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:2208.11735 · Physical review. D/Physical review. D.

Hartree-Fock approach to dynamical mass generation in the generalized ( 2+1 )-dimensional Thirring model

T. G. Khunjua, К. Г. Клименко, Roman N. Zhokhov

Published Oct 10, 2022• 3 citations
arXivHTMLPDF

Abstract

The ($2+1$)-dimensional generalized massless Thirring model with four-component Fermi fields is investigated by the Hartree-Fock method. The Lagrangian of this model is constructed from two different four-fermion structures. One of them takes into account the vector $\ifmmode\times\else\texttimes\fi{}$ vector channel of fermion interaction with coupling constant ${G}_{v}$, the other, the scalar $\ifmmode\times\else\texttimes\fi{}$ scalar channel with coupling ${G}_{s}$. At some relation between bare couplings ${G}_{s}$ and ${G}_{v}$, the Hartree-Fock equation for self-energy of fermions can be renormalized, and dynamical generation of the Dirac and Haldane fermion masses is possible. As a result, a phase portrait of the model consists of two nontrivial phases. In the first one the chiral symmetry is spontaneously broken due to dynamical appearance of the Dirac mass term, while in the second phase a spontaneous breaking of the spatial parity $\mathcal{P}$ is induced by a Haldane mass term. It is shown that in the particular case of a pure Thirring model, i.e., at ${G}_{s}=0$, the ground state of the system is indeed a mixture of these phases. Moreover, it was found that dynamical generation of fermion masses is possible for any finite number of Fermi fields.

DOI
10.1103/physrevd.106.085002

Related papers

Itinerant quantum multicriticality of two-dimensional Dirac fermions

Bitan Roy, Pallab Goswami, Vladimir Juričić

We analyze emergent quantum multicriticality for strongly interacting, massless Dirac fermions in two spatial dimensions ($d=2$) within the framework of Gross-Neveu-Yukawa models, by considering the competing order parameters that give rise to fully gapped (insulating or superconducting) ground states. We focus only on those competing orders which can be rotated into each other by generators of an exact or emergent chiral symmetry of massless Dirac fermions, and break $\text{O}({S}_{1})$ and $\text{O}({S}_{2})$ symmetries in the ordered phase. Performing a renormalization-group analysis by using the $\ensuremath{\epsilon}=(3\ensuremath{-}d)$ expansion scheme, we show that all the coupling constants in the critical hyperplane flow toward a new attractive fixed point, supporting an enlarged $\text{O}({S}_{1}+{S}_{2})$ chiral symmetry. Such a fixed point acts as an exotic quantum multicritical point (MCP), governing the continuous semimetal-insulator as well as insulator-insulator (for example, antiferromagnet to valence bond solid) quantum phase transitions. In comparison with the lower symmetric semimetal-insulator quantum critical points, possessing either $\text{O}({S}_{1})$ or $\text{O}({S}_{2})$ chiral symmetry, the MCP displays enhanced correlation length exponents, and anomalous scaling dimensions for both fermionic and bosonic fields. We discuss the scaling properties of the ratio of bosonic and fermionic masses, and the increased dc resistivity at the MCP. By computing the scaling dimensions of different local fermion bilinears in the particle-hole channel, we establish that most of the four fermion operators or generalized density-density correlation functions display faster power-law decays at the MCP compared to the free fermion and lower symmetric itinerant quantum critical points. Possible generalization of this scenario to higher-dimensional Dirac fermions is also outlined.

arXiv:1712.05400•Physical review. B./Physical review. B•May 11, 2018•22 citations

Quantum Monte Carlo Simulation of the Chiral Heisenberg Gross-Neveu-Yukawa Phase Transition with a Single Dirac Cone

Thomas C. Lang, Andreas Martin Läuchli

We present quantum Monte Carlo simulations for the chiral Heisenberg Gross-Neveu-Yukawa quantum phase transition of relativistic fermions with N=4 Dirac spinor components subject to a repulsive, local four fermion interaction in (2+1)D. Here we employ a two-dimensional lattice Hamiltonian with a single, spin-degenerate Dirac cone, which exactly reproduces a linear energy-momentum relation for all finite size lattice momenta in the absence of interactions. This allows us to significantly reduce finite size corrections compared to the widely studied honeycomb and π-flux lattices. A Hubbard term dynamically generates a mass beyond a critical coupling of U_{c}=6.76(1) as the system acquires antiferromagnetic order and SU(2) spin rotational symmetry is spontaneously broken. At the quantum phase transition, we extract a self-consistent set of critical exponents ν=0.98(1), η_{ϕ}=0.53(1), η_{ψ}=0.18(1), and β=0.75(1). We provide evidence for the continuous degradation of the quasiparticle weight of the fermionic excitations as the critical point is approached from the semimetallic phase. Finally, we study the effective "speed of light" of the low-energy relativistic description, which depends on the interaction U, but is expected to be regular across the quantum phase transition. We illustrate that the strongly coupled bosonic and fermionic excitations share a common velocity at the critical point.

arXiv:1808.01230•Physical Review Letters•Sep 25, 2019•73 citations

Supersymmetry breaking and Nambu-Goldstone fermions in an extended Nicolai model

Noriaki Sannomiya, Hosho Katsura, Yu Nakayama

We study a model of interacting spinless fermions in a one dimensional lattice with supersymmetry (SUSY). The Hamiltonian is given by the anticommutator of two supercharges $Q$ and ${Q}^{\ifmmode\dagger\else\textdagger\fi{}}$, each of which is comprised solely of fermion operators and possesses one adjustable parameter $g$. When the parameter $g$ vanishes, the model is identical to the one studied by Nicolai [H. Nicolai, J. Phys. A 9, 1497 (1976)], where the zero-energy ground state is exponentially degenerate. On the other hand, in the large-$g$ limit the model reduces to the free-fermion chain with a four-fold degenerate ground state. We show that for finite chains SUSY is spontaneously broken when $g>0$. We also rigorously prove that for sufficiently large $g$ the ground-state energy density is nonvanishing in the infinite-volume limit. We further analyze the nature of the low-energy excitations by employing various techniques such as rigorous inequalities, exact numerical diagonalization, and renormalization group method with bosonization. The analysis reveals that the low-energy excitations are described by massless Dirac fermions (or Thirring fermions more generally), which can be thought of as Nambu-Goldstone fermions from the spontaneous SUSY breaking.

arXiv:1606.03947•Physical review. D/Physical review. D.•Aug 25, 2016•17 citations

Symmetric mass generation of interacting chiral fermions on a one-dimensional lattice without fermion doubling

V. A. Zakharov, Atsushi Ueda, Frank Verstraete +1 more

Symmetric mass generation is the interaction-induced opening of a fermion gap without spontaneous symmetry breaking. The anomaly-free 3-4-5-0 model of Wang and Wen provides a minimal one-dimensional setting for this phenomenon, but a direct lattice realization faces two obstacles: fermion doubling for local chiral discretizations and perturbative irrelevance of the six-fermion gapping interaction. We address both obstacles. First, we formulate the model on a strictly one-dimensional tangent-fermion lattice, where a nonlocal hopping produces a single chiral branch without a mirror partner while retaining an efficient tensor-network representation. Second, we add a Hubbard-type density-density interaction (Luttinger parameter $K$) that reduces the scaling dimension of the 3-4-5-0 interaction from $5$ to $5K$, making it relevant for $K<2/5$. Density-matrix renormalization group calculations show the opening of an excitation gap in this regime without the appearance of a degenerate ground state, the hallmark of symmetric mass generation.

hep-thCross-listedarXiv:2606.24713•Physical Review Research•Jun 24, 2026•0 citations

Dynamic mass generation and topological order in overscreened Kondo lattices

Ge Yang, Yashar Komijani

Multichannel Kondo lattice models are examples of strongly correlated electronic systems that exhibit non-Fermi-liquid behavior due to the presence of a continuous channel symmetry. Mean-field analyses have predicted that these systems undergo channel symmetry breaking at low temperature. We use the dynamical large- N technique to study temporal and spatial fluctuations of the multichannel Kondo model on a honeycomb lattice and find that this prediction is not generally true. Rather, we find a ( 2+1 )-dimensional conformally invariant fixed point, governed by critical exponents that are found numerically. When we break time-reversal symmetry by adding a Haldane mass to the conduction electrons, three phases, separated by continuous transitions, are discernible: one characterized by dynamic mass generation and spontaneous breaking of the channel symmetry, one where topological defects restore channel symmetry but preserve the gap, and one with a Kondo-coupled chiral spin liquid. We argue that the last phase is a fractional Chern insulator with anyonic excitations. Published by the American Physical Society 2024

arXiv:2303.03356•Physical Review Research•Mar 6, 2024•6 citations

Tensorial Gross-Neveu models

Dario Benedetti, Sylvain Carrozza, Razvan Gurău +1 more

We define and study various tensorial generalizations of the Gross-Neveu model in two dimensions, that is, models with four-fermion interactions and G3 symmetry, where we take either G = U(N) or G = O(N). Such models can also be viewed as two-dimensional generalizations of the Sachdev-Ye-Kitaev model, or more precisely of its tensorial counterpart introduced by Klebanov and Tarnopolsky, which is in part our motivation for studying them. Using the Schwinger-Dyson equations at large-N, we discuss the phenomenon of dynamical mass generation and possible combinations of couplings to avoid it. For the case G = U(N),we introduce an intermediate field representation and perform a stability analysis of the vacua. It turns out that the only apparently viable combination of couplings that avoids mass generation corresponds to an unstable vacuum. The stable vacuum breaks U(N)3 invariance, in contradiction with the Coleman-Mermin-Wagner theorem, but this is an artifact of the large-N expansion, similar to the breaking of continuous chiral symmetry in the chiral Gross-Neveu model.

arXiv:1710.10253•Journal of High Energy Physics•Jan 1, 2018•41 citations
The (2+12+12+1)-dimensional generalized massless Thirring model with four-component Fermi fields is investigated by the Hartree-Fock method. The Lagrangian of this model is constructed from two different four-fermion structures. One of them takes into account the vector \ifmmode×\else\texttimes\fi\ifmmode\times\else\texttimes\fi{}\ifmmode×\else\texttimes\fi vector channel of fermion interaction with coupling constant Gv{G}_{v}Gv​, the other, the scalar \ifmmode×\else\texttimes\fi\ifmmode\times\else\texttimes\fi{}\ifmmode×\else\texttimes\fi scalar channel with coupling Gs{G}_{s}Gs​. At some relation between bare couplings Gs{G}_{s}Gs​ and Gv{G}_{v}Gv​, the Hartree-Fock equation for self-energy of fermions can be renormalized, and dynamical generation of the Dirac and Haldane fermion masses is possible. As a result, a phase portrait of the model consists of two nontrivial phases. In the first one the chiral symmetry is spontaneously broken due to dynamical appearance of the Dirac mass term, while in the second phase a spontaneous breaking of the spatial parity P\mathcal{P}P is induced by a Haldane mass term. It is shown that in the particular case of a pure Thirring model, i.e., at Gs=0{G}_{s}=0Gs​=0, the ground state of the system is indeed a mixture of these phases. Moreover, it was found that dynamical generation of fermion masses is possible for any finite number of Fermi fields.
We analyze emergent quantum multicriticality for strongly interacting, massless Dirac fermions in two spatial dimensions (d=2d=2d=2) within the framework of Gross-Neveu-Yukawa models, by considering the competing order parameters that give rise to fully gapped (insulating or superconducting) ground states. We focus only on those competing orders which can be rotated into each other by generators of an exact or emergent chiral symmetry of massless Dirac fermions, and break O(S1)\text{O}({S}_{1})O(S1​) and O(S2)\text{O}({S}_{2})O(S2​) symmetries in the ordered phase. Performing a renormalization-group analysis by using the $\ensuremath{\epsilon}=(3\ensuremath{-}d)$ expansion scheme, we show that all the coupling constants in the critical hyperplane flow toward a new attractive fixed point, supporting an enlarged O(S1+S2)\text{O}({S}_{1}+{S}_{2})O(S1​+S2​) chiral symmetry. Such a fixed point acts as an exotic quantum multicritical point (MCP), governing the continuous semimetal-insulator as well as insulator-insulator (for example, antiferromagnet to valence bond solid) quantum phase transitions. In comparison with the lower symmetric semimetal-insulator quantum critical points, possessing either O(S1)\text{O}({S}_{1})O(S1​) or O(S2)\text{O}({S}_{2})O(S2​) chiral symmetry, the MCP displays enhanced correlation length exponents, and anomalous scaling dimensions for both fermionic and bosonic fields. We discuss the scaling properties of the ratio of bosonic and fermionic masses, and the increased dc resistivity at the MCP. By computing the scaling dimensions of different local fermion bilinears in the particle-hole channel, we establish that most of the four fermion operators or generalized density-density correlation functions display faster power-law decays at the MCP compared to the free fermion and lower symmetric itinerant quantum critical points. Possible generalization of this scenario to higher-dimensional Dirac fermions is also outlined.
We present quantum Monte Carlo simulations for the chiral Heisenberg Gross-Neveu-Yukawa quantum phase transition of relativistic fermions with N=4 Dirac spinor components subject to a repulsive, local four fermion interaction in (2+1)D. Here we employ a two-dimensional lattice Hamiltonian with a single, spin-degenerate Dirac cone, which exactly reproduces a linear energy-momentum relation for all finite size lattice momenta in the absence of interactions. This allows us to significantly reduce finite size corrections compared to the widely studied honeycomb and π-flux lattices. A Hubbard term dynamically generates a mass beyond a critical coupling of U_{c}=6.76(1) as the system acquires antiferromagnetic order and SU(2) spin rotational symmetry is spontaneously broken. At the quantum phase transition, we extract a self-consistent set of critical exponents ν=0.98(1), η_{ϕ}=0.53(1), η_{ψ}=0.18(1), and β=0.75(1). We provide evidence for the continuous degradation of the quasiparticle weight of the fermionic excitations as the critical point is approached from the semimetallic phase. Finally, we study the effective "speed of light" of the low-energy relativistic description, which depends on the interaction U, but is expected to be regular across the quantum phase transition. We illustrate that the strongly coupled bosonic and fermionic excitations share a common velocity at the critical point.
We study a model of interacting spinless fermions in a one dimensional lattice with supersymmetry (SUSY). The Hamiltonian is given by the anticommutator of two supercharges QQQ and ${Q}^{\ifmmode\dagger\else\textdagger\fi{}}$, each of which is comprised solely of fermion operators and possesses one adjustable parameter ggg. When the parameter ggg vanishes, the model is identical to the one studied by Nicolai [H. Nicolai, J. Phys. A 9, 1497 (1976)], where the zero-energy ground state is exponentially degenerate. On the other hand, in the large-ggg limit the model reduces to the free-fermion chain with a four-fold degenerate ground state. We show that for finite chains SUSY is spontaneously broken when g>0g>0g>0. We also rigorously prove that for sufficiently large ggg the ground-state energy density is nonvanishing in the infinite-volume limit. We further analyze the nature of the low-energy excitations by employing various techniques such as rigorous inequalities, exact numerical diagonalization, and renormalization group method with bosonization. The analysis reveals that the low-energy excitations are described by massless Dirac fermions (or Thirring fermions more generally), which can be thought of as Nambu-Goldstone fermions from the spontaneous SUSY breaking.
Symmetric mass generation is the interaction-induced opening of a fermion gap without spontaneous symmetry breaking. The anomaly-free 3-4-5-0 model of Wang and Wen provides a minimal one-dimensional setting for this phenomenon, but a direct lattice realization faces two obstacles: fermion doubling for local chiral discretizations and perturbative irrelevance of the six-fermion gapping interaction. We address both obstacles. First, we formulate the model on a strictly one-dimensional tangent-fermion lattice, where a nonlocal hopping produces a single chiral branch without a mirror partner while retaining an efficient tensor-network representation. Second, we add a Hubbard-type density-density interaction (Luttinger parameter KKK) that reduces the scaling dimension of the 3-4-5-0 interaction from 555 to 5K5K5K, making it relevant for K<2/5K<2/5K<2/5. Density-matrix renormalization group calculations show the opening of an excitation gap in this regime without the appearance of a degenerate ground state, the hallmark of symmetric mass generation.
Multichannel Kondo lattice models are examples of strongly correlated electronic systems that exhibit non-Fermi-liquid behavior due to the presence of a continuous channel symmetry. Mean-field analyses have predicted that these systems undergo channel symmetry breaking at low temperature. We use the dynamical large- N technique to study temporal and spatial fluctuations of the multichannel Kondo model on a honeycomb lattice and find that this prediction is not generally true. Rather, we find a ( 2+1 )-dimensional conformally invariant fixed point, governed by critical exponents that are found numerically. When we break time-reversal symmetry by adding a Haldane mass to the conduction electrons, three phases, separated by continuous transitions, are discernible: one characterized by dynamic mass generation and spontaneous breaking of the channel symmetry, one where topological defects restore channel symmetry but preserve the gap, and one with a Kondo-coupled chiral spin liquid. We argue that the last phase is a fractional Chern insulator with anyonic excitations. Published by the American Physical Society 2024
We define and study various tensorial generalizations of the Gross-Neveu model in two dimensions, that is, models with four-fermion interactions and G3 symmetry, where we take either G = U(N) or G = O(N). Such models can also be viewed as two-dimensional generalizations of the Sachdev-Ye-Kitaev model, or more precisely of its tensorial counterpart introduced by Klebanov and Tarnopolsky, which is in part our motivation for studying them. Using the Schwinger-Dyson equations at large-N, we discuss the phenomenon of dynamical mass generation and possible combinations of couplings to avoid it. For the case G = U(N),we introduce an intermediate field representation and perform a stability analysis of the vacua. It turns out that the only apparently viable combination of couplings that avoids mass generation corresponds to an unstable vacuum. The stable vacuum breaks U(N)3 invariance, in contradiction with the Coleman-Mermin-Wagner theorem, but this is an artifact of the large-N expansion, similar to the breaking of continuous chiral symmetry in the chiral Gross-Neveu model.