We summarize a study of an Abelian gauge theory in 2+1 dimensions, the gauge field being coupled to nonrelativistic Fermions. The Action for the gauge field is a combination of the Maxwell term and a Chern-Simons (CS) term. We study the limit of vanishing Fermions ’ charge, keeping fixed the gauge field mass induced by the CS term. By considering a closed surface, in particular a torus to keep translational invariance, we show that the Fermions do not decouple completely from the gauge field, and in fact they behave as Anyons treated in the-translationally invariant formulation of the- Mean Field approximation. We describe the exact solution of this limiting case. This lecture is based on ref.[1] and on previous work done on the problem of a translational invariant formulation of the Anyon mean field theory and its properties with respect to superfluidity and possibly superconductivity (for a general review of the previous work see ref.[2]). Actually, to discuss those issues in a proper way it is necessary to introduce a suitable mathematical formalism:We summarize a study of an Abelian gauge theory in 2+1 dimensions, the gauge field being coupled to nonrelativistic Fermions. The Action for the gauge field is a combination of the Maxwell term and a Chern-Simons (CS) term. We study the limit of vanishing Fermions ’ charge, keeping fixed the gauge field mass induced by the CS term. By considering a closed surface, in particular a torus to keep translational invariance, we show that the Fermions do not decouple completely from the gauge field, and in fact they behave as Anyons treated in the-translationally invariant formulation of the- Mean Field approximation. We describe the exact solution of this limiting case. This lecture is based on ref.[1] and on previous work done on the problem of a translational invariant formulation of the Anyon mean field theory and its properties with respect to superfluidity and possibly superconductivity (for a general review of the previous work see ref.[2]). Actually, to discuss those issues in a proper way it is necessary to introduce a suitable mathematical formalism:

Jan 1, 20080 citations

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

Tohru Koma

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.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.

arXiv:2205.00835May 2, 20220 citations

Bosonization with a background U(1) gauge field

Yuan Yao, Yoshiki Fukusumi

Bosonization is one of the most significant frameworks to analyze fermionic systems. In this work, we propose a new bosonization of Dirac fermion coupled with $U(1)$ background gauge field. Our new bosonization is consistent with gauge invariance, global chiral anomaly matching and fermion-boson operator correspondence, either of which is not satisfied by previously developed bosonizations. The bilinear Dirac-mass term condensation paradox and its generalized form are resolved by our bosonization. This new bosonization approach to interacting systems also correctly captures the conformal characters of a significant class of critical lattice models, such as conformal dimensions and the conformal anomaly of XXZ spin chain with twisted boundary condition. Our work clarifies the equivalence of fermionic flux insertion and bosonic background charge insertion for two-dimensional conformal field theory.Bosonization is one of the most significant frameworks to analyze fermionic systems. In this work, we propose a new bosonization of Dirac fermion coupled with $U(1)$ background gauge field. Our new bosonization is consistent with gauge invariance, global chiral anomaly matching and fermion-boson operator correspondence, either of which is not satisfied by previously developed bosonizations. The bilinear Dirac-mass term condensation paradox and its generalized form are resolved by our bosonization. This new bosonization approach to interacting systems also correctly captures the conformal characters of a significant class of critical lattice models, such as conformal dimensions and the conformal anomaly of XXZ spin chain with twisted boundary condition. Our work clarifies the equivalence of fermionic flux insertion and bosonic background charge insertion for two-dimensional conformal field theory.

arXiv:1902.06584Physical review. B./Physical review. BAug 2, 201910 citations

Interacting Dirac Fermions in (3+1) Dimensions

Hiroki Isobe

Springer thesesJan 1, 20170 citations

π

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

\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.

Bosonization is one of the most significant frameworks to analyze fermionic systems. In this work, we propose a new bosonization of Dirac fermion coupled with

U(1)

background gauge field. Our new bosonization is consistent with gauge invariance, global chiral anomaly matching and fermion-boson operator correspondence, either of which is not satisfied by previously developed bosonizations. The bilinear Dirac-mass term condensation paradox and its generalized form are resolved by our bosonization. This new bosonization approach to interacting systems also correctly captures the conformal characters of a significant class of critical lattice models, such as conformal dimensions and the conformal anomaly of XXZ spin chain with twisted boundary condition. Our work clarifies the equivalence of fermionic flux insertion and bosonic background charge insertion for two-dimensional conformal field theory.