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

APS

Mutual Chern-Simons Landau-Ginzburg theory for continuous quantum phase transition of Z2 topological order

Su-Peng Kou, Yu Jing, Xiao-Gang Wen

Published Sep 1, 2009• 1 citation
OpenAlex

Abstract

In this paper, we develop an effective mutual Chern-Simons Landau-Ginzburg (MCSLG) theory to describe the continuous topological quantum phase transition (TQPT). In particular, we consider the TQPT between a spin-polarized phase (a state without topological order) and a Z[subscript 2] topologically ordered state. The TQPT is not induced by spontaneous symmetry breaking. Instead the [subscript 2]topological order is broken down by the condensation of [subscript 2] charged quasiparticles. By generalizing the hierarchy theory of fractional quantum Hall effect to [subscript 2] topological order, we show that the TQPT belongs to the universal class of three-dimensional Ising phase transition. In the end, we applied the MCSLG theory to the toric code model.

Related papers

Mutual Chern-Simons Landau-Ginzburg theory for continuous quantum phase transition of Z2 topological order

Su-Peng Kou, Jing Yu, Xiao-Gang Wen

In this paper, we develop an effective mutual Chern-Simons Landau-Ginzburg (MCSLG) theory to describe the continuous topological quantum phase transition (TQPT). In particular, we consider the TQPT between a spin-polarized phase (a state without topological order) and a ${Z}_{2}$ topologically ordered state. The TQPT is not induced by spontaneous symmetry breaking. Instead the ${Z}_{2}$ topological order is broken down by the condensation of ${Z}_{2}$ charged quasiparticles. By generalizing the hierarchy theory of fractional quantum Hall effect to ${Z}_{2}$ topological order, we show that the TQPT belongs to the universal class of three-dimensional Ising phase transition. In the end, we applied the MCSLG theory to the toric code model.

Physical Review B•Sep 3, 2009•16 citations

Topological quantum phase transition in the transverse Wen-plaquette model

Jing Yu, Su-Peng Kou, Xiao-Gang Wen

In this paper we study the quantum phase transition between a quantum state with topological order and that without topological order (TQPT) in the transverse Wen-plaquette model. By mapping the transverse Wen-plaquette model onto the d =1 Ising model in a transverse field, some exact results of the TQPT are obtained. The transverse Wen-plaquette model undergoes a continuous TQPT which is characterized in terms of the expectation values of string operators. In particular, the duality relationship between open-string and closed-string is explored —the expectation value of closed-string operator and that of open-string operator play the role of "order parameter" and "disorder order parameter" of the topological order, respectively. At the critical point the "condensation" of both Z 2 vortex and Z 2 charge indicates that the TQPT is a new type of phase transition beyond the Landau-Ginzburg-Wilson paradigm.

Europhysics Letters (EPL)•Sep 18, 2008•45 citations

Quantum critical phase transition between two topologically ordered phases in the Ising toric code bilayer

Raymond Wiedmann, Lea Lenke, Matthias R. Walther +2 more

We demonstrate that two toric code layers on the square lattice coupled by an Ising interaction display two distinct phases with intrinsic topological order. The second-order quantum phase transition between the weakly coupled ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ and the strongly coupled ${\mathbb{Z}}_{2}$ topological order can be described by the condensation of bosonic quasiparticles from both sides and belongs to the three-dimensional ${\mathrm{Ising}}^{*}$ universality class. This can be shown by an exact duality transformation to the transverse-field Ising model on the square lattice, which builds on the existence of an extensive number of local ${\mathbb{Z}}_{2}$ conserved parities. These conserved quantities correspond to the product of two adjacent star operators on different layers. Notably, we show that the low-energy effective model derived about the limit of large Ising coupling is given by an effective single-layer toric code in terms of the conserved quantities of the Ising toric code bilayer. The two topological phases are further characterized by the topological entanglement entropy which serves as a nonlocal order parameter.

arXiv:2010.05982•Physical review. B./Physical review. B•Dec 17, 2020•19 citations

Mutual Chern-Simons theory for Z2 topological order

Su-Peng Kou, Michael Levin, Xiao-Gang Wen

We study several different ${Z}_{2}$ topological ordered states in frustrated spin systems. The effective theories for those different ${Z}_{2}$ topological orders all have the same form---a ${Z}_{2}$ gauge theory which can also be written as a mutual $U(1)\ifmmode\times\else\texttimes\fi{}U(1)$ Chern-Simons theory. However, we find that the different ${Z}_{2}$ topological orders are reflected in different projective realizations of lattice symmetry in the same effective mutual Chern-Simons theory. This result is obtained by comparing the ground-state degeneracy, the ground-state quantum numbers, the gapless edge state, and the projective symmetry group of quasiparticles calculated from the slave-particle theory and from the effective mutual Chern-Simons theories. Our study reveals intricate relations between topological order and symmetry.

arXiv:0803.2300•Physical Review B•Oct 31, 2008•90 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.

arXiv:2207.02701•Physical review. B./Physical review. B•Dec 6, 2022•5 citations

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
In this paper, we develop an effective mutual Chern-Simons Landau-Ginzburg (MCSLG) theory to describe the continuous topological quantum phase transition (TQPT). In particular, we consider the TQPT between a spin-polarized phase (a state without topological order) and a Z[subscript 2] topologically ordered state. The TQPT is not induced by spontaneous symmetry breaking. Instead the [subscript 2]topological order is broken down by the condensation of [subscript 2] charged quasiparticles. By generalizing the hierarchy theory of fractional quantum Hall effect to [subscript 2] topological order, we show that the TQPT belongs to the universal class of three-dimensional Ising phase transition. In the end, we applied the MCSLG theory to the toric code model.
In this paper, we develop an effective mutual Chern-Simons Landau-Ginzburg (MCSLG) theory to describe the continuous topological quantum phase transition (TQPT). In particular, we consider the TQPT between a spin-polarized phase (a state without topological order) and a Z2{Z}_{2}Z2​ topologically ordered state. The TQPT is not induced by spontaneous symmetry breaking. Instead the Z2{Z}_{2}Z2​ topological order is broken down by the condensation of Z2{Z}_{2}Z2​ charged quasiparticles. By generalizing the hierarchy theory of fractional quantum Hall effect to Z2{Z}_{2}Z2​ topological order, we show that the TQPT belongs to the universal class of three-dimensional Ising phase transition. In the end, we applied the MCSLG theory to the toric code model.
In this paper we study the quantum phase transition between a quantum state with topological order and that without topological order (TQPT) in the transverse Wen-plaquette model. By mapping the transverse Wen-plaquette model onto the d =1 Ising model in a transverse field, some exact results of the TQPT are obtained. The transverse Wen-plaquette model undergoes a continuous TQPT which is characterized in terms of the expectation values of string operators. In particular, the duality relationship between open-string and closed-string is explored —the expectation value of closed-string operator and that of open-string operator play the role of "order parameter" and "disorder order parameter" of the topological order, respectively. At the critical point the "condensation" of both Z 2 vortex and Z 2 charge indicates that the TQPT is a new type of phase transition beyond the Landau-Ginzburg-Wilson paradigm.
We demonstrate that two toric code layers on the square lattice coupled by an Ising interaction display two distinct phases with intrinsic topological order. The second-order quantum phase transition between the weakly coupled ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ and the strongly coupled Z2{\mathbb{Z}}_{2}Z2​ topological order can be described by the condensation of bosonic quasiparticles from both sides and belongs to the three-dimensional Ising∗{\mathrm{Ising}}^{*}Ising∗ universality class. This can be shown by an exact duality transformation to the transverse-field Ising model on the square lattice, which builds on the existence of an extensive number of local Z2{\mathbb{Z}}_{2}Z2​ conserved parities. These conserved quantities correspond to the product of two adjacent star operators on different layers. Notably, we show that the low-energy effective model derived about the limit of large Ising coupling is given by an effective single-layer toric code in terms of the conserved quantities of the Ising toric code bilayer. The two topological phases are further characterized by the topological entanglement entropy which serves as a nonlocal order parameter.
We study several different Z2{Z}_{2}Z2​ topological ordered states in frustrated spin systems. The effective theories for those different Z2{Z}_{2}Z2​ topological orders all have the same form---a Z2{Z}_{2}Z2​ gauge theory which can also be written as a mutual $U(1)\ifmmode\times\else\texttimes\fi{}U(1)$ Chern-Simons theory. However, we find that the different Z2{Z}_{2}Z2​ topological orders are reflected in different projective realizations of lattice symmetry in the same effective mutual Chern-Simons theory. This result is obtained by comparing the ground-state degeneracy, the ground-state quantum numbers, the gapless edge state, and the projective symmetry group of quasiparticles calculated from the slave-particle theory and from the effective mutual Chern-Simons theories. Our study reveals intricate relations between topological order and symmetry.
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 SO(N)\text{SO}(N)SO(N) gauge fields with a Chern-Simons term. We further show that Euclidean Majorana zero modes bound to Z2{\mathbb{Z}}_{2}Z2​ monopole-instantons in the emergent non-Abelian gauge field are key to understanding spontaneous symmetry breaking in the ordered phase.
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.