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

Physical review. B./Physical review. B

Fermionic interpretation of the quantum phase transition in XXZ magnets

Tao Yang, Bangming Wang, Peng Song, Rui Wang, Baigeng Wang

Published Oct 31, 2022• 1 citation
Publisher

Abstract

Quantum phase transitions in quantum magnets constitute an important field that has attracted great interests. Although a variety of analytic and numerical methods have been introduced in this direction, faithful fermionic descriptions are still desired because they can transform the spin models to a form tractable by conventional many-body techniques, yielding more transparent physical pictures. We generalize the Chern-Simons fermionization approach and apply it to the XXZ quantum magnets. After fermionzation, the Ising interaction of the XXZ spin model leads to a fermion-fermion interaction. We show that the additional fermion-fermion interaction generates an interesting phase transition between two fermionic ordered states, i.e., the Chern-Simons superconductor and the Chern-Simons exciton insulator. We also demonstrate that this transition in the fermionic language essentially describes the quantum phase transition between the planar and out-of-plane N\'eel orders in the spin picture. The fermionic mean-field theory further leads to a nonlinear $\ensuremath{\sigma}$ model that describes the quantum phase transition, which is further supported by our density matrix renormalization group calculations in the original spin model. Our work introduces a fermionic interpretation of quantum phase transitions and advances the existing knowledge of quantum magnets.

DOI
10.1103/physrevb.106.144438

Related papers

Emergent topological orders and phase transitions in lattice Chern-Simons theory of quantum magnets

Rui Wang, Z. Y. Xie, Baigeng Wang +1 more

Topological phase transitions involving intrinsic topological orders are usually characterized by qualitative changes of ground state quantum entanglement, which cannot be described by conventional mean-field theories with local order parameters. Here, we apply the lattice Chern-Simons theory to study frustrated quantum magnets and show that the conventional concepts, such as the order parameter and symmetry breaking, can still play a crucial role in certain topological phase transitions. The lattice Chern-Simons representation establishes a nonlocal mapping from quantum spin models to interacting spinless Dirac fermions. We show that breaking certain emergent symmetries of the fermionic theory could provide a unified approach to describing both magnetic and topological orders, as well as the topological phase transitions between them. We apply this method to the perturbed spin-1/2 ${J}_{1}\ensuremath{-}{J}_{2}$ XY model on the honeycomb lattice and predict a nonuniform chiral spin liquid ground state in the strong frustration region. This is further verified by our high-precision tensor network calculations. These results suggest that the lattice Chern-Simons theory can simplify the complicated topological phase transitions to effective mean-field theories in terms of fermionic degrees of freedom, which lead to different understandings that help to understand the frustrated quantum magnets.

Physical review. B./Physical review. B•Sep 29, 2022•20 citations

Chern-Simons superconductors and their instabilities

Rui Wang, Baigeng Wang, Tigran Sedrakyan

Two-dimensional quantum antiferromagnets host rich physics, including long-range ordering, high-${T}_{c}$ superconductivity, quantum spin-liquid behavior, topological ordering, a variety of other exotic phases, and quantum criticalities. Frustrating perturbations in antiferromagnets may give rise to strong quantum fluctuations, challenging the theoretical understanding of the many-body ground state. Here we develop a method to describe the quantum antiferromagnets using fermionic degrees of freedom. The method is based on a formally exact mapping between spin-exchange models and theories describing fermionic matter with the emergent $\text{U}(1)$ Chern-Simons gauge field. For the planar N\'eel state, this mapping self-consistently generates the Chern-Simons superconductor mean-field ground state of introduced spinless fermions. We systematically compare the Chern-Simons superconductor state with the planar N\'eel state at the level of collective modes as well as order parameters. We reveal qualitative and quantitative correspondences between these two states. We demonstrate that such a construction using the fractionalized excitations and Chern-Simons gauge field can not only describe the N\'eel order, but can also be applied to study quantum spin liquids. Furthermore, we show that the confinement-deconfinement transitions from the N\'eel order to quantum spin liquids are signaled and characterized by the instabilities of Chern-Simons superconductors, driven by strong frustration. The results suggest observing and classifying the descendants of antiferromagnets, including other ordered states and unconventional superconductors, as well as emergent quantum spin liquids.

arXiv:2010.10067•Physical review. B./Physical review. B•Feb 4, 2022•14 citations

Chern-Simons fermionization approach to two-dimensional quantum magnets: Implications for antiferromagnetic magnons and unconventional quantum phase transitions

Rui Wang, Baigeng Wang, Tigran Sedrakyan

We develop an approach to describe antiferromagnetic magnons on a bipartite lattice supporting the N\'eel state using fractionalized degrees of freedom typically inherent to quantum spin liquids. In particular, we consider a long-range magnetically ordered state of interacting two-dimensional quantum spin-$1/2$ XY models using the Chern-Simons (CS) fermion representation of interacting spins. The interaction leads to Cooper instability and pairing of CS fermions, and to CS superconductivity which spontaneously violates the continuous $\mathrm{U}(1)$ symmetry generating a linearly dispersing gapless Nambu-Goldstone mode due to phase fluctuations. We evaluate this mode and show that it is in high-precision agreement with magnons of the corresponding N\'eel antiferromagnet irrespective to the lattice symmetry. Using the fermion formulation of the system with frustration, we show that the competing interactions emerge in the form of long-range interaction vertices mediated by the CS gauge field, which are responsible for restoring the continuous symmetry at sufficiently strong frustration. We identify these new interaction vertices and discuss their implications for unconventional phase transitions. We also apply the proposed theory to a model of anyons that can be tuned continuously from fermions to bosons, and discuss the results.

arXiv:1709.00060•Physical review. B./Physical review. B•Aug 2, 2018•15 citations

Theory of competing Chern-Simons orders and emergent phase transitions

Rui Wang, Z. Y. Xie, Baigeng Wang +1 more

Based on the Chern-Simons fermionization of spin-1/2 operators, we propose a systematic framework to investigate the competition between emergent phases in frustrated two-dimensional XY quantum magnets. Application of the method to the antiferromagnetic honeycomb spin-1/2 $J_1$-$J_2$ XY model reveals an unconventional phase transition between two Chern-Simons orders: the Chern-Simons superconductor and the exciton insulator of Chern-Simons fermions. We show that in the spin language, this transition translates to the transition from the planar Néel state to the non-uniform chiral spin-liquid that was proposed earlier in the literature. Namely, the Chern-Simons superconductor describes the planar Néel state, while the Chern-Simons exciton insulator corresponds to the non-uniform chiral spin-liquid. These results are further confirmed by our high-precision tensor network calculation, which provides the first numerical evidence for the transition from Néel order to a non-uniform chiral spin-liquid. We argue that the developed method can be applied to other frustrated quantum magnets of XXZ type and can detect unconventional phase transitions.

arXiv:2101.04864•Jan 13, 2021•0 citations

Order And Disorder In Spin Systems

Yury Kiselev

The study of various types of orders and phases of matter in magnetic systems is extremely popular theme in condensed matter physics. Not only frequently applicable in real world, quantum magnetism provides a fruitful and at the same time frequently feasible framework to search for new phases of matter. In this work we study quantum and associated classical phase transitions of spin systems by combinations of numerical and analytical methods.In Chapter 1 we study quantum paramagnets that are a generalization of antiferromagnetic AKLT [2] construction for the spin group SU(N). We map the so called simplex solid state ansatz wavefunction into a partition function of the classical system on the same lattice using coherent state representation. Then we study phase transitions of that classical models via classical Monte Carlo methods to determine properties of the original quantum simplex solid state [3].In Chapter 2 we are interested in the O(2) model in (2+1)-dimensional system with the dilution, i.e. when some sites are removed from the system. This relativistic O(2) model originates from the description of a transition between superfluid and Mott insulator phases. We study critical properties of the transition using a recently invented [4] worm algorithm.In Chapter 3 we are studying large-N expansion for the symplectic spin group Sp(N). When the original SU(2) spin model proves to be too difficult to be solved, for example at low temperatures, one of the alternative ways is to consider a system based on a more general spin group, for example SU(N) or Sp(N). Results often remain relevant for the original SU(2) group. We are using the mean-field theory approach to find phase transitions associated with the cubic Sp(N) system at large N.Another fundamental question of interest is how different systems behave when they are periodically driven by some external field. Such systems are called Floquet systems, and we will study them in Chapter 4. We will consider interacting and non-interacting systems, that might have a non-conventional type of ordering, topological order.

•Jan 1, 2017•0 citations

Jordan-Wigner fermionization of quantum spin systems on arbitrary two-dimensional lattices: A mutual Chern-Simons approach

Jagannath Das, Aman Kumar, Avijit Maity +1 more

A variety of analytical approaches have been developed for the study of quantum spin systems in two dimensions, the notable ones being spin-waves, slave boson/fermion parton constructions, and for lattices with one-to-one local correspondence of faces and vertices, the two-dimensional (2D) Jordan-Wigner (JW) fermionization. Field-theoretically, JW fermionization is implemented through Chern-Simons (CS) flux attachment. For a correct fermionization of lattice quantum spin-$1/2$ magnets, it is necessary that the fermions obey mutual bosonic (anyonic) statistics under exchange---this is not possible to implement on arbitrary 2D lattices if fermionic matter couples only to the lattice gauge fields. Enlarging the gauge degrees of freedom to include the dual lattice allows the construction of consistent mutual Chern-Simons field theories. Here we propose a mutual CS theory where the microscopic (spin) degrees of freedom are represented as lattice fermionic matter additionally coupled to specific combinations of dual lattice gauge fields that depend on the local geometry. We illustrate the use of this method for understanding the properties of a honeycomb Kitaev model subjected to a strong Zeeman field in the $z$-direction. Our CS gauge theory framework provides an understanding of why the topological phase is degraded at lower (higher) critical fields for the ferromagnetic (antiferromagnetic) Kitaev interaction. Additionally, we observe an effectively one-dimensional character of the low excitations at higher fields in the $z$-direction, which we also confirm by spin-wave calculations.

arXiv:2210.07718•Physical review. B./Physical review. B•Aug 7, 2023•3 citations
Quantum phase transitions in quantum magnets constitute an important field that has attracted great interests. Although a variety of analytic and numerical methods have been introduced in this direction, faithful fermionic descriptions are still desired because they can transform the spin models to a form tractable by conventional many-body techniques, yielding more transparent physical pictures. We generalize the Chern-Simons fermionization approach and apply it to the XXZ quantum magnets. After fermionzation, the Ising interaction of the XXZ spin model leads to a fermion-fermion interaction. We show that the additional fermion-fermion interaction generates an interesting phase transition between two fermionic ordered states, i.e., the Chern-Simons superconductor and the Chern-Simons exciton insulator. We also demonstrate that this transition in the fermionic language essentially describes the quantum phase transition between the planar and out-of-plane N'eel orders in the spin picture. The fermionic mean-field theory further leads to a nonlinear \ensuremathσ\ensuremath{\sigma}\ensuremathσ model that describes the quantum phase transition, which is further supported by our density matrix renormalization group calculations in the original spin model. Our work introduces a fermionic interpretation of quantum phase transitions and advances the existing knowledge of quantum magnets.
Topological phase transitions involving intrinsic topological orders are usually characterized by qualitative changes of ground state quantum entanglement, which cannot be described by conventional mean-field theories with local order parameters. Here, we apply the lattice Chern-Simons theory to study frustrated quantum magnets and show that the conventional concepts, such as the order parameter and symmetry breaking, can still play a crucial role in certain topological phase transitions. The lattice Chern-Simons representation establishes a nonlocal mapping from quantum spin models to interacting spinless Dirac fermions. We show that breaking certain emergent symmetries of the fermionic theory could provide a unified approach to describing both magnetic and topological orders, as well as the topological phase transitions between them. We apply this method to the perturbed spin-1/2 ${J}_{1}\ensuremath{-}{J}_{2}$ XY model on the honeycomb lattice and predict a nonuniform chiral spin liquid ground state in the strong frustration region. This is further verified by our high-precision tensor network calculations. These results suggest that the lattice Chern-Simons theory can simplify the complicated topological phase transitions to effective mean-field theories in terms of fermionic degrees of freedom, which lead to different understandings that help to understand the frustrated quantum magnets.
Two-dimensional quantum antiferromagnets host rich physics, including long-range ordering, high-Tc{T}_{c}Tc​ superconductivity, quantum spin-liquid behavior, topological ordering, a variety of other exotic phases, and quantum criticalities. Frustrating perturbations in antiferromagnets may give rise to strong quantum fluctuations, challenging the theoretical understanding of the many-body ground state. Here we develop a method to describe the quantum antiferromagnets using fermionic degrees of freedom. The method is based on a formally exact mapping between spin-exchange models and theories describing fermionic matter with the emergent U(1)\text{U}(1)U(1) Chern-Simons gauge field. For the planar N\'eel state, this mapping self-consistently generates the Chern-Simons superconductor mean-field ground state of introduced spinless fermions. We systematically compare the Chern-Simons superconductor state with the planar N\'eel state at the level of collective modes as well as order parameters. We reveal qualitative and quantitative correspondences between these two states. We demonstrate that such a construction using the fractionalized excitations and Chern-Simons gauge field can not only describe the N\'eel order, but can also be applied to study quantum spin liquids. Furthermore, we show that the confinement-deconfinement transitions from the N\'eel order to quantum spin liquids are signaled and characterized by the instabilities of Chern-Simons superconductors, driven by strong frustration. The results suggest observing and classifying the descendants of antiferromagnets, including other ordered states and unconventional superconductors, as well as emergent quantum spin liquids.
We develop an approach to describe antiferromagnetic magnons on a bipartite lattice supporting the N\'eel state using fractionalized degrees of freedom typically inherent to quantum spin liquids. In particular, we consider a long-range magnetically ordered state of interacting two-dimensional quantum spin-1/21/21/2 XY models using the Chern-Simons (CS) fermion representation of interacting spins. The interaction leads to Cooper instability and pairing of CS fermions, and to CS superconductivity which spontaneously violates the continuous U(1)\mathrm{U}(1)U(1) symmetry generating a linearly dispersing gapless Nambu-Goldstone mode due to phase fluctuations. We evaluate this mode and show that it is in high-precision agreement with magnons of the corresponding N\'eel antiferromagnet irrespective to the lattice symmetry. Using the fermion formulation of the system with frustration, we show that the competing interactions emerge in the form of long-range interaction vertices mediated by the CS gauge field, which are responsible for restoring the continuous symmetry at sufficiently strong frustration. We identify these new interaction vertices and discuss their implications for unconventional phase transitions. We also apply the proposed theory to a model of anyons that can be tuned continuously from fermions to bosons, and discuss the results.
Based on the Chern-Simons fermionization of spin-1/2 operators, we propose a systematic framework to investigate the competition between emergent phases in frustrated two-dimensional XY quantum magnets. Application of the method to the antiferromagnetic honeycomb spin-1/2 J1J_1J1​-J2J_2J2​ XY model reveals an unconventional phase transition between two Chern-Simons orders: the Chern-Simons superconductor and the exciton insulator of Chern-Simons fermions. We show that in the spin language, this transition translates to the transition from the planar Néel state to the non-uniform chiral spin-liquid that was proposed earlier in the literature. Namely, the Chern-Simons superconductor describes the planar Néel state, while the Chern-Simons exciton insulator corresponds to the non-uniform chiral spin-liquid. These results are further confirmed by our high-precision tensor network calculation, which provides the first numerical evidence for the transition from Néel order to a non-uniform chiral spin-liquid. We argue that the developed method can be applied to other frustrated quantum magnets of XXZ type and can detect unconventional phase transitions.
The study of various types of orders and phases of matter in magnetic systems is extremely popular theme in condensed matter physics. Not only frequently applicable in real world, quantum magnetism provides a fruitful and at the same time frequently feasible framework to search for new phases of matter. In this work we study quantum and associated classical phase transitions of spin systems by combinations of numerical and analytical methods.In Chapter 1 we study quantum paramagnets that are a generalization of antiferromagnetic AKLT [2] construction for the spin group SU(N). We map the so called simplex solid state ansatz wavefunction into a partition function of the classical system on the same lattice using coherent state representation. Then we study phase transitions of that classical models via classical Monte Carlo methods to determine properties of the original quantum simplex solid state [3].In Chapter 2 we are interested in the O(2) model in (2+1)-dimensional system with the dilution, i.e. when some sites are removed from the system. This relativistic O(2) model originates from the description of a transition between superfluid and Mott insulator phases. We study critical properties of the transition using a recently invented [4] worm algorithm.In Chapter 3 we are studying large-N expansion for the symplectic spin group Sp(N). When the original SU(2) spin model proves to be too difficult to be solved, for example at low temperatures, one of the alternative ways is to consider a system based on a more general spin group, for example SU(N) or Sp(N). Results often remain relevant for the original SU(2) group. We are using the mean-field theory approach to find phase transitions associated with the cubic Sp(N) system at large N.Another fundamental question of interest is how different systems behave when they are periodically driven by some external field. Such systems are called Floquet systems, and we will study them in Chapter 4. We will consider interacting and non-interacting systems, that might have a non-conventional type of ordering, topological order.
A variety of analytical approaches have been developed for the study of quantum spin systems in two dimensions, the notable ones being spin-waves, slave boson/fermion parton constructions, and for lattices with one-to-one local correspondence of faces and vertices, the two-dimensional (2D) Jordan-Wigner (JW) fermionization. Field-theoretically, JW fermionization is implemented through Chern-Simons (CS) flux attachment. For a correct fermionization of lattice quantum spin-1/21/21/2 magnets, it is necessary that the fermions obey mutual bosonic (anyonic) statistics under exchange---this is not possible to implement on arbitrary 2D lattices if fermionic matter couples only to the lattice gauge fields. Enlarging the gauge degrees of freedom to include the dual lattice allows the construction of consistent mutual Chern-Simons field theories. Here we propose a mutual CS theory where the microscopic (spin) degrees of freedom are represented as lattice fermionic matter additionally coupled to specific combinations of dual lattice gauge fields that depend on the local geometry. We illustrate the use of this method for understanding the properties of a honeycomb Kitaev model subjected to a strong Zeeman field in the zzz-direction. Our CS gauge theory framework provides an understanding of why the topological phase is degraded at lower (higher) critical fields for the ferromagnetic (antiferromagnetic) Kitaev interaction. Additionally, we observe an effectively one-dimensional character of the low excitations at higher fields in the zzz-direction, which we also confirm by spin-wave calculations.