Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2006.10081 · Physical review. B./Physical review. B

Phases and quantum phase transitions in an anisotropic ferromagnetic Kitaev-Heisenberg- Γ magnet

Animesh Nanda, Kusum Dhochak, Subhro Bhattacharjee

Published Dec 11, 2020• 12 citations
arXivHTMLPDF

Abstract

We study the spin-$1/2$ ferromagnetic Heisenberg-Kitaev-$\mathrm{\ensuremath{\Gamma}}$ model in the anisotropic (Toric code) limit to reveal the nature of the quantum phase transition between the gapped ${Z}_{2}$ quantum spin liquid and a spin-ordered phase (driven by Heisenberg interactions) as well as a trivial paramagnet (driven by pseudodipolar interactions $\mathrm{\ensuremath{\Gamma}}$). The transitions are obtained by a simultaneous condensation of the Ising electric and magnetic charges---the fractionalized excitations of the ${Z}_{2}$ quantum spin liquid. Both these transitions can be continuous and are examples of deconfined quantum critical points. Crucial to our calculations are the symmetry implementations on the soft electric and magnetic modes that become critical. In particular, we find strong constraints on the structure of the critical theory arising from time reversal and lattice translation symmetries with the latter acting as an anyon permutation symmetry that endows the critical theory with a manifestly self-dual structure. We find that the transition between the quantum spin liquid and the spin-ordered phase belongs to a self-dual modified Abelian Higgs field theory while that between the spin liquid and the trivial paramagnet belongs to a self-dual ${Z}_{2}$ gauge theory. We also study the effect of an external Zeeman field to show an interesting similarity between the polarized paramagnet obtained due to the Zeeman field and the trivial paramagnet driven pseudodipolar interactions. Interestingly, both the spin liquid and the spin-ordered phases have easily identifiable counterparts in the isotropic limit and the present calculations may shed insights into the corresponding transitions in the material relevant isotropic limit.

DOI
10.1103/physrevb.102.235124

Related papers

Phases and quantum phase transitions in the anisotropic antiferromagnetic Kitaev-Heisenberg- Γ magnet

Animesh Nanda, Adhip Agarwala, Subhro Bhattacharjee

We study the Kitaev-Heisenberg-$\mathrm{\ensuremath{\Gamma}}$ model with antiferromagnetic Kitaev exchanges in the strong anisotropic (toric code) limit to understand the phases and the intervening phase transitions between the gapped ${Z}_{2}$ quantum spin liquid and the spin-ordered (in the Heisenberg limit) as well as paramagnetic phases (in the pseudodipolar, $\mathrm{\ensuremath{\Gamma}}$, limit). We find that the paramagnetic phase obtained in the large $\mathrm{\ensuremath{\Gamma}}$ limit has no topological entanglement entropy and is proximate to a gapless critical point of a system described by equal superposition of differently oriented stacked one-dimensional ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry protected topological phases. Using a combination of exact diagonalization calculations and field theoretic analysis we map out the phases and phase transitions to reveal the complete phase diagram as a function of the Heisenberg, the Kitaev, and the pseudodipolar interactions. Our work shows a rich plethora of unconventional phases and phase transitions and provides a comprehensive understanding of the physics of anisotropic Kitaev-Heisenberg-$\mathrm{\ensuremath{\Gamma}}$ systems along with our recent paper [Phys. Rev. B 102, 235124 (2020)] where the ferromagnetic Kitaev exchange was studied.

arXiv:2107.09697•Physical review. B./Physical review. B•Nov 9, 2021•9 citations

Anyon dynamics in field-driven phases of the anisotropic Kitaev model

Shi Feng, Adhip Agarwala, Subhro Bhattacharjee +1 more

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 ${Z}_{2}$ fluxes and gapless linearly dispersing Majorana fermions in the isotropic limit (${K}_{x}={K}_{y}={K}_{z}$). We explore the phase diagram along two axes, an external magnetic field $h$, applied out of plane of the honeycomb, and anisotropic interactions ${K}_{z}$ larger than the other two (${K}_{x}={K}_{y}\ensuremath{\equiv}K$). For ${K}_{z}/K\ensuremath{\gg}2$ and $h=0$, the matter Majorana fermions have the largest gap, and the system is described by a gapped ${Z}_{2}$ 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 ($e$) and magnetic ($m$) charges, disperses along a specific one-dimensional direction that provides a fingerprint of fractionalization. At lower ${K}_{z}$ 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.

arXiv:2206.12990•Physical review. B./Physical review. B•Jul 27, 2023•18 citations

Liquid-Liquid Transition in Kitaev Magnets Driven by Spin Fractionalization

Joji Nasu, Yasuyuki Kato, Junki Yoshitake +2 more

While phase transitions between magnetic analogs of three states of matter -- a long-range ordered state, paramagnet, and spin liquid -- have been extensively studied, the possibility of liquid-liquid transitions, namely, between different spin liquids, remains elusive. By introducing the Ising coupling into the honeycomb Kitaev model with bond asymmetry, we discover that the Kitaev spin liquid becomes a bond-nematic quantum paramagnet before magnetically ordered. The phase transition between the two liquid-like states with different topological nature is of first order, driven by delocalization of the $Z_2$ gauge fluxes, and persists to a critical point at finite temperature located inside the regime where quantum spins are fractionalized. It is suggested that similar transitions may occur in other perturbed Kitaev magnets with bond asymmetry.

Bulletin of the American Physical Society•Mar 13, 2017•1 citation

Quantum spin liquid ground states of the Heisenberg-Kitaev model on the triangular lattice

Pavel Kos, Matthias Punk

We study quantum disordered ground states of the two-dimensional Heisenberg-Kitaev model on the triangular lattice using a Schwinger boson approach. Our aim is to identify and characterize potential gapped quantum spin liquid phases that are stabilized by anisotropic Kitaev interactions. For antiferromagnetic Heisenberg and Kitaev couplings and sufficiently small spin $S$, we find three different symmetric ${Z}_{2}$ spin liquid phases, separated by two continuous quantum phase transitions. Interestingly, the gap of elementary excitations remains finite throughout the transitions. The first spin liquid phase corresponds to the well-known zero-flux state in the Heisenberg limit, which is stable with respect to small Kitaev couplings and develops ${120}^{\ensuremath{\circ}}$ order in the semiclassical limit at large $S$. In the opposite Kitaev limit, we find a different spin liquid ground state, which is a quantum disordered version of a magnetically ordered state with antiferromagnetic chains, in accordance with results in the classical limit. Finally, at intermediate couplings, we find a spin liquid state with unusual spin correlations. Upon spinon condensation, this state develops Bragg peaks at incommensurate momenta in close analogy to the magnetically ordered Z2 vortex crystal phase, which has been analyzed in recent theoretical works.

arXiv:1611.05454•Physical review. B./Physical review. B•Jan 20, 2017•26 citations

Majorana-fermion mean field theories of Kitaev quantum spin liquids

Shahnam Ghanbari Saheli, Jennifer Shu-Jen Lin, Huanzhi Hu +1 more

We determine the phase diagrams of anisotropic Kitaev-Heisenberg models on the honeycomb lattice using parton mean-field theories based on different Majorana fermion representations of the S=1/2 spin operators. First, we use a two-dimensional Jordan-Wigner transformation (JWT) involving a semi-infinite snake string operator. To ensure that the fermionized Hamiltonian remains local, we consider the limit of extreme Ising exchange anisotropy in the Heisenberg sector. Second, we use the conventional Kitaev representation in terms of four Majorana fermions subject to local constraints, which we enforce through Lagrange multipliers. For both representations, we self-consistently decouple the interaction terms in the bond and magnetization channels and determine the phase diagrams as a function of the anisotropy of the Kitaev couplings and the relative strength of the Ising exchange. While both mean-field theories produce identical phase boundaries for the topological phase transition between the gapless and gapped Kitaev quantum spin liquids, the JWT fails to correctly describe the magnetic instability and finite-temperature behavior. Our results show that the magnetic phase transition is first order at low temperatures but becomes continuous above a certain temperature. At this energy scale, we also observe a finite-temperature crossover on the quantum-spin-liquid side, from a fractionalized paramagnet at low temperatures, in which gapped flux excitations are frozen out, to a conventional paramagnet at high temperatures. Published by the American Physical Society 2024

arXiv:2310.10230•Physical review. B./Physical review. B•Jan 9, 2024•7 citations

Phases and phase transitions of a perturbed Kekulé-Kitaev model

Eoin Quinn, Subhro Bhattacharjee, Roderich Moessner

We study the quantum-spin-liquid phase in a variant of the Kitaev model where the bonds of the honeycomb lattice are distributed in a Kekul\'e pattern. The system supports gapped and gapless ${Z}_{2}$ quantum spin liquids with interesting differences from the original Kitaev model, the most notable being a gapped ${Z}_{2}$ spin liquid on a kagome lattice. Perturbing the exactly solvable model with antiferromagnetic Heisenberg perturbations, we find a magnetically ordered phase stabilized by a quantum ``order by disorder'' mechanism, as well as an exotic continuous quantum phase transition between the topological spin liquid and this magnetically ordered phase. Using a combination of field theory and Monte Carlo simulations, we find that the transition likely belongs to the $3D\text{\ensuremath{-}}XY\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ universality class.

arXiv:1501.07582•Physical Review B•Apr 16, 2015•13 citations
We study the spin-1/21/21/2 ferromagnetic Heisenberg-Kitaev-\ensuremathΓ\mathrm{\ensuremath{\Gamma}}\ensuremathΓ model in the anisotropic (Toric code) limit to reveal the nature of the quantum phase transition between the gapped Z2{Z}_{2}Z2​ quantum spin liquid and a spin-ordered phase (driven by Heisenberg interactions) as well as a trivial paramagnet (driven by pseudodipolar interactions \ensuremathΓ\mathrm{\ensuremath{\Gamma}}\ensuremathΓ). The transitions are obtained by a simultaneous condensation of the Ising electric and magnetic charges---the fractionalized excitations of the Z2{Z}_{2}Z2​ quantum spin liquid. Both these transitions can be continuous and are examples of deconfined quantum critical points. Crucial to our calculations are the symmetry implementations on the soft electric and magnetic modes that become critical. In particular, we find strong constraints on the structure of the critical theory arising from time reversal and lattice translation symmetries with the latter acting as an anyon permutation symmetry that endows the critical theory with a manifestly self-dual structure. We find that the transition between the quantum spin liquid and the spin-ordered phase belongs to a self-dual modified Abelian Higgs field theory while that between the spin liquid and the trivial paramagnet belongs to a self-dual Z2{Z}_{2}Z2​ gauge theory. We also study the effect of an external Zeeman field to show an interesting similarity between the polarized paramagnet obtained due to the Zeeman field and the trivial paramagnet driven pseudodipolar interactions. Interestingly, both the spin liquid and the spin-ordered phases have easily identifiable counterparts in the isotropic limit and the present calculations may shed insights into the corresponding transitions in the material relevant isotropic limit.
We study the Kitaev-Heisenberg-$\mathrm{\ensuremath{\Gamma}}$ model with antiferromagnetic Kitaev exchanges in the strong anisotropic (toric code) limit to understand the phases and the intervening phase transitions between the gapped Z2{Z}_{2}Z2​ quantum spin liquid and the spin-ordered (in the Heisenberg limit) as well as paramagnetic phases (in the pseudodipolar, $\mathrm{\ensuremath{\Gamma}}$, limit). We find that the paramagnetic phase obtained in the large $\mathrm{\ensuremath{\Gamma}}$ limit has no topological entanglement entropy and is proximate to a gapless critical point of a system described by equal superposition of differently oriented stacked one-dimensional ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry protected topological phases. Using a combination of exact diagonalization calculations and field theoretic analysis we map out the phases and phase transitions to reveal the complete phase diagram as a function of the Heisenberg, the Kitaev, and the pseudodipolar interactions. Our work shows a rich plethora of unconventional phases and phase transitions and provides a comprehensive understanding of the physics of anisotropic Kitaev-Heisenberg-$\mathrm{\ensuremath{\Gamma}}$ systems along with our recent paper [Phys. Rev. B 102, 235124 (2020)] where the ferromagnetic Kitaev exchange was studied.
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.
While phase transitions between magnetic analogs of three states of matter -- a long-range ordered state, paramagnet, and spin liquid -- have been extensively studied, the possibility of liquid-liquid transitions, namely, between different spin liquids, remains elusive. By introducing the Ising coupling into the honeycomb Kitaev model with bond asymmetry, we discover that the Kitaev spin liquid becomes a bond-nematic quantum paramagnet before magnetically ordered. The phase transition between the two liquid-like states with different topological nature is of first order, driven by delocalization of the Z2Z_2Z2​ gauge fluxes, and persists to a critical point at finite temperature located inside the regime where quantum spins are fractionalized. It is suggested that similar transitions may occur in other perturbed Kitaev magnets with bond asymmetry.
We study quantum disordered ground states of the two-dimensional Heisenberg-Kitaev model on the triangular lattice using a Schwinger boson approach. Our aim is to identify and characterize potential gapped quantum spin liquid phases that are stabilized by anisotropic Kitaev interactions. For antiferromagnetic Heisenberg and Kitaev couplings and sufficiently small spin SSS, we find three different symmetric Z2{Z}_{2}Z2​ spin liquid phases, separated by two continuous quantum phase transitions. Interestingly, the gap of elementary excitations remains finite throughout the transitions. The first spin liquid phase corresponds to the well-known zero-flux state in the Heisenberg limit, which is stable with respect to small Kitaev couplings and develops ${120}^{\ensuremath{\circ}}$ order in the semiclassical limit at large SSS. In the opposite Kitaev limit, we find a different spin liquid ground state, which is a quantum disordered version of a magnetically ordered state with antiferromagnetic chains, in accordance with results in the classical limit. Finally, at intermediate couplings, we find a spin liquid state with unusual spin correlations. Upon spinon condensation, this state develops Bragg peaks at incommensurate momenta in close analogy to the magnetically ordered Z2 vortex crystal phase, which has been analyzed in recent theoretical works.
We determine the phase diagrams of anisotropic Kitaev-Heisenberg models on the honeycomb lattice using parton mean-field theories based on different Majorana fermion representations of the S=1/2 spin operators. First, we use a two-dimensional Jordan-Wigner transformation (JWT) involving a semi-infinite snake string operator. To ensure that the fermionized Hamiltonian remains local, we consider the limit of extreme Ising exchange anisotropy in the Heisenberg sector. Second, we use the conventional Kitaev representation in terms of four Majorana fermions subject to local constraints, which we enforce through Lagrange multipliers. For both representations, we self-consistently decouple the interaction terms in the bond and magnetization channels and determine the phase diagrams as a function of the anisotropy of the Kitaev couplings and the relative strength of the Ising exchange. While both mean-field theories produce identical phase boundaries for the topological phase transition between the gapless and gapped Kitaev quantum spin liquids, the JWT fails to correctly describe the magnetic instability and finite-temperature behavior. Our results show that the magnetic phase transition is first order at low temperatures but becomes continuous above a certain temperature. At this energy scale, we also observe a finite-temperature crossover on the quantum-spin-liquid side, from a fractionalized paramagnet at low temperatures, in which gapped flux excitations are frozen out, to a conventional paramagnet at high temperatures. Published by the American Physical Society 2024
We study the quantum-spin-liquid phase in a variant of the Kitaev model where the bonds of the honeycomb lattice are distributed in a Kekul\'e pattern. The system supports gapped and gapless Z2{Z}_{2}Z2​ quantum spin liquids with interesting differences from the original Kitaev model, the most notable being a gapped Z2{Z}_{2}Z2​ spin liquid on a kagome lattice. Perturbing the exactly solvable model with antiferromagnetic Heisenberg perturbations, we find a magnetically ordered phase stabilized by a quantum ``order by disorder'' mechanism, as well as an exotic continuous quantum phase transition between the topological spin liquid and this magnetically ordered phase. Using a combination of field theory and Monte Carlo simulations, we find that the transition likely belongs to the $3D\text{\ensuremath{-}}XY\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ universality class.