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:2204.12859 · Physical review. B./Physical review. B

Negative sign free formulations of generalized Kitaev models with higher symmetries

Toshihiro Sato, Fakher F. Assaad

Published Oct 4, 2022• 0 citations
arXivHTMLPDF

Abstract

We provide a negative sign free formulation of the auxiliary field quantum Monte Carlo algorithm for generalized Kitaev models with higher symmetries. Our formulation is based on the Abrikosov fermion representation of the spin-1/2 degree of freedom and the phase pinning approach [T. Sato and F. F. Assaad, Phys. Rev. B 104, L081106 (2021)]. Enhancing the number of fermion flavors or orbitals from 1 to $N$ allows one to generalize the inherent ${Z}_{2}$ global symmetry to ${Z}_{2}\ifmmode\times\else\texttimes\fi{}\mathrm{S}\mathrm{U}{(N)}_{o}$. Using this general approach, we study the ${Z}_{2}\ifmmode\times\else\texttimes\fi{}\mathrm{S}\mathrm{U}{(2)}_{o}$ Kitaev-Heisenberg model reflecting the competition between the isotropic Heisenberg exchange and Kitaev-type bond-directional exchange interactions. We show that the symmetry enhancement provides a path to escape frustration and that the spin liquid phases in the original ${Z}_{2}$-symmetric model are not present in this model. Nevertheless, the ground-state phase diagram is extremely rich and has points with higher global and local continuous symmetries as well as deconfined quantum critical points.

DOI
10.1103/physrevb.106.155110

Related papers

Quantum Monte Carlo simulation of generalized Kitaev models

Toshihiro Sato, Fakher F. Assaad

Frustrated spin systems generically suffer from the negative sign problem inherent in Monte Carlo methods. Since the severity of this problem is formulation dependent, optimization strategies can be put forward. We introduce a phase pinning approach in the realm of the auxiliary field quantum Monte Carlo algorithm. If we can find an antiunitary operator that commutes with the one-body Hamiltonian coupled to the auxiliary field, then the phase of the action is pinned to 0 and $\ensuremath{\pi}$. For generalized Kitaev models, we can successfully apply this strategy and observe a remarkable improvement of the average sign. We use this method to study the thermodynamical and dynamical properties of the Kitaev-Heisenberg model down to temperatures corresponding to half of the exchange coupling constant. Our dynamical data reveal finite temperature properties of ordered and spin-liquid phases inherent in this model.

arXiv:2012.12283•Physical review. B./Physical review. B•Aug 10, 2021•25 citations

Thermodynamics of Kitaev Spin Liquids

Tim Eschmann

In the field of frustrated magnetism, Kitaev models provide prototypical examples of exactly solvable quantum spin liquids, in which the spin degrees of freedom fractionalize into Majorana fermions coupled to an emergent Z2 gauge field. While the ground states of these models are already well understood as exhibiting different kinds of Majorana (semi)metals, it has still been an open task to achieve a detailed understanding of the thermodynamics, and, namely, the low-temperature ordering of the gauge field into different Z2 flux configurations. In this thesis, we investigate the thermodynamics of Kitaev systems in two and three spatial dimensions with sign-problem-free quantum Monte Carlo simulations. In a first study of elementary 3D Kitaev models, we verify that the ground state Z2 flux sectors of these systems are entirely determined by their elementary plaquette length – a result which shows the validity of Lieb’s theorem for lattice geometries which lack the geometric requirements for its proof. We closely investigate the low-temperature phase transition associated with gauge-ordering, which is a particular realization of an inverted Ising phase transition, as it occurs in general lattice Z2 gauge theories. Our results corroborate the understanding of this transition as separating different regimes in terms of vison-loop excitations, by showing a clear correlation between the critical temperature and the vison gap. We also introduce the concept of “gauge frustration”, which, for a particular 3D Kitaev model, leads to the suppression of the phase transition and a more complex physical behavior at low temperatures. A second quantum Monte Carlo study focuses on a generalized 2D Kitaev model on a five-coordinated lattice system. Here, the gauge ordering is accompanied by the spontaneous breaking of time-reversal symmetry, a scenario which not only allows for a variety of topological ground states, but also for the occurrence of a phase transition in two spatial dimensions. We show that this model exhibits such a transition at a particularly high temperature scale. Moreover, the system possesses a number of phases where the Z2 fluxes are only partially ordered.

•Jul 27, 2020•1 citation

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

Three-Dimensional Kitaev Spin Liquids

Kevin M. O'Brien

The formation of quantum spin liquids in frustrated magnets represents an exciting possibility due to the rather exotic features they harbor, including fractionalized excitations and emergent gauge fields. \nUnfortunately, they are notoriously difficult to study as there are often no good analytical methods available and quantum Monte Carlo simulations are hindered by the negative sign problem. \nThe Kitaev honeycomb model is a notable exception of a frustrated quantum model which is exactly solvable and which hosts a number of distinct quantum spin liquid ground states. \nAs such, it allows for a rare opportunity to study the physics of spin liquids with full analytical control. \n \nIn this thesis, we study the fractionalization of spin-1/2 moments into Majorana fermions and an emergent Z2 gauge field in a generalization of the Kitaev honeycomb model to a number of three-dimensional lattices. \nWhile the excitations of the gauge field are always gapped, the fermionic quasiparticles may exhibit a gapless dispersion, forming fully two-dimensional Fermi surfaces, symmetry protected nodal lines, or topological Weyl nodes. \nWe show that one can deduce rather general constraints on the possible gapless excitations by making use of an object called the projective symmetry group. \nIn doing so we provide a scheme for classifying the various gapless Kitaev spin liquids. \nA thorough analysis is carried out for a number of these spin liquids, primarily investigating the stability of the gapless modes and the novel features resulting from their sometimes non-trivial topology, as well as their effects on certain equal-time correlation functions.

•Jul 4, 2019•0 citations

Functional renormalization group for frustrated magnets with nondiagonal spin interactions

Finn Lasse Buessen, Vincent Noculak, Simon Trebst +1 more

In the field of quantum magnetism, the advent of numerous spin-orbit assisted Mott insulating compounds, such as the family of Kitaev materials, has led to a growing interest in studying general spin models with nondiagonal interactions that do not retain the SU(2) invariance of the underlying spin degrees of freedom. However, the exchange frustration arising from these nondiagonal and often bond-directional interactions for two- and three-dimensional lattice geometries poses a serious challenge for numerical many-body simulation techniques. In this paper, we present an extended formulation of the pseudofermion functional renormalization group that is capable of capturing the physics of frustrated quantum magnets with generic (diagonal and off-diagonal) two-spin interaction terms. Based on a careful symmetry analysis of the underlying flow equations, we reveal that the computational complexity grows only moderately, as compared to models with only diagonal interaction terms. We apply the formalism to a kagome antiferromagnet which is augmented by general in-plane and out-of-plane Dzyaloshinskii-Moriya (DM) interactions, as argued to be present in the spin liquid candidate material herbertsmithite. We calculate the complete ground state phase diagram in the strength of in-plane and out-of-plane DM couplings, and discuss the extended stability of the spin liquid of the unperturbed kagome antiferromagnet in the presence of these couplings.

arXiv:1905.04190•Physical review. B./Physical review. B•Sep 27, 2019•29 citations

Field-induced phase transitions in the Kitaev-Heisenberg model: A sign-problem-free quantum Monte Carlo study and possible application to α − RuCl 3

Xiangbin Zou, Shuo Liu, Wenan Guo +1 more

The frustrated magnet $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$ is one of the prime candidates for realizing a Kitaev quantum spin liquid (QSL). However, the existence of a field-induced intermediate QSL phase in this material remains under debate. Here, we employ sign-free numerically exact quantum Monte Carlo simulations to investigate the Kitaev-Heisenberg (KH) model on the honeycomb lattice with $K=\ensuremath{-}2J$ under an applied magnetic field along the $z$ direction. Our findings reveal that the system undergoes a direct quantum phase transition from a zigzag magnetically ordered phase to a spin-polarized phase at zero temperature, which belongs to the 3D XY universality class. At finite temperatures, a Berezinskii-Kosterlitz-Thouless transition line separates the spin-polarized phase from a quasi-long-range ordered state, eventually terminating at the quantum critical point. Our results convincingly show that there is no intermediate QSL phase in the KH model with a $z$-direction magnetic field, which we believe will shed important light on understanding experimental observations in $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$.

arXiv:2501.12437•Physical review. B./Physical review. B•Sep 8, 2025•2 citations
We provide a negative sign free formulation of the auxiliary field quantum Monte Carlo algorithm for generalized Kitaev models with higher symmetries. Our formulation is based on the Abrikosov fermion representation of the spin-1/2 degree of freedom and the phase pinning approach [T. Sato and F. F. Assaad, Phys. Rev. B 104, L081106 (2021)]. Enhancing the number of fermion flavors or orbitals from 1 to NNN allows one to generalize the inherent Z2{Z}_{2}Z2​ global symmetry to Z2\ifmmode×\else\texttimes\fiSU(N)o{Z}_{2}\ifmmode\times\else\texttimes\fi{}\mathrm{S}\mathrm{U}{(N)}_{o}Z2​\ifmmode×\else\texttimes\fiSU(N)o​. Using this general approach, we study the Z2\ifmmode×\else\texttimes\fiSU(2)o{Z}_{2}\ifmmode\times\else\texttimes\fi{}\mathrm{S}\mathrm{U}{(2)}_{o}Z2​\ifmmode×\else\texttimes\fiSU(2)o​ Kitaev-Heisenberg model reflecting the competition between the isotropic Heisenberg exchange and Kitaev-type bond-directional exchange interactions. We show that the symmetry enhancement provides a path to escape frustration and that the spin liquid phases in the original Z2{Z}_{2}Z2​-symmetric model are not present in this model. Nevertheless, the ground-state phase diagram is extremely rich and has points with higher global and local continuous symmetries as well as deconfined quantum critical points.
Frustrated spin systems generically suffer from the negative sign problem inherent in Monte Carlo methods. Since the severity of this problem is formulation dependent, optimization strategies can be put forward. We introduce a phase pinning approach in the realm of the auxiliary field quantum Monte Carlo algorithm. If we can find an antiunitary operator that commutes with the one-body Hamiltonian coupled to the auxiliary field, then the phase of the action is pinned to 0 and $\ensuremath{\pi}$. For generalized Kitaev models, we can successfully apply this strategy and observe a remarkable improvement of the average sign. We use this method to study the thermodynamical and dynamical properties of the Kitaev-Heisenberg model down to temperatures corresponding to half of the exchange coupling constant. Our dynamical data reveal finite temperature properties of ordered and spin-liquid phases inherent in this model.
In the field of frustrated magnetism, Kitaev models provide prototypical examples of exactly solvable quantum spin liquids, in which the spin degrees of freedom fractionalize into Majorana fermions coupled to an emergent Z2 gauge field. While the ground states of these models are already well understood as exhibiting different kinds of Majorana (semi)metals, it has still been an open task to achieve a detailed understanding of the thermodynamics, and, namely, the low-temperature ordering of the gauge field into different Z2 flux configurations. In this thesis, we investigate the thermodynamics of Kitaev systems in two and three spatial dimensions with sign-problem-free quantum Monte Carlo simulations. In a first study of elementary 3D Kitaev models, we verify that the ground state Z2 flux sectors of these systems are entirely determined by their elementary plaquette length – a result which shows the validity of Lieb’s theorem for lattice geometries which lack the geometric requirements for its proof. We closely investigate the low-temperature phase transition associated with gauge-ordering, which is a particular realization of an inverted Ising phase transition, as it occurs in general lattice Z2 gauge theories. Our results corroborate the understanding of this transition as separating different regimes in terms of vison-loop excitations, by showing a clear correlation between the critical temperature and the vison gap. We also introduce the concept of “gauge frustration”, which, for a particular 3D Kitaev model, leads to the suppression of the phase transition and a more complex physical behavior at low temperatures. A second quantum Monte Carlo study focuses on a generalized 2D Kitaev model on a five-coordinated lattice system. Here, the gauge ordering is accompanied by the spontaneous breaking of time-reversal symmetry, a scenario which not only allows for a variety of topological ground states, but also for the occurrence of a phase transition in two spatial dimensions. We show that this model exhibits such a transition at a particularly high temperature scale. Moreover, the system possesses a number of phases where the Z2 fluxes are only partially ordered.
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 formation of quantum spin liquids in frustrated magnets represents an exciting possibility due to the rather exotic features they harbor, including fractionalized excitations and emergent gauge fields. \nUnfortunately, they are notoriously difficult to study as there are often no good analytical methods available and quantum Monte Carlo simulations are hindered by the negative sign problem. \nThe Kitaev honeycomb model is a notable exception of a frustrated quantum model which is exactly solvable and which hosts a number of distinct quantum spin liquid ground states. \nAs such, it allows for a rare opportunity to study the physics of spin liquids with full analytical control. \n \nIn this thesis, we study the fractionalization of spin-1/2 moments into Majorana fermions and an emergent Z2 gauge field in a generalization of the Kitaev honeycomb model to a number of three-dimensional lattices. \nWhile the excitations of the gauge field are always gapped, the fermionic quasiparticles may exhibit a gapless dispersion, forming fully two-dimensional Fermi surfaces, symmetry protected nodal lines, or topological Weyl nodes. \nWe show that one can deduce rather general constraints on the possible gapless excitations by making use of an object called the projective symmetry group. \nIn doing so we provide a scheme for classifying the various gapless Kitaev spin liquids. \nA thorough analysis is carried out for a number of these spin liquids, primarily investigating the stability of the gapless modes and the novel features resulting from their sometimes non-trivial topology, as well as their effects on certain equal-time correlation functions.
In the field of quantum magnetism, the advent of numerous spin-orbit assisted Mott insulating compounds, such as the family of Kitaev materials, has led to a growing interest in studying general spin models with nondiagonal interactions that do not retain the SU(2) invariance of the underlying spin degrees of freedom. However, the exchange frustration arising from these nondiagonal and often bond-directional interactions for two- and three-dimensional lattice geometries poses a serious challenge for numerical many-body simulation techniques. In this paper, we present an extended formulation of the pseudofermion functional renormalization group that is capable of capturing the physics of frustrated quantum magnets with generic (diagonal and off-diagonal) two-spin interaction terms. Based on a careful symmetry analysis of the underlying flow equations, we reveal that the computational complexity grows only moderately, as compared to models with only diagonal interaction terms. We apply the formalism to a kagome antiferromagnet which is augmented by general in-plane and out-of-plane Dzyaloshinskii-Moriya (DM) interactions, as argued to be present in the spin liquid candidate material herbertsmithite. We calculate the complete ground state phase diagram in the strength of in-plane and out-of-plane DM couplings, and discuss the extended stability of the spin liquid of the unperturbed kagome antiferromagnet in the presence of these couplings.
The frustrated magnet $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$ is one of the prime candidates for realizing a Kitaev quantum spin liquid (QSL). However, the existence of a field-induced intermediate QSL phase in this material remains under debate. Here, we employ sign-free numerically exact quantum Monte Carlo simulations to investigate the Kitaev-Heisenberg (KH) model on the honeycomb lattice with $K=\ensuremath{-}2J$ under an applied magnetic field along the zzz direction. Our findings reveal that the system undergoes a direct quantum phase transition from a zigzag magnetically ordered phase to a spin-polarized phase at zero temperature, which belongs to the 3D XY universality class. At finite temperatures, a Berezinskii-Kosterlitz-Thouless transition line separates the spin-polarized phase from a quasi-long-range ordered state, eventually terminating at the quantum critical point. Our results convincingly show that there is no intermediate QSL phase in the KH model with a zzz-direction magnetic field, which we believe will shed important light on understanding experimental observations in $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$.