arXiv:2003.05645 · Physical review. B./Physical review. B

Topological phase transition to Abelian anyon phases due to off-diagonal exchange interaction in the Kitaev spin liquid state

Daichi Takikawa, Satoshi Fujimoto

Published Nov 9, 2020• 17 citations

Abstract

We investigate how the Kitaev spin liquid state described by Majorana fermions coupled with ${Z}_{2}$ gauge fields is affected by non-Kitaev interactions which exist in real candidate materials of the Kitaev magnet. It is found that the off-diagonal exchange interaction referred to as the ${\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}$ term dramatically changes the Majorana band structure in the case of the antiferromagnetic Kitaev interaction and gives rise to a topological phase transition from a non-Abelian topological phase with the Chern number equal to $\ifmmode\pm\else\textpm\fi{}1$ to an Abelian phase with the Chern number equal to $\ifmmode\pm\else\textpm\fi{}2$, in which the ${Z}_{2}$ vortices behave as Abelian anyons. On the other hand, other non-Kitaev interactions such as the Heisenberg interaction and the $\mathrm{\ensuremath{\Gamma}}$ term only affect the bandwidth of the Majorana band as long as the spin liquid state is not destabilized.

DOI: 10.1103/physrevb.102.174414

Related papers

Successive Majorana topological transitions driven by a magnetic field in the Kitaev model

Joji Nasu, Yasuyuki Kato, Yoshitomo Kamiya +1 more

arXiv:1805.02423Physical review. B./Physical review. BAug 30, 201872 citations

Topological nematic phase transition in Kitaev magnets under applied magnetic fields

Masahiro O. Takahashi, Masahiko G. Yamada, Daichi Takikawa +2 more

arXiv:2101.05959Physical Review ResearchJun 7, 202112 citations

Mixed-State Topological Transition and Emergent Fractionalized Supersymmetry in the Non-Abelian Kitaev Spin Liquid

Zhou-Quan Wan, Hong Yao, Shao-Kai Jian

cond-mat.str-elCross-listedarXiv:2610.38741Oct 30, 20260 citations

Continuous transition between Ising magnetic order and a chiral spin liquid

G. Shankar, Chien-Hung Lin, Joseph Maciejko

arXiv:2207.02701Physical review. B./Physical review. BDec 6, 20225 citations

Microscopic theory of field-tuned topological transitions in the Kitaev honeycomb model

Jagannath Das, Aman Kumar, Vikram Tripathi

arXiv:2401.12750Physical review. B./Physical review. BJan 21, 20251 citation

Spin correlation and Majorana spectrum in chiral spin liquids in a decorated-honeycomb Kitaev model

譲治那須, Joji Nasu, 幸年求 +1 more

Feb 1, 20160 citations

arXiv:2003.05645 · Physical review. B./Physical review. B

Topological phase transition to Abelian anyon phases due to off-diagonal exchange interaction in the Kitaev spin liquid state

Daichi Takikawa, Satoshi Fujimoto

Published Nov 9, 2020• 17 citations

arXiv HTML PDF

Abstract

We investigate how the Kitaev spin liquid state described by Majorana fermions coupled with

{Z}_{2}

gauge fields is affected by non-Kitaev interactions which exist in real candidate materials of the Kitaev magnet. It is found that the off-diagonal exchange interaction referred to as the

{\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}

term dramatically changes the Majorana band structure in the case of the antiferromagnetic Kitaev interaction and gives rise to a topological phase transition from a non-Abelian topological phase with the Chern number equal to

\ifmmode\pm\else\textpm\fi{}1

to an Abelian phase with the Chern number equal to

\ifmmode\pm\else\textpm\fi{}2

, in which the

{Z}_{2}

vortices behave as Abelian anyons. On the other hand, other non-Kitaev interactions such as the Heisenberg interaction and the

\mathrm{\ensuremath{\Gamma}}

term only affect the bandwidth of the Majorana band as long as the spin liquid state is not destabilized.

DOI: 10.1103/physrevb.102.174414

Related papers

Successive Majorana topological transitions driven by a magnetic field in the Kitaev model

Joji Nasu, Yasuyuki Kato, Yoshitomo Kamiya +1 more

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 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.

arXiv:1805.02423Physical review. B./Physical review. BAug 30, 201872 citations

Topological nematic phase transition in Kitaev magnets under applied magnetic fields

Masahiro O. Takahashi, Masahiko G. Yamada, Daichi Takikawa +2 more

We propose a scenario of realizing the toric code phase, which can be potentially utilized for fault-tolerant quantum computation, in candidate materials of Kitaev magnets.It is demonstrated that four-body interactions among Majorana fermions in the Kitaev spin liquid state, which are induced by applied magnetic fields as well as non-Kitaev-type exchange interactions, trigger a nematic phase transition of Majorana bonds without magnetic orders, accompanying the change of the Chern number from ±1 to zero.This gapful spin liquid state with zero Chern number is simply the toric code phase.Our result potentially explains the topological nematic transition recently observed in α-RuCl 3 via heat capacity measurements (O.Tanaka et al.,We propose a scenario of realizing the toric code phase, which can be potentially utilized for fault-tolerant quantum computation, in candidate materials of Kitaev magnets.It is demonstrated that four-body interactions among Majorana fermions in the Kitaev spin liquid state, which are induced by applied magnetic fields as well as non-Kitaev-type exchange interactions, trigger a nematic phase transition of Majorana bonds without magnetic orders, accompanying the change of the Chern number from ±1 to zero.This gapful spin liquid state with zero Chern number is simply the toric code phase.Our result potentially explains the topological nematic transition recently observed in α-RuCl 3 via heat capacity measurements (O.Tanaka et al.,

arXiv:2101.05959Physical Review ResearchJun 7, 202112 citations

Mixed-State Topological Transition and Emergent Fractionalized Supersymmetry in the Non-Abelian Kitaev Spin Liquid

Zhou-Quan Wan, Hong Yao, Shao-Kai Jian

We study a mixed-state topological transition in the non-Abelian Kitaev spin liquid driven by a bond-dephasing channel. In the vectorized picture, bond dephasing maps onto interactions between two time-reversed fractionalized Majorana systems coupled to gapped $\mathbb Z_2$ gauge fields, enabling sign-problem-free Monte Carlo simulations. We show that increasing dephasing condenses an inter-copy fermion pair, converting ${\rm Ising}\boxtimes\overline{\rm{Ising}}$ to toric-code order in the Choi state and yielding intrinsically mixed-state topological order upon unfolding. More interestingly, at a temporal interface between topological and trivial Kitaev states, a Chern-number mismatch enforces a gapless Majorana mode. Dephasing then drives a mixed-state topological transition at which the gapless mode develops emergent $\mathcal N=1$ supersymmetry. Our Monte Carlo simulations yield $ν=1.27(3)$ and $η_b\simeqη_f\simeq0.40$ , consistent with the exact supersymmetric values $5/4$ and $2/5$ . The supersymmetric partners are fractionalized and appear physically as distinct gauge-invariant spin strings, providing concrete observables for future experiments.We study a mixed-state topological transition in the non-Abelian Kitaev spin liquid driven by a bond-dephasing channel. In the vectorized picture, bond dephasing maps onto interactions between two time-reversed fractionalized Majorana systems coupled to gapped $\mathbb Z_2$ gauge fields, enabling sign-problem-free Monte Carlo simulations. We show that increasing dephasing condenses an inter-copy fermion pair, converting ${\rm Ising}\boxtimes\overline{\rm{Ising}}$ to toric-code order in the Choi state and yielding intrinsically mixed-state topological order upon unfolding. More interestingly, at a temporal interface between topological and trivial Kitaev states, a Chern-number mismatch enforces a gapless Majorana mode. Dephasing then drives a mixed-state topological transition at which the gapless mode develops emergent $\mathcal N=1$ supersymmetry. Our Monte Carlo simulations yield $ν=1.27(3)$ and $η_b\simeqη_f\simeq0.40$, consistent with the exact supersymmetric values $5/4$ and $2/5$. The supersymmetric partners are fractionalized and appear physically as distinct gauge-invariant spin strings, providing concrete observables for future experiments.

cond-mat.str-elCross-listedarXiv:2610.38741Oct 30, 20260 citations

Continuous transition between Ising magnetic order and a chiral spin liquid

G. Shankar, Chien-Hung Lin, Joseph Maciejko

The competition between fractionalized spin-liquid states and magnetically ordered phases is an important paradigm in frustrated magnetism. Spin-orbit coupled Mott insulators with Ising-like magnetic anisotropies, such as Kitaev materials, are a particularly rich playground to explore this competition. In this work, we use effective field theory methods to show that a direct quantum phase transition can occur in two-dimensional (2D) Ising spin systems between a topologically ordered chiral spin liquid and a phase with magnetic long-range order. Such a transition can be protected by lattice symmetries and is described by a theory of massless Majorana fields coupled to non-Abelian $\text{SO}(N)$ gauge fields with a Chern-Simons term. We further show that Euclidean Majorana zero modes bound to ${\mathbb{Z}}_{2}$ monopole-instantons in the emergent non-Abelian gauge field are key to understanding spontaneous symmetry breaking in the ordered phase.The competition between fractionalized spin-liquid states and magnetically ordered phases is an important paradigm in frustrated magnetism. Spin-orbit coupled Mott insulators with Ising-like magnetic anisotropies, such as Kitaev materials, are a particularly rich playground to explore this competition. In this work, we use effective field theory methods to show that a direct quantum phase transition can occur in two-dimensional (2D) Ising spin systems between a topologically ordered chiral spin liquid and a phase with magnetic long-range order. Such a transition can be protected by lattice symmetries and is described by a theory of massless Majorana fields coupled to non-Abelian $\text{SO}(N)$ gauge fields with a Chern-Simons term. We further show that Euclidean Majorana zero modes bound to ${\mathbb{Z}}_{2}$ monopole-instantons in the emergent non-Abelian gauge field are key to understanding spontaneous symmetry breaking in the ordered phase.

arXiv:2207.02701Physical review. B./Physical review. BDec 6, 20225 citations

Microscopic theory of field-tuned topological transitions in the Kitaev honeycomb model

Jagannath Das, Aman Kumar, Vikram Tripathi

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 $c=1/2,$ 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 $c=1/2$ 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.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 $c=1/2,$ 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 $c=1/2$ 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.

arXiv:2401.12750Physical review. B./Physical review. BJan 21, 20251 citation

Spin correlation and Majorana spectrum in chiral spin liquids in a decorated-honeycomb Kitaev model

譲治那須, Joji Nasu, 幸年求 +1 more

Temperature evolution of the spin correlation and excitation spectrum is investigated for a Kitaev model defined on a decorated honeycomb lattice by using the quantum Monte Carlo simulation in the Majorana fermion representation. The ground state of this quantum spin model is given by two kinds of chiral spin liquids: one is topologically trivial with Abelian anyon excitations, and the other is topologically nontrivial with non-Abelian anyon excitations. While lowering temperature, the model exhibits several crossovers in the paramagnetic state, which originate from the fractionalization of quantum spins into Majorana fermions, in addition to a phase transition associated with time reversal symmetry breaking. We show that the spin correlation develops around the crossover temperatures, whereas it shows a slight change at the critical temperature, as in other Kitaev-type models. We also calculate the excitation spectrum in terms of Majorana fermions, and find that the excitation gap in the non-Abelian phase is fragile against thermal fluctuations of the $Z_2$ fluxes, while that in the Abelian phase remains open.Temperature evolution of the spin correlation and excitation spectrum is investigated for a Kitaev model defined on a decorated honeycomb lattice by using the quantum Monte Carlo simulation in the Majorana fermion representation. The ground state of this quantum spin model is given by two kinds of chiral spin liquids: one is topologically trivial with Abelian anyon excitations, and the other is topologically nontrivial with non-Abelian anyon excitations. While lowering temperature, the model exhibits several crossovers in the paramagnetic state, which originate from the fractionalization of quantum spins into Majorana fermions, in addition to a phase transition associated with time reversal symmetry breaking. We show that the spin correlation develops around the crossover temperatures, whereas it shows a slight change at the critical temperature, as in other Kitaev-type models. We also calculate the excitation spectrum in terms of Majorana fermions, and find that the excitation gap in the non-Abelian phase is fragile against thermal fluctuations of the $Z_2$ fluxes, while that in the Abelian phase remains open.

Feb 1, 20160 citations

We investigate how the Kitaev spin liquid state described by Majorana fermions coupled with

{Z}_{2}

gauge fields is affected by non-Kitaev interactions which exist in real candidate materials of the Kitaev magnet. It is found that the off-diagonal exchange interaction referred to as the

{\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}

\ifmmode\pm\else\textpm\fi{}1

to an Abelian phase with the Chern number equal to

\ifmmode\pm\else\textpm\fi{}2

, in which the

{Z}_{2}

vortices behave as Abelian anyons. On the other hand, other non-Kitaev interactions such as the Heisenberg interaction and the

\mathrm{\ensuremath{\Gamma}}

term only affect the bandwidth of the Majorana band as long as the spin liquid state is not destabilized.

\mathbb Z_2

gauge fields, enabling sign-problem-free Monte Carlo simulations. We show that increasing dephasing condenses an inter-copy fermion pair, converting

{\rm Ising}\boxtimes\overline{\rm{Ising}}

to toric-code order in the Choi state and yielding intrinsically mixed-state topological order upon unfolding. More interestingly, at a temporal interface between topological and trivial Kitaev states, a Chern-number mismatch enforces a gapless Majorana mode. Dephasing then drives a mixed-state topological transition at which the gapless mode develops emergent

\mathcal N=1

supersymmetry. Our Monte Carlo simulations yield

ν=1.27(3)

and

η_b\simeqη_f\simeq0.40

, consistent with the exact supersymmetric values

5/4

and

2/5

. The supersymmetric partners are fractionalized and appear physically as distinct gauge-invariant spin strings, providing concrete observables for future experiments.

\text{SO}(N)

gauge fields with a Chern-Simons term. We further show that Euclidean Majorana zero modes bound to

{\mathbb{Z}}_{2}

monopole-instantons in the emergent non-Abelian gauge field are key to understanding spontaneous symmetry breaking in the ordered phase.

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=1/2

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.

Z_2

fluxes, while that in the Abelian phase remains open.