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:1712.07808 · Physical review. D/Physical review. D.

Berezinskii-Kosterlitz-Thouless transition in lattice Schwinger model with one flavor of Wilson fermion

Yuya Shimizu, Y. Kuramashi

Published Feb 6, 2018• 65 citations
arXivHTMLPDF

Abstract

We have made a detailed study of the phase structure for the lattice Schwinger model with one flavor of Wilson fermion on the $(m,g)$ plane. For numerical investigation, we develop a decorated tensor renormalization method for lattice gauge theories with fermions incorporating the Grassmann tensor renormalization. Our algorithm manifestly preserves rotation and reflection symmetries. We find not only a parity-broken phase but also a Berezinskii-Kosterlitz-Thouless (BKT) transition by evaluating the central charge and an expectation value of a projection operator into the parity-odd subspace. The BKT phase boundaries converge into the degenerated doubler pole $(m,g)=(\ensuremath{-}2,0)$, while the parity-breaking transition line ends at the physical pole $(m,g)=(0,0)$. In addition, our analysis of scaling dimensions indicates that a conformal field theory with SU(2) symmetry arises on the line of $m=\ensuremath{-}2$.

DOI
10.1103/physrevd.97.034502

Related papers

Phase diagram of the single-flavor Gross-Neveu-Wilson model from the Grassmann corner transfer matrix renormalization group

Jian-Gang Kong, Shinichiro Akiyama, Tao Shi +1 more

We investigate the phase structure of the ( 1 + 1 )-dimensional single-flavor Gross–Neveu model with Wilson fermions using the Grassmann corner transfer matrix renormalization group (CTMRG). The path integral is formulated as a two-dimensional Grassmann tensor network and approximately contracted by the Grassmann CTMRG algorithm. We investigate the phase diagram by varying the fermion mass and the four-fermion coupling, using the pseudoscalar condensate as an order parameter for the Z 2 parity symmetry breaking phase. The universality classes of the phase boundaries are identified through the central charge c obtained via scaling analysis of the entanglement entropy. Furthermore, we extract the quantity related to the entanglement spectrum from the converged CTMRG environments, allowing us to distinguish the topological insulator phase and the trivial phase. The resulting phase structure suggests that the Aoki phase is separated from the other phases by critical lines characterized by c = 1 / 2 , while the critical lines with c = 1 separate the topological insulating and trivial phases. Our numerical results also indicate that the Aoki phase does not persist in the strong-coupling regime for the single-flavor theory.

arXiv:2602.21705•Physical review. D/Physical review. D.•Apr 27, 2026•0 citations

Duality, Gauge Symmetries, Renormalization Groups and the BKT Transition

Jorge V. José

In this chapter, I will briefly review, from my own perspective, the situation within theoretical physics at the beginning of the 1970s, and the advances that played an important role in providing a solid theoretical and experimental foundation for the Berezinskii–Kosterlitz–Thouless theory (BKT). Over this period, it became clear that the Abelian gauge symmetry of the 2D-XY model had to be preserved to get the right phase structure of the model. In previous analyses, this symmetry was broken when using low order calculational approximations. Duality transformations at that time for two-dimensional models with compact gauge symmetries were introduced by José, Kadanoff, Nelson and Kirkpatrick (JKKN). Their goal was to analyze the phase structure and excitations of XY and related models, including symmetry breaking fields which are experimentally important. In a separate context, Migdal had earlier developed an approximate Renormalization Group (RG) algorithm to implement Wilson’s RG for lattice gauge theories. Although Migdal’s RG approach, later extended by Kadanoff, did not produce a true phase transition for the XY model, it almost did asymptotically in terms of a non-perturbative expansion in the coupling constant with an essential singularity. Using these advances, including work done on instantons (vortices), JKKN analyzed the behavior of the spin–spin correlation functions of the 2D XY-model in terms of an expansion in temperature and vortex-pair fugacity. Their analysis led to a perturbative derivation of RG equations for the XY model which are the same as those first derived by Kosterlitz for the two-dimensional Coulomb gas. JKKN’s results gave a theoretical formulation foundation and justification for BKT’s sound physical assumptions and for the validity of their calculational approximations that were, in principle, strictly valid only at very low temperatures, away from the critical [Formula: see text] temperature. The theoretical predictions were soon tested successfully against experimental results on superfluid helium films. The success of the BKT theory also gave one of the first quantitative proofs of the validity of the RG theory.

International Journal of Modern Physics B•Jan 20, 2017•4 citations

Quantum Berezinskii-Kosterltz-Thouless Transition for Topological Insulator : A Detail Study of Renormalization Group Theory

Ranjith Kumar R, S Rahul, Surya Narayan +1 more

We present extensive calculation and the results of quantum Berezinskii-Kosterlitz-Thouless (BKT) transition for interacting helical liquid system. This system shows the quantum BKT transition for the different physical situation which we present by two model Hamiltonians. We derive the renormalization group (RG) equations explicitly and present the flow lines behavior. We also present RG flow lines based on the exact solution. We observe that the Majorana fermion zero modes physics and the gapped Ising phase of the system. We also observe that the evidence of gapless Luttinger liquid phase as a common phase for both quantum BKT transitions.

•Oct 2, 2018•0 citations

Berezinskii-Kosterlitz-Thouless quantum transition in 2 dimensions

M. C. Diamantini, Carlo A. Trugenberger, V. M. Vinokur

The Berezinskii-Kosterlitz-Thouless (BKT) transition is the prototype of a phase transition driven by the formation and interaction of topological defects in two-dimensional (2D) systems. In typical models these are vortices: above a transition temperature $T_{\rm BKT}$ vortices are free, below this transition temperature they get confined. In this work we extend the concept of BKT transition to quantum systems in two dimensions. In particular, we demonstrate that a zero-temperature quantum BKT phase transition, driven by a coupling constant can occur in 2D models governed by an effective gauge field theory with a diverging dielectric constant. One particular example is that of a compact U(1) gauge theory with a diverging dielectric constant, where the quantum BKT transition is induced by non-relativistic, purely 2D magnetic monopoles, which can be viewed also as electric vortices. These quantum BKT transitions have the same diverging exponent $z$ as the quantum Griffiths transition but have nothing to do with disorder.

arXiv:2510.06682•Oct 8, 2025•0 citations

Berezinskii–Kosterlitz–Thouless Quantum Transition in Two Dimensions

M. C. Diamantini, Carlo A. Trugenberger, V. M. Vinokur

The Berezinskii–Kosterlitz–Thouless (BKT) transition is the prototype of a phase transition driven by the formation and interaction of topological defects in two-dimensional (2D) systems. In typical models, these are vortices: above a transition temperature TBKT, vortices are free; below this transition temperature, they get confined. In this work, we extend the concept of BKT transition to quantum systems in two dimensions. In particular, we demonstrate that a zero-temperature quantum BKT phase transition driven by a coupling constant can occur in 2D models governed by an effective gauge field theory with a diverging dielectric constant. One particular example is that of a compact U(1) gauge theory with a diverging dielectric constant, where the quantum BKT transition is induced by non-relativistic, purely 2D magnetic monopoles, which can be viewed also as electric vortices. These quantum BKT transitions have the same diverging exponent z as the quantum Griffiths transition but are not related to disorder.

Materials•Feb 26, 2026•2 citations

Fermion discretization effects in the two-flavor lattice Schwinger model: A study with matrix product states

Tim Schwägerl, Karl Jansen, Stefan Kühn

We present a comprehensive tensor network study of staggered, Wilson, and twisted mass fermions in the Hamiltonian formulation, using the massive two-flavor Schwinger model as a benchmark. Particular emphasis is placed on twisted mass fermions, whose properties in this context have not been systematically explored before. We confirm the expected O ( a ) improvement in the free theory and observe that this improvement persists in the interacting case. By leveraging an electric-field-based method for mass renormalization, we reliably tune to maximal twist and establish the method’s applicability in the two-flavor model. Once mass renormalization is included, the pion mass exhibits rapid convergence to the continuum limit. Finite-volume effects are addressed using two complementary approaches: dispersion relation fits and finite-volume scaling. Our results show excellent agreement with semiclassical predictions and reveal a milder volume dependence for twisted mass fermions compared to staggered and Wilson discretizations. In addition, we observe clear isospin-breaking effects, suggesting intriguing parallels with lattice QCD. These findings highlight the advantages of twisted mass fermions for Hamiltonian simulations and motivate their further exploration—particularly in view of future applications to higher-dimensional lattice gauge theories.

arXiv:2509.02329•Physical review. D/Physical review. D.•Oct 16, 2025•1 citation
We have made a detailed study of the phase structure for the lattice Schwinger model with one flavor of Wilson fermion on the (m,g)(m,g)(m,g) plane. For numerical investigation, we develop a decorated tensor renormalization method for lattice gauge theories with fermions incorporating the Grassmann tensor renormalization. Our algorithm manifestly preserves rotation and reflection symmetries. We find not only a parity-broken phase but also a Berezinskii-Kosterlitz-Thouless (BKT) transition by evaluating the central charge and an expectation value of a projection operator into the parity-odd subspace. The BKT phase boundaries converge into the degenerated doubler pole (m,g)=(\ensuremath−2,0)(m,g)=(\ensuremath{-}2,0)(m,g)=(\ensuremath−2,0), while the parity-breaking transition line ends at the physical pole (m,g)=(0,0)(m,g)=(0,0)(m,g)=(0,0). In addition, our analysis of scaling dimensions indicates that a conformal field theory with SU(2) symmetry arises on the line of m=\ensuremath−2m=\ensuremath{-}2m=\ensuremath−2.
We investigate the phase structure of the ( 1 + 1 )-dimensional single-flavor Gross–Neveu model with Wilson fermions using the Grassmann corner transfer matrix renormalization group (CTMRG). The path integral is formulated as a two-dimensional Grassmann tensor network and approximately contracted by the Grassmann CTMRG algorithm. We investigate the phase diagram by varying the fermion mass and the four-fermion coupling, using the pseudoscalar condensate as an order parameter for the Z 2 parity symmetry breaking phase. The universality classes of the phase boundaries are identified through the central charge c obtained via scaling analysis of the entanglement entropy. Furthermore, we extract the quantity related to the entanglement spectrum from the converged CTMRG environments, allowing us to distinguish the topological insulator phase and the trivial phase. The resulting phase structure suggests that the Aoki phase is separated from the other phases by critical lines characterized by c = 1 / 2 , while the critical lines with c = 1 separate the topological insulating and trivial phases. Our numerical results also indicate that the Aoki phase does not persist in the strong-coupling regime for the single-flavor theory.
In this chapter, I will briefly review, from my own perspective, the situation within theoretical physics at the beginning of the 1970s, and the advances that played an important role in providing a solid theoretical and experimental foundation for the Berezinskii–Kosterlitz–Thouless theory (BKT). Over this period, it became clear that the Abelian gauge symmetry of the 2D-XY model had to be preserved to get the right phase structure of the model. In previous analyses, this symmetry was broken when using low order calculational approximations. Duality transformations at that time for two-dimensional models with compact gauge symmetries were introduced by José, Kadanoff, Nelson and Kirkpatrick (JKKN). Their goal was to analyze the phase structure and excitations of XY and related models, including symmetry breaking fields which are experimentally important. In a separate context, Migdal had earlier developed an approximate Renormalization Group (RG) algorithm to implement Wilson’s RG for lattice gauge theories. Although Migdal’s RG approach, later extended by Kadanoff, did not produce a true phase transition for the XY model, it almost did asymptotically in terms of a non-perturbative expansion in the coupling constant with an essential singularity. Using these advances, including work done on instantons (vortices), JKKN analyzed the behavior of the spin–spin correlation functions of the 2D XY-model in terms of an expansion in temperature and vortex-pair fugacity. Their analysis led to a perturbative derivation of RG equations for the XY model which are the same as those first derived by Kosterlitz for the two-dimensional Coulomb gas. JKKN’s results gave a theoretical formulation foundation and justification for BKT’s sound physical assumptions and for the validity of their calculational approximations that were, in principle, strictly valid only at very low temperatures, away from the critical [Formula: see text] temperature. The theoretical predictions were soon tested successfully against experimental results on superfluid helium films. The success of the BKT theory also gave one of the first quantitative proofs of the validity of the RG theory.
We present extensive calculation and the results of quantum Berezinskii-Kosterlitz-Thouless (BKT) transition for interacting helical liquid system. This system shows the quantum BKT transition for the different physical situation which we present by two model Hamiltonians. We derive the renormalization group (RG) equations explicitly and present the flow lines behavior. We also present RG flow lines based on the exact solution. We observe that the Majorana fermion zero modes physics and the gapped Ising phase of the system. We also observe that the evidence of gapless Luttinger liquid phase as a common phase for both quantum BKT transitions.
The Berezinskii-Kosterlitz-Thouless (BKT) transition is the prototype of a phase transition driven by the formation and interaction of topological defects in two-dimensional (2D) systems. In typical models these are vortices: above a transition temperature TBKTT_{\rm BKT}TBKT​ vortices are free, below this transition temperature they get confined. In this work we extend the concept of BKT transition to quantum systems in two dimensions. In particular, we demonstrate that a zero-temperature quantum BKT phase transition, driven by a coupling constant can occur in 2D models governed by an effective gauge field theory with a diverging dielectric constant. One particular example is that of a compact U(1) gauge theory with a diverging dielectric constant, where the quantum BKT transition is induced by non-relativistic, purely 2D magnetic monopoles, which can be viewed also as electric vortices. These quantum BKT transitions have the same diverging exponent zzz as the quantum Griffiths transition but have nothing to do with disorder.
The Berezinskii–Kosterlitz–Thouless (BKT) transition is the prototype of a phase transition driven by the formation and interaction of topological defects in two-dimensional (2D) systems. In typical models, these are vortices: above a transition temperature TBKT, vortices are free; below this transition temperature, they get confined. In this work, we extend the concept of BKT transition to quantum systems in two dimensions. In particular, we demonstrate that a zero-temperature quantum BKT phase transition driven by a coupling constant can occur in 2D models governed by an effective gauge field theory with a diverging dielectric constant. One particular example is that of a compact U(1) gauge theory with a diverging dielectric constant, where the quantum BKT transition is induced by non-relativistic, purely 2D magnetic monopoles, which can be viewed also as electric vortices. These quantum BKT transitions have the same diverging exponent z as the quantum Griffiths transition but are not related to disorder.
We present a comprehensive tensor network study of staggered, Wilson, and twisted mass fermions in the Hamiltonian formulation, using the massive two-flavor Schwinger model as a benchmark. Particular emphasis is placed on twisted mass fermions, whose properties in this context have not been systematically explored before. We confirm the expected O ( a ) improvement in the free theory and observe that this improvement persists in the interacting case. By leveraging an electric-field-based method for mass renormalization, we reliably tune to maximal twist and establish the method’s applicability in the two-flavor model. Once mass renormalization is included, the pion mass exhibits rapid convergence to the continuum limit. Finite-volume effects are addressed using two complementary approaches: dispersion relation fits and finite-volume scaling. Our results show excellent agreement with semiclassical predictions and reveal a milder volume dependence for twisted mass fermions compared to staggered and Wilson discretizations. In addition, we observe clear isospin-breaking effects, suggesting intriguing parallels with lattice QCD. These findings highlight the advantages of twisted mass fermions for Hamiltonian simulations and motivate their further exploration—particularly in view of future applications to higher-dimensional lattice gauge theories.