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

International Journal of Modern Physics A

GAUGE INVARIANCE AND BROKEN SYMMETRIES IN ANYON SUPERFLUIDS

D. Boyanovsky

Published Sep 30, 1992• 12 citations
Publisher

Abstract

We review aspects of broken symmetry and the nature of long range order in theories of anyons starting with bosons with a statistical interaction. We introduce a novel gauge invariant quantization scheme that allows the identification of local and gauge invariant order parameters. The connection between spin and statistics is reviewed and the consequences of broken symmetries in the anyon representation are discussed. An anyon gas is studied in the Bogoliubov approximation, it is determined that the ground state is a condensate of charge-flux composites with “quasi-long-range order” at zero temperature, a “weak” gap in the spectrum and finite helicity modulus. The system is disordered at nonzero temperatures. The disorder is not caused by Goldstone bosons but by the strong infrared behavior arising from the Coulomb interaction induced by the long-range statistical interaction. The properties of topological vortices in nonrelativistic and in relativistic Landau-Ginzburg theories are studied in detail. We study the physics of the mean-field ansatz and quasi-long range order in a simple exactly soluble relativistic model. This model exhibits a novel phenomenon of charge redistribution to the boundaries and restoration of translational invariance in the infinite volume limit. It also illuminates the physics of quasi-long-range order with a gap in the spectrum, statistical charge polarization by external magnetic fields and the role of “large” gauge transformations.

DOI
10.1142/s0217751x92002696

Related papers

Long-range order of two interacting anyon-gas layers of opposite statistical charge

C. A. A. de Carvalho, D. Boyanovsky

We study aspects of long-range order in a model consisting of two interacting layers of anyon gases with opposite statistical charge. We start with bosons with a statistical interaction and second quantize the theory using an approach that uses a local and gauge-invariant order parameter. We then study this model in the Bogoliubov approximation. We explicitly compute the depletion from the condensate and gauge-invariant correlation functions. We show that the ground state is a ``weak'' condensate of charge-flux composites with ``nearly long-range order.'' The condensate becomes statistically neutral in the thermodynamic limit. It is pointed out that the ``nearly long-range order'' at zero temperature has its origins in the logarithmic infrared singularities arising from the two-dimensional Coulomb repulsion induced by the long-range statistical interaction. The quasiparticle spectrum consists of one massless mode, and a plasmon mode with a weak gap. The system is quasiordered at any nonzero temperature, but we present evidence for a finite helicity modulus.

Physical review. B, Condensed matter•Jan 1, 1993•7 citations

ON THE MEAN FIELD ANSATZ IN ANYON THEORIES

D. Boyanovsky

We study the physics of the mean field ansatz in a simplified anyon model. The model is solved exactly in a finite geometry with particular attention to boundary conditions compatible with a mean statistical magnetic field. The spectrum consists of bosonic plasma-like density fluctuations with a gap. We find that the repulsive Coulomb interactions introduced by the fluctuations around the mean field force the statistical charge and magnetic field to become localized near the boundaries. Translational invariance, broken by the mean field, is restored in the infinite volume limit as the statistical charge and magnetic field escape to infinity. The polarization effects of an external magnetic field and the role of “large” gauge transformations are also studied.

International Journal of Modern Physics A•Jul 30, 1992•1 citation

Two-dimensional Bose liquid with strong gauge-field interaction

M. V. Feigel’man, V. B. Geshkenbeǐn, L. B. Ioffe +1 more

Two unrelated problems can be reduced to a model of a Bose gas interacting with a gauge field: (i) the effect of thermal fluctuations on a system of vortices in bulk superconductors in fields ${\mathit{H}}_{\mathit{c}1}$\ensuremath{\ll}H\ensuremath{\ll}${\mathit{H}}_{\mathit{c}2}$, and (ii) charged, spinless excitations in two-dimensional (2D) strongly correlated electron systems. Both problems are important for the theory of high-temperature superconductors. We study this model in three regimes: at finite temperatures, assuming that the gauge field is purely transverse; at T=0, for the purely static (2D Coulomb) interaction; and at T=0, for a weak Coulomb interaction and a strong transverse one. Transverse interactions suppress the temperature of the superfluid transition significantly. A sufficiently strong transvese interaction is shown to produce a phase separation as the temperature decreases (in the absence of Coulomb repulsion). If there is Coulomb repulsion, the ground state does not have off-diagonal long-range order but the superfluid density is not zero unless the Coulomb constant exceeds a critical value. Sufficiently strong coupling to the transverse field destroys superfluidity as well. In the normal state formed at large couplings, the translational invariance is intact. We propose a bosonic ground state that is not superfluid at T=0. We discuss the implications of these results both for vortex liquids and strongly correlated electron systems.

Physical review. B, Condensed matter•Dec 1, 1993•82 citations

Theory of weak symmetry breaking of translations in Z2 topologically ordered states and its relation to topological superconductivity from an exact lattice Z2 charge-flux attachment

Peng Rao, Inti Sodemann

We study Z 2 topologically ordered states enriched by translational symmetry by employing a recently developed two-dimensional (2D) bosonization approach that implements an exact Z 2 charge-flux attachment in the lattice.Such states can display "weak symmetry breaking" of translations, in which both the Hamiltonian and ground state remain fully translational invariant but the symmetry is "broken" by its anyon quasiparticles, in the sense that its action maps them into a different superselection sector.We demonstrate that this phenomenon occurs when the fermionic spinons form a weak topological superconductor in the form of a 2D stack of one-dimensional Kitaev wires, leading to the amusing property that there is no local operator that can transport the π -flux quasiparticle across a single Kitaev wire of fermonic spinons without paying an energy gap in spite of the vacuum remaining fully translational invariant.We explain why this phenomenon occurs hand in hand with other previously identified peculiar features such as ground-state degeneracy dependence on the size of the torus and the appearance of dangling boundary Majorana modes in certain Z 2 topologically ordered states.Moreover, by extending the Z 2 charge-flux attachment to open lattices and cylinders, we construct a plethora of exactly solvable models providing an exact description of their dispersive Majorana gapless boundary modes.We also review the Z × (Z 2 ) 3 classification of 2D BdG Hamiltonians (class D) enriched by translational symmetry and provide arguments on its robust stability against interactions and self-averaging disorder that preserve translational symmetry.

arXiv:2012.12280•Physical Review Research•May 14, 2021•18 citations

Landau-Ginzburg Paradigm of Topological Phases

Y. B. Zhao, Yidun Wan

Topologically ordered matter phases have been regarded as beyond the Landau-Ginzburg symmetry breaking paradigm of matter phases. Recent studies of anyon condensation in topological phases, however, may fit topological phases back in the Landau-Ginzburg paradigm. To truly do so, we realized that the string-net model of topological phases is in fact an effective lattice gauge theory coupled with anyonic matter once two modifications are made: (1) We reinterpret anyons as matter fields coupled to lattice gauge fields, thus extending the HGW model to a genuine Hamiltonian lattice gauge theory. (2) By explicitly incorporating the internal degrees of freedom of anyons, we construct an enlarged Hilbert space that supports well-defined gauge transformations and covariant coupling, restoring the analogy with conventional lattice gauge field theory. In this modified string-net model, topological phase transitions induced by anyon condensation and their consequent phenomena, such as order parameter fields, coherent states, Goldstone modes, and gapping gauge degrees of freedom, can be formulated exactly as Landau's effective theory of the Higgs mechanism. To facilitate the understanding, we also compare anyon condensation to/with the Higgs boson condensation in the electroweak theory and the Cooper pair condensation.

arXiv:2506.05319•Jun 5, 2025•1 citation

ANYONIZATION AND SUPERFLUIDITY OF LATTICE CHERN-SIMONS THEORY

D. ELIEZER, Gordon Walter Semenoff, Shiying Wu

We prove, working in the Hamiltonian formalism, that a U(1) Chern-Simons theory coupled to fermions on a lattice can be mapped exactly onto a theory of interacting lattice anyons. This map does not involve any singular gauge transformations, and is everywhere well defined. We also prove that, when the statistics parameter is an odd integer so that the anyons are bosons, the ground state, which consists of a condensate of bound pairs of flux tubes and fermions, breaks phase invariance. The ensuing long range order implies that the system is an unconventional superfluid.

Modern Physics Letters A•Feb 28, 1992•16 citations
We review aspects of broken symmetry and the nature of long range order in theories of anyons starting with bosons with a statistical interaction. We introduce a novel gauge invariant quantization scheme that allows the identification of local and gauge invariant order parameters. The connection between spin and statistics is reviewed and the consequences of broken symmetries in the anyon representation are discussed. An anyon gas is studied in the Bogoliubov approximation, it is determined that the ground state is a condensate of charge-flux composites with “quasi-long-range order” at zero temperature, a “weak” gap in the spectrum and finite helicity modulus. The system is disordered at nonzero temperatures. The disorder is not caused by Goldstone bosons but by the strong infrared behavior arising from the Coulomb interaction induced by the long-range statistical interaction. The properties of topological vortices in nonrelativistic and in relativistic Landau-Ginzburg theories are studied in detail. We study the physics of the mean-field ansatz and quasi-long range order in a simple exactly soluble relativistic model. This model exhibits a novel phenomenon of charge redistribution to the boundaries and restoration of translational invariance in the infinite volume limit. It also illuminates the physics of quasi-long-range order with a gap in the spectrum, statistical charge polarization by external magnetic fields and the role of “large” gauge transformations.
We study aspects of long-range order in a model consisting of two interacting layers of anyon gases with opposite statistical charge. We start with bosons with a statistical interaction and second quantize the theory using an approach that uses a local and gauge-invariant order parameter. We then study this model in the Bogoliubov approximation. We explicitly compute the depletion from the condensate and gauge-invariant correlation functions. We show that the ground state is a ``weak'' condensate of charge-flux composites with ``nearly long-range order.'' The condensate becomes statistically neutral in the thermodynamic limit. It is pointed out that the ``nearly long-range order'' at zero temperature has its origins in the logarithmic infrared singularities arising from the two-dimensional Coulomb repulsion induced by the long-range statistical interaction. The quasiparticle spectrum consists of one massless mode, and a plasmon mode with a weak gap. The system is quasiordered at any nonzero temperature, but we present evidence for a finite helicity modulus.
We study the physics of the mean field ansatz in a simplified anyon model. The model is solved exactly in a finite geometry with particular attention to boundary conditions compatible with a mean statistical magnetic field. The spectrum consists of bosonic plasma-like density fluctuations with a gap. We find that the repulsive Coulomb interactions introduced by the fluctuations around the mean field force the statistical charge and magnetic field to become localized near the boundaries. Translational invariance, broken by the mean field, is restored in the infinite volume limit as the statistical charge and magnetic field escape to infinity. The polarization effects of an external magnetic field and the role of “large” gauge transformations are also studied.
Two unrelated problems can be reduced to a model of a Bose gas interacting with a gauge field: (i) the effect of thermal fluctuations on a system of vortices in bulk superconductors in fields Hc1{\mathit{H}}_{\mathit{c}1}Hc1​\ensuremath{\ll}H\ensuremath{\ll}Hc2{\mathit{H}}_{\mathit{c}2}Hc2​, and (ii) charged, spinless excitations in two-dimensional (2D) strongly correlated electron systems. Both problems are important for the theory of high-temperature superconductors. We study this model in three regimes: at finite temperatures, assuming that the gauge field is purely transverse; at T=0, for the purely static (2D Coulomb) interaction; and at T=0, for a weak Coulomb interaction and a strong transverse one. Transverse interactions suppress the temperature of the superfluid transition significantly. A sufficiently strong transvese interaction is shown to produce a phase separation as the temperature decreases (in the absence of Coulomb repulsion). If there is Coulomb repulsion, the ground state does not have off-diagonal long-range order but the superfluid density is not zero unless the Coulomb constant exceeds a critical value. Sufficiently strong coupling to the transverse field destroys superfluidity as well. In the normal state formed at large couplings, the translational invariance is intact. We propose a bosonic ground state that is not superfluid at T=0. We discuss the implications of these results both for vortex liquids and strongly correlated electron systems.
We study Z 2 topologically ordered states enriched by translational symmetry by employing a recently developed two-dimensional (2D) bosonization approach that implements an exact Z 2 charge-flux attachment in the lattice.Such states can display "weak symmetry breaking" of translations, in which both the Hamiltonian and ground state remain fully translational invariant but the symmetry is "broken" by its anyon quasiparticles, in the sense that its action maps them into a different superselection sector.We demonstrate that this phenomenon occurs when the fermionic spinons form a weak topological superconductor in the form of a 2D stack of one-dimensional Kitaev wires, leading to the amusing property that there is no local operator that can transport the π -flux quasiparticle across a single Kitaev wire of fermonic spinons without paying an energy gap in spite of the vacuum remaining fully translational invariant.We explain why this phenomenon occurs hand in hand with other previously identified peculiar features such as ground-state degeneracy dependence on the size of the torus and the appearance of dangling boundary Majorana modes in certain Z 2 topologically ordered states.Moreover, by extending the Z 2 charge-flux attachment to open lattices and cylinders, we construct a plethora of exactly solvable models providing an exact description of their dispersive Majorana gapless boundary modes.We also review the Z × (Z 2 ) 3 classification of 2D BdG Hamiltonians (class D) enriched by translational symmetry and provide arguments on its robust stability against interactions and self-averaging disorder that preserve translational symmetry.
Topologically ordered matter phases have been regarded as beyond the Landau-Ginzburg symmetry breaking paradigm of matter phases. Recent studies of anyon condensation in topological phases, however, may fit topological phases back in the Landau-Ginzburg paradigm. To truly do so, we realized that the string-net model of topological phases is in fact an effective lattice gauge theory coupled with anyonic matter once two modifications are made: (1) We reinterpret anyons as matter fields coupled to lattice gauge fields, thus extending the HGW model to a genuine Hamiltonian lattice gauge theory. (2) By explicitly incorporating the internal degrees of freedom of anyons, we construct an enlarged Hilbert space that supports well-defined gauge transformations and covariant coupling, restoring the analogy with conventional lattice gauge field theory. In this modified string-net model, topological phase transitions induced by anyon condensation and their consequent phenomena, such as order parameter fields, coherent states, Goldstone modes, and gapping gauge degrees of freedom, can be formulated exactly as Landau's effective theory of the Higgs mechanism. To facilitate the understanding, we also compare anyon condensation to/with the Higgs boson condensation in the electroweak theory and the Cooper pair condensation.
We prove, working in the Hamiltonian formalism, that a U(1) Chern-Simons theory coupled to fermions on a lattice can be mapped exactly onto a theory of interacting lattice anyons. This map does not involve any singular gauge transformations, and is everywhere well defined. We also prove that, when the statistics parameter is an odd integer so that the anyons are bosons, the ground state, which consists of a condensate of bound pairs of flux tubes and fermions, breaks phase invariance. The ensuing long range order implies that the system is an unconventional superfluid.