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:cond-mat/0410391 · Physical Review B

Mutual Chern-Simons effective theory of doped antiferromagnets

Su-Peng Kou, Xiao-Liang Qi, Zheng-Yu Weng

Published Jun 6, 2005• 47 citations
arXivHTMLPDF

Abstract

A mutual Chern-Simons Lagrangian is derived as a minimal field theory description of the phase-string model for doped antiferromagnets. Such an effective Lagrangian is shown to retain the full symmetries of parity, time reversal, and global SU(2) spin rotation, in contrast to conventional Chern-Simons theories where first two symmetries are usually broken. Two ordered phases---i.e., antiferromagnetic and superconducting states---are found at low temperatures as characterized by ``dual'' Meissner effects and dual-flux-quantization conditions due to the mutual Chern-Simons gauge structure. A ``dual'' confinement in charge and spin degrees of freedom occurs such that no true spin-charge separation is present in these ordered phases, but the spin-charge separation/deconfinement serves as a driving force in the unconventional phase transitions of these ordered states to disordered states.

DOI
10.1103/physrevb.71.235102

Related papers

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

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

Superconductivity in mutual Chern-Simons gauge theory

Peng Ye, Long Zhang, Zheng-Yu Weng

In this work, we present a topological characterization of superconductivity in a prototype electron fractionalization model for doped Mott insulators. In this model, spinons and holons are coupled via the mutual Chern-Simons gauge fields. We obtain a low-lying effective description of the collective current fluctuations by integrating out the matter fields, which replaces the conventional Ginzburg-Landau action to describe the generalized rigidity of superconductivity. The superconducting phase coherence is essentially characterized by a topological order parameter related to a Gaussian linking number, and an experiment is proposed to probe this topological property. We further show that a gauge-neutral fermionic mode can naturally emerge in this model, which behaves like a Bogoliubov quasiparticle.

arXiv:1110.0125•Physical Review B•May 29, 2012•8 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

Phase string effect and mutual Chern-Simons theory of Hubbard model

Zhang Long, Zheng-Yu Weng

The fermion sign plays a dominant role in Fermi liquid theory. However, in Mott insulators, the strong Coulomb interaction suppresses the charge fluctuations and eliminates the fermion signs due to electron permutation. In this article, we first review the phase string theory of the Hubbard model for a bipartite lattice, which unifies the Fermi liquid at weak coupling and the antiferromagnetic Mott insulator at strong coupling. We first derive the exact sign structure of the Hubbard model for an arbitrary Coulomb interaction U. In small U limit, the conventional fermion sign is restored, while at large U limit, it leads to the phase string sign structure of the t-J model. For half filling, we construct an electron fractionalization representation, in which chargons and spinons are coupled to each other via emergent mutual Chern-Simons gauge fields. The corresponding ground state ansatz and low energy effective theory capture the ground state phase diagram of the Hubbard model qualitatively. For weak coupling regime, the Fermi liquid quasiparticle is formed by the bound state of a chargon and a spinon, and the long range phase coherence is determined by the background spin correlation. The Mott transition can be realized either by forming the chargon gap or by condensing the background spinons.

Acta Physica Sinica•Jan 1, 2015•0 citations

U(1)×SU(2) Chern-Simons gauge theory of underdoped cuprate superconductors

P. A. Marchetti, Zhao-Bin Su, Lu Yu

The Chern-Simons bosonization with $\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$ gauge field is applied to the two-dimensional $t\ensuremath{-}J$ model in the limit $t\ensuremath{\gg}J,$ to study the normal-state properties of underdoped cuprate superconductors. We prove the existence of an upper bound on the partition function for holons in a spinon background, and we find the optimal spinon configuration saturating the upper bound on average---a coexisting flux phase and $s+id$-like resonating-valence-bond state. After neglecting the feedback of holon fluctuations on the U(1) field B and spinon fluctuations on the SU(2) field $V$, the holon field is a fermion and the spinon field is a hard-core boson. Within this approximation we show that the B field produces a \ensuremath{\pi} flux phase for the holons, converting them into Dirac-like fermions, while the $V$ field, taking into account the feedback of holons produces a gap for the spinons vanishing in the zero-doping limit. The nonlinear-\ensuremath{\sigma} model with a mass term describes the crossover from the short-ranged antiferromagnetic (AF) state in doped samples to long-range AF order in reference compounds. Moreover, we derive a low-energy effective action in terms of spinons, holons and a self-generated U(1) gauge field. Neglecting the gauge fluctuations, the holons are described by the Fermi-liquid theory with a Fermi surface consisting of four ``half-pockets'' centered at (\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2) and one reproduces the results for the electron spectral function obtained in the mean-field approximation, in agreement with the photoemission data on underdoped cuprates. The gauge fluctuations are not confining due to coupling to holons, but nevertheless yield an attractive interaction between spinons and holons leading to a bound state with electron quantum numbers. The renormalization effects due to gauge fluctuations give rise to non-Fermi-liquid behavior for the composite electron, in certain temperature range showing the linear in T resistivity. This formalism provides a new interpretation of the spin gap in the underdoped superconductors (mainly due to the short-ranged AF order) and predicts that the minimal gap for the physical electron is proportional to the square root of the doping concentration. Therefore the gap does not vanish in any direction. All these predictions can be checked explicitly in experiment.

arXiv:cond-mat/9805191•Physical review. B, Condensed matter•Sep 1, 1998•36 citations
A mutual Chern-Simons Lagrangian is derived as a minimal field theory description of the phase-string model for doped antiferromagnets. Such an effective Lagrangian is shown to retain the full symmetries of parity, time reversal, and global SU(2) spin rotation, in contrast to conventional Chern-Simons theories where first two symmetries are usually broken. Two ordered phases---i.e., antiferromagnetic and superconducting states---are found at low temperatures as characterized by dual'' Meissner effects and dual-flux-quantization conditions due to the mutual Chern-Simons gauge structure. A dual'' confinement in charge and spin degrees of freedom occurs such that no true spin-charge separation is present in these ordered phases, but the spin-charge separation/deconfinement serves as a driving force in the unconventional phase transitions of these ordered states to disordered states.
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.
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.
In this work, we present a topological characterization of superconductivity in a prototype electron fractionalization model for doped Mott insulators. In this model, spinons and holons are coupled via the mutual Chern-Simons gauge fields. We obtain a low-lying effective description of the collective current fluctuations by integrating out the matter fields, which replaces the conventional Ginzburg-Landau action to describe the generalized rigidity of superconductivity. The superconducting phase coherence is essentially characterized by a topological order parameter related to a Gaussian linking number, and an experiment is proposed to probe this topological property. We further show that a gauge-neutral fermionic mode can naturally emerge in this model, which behaves like a Bogoliubov quasiparticle.
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.
The fermion sign plays a dominant role in Fermi liquid theory. However, in Mott insulators, the strong Coulomb interaction suppresses the charge fluctuations and eliminates the fermion signs due to electron permutation. In this article, we first review the phase string theory of the Hubbard model for a bipartite lattice, which unifies the Fermi liquid at weak coupling and the antiferromagnetic Mott insulator at strong coupling. We first derive the exact sign structure of the Hubbard model for an arbitrary Coulomb interaction U. In small U limit, the conventional fermion sign is restored, while at large U limit, it leads to the phase string sign structure of the t-J model. For half filling, we construct an electron fractionalization representation, in which chargons and spinons are coupled to each other via emergent mutual Chern-Simons gauge fields. The corresponding ground state ansatz and low energy effective theory capture the ground state phase diagram of the Hubbard model qualitatively. For weak coupling regime, the Fermi liquid quasiparticle is formed by the bound state of a chargon and a spinon, and the long range phase coherence is determined by the background spin correlation. The Mott transition can be realized either by forming the chargon gap or by condensing the background spinons.
The Chern-Simons bosonization with $\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$ gauge field is applied to the two-dimensional $t\ensuremath{-}J$ model in the limit $t\ensuremath{\gg}J,$ to study the normal-state properties of underdoped cuprate superconductors. We prove the existence of an upper bound on the partition function for holons in a spinon background, and we find the optimal spinon configuration saturating the upper bound on average---a coexisting flux phase and s+ids+ids+id-like resonating-valence-bond state. After neglecting the feedback of holon fluctuations on the U(1) field B and spinon fluctuations on the SU(2) field VVV, the holon field is a fermion and the spinon field is a hard-core boson. Within this approximation we show that the B field produces a \ensuremath{\pi} flux phase for the holons, converting them into Dirac-like fermions, while the VVV field, taking into account the feedback of holons produces a gap for the spinons vanishing in the zero-doping limit. The nonlinear-\ensuremath{\sigma} model with a mass term describes the crossover from the short-ranged antiferromagnetic (AF) state in doped samples to long-range AF order in reference compounds. Moreover, we derive a low-energy effective action in terms of spinons, holons and a self-generated U(1) gauge field. Neglecting the gauge fluctuations, the holons are described by the Fermi-liquid theory with a Fermi surface consisting of four ``half-pockets'' centered at (\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2) and one reproduces the results for the electron spectral function obtained in the mean-field approximation, in agreement with the photoemission data on underdoped cuprates. The gauge fluctuations are not confining due to coupling to holons, but nevertheless yield an attractive interaction between spinons and holons leading to a bound state with electron quantum numbers. The renormalization effects due to gauge fluctuations give rise to non-Fermi-liquid behavior for the composite electron, in certain temperature range showing the linear in T resistivity. This formalism provides a new interpretation of the spin gap in the underdoped superconductors (mainly due to the short-ranged AF order) and predicts that the minimal gap for the physical electron is proportional to the square root of the doping concentration. Therefore the gap does not vanish in any direction. All these predictions can be checked explicitly in experiment.