obtain a phase diagram of this ${J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{\ensuremath{\chi}}$ system, where different ground states are distinguished based on the gap and the nature of excitation spectrum, topological invariant of the excitations, the nature of spin-spin correlation, and the symmetries of the mean-field parameters. The chiral ${\mathbb{Z}}_{2}$ state is characterized by the nontrivial Chern number of the excitation bands and lack of long-range magnetic order, which leads to a large thermal Hall coefficient.We theoretically investigate, within Schwinger-boson mean-field theory, the transition from a gapped ${\mathbb{Z}}_{2}$ quantum spin liquid, in a ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg spin-$\frac{1}{2}$ system in a honeycomb lattice, to a chiral ${\mathbb{Z}}_{2}$ spin-liquid phase under the presence of time-reversal symmetry-breaking scalar chiral interaction (with amplitude ${J}_{\ensuremath{\chi}}$). We numerically obtain a phase diagram of this ${J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{\ensuremath{\chi}}$ system, where different ground states are distinguished based on the gap and the nature of excitation spectrum, topological invariant of the excitations, the nature of spin-spin correlation, and the symmetries of the mean-field parameters. The chiral ${\mathbb{Z}}_{2}$ state is characterized by the nontrivial Chern number of the excitation bands and lack of long-range magnetic order, which leads to a large thermal Hall coefficient.

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Motivated by recent transport measurements in high- ${T}_{c}$ 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 $T>0$ , also show a similarly enhanced thermal Hall conductivity.Motivated by recent transport measurements in high-${T}_{c}$ 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 $T>0$, also show a similarly enhanced thermal Hall conductivity.

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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

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_{xIn 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

Sep 7, 20260 citations

Quantum thermal Hall effect of chiral spinons on a kagome strip

Pavel Tikhonov, Efrat Shimshoni

We develop a theory for the thermal Hall coefficient in a spin- $\frac{1}{2}$ system on a strip of kagome lattice, where a chiral spin-interaction term is present. To this end, we model the kagome strip as a three-leg $XXZ$ spin-ladder, and use bosonization to derive a low-energy theory for the spinons in this system. Introducing further a Dzyaloshinskii-Moriya interaction ( $D$ ) and a tunable magnetic field ( $B$ ), we identify three distinct $B$ -dependent quantum phases: a valence-bond crystal (VBC), a ``metallic'' spin liquid (MSL), and a chiral spin liquid (CSL). In the presence of a temperature difference $\mathrm{\ensuremath{\Delta}}T$ between the top and the bottom edges of the strip, we evaluate the net heat current ${J}_{h}$ along the strip, and consequently the thermal Hall conductivity ${\ensuremath{\kappa}}_{xy}$. We find that the VBC-MSL-CSL transitions are accompanied by a pronounced qualitative change in the behavior of ${\ensuremath{\kappa}}_{xy}$ as a function of $B$ . In particular, analogously to the quantum Hall effect, ${\ensuremath{\kappa}}_{xy}$ in the CSL phase exhibits a quantized plateau centered around a commensurate value of the spinon filling factor ${\ensuremath{\nu}}_{s}\ensuremath{\propto}B/D$.We develop a theory for the thermal Hall coefficient in a spin-$\frac{1}{2}$ system on a strip of kagome lattice, where a chiral spin-interaction term is present. To this end, we model the kagome strip as a three-leg $XXZ$ spin-ladder, and use bosonization to derive a low-energy theory for the spinons in this system. Introducing further a Dzyaloshinskii-Moriya interaction ($D$) and a tunable magnetic field ($B$), we identify three distinct $B$-dependent quantum phases: a valence-bond crystal (VBC), a ``metallic'' spin liquid (MSL), and a chiral spin liquid (CSL). In the presence of a temperature difference $\mathrm{\ensuremath{\Delta}}T$ between the top and the bottom edges of the strip, we evaluate the net heat current ${J}_{h}$ along the strip, and consequently the thermal Hall conductivity ${\ensuremath{\kappa}}_{xy}$. We find that the VBC-MSL-CSL transitions are accompanied by a pronounced qualitative change in the behavior of ${\ensuremath{\kappa}}_{xy}$ as a function of $B$. In particular, analogously to the quantum Hall effect, ${\ensuremath{\kappa}}_{xy}$ in the CSL phase exhibits a quantized plateau centered around a commensurate value of the spinon filling factor ${\ensuremath{\nu}}_{s}\ensuremath{\propto}B/D$.

arXiv:1903.00246Physical review. B./Physical review. BMay 28, 20194 citations

Thermal Hall effect and neutral spinons in a doped Mott insulator

Zhi-Jian Song, Jiaxin Zhang, Zheng-Yu Weng

In the pseudogap phase of the cuprate, a thermal Hall response of neutral objects has been recently detected experimentally, which continuously persists into the antiferromagnetic insulating phase. In this paper, we study the transport properties of neutral spinons as the elementary excitation of a doped Mott insulator, which is governed by a mutual Chern-Simons topological gauge structure. We show that such a chiral spinon as a composite of an S = 1 / 2 spin sitting at the core of a supercurrent vortex, can contribute to the thermal Hall effect, thermopower, and Hall effect due to its intrinsic transverse (cyclotron) motion under internal fictitious fluxes. In particular, the magnitudes of the transport coefficients are phenomenologically determined by two basic parameters: the doping concentration and T c , quantitatively consistent with the experimental measurements including the signs and qualitative temperature and magnetic-field dependence. Combined with the predictions of the spinon longitudinal transport properties, including the Nernst and spin Hall effects, a phenomenological description of the pseudogap phase is established as characterized by the neutral spinon excitations, which eventually become “confined” with an intrinsic superconducting transition at T c . Finally, within this theoretical framework, the “order to order” phase transition between the superconducting and antiferromagnetic insulating phases are briefly discussed, with the thermal Hall monotonically increasing into the latter. Published by the American Physical Society 2024In the pseudogap phase of the cuprate, a thermal Hall response of neutral objects has been recently detected experimentally, which continuously persists into the antiferromagnetic insulating phase. In this paper, we study the transport properties of neutral spinons as the elementary excitation of a doped Mott insulator, which is governed by a mutual Chern-Simons topological gauge structure. We show that such a chiral spinon as a composite of an S = 1 / 2 spin sitting at the core of a supercurrent vortex, can contribute to the thermal Hall effect, thermopower, and Hall effect due to its intrinsic transverse (cyclotron) motion under internal fictitious fluxes. In particular, the magnitudes of the transport coefficients are phenomenologically determined by two basic parameters: the doping concentration and T c , quantitatively consistent with the experimental measurements including the signs and qualitative temperature and magnetic-field dependence. Combined with the predictions of the spinon longitudinal transport properties, including the Nernst and spin Hall effects, a phenomenological description of the pseudogap phase is established as characterized by the neutral spinon excitations, which eventually become “confined” with an intrinsic superconducting transition at T c . Finally, within this theoretical framework, the “order to order” phase transition between the superconducting and antiferromagnetic insulating phases are briefly discussed, with the thermal Hall monotonically increasing into the latter. Published by the American Physical Society 2024

arXiv:2309.05711Physical Review ResearchJun 26, 20244 citations

0

-flux,

π

-flux, and chiral

π/2

-flux sectors have been proposed. In this paper, we compare self-consistent saddle-point solutions in these three sectors for a frustrated

J_1-J_2

0

-flux response is positive and the

π

-flux response is negative for

D>0

and

h>0

under the sign conventions used in this paper, whereas the

π/2

-flux response changes sign with increasing temperature. We further show that a projective symmetry combining a sixfold rotation with time reversal forces the zero-field response of a fixed chiral

π/2

-flux domain to vanish. Within the mean-field regime examined in this paper, the sign and temperature dependence of

κ_{xy}/T

therefore provide a flux-sensitive transport signature.

We theoretically investigate, within Schwinger-boson mean-field theory, the transition from a gapped

{\mathbb{Z}}_{2}

quantum spin liquid, in a ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg spin-

\frac{1}{2}

system in a honeycomb lattice, to a chiral

{\mathbb{Z}}_{2}

spin-liquid phase under the presence of time-reversal symmetry-breaking scalar chiral interaction (with amplitude ${J}_{\ensuremath{\chi}}$). We numerically obtain a phase diagram of this ${J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{\ensuremath{\chi}}$ system, where different ground states are distinguished based on the gap and the nature of excitation spectrum, topological invariant of the excitations, the nature of spin-spin correlation, and the symmetries of the mean-field parameters. The chiral

{\mathbb{Z}}_{2}

state is characterized by the nontrivial Chern number of the excitation bands and lack of long-range magnetic order, which leads to a large thermal Hall coefficient.

Motivated by recent transport measurements in high-

{T}_{c}

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>0

, also show a similarly enhanced thermal Hall conductivity.

{\mathrm{CuO}}_{2}

planes of the

{\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{6}

compound. We describe the columnar valence bond solid phase within the bond-operator formalism, which allows us to map the Heisenberg model into an effective interacting boson model written in terms of triplet operators. The effective boson model is studied within the harmonic approximation, and the triplon excitation bands of the columnar valence bond solid phase are determined. We then calculate the Berry curvature and the Chern numbers of the triplon excitation bands and, finally, determine the thermal Hall conductivity due to triplons as a function of the temperature. We find that the Dzyaloshinskii-Moriya interaction yields a finite Berry curvature for the triplon bands, but the corresponding Chern numbers vanish. Although the triplon excitations are topologically trivial, the thermal Hall conductivity of the columnar valence bond solid phase in the square lattice antiferromagnet is finite at low temperatures. Our results complement a previous study by Samajdar et al., Phys. Rev. B 99, 165126 (2019) concerning the thermal Hall effect due to spinons of a spin-liquid phase on a square lattice. We also comment on the relations of our results with a no-go condition for a thermal Hall effect previously derived for ordered magnets by Katsura et al., Phys. Rev. Lett. 104, 066403 (2010).

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

We develop a theory for the thermal Hall coefficient in a spin-

\frac{1}{2}

system on a strip of kagome lattice, where a chiral spin-interaction term is present. To this end, we model the kagome strip as a three-leg

XXZ

spin-ladder, and use bosonization to derive a low-energy theory for the spinons in this system. Introducing further a Dzyaloshinskii-Moriya interaction (

D

) and a tunable magnetic field (

B

), we identify three distinct

B

-dependent quantum phases: a valence-bond crystal (VBC), a ``metallic'' spin liquid (MSL), and a chiral spin liquid (CSL). In the presence of a temperature difference $\mathrm{\ensuremath{\Delta}}T$ between the top and the bottom edges of the strip, we evaluate the net heat current

{J}_{h}

along the strip, and consequently the thermal Hall conductivity ${\ensuremath{\kappa}}_{xy}$. We find that the VBC-MSL-CSL transitions are accompanied by a pronounced qualitative change in the behavior of ${\ensuremath{\kappa}}_{xy}$ as a function of

B

. In particular, analogously to the quantum Hall effect, ${\ensuremath{\kappa}}_{xy}$ in the CSL phase exhibits a quantized plateau centered around a commensurate value of the spinon filling factor ${\ensuremath{\nu}}_{s}\ensuremath{\propto}B/D$.