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:2205.00835

$π$ Flux Phase and Superconductivity for Lattice Fermions Coupled to Classical Gauge Fields

Tohru Koma

Published May 2, 2022• 0 citations
arXivHTMLPDF

Abstract

We study superconducting lattice fermions coupled to classical gauge fields. Namely, without the gauge fields, the lattice fermions show superconducting long-range order. In the previous paper, the existence of the long-range order was proved by the present author. This paper is the continuing part of the previous one. More precisely, the interactions between fermions were assumed to be a Bardeen-Cooper-Schrieffer-type pairing which is a nearest-neighbour two-body interaction on the hypercubic lattice $\mathbb{Z}^d$ with the dimension $d\ge 3$. In the present paper, we deal with the corresponding gauged model by introducing classical U(1) gauge fields. We prove that a certain configuration of gauge fields which yields the $π$ flux for all the plaquettes in the lattice minimizes the ground-state energy of the fermion system. Since this configuration of the gauge fields exactly coincides with the hopping amplitudes of the Hamiltonian of the previous paper, the ground state of the whole system shows superconducting long-range order. Any configuration of gauge fields which is gauge-equivalent to this special configuration of the gauge fields also yields another ground state. However, if the Cooper pair correlations are averaged over the gauge-equivalent class, then they are vanishing except for the on-site correlation because the Cooper pair correlations are not necessarily gauge invariant. On the other hand, the string Cooper pair correlations which are gauge invariant always show the same long-range order for any configuration of gauge fields in the gauge-equivalent class.

DOI
10.48550/arxiv.2205.00835

Related papers

Decay of superconducting correlations for gauged electrons in dimensions D ≤ 4

Yasuhiro Tada, Tohru Koma

We study lattice superconductors coupled to gauge fields, such as an attractive Hubbard model in electromagnetic fields, with a standard gauge fixing. We prove upper bounds for a two-point Cooper pair correlation at finite temperatures in spatial dimensions D ≤ 4. The upper bounds decay exponentially in three dimensions and by power law in four dimensions. These imply the absence of the superconducting long-range order for the Cooper pair amplitude as a consequence of fluctuations of the gauge fields. Since our results hold for the gauge fixing Hamiltonian, they cannot be obtained as a corollary of Elitzur’s theorem.

arXiv:1612.00619•Journal of Mathematical Physics•Mar 1, 2018•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

Superconductivity and low energy excitations in an attractive Hubbard model

Yukimi Goto, Tohru Koma, Hironobu Yoshida

Abstract We study an attractive Hubbard model on bipartite lattices. In the grand canonical formalism, we prove the existence of superconducting long-range order in the ground state on Lieb lattices with the chemical potential corresponding to half filling. We also study the low-energy excitations above several ground states for the translationally invariant Hamiltonian. We prove the following: (i) The pairing excitations are gapless above the ground states when the number of fermions deviates from that of the half filling by the order of the volume. (ii) A certain class of single-fermion excitations shows a non-vanishing spectral gap above the ground state with an even number of fermions in a strong coupling and low-density regime.

arXiv:2509.19780•Physica Scripta•Jul 21, 2026•0 citations

Stability of the $$\pi $$-Flux Phase for $$\mathbb {Z}_{2}$$ Lattice Gauge Theory Coupled to Fermionic Matter

Leonardo Goller, Marcello Porta

Abstract We consider the two-dimensional $$\mathbbm {Z}_{2}$$ Z 2 lattice gauge theory coupled to fermionic matter. In absence of electric fields, we prove that, at half-filling, the ground state of the gauge theory coincides with the $$\pi $$ π -flux phase, associated with magnetic flux equal to $$\pi $$ π in every elementary lattice plaquette, provided the fermionic hopping is large enough. This proves in particular the semimetallic behavior of the ground state of the model. Furthermore, we compute the magnetic susceptibility of the gauge theory, and we prove that it is given by the one of massless 2 d Dirac fermions, thus rigorously justifying recent numerical simulations. The proof is based on reflection positivity and chessboard estimates, and on lattice conservation laws for the computation of the transport coefficient.

arXiv:2501.10065•Journal of Statistical Physics•May 22, 2026•0 citations

Cooper pair insulators and theory of correlated superconductors

Predrag Nikolić, Zlatko Tešanović

The pseudogap state of cuprate high-temperature superconductors has been often viewed as either a yet unknown competing order or a precursor state to superconductivity. While awaiting the resolution of the pseudogap problem in cuprates, we demonstrate that local pairing fluctuations, vortex liquid dynamics, and other precursor phenomena can emerge quite generally whenever fermionic excitations remain gapped across the superconducting transition, regardless of the gap origin. Our choice of a tractable model is a lattice band insulator with short-range attractive interactions between fermions in the $s$-wave channel. An effective crossover between Bardeen-Cooper-Schrieffer (BCS) and Bose-Einstein condensate (BEC) regimes can be identified in any band insulator above two dimensions, while in two dimensions only the BEC regime exists. The superconducting transition is ``unconventional'' (non-pair-breaking) in the BEC regime, identified by either the bosonic mean field or $\mathit{XY}$ universality class. The insulator adjacent to the superconductor in the BEC regime is a bosonic Mott insulator of Cooper pairs, which may be susceptible to charge density wave ordering. We construct a function of the many-body excitation spectrum whose nonanalytic changes define a sharp distinction between band and Mott insulators. The corresponding ``second-order transition'' can be observed out of equilibrium by driving a Cooper pair laser in the Mott insulator. We explicitly show that the gap for charged bosonic excitations lies below the threshold for Cooper pair breakup in any BEC regime, despite quantum fluctuations. Our discussion ends with a view of possible consequences for cuprates, where antinodal pair dynamics has certain features in common with our simple $s$-wave picture.

arXiv:1008.4369•Physical Review B•Feb 2, 2011•16 citations

Cooper Pair Formation in U(1) Gauge Theory of High Temperature Superconductivity

Atsuya Kumagai, Masahiko Hayashi, Hiromichi Ebisawa

We study the two-dimensional spin-charge separated Ginzburg-Landau theory containing U(1) gauge interactions as a semi-phenomenological model describing fluctuating condensates in high temperature superconductivity. Transforming the original GL action, we abstract the effective action of Cooper pair. Especially, we clarify how Cooper pair correlation evolves in the normal state from the point of view of spin-charge separation. Furthermore, we point out how Cooper pair couples to gauge field in a gauge-invariant way, stressing the insensitivity of Cooper pair to infrared gauge field fluctuation.

arXiv:cond-mat/0006097•Journal of the Physical Society of Japan•Feb 15, 2001•0 citations
πππ Flux Phase and Superconductivity for Lattice Fermions Coupled to Classical Gauge Fields
We study superconducting lattice fermions coupled to classical gauge fields. Namely, without the gauge fields, the lattice fermions show superconducting long-range order. In the previous paper, the existence of the long-range order was proved by the present author. This paper is the continuing part of the previous one. More precisely, the interactions between fermions were assumed to be a Bardeen-Cooper-Schrieffer-type pairing which is a nearest-neighbour two-body interaction on the hypercubic lattice Zd\mathbb{Z}^dZd with the dimension d≥3d\ge 3d≥3. In the present paper, we deal with the corresponding gauged model by introducing classical U(1) gauge fields. We prove that a certain configuration of gauge fields which yields the πππ flux for all the plaquettes in the lattice minimizes the ground-state energy of the fermion system. Since this configuration of the gauge fields exactly coincides with the hopping amplitudes of the Hamiltonian of the previous paper, the ground state of the whole system shows superconducting long-range order. Any configuration of gauge fields which is gauge-equivalent to this special configuration of the gauge fields also yields another ground state. However, if the Cooper pair correlations are averaged over the gauge-equivalent class, then they are vanishing except for the on-site correlation because the Cooper pair correlations are not necessarily gauge invariant. On the other hand, the string Cooper pair correlations which are gauge invariant always show the same long-range order for any configuration of gauge fields in the gauge-equivalent class.
We study lattice superconductors coupled to gauge fields, such as an attractive Hubbard model in electromagnetic fields, with a standard gauge fixing. We prove upper bounds for a two-point Cooper pair correlation at finite temperatures in spatial dimensions D ≤ 4. The upper bounds decay exponentially in three dimensions and by power law in four dimensions. These imply the absence of the superconducting long-range order for the Cooper pair amplitude as a consequence of fluctuations of the gauge fields. Since our results hold for the gauge fixing Hamiltonian, they cannot be obtained as a corollary of Elitzur’s theorem.
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.
Abstract We study an attractive Hubbard model on bipartite lattices. In the grand canonical formalism, we prove the existence of superconducting long-range order in the ground state on Lieb lattices with the chemical potential corresponding to half filling. We also study the low-energy excitations above several ground states for the translationally invariant Hamiltonian. We prove the following: (i) The pairing excitations are gapless above the ground states when the number of fermions deviates from that of the half filling by the order of the volume. (ii) A certain class of single-fermion excitations shows a non-vanishing spectral gap above the ground state with an even number of fermions in a strong coupling and low-density regime.
Stability of the π\pi π-Flux Phase for Z2\mathbb {Z}_{2}Z2​ Lattice Gauge Theory Coupled to Fermionic Matter
Abstract We consider the two-dimensional $$\mathbbm {Z}_{2}$$ Z 2 lattice gauge theory coupled to fermionic matter. In absence of electric fields, we prove that, at half-filling, the ground state of the gauge theory coincides with the π\pi π π -flux phase, associated with magnetic flux equal to π\pi π π in every elementary lattice plaquette, provided the fermionic hopping is large enough. This proves in particular the semimetallic behavior of the ground state of the model. Furthermore, we compute the magnetic susceptibility of the gauge theory, and we prove that it is given by the one of massless 2 d Dirac fermions, thus rigorously justifying recent numerical simulations. The proof is based on reflection positivity and chessboard estimates, and on lattice conservation laws for the computation of the transport coefficient.
The pseudogap state of cuprate high-temperature superconductors has been often viewed as either a yet unknown competing order or a precursor state to superconductivity. While awaiting the resolution of the pseudogap problem in cuprates, we demonstrate that local pairing fluctuations, vortex liquid dynamics, and other precursor phenomena can emerge quite generally whenever fermionic excitations remain gapped across the superconducting transition, regardless of the gap origin. Our choice of a tractable model is a lattice band insulator with short-range attractive interactions between fermions in the sss-wave channel. An effective crossover between Bardeen-Cooper-Schrieffer (BCS) and Bose-Einstein condensate (BEC) regimes can be identified in any band insulator above two dimensions, while in two dimensions only the BEC regime exists. The superconducting transition is ``unconventional'' (non-pair-breaking) in the BEC regime, identified by either the bosonic mean field or XY\mathit{XY}XY universality class. The insulator adjacent to the superconductor in the BEC regime is a bosonic Mott insulator of Cooper pairs, which may be susceptible to charge density wave ordering. We construct a function of the many-body excitation spectrum whose nonanalytic changes define a sharp distinction between band and Mott insulators. The corresponding ``second-order transition'' can be observed out of equilibrium by driving a Cooper pair laser in the Mott insulator. We explicitly show that the gap for charged bosonic excitations lies below the threshold for Cooper pair breakup in any BEC regime, despite quantum fluctuations. Our discussion ends with a view of possible consequences for cuprates, where antinodal pair dynamics has certain features in common with our simple sss-wave picture.
We study the two-dimensional spin-charge separated Ginzburg-Landau theory containing U(1) gauge interactions as a semi-phenomenological model describing fluctuating condensates in high temperature superconductivity. Transforming the original GL action, we abstract the effective action of Cooper pair. Especially, we clarify how Cooper pair correlation evolves in the normal state from the point of view of spin-charge separation. Furthermore, we point out how Cooper pair couples to gauge field in a gauge-invariant way, stressing the insensitivity of Cooper pair to infrared gauge field fluctuation.