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

Leonardo Goller, Marcello Porta

Abstract

We consider the two-dimensional $\mathbb{Z}_{2}$ Ising 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 $π$-flux phase, associated with magnetic flux equal to $π$ 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 $2d$ Dirac fermions, thus rigorously justifying recent numerical computations. The proof is based on reflection positivity and chessboard estimates, and on lattice conservation laws for the computation of the transport coefficient.

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

Abstract

We consider the two-dimensional Ising 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 -flux phase, associated with magnetic flux equal to 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 Dirac fermions, thus rigorously justifying recent numerical computations. The proof is based on reflection positivity and chessboard estimates, and on lattice conservation laws for the computation of the transport coefficient.
Paper Structure (18 sections, 11 theorems, 254 equations, 10 figures)

This paper contains 18 sections, 11 theorems, 254 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. The model
  4. Pure gauge sector
  5. Coupling with fermionic matter
  6. The Hamiltonian and the Gibbs state
  7. Results
  8. Stability of the $\pi$-flux phase
  9. Susceptibility
  10. Reflection positivity and chessboard estimates
  11. Reflection positivity and the $\pi$-flux phase
  12. Analysis of the $\pi$-flux phases
  13. Chessboard estimates
  14. Proof of Theorem \ref{['thm:main']}
  15. Proof of Proposition \ref{['prp:susc']}
  16. ...and 3 more sections

Key Result

Proposition 2.5

If $\{\sigma_{x,\mu}\}$ and $\{\sigma'_{x,\mu}\}$ are gauge equivalent, there exists a unitary operator $U: \mathcal{F}_{L} \to \mathcal{F}_{L}$ such that:

Figures (10)

  • Figure 1: Graphical representation of the Star Operator.
  • Figure 2: Graphical representation of operators introduced in Example \ref{['ex:op']}.
  • Figure 3: Graphical representation of $W_{x,x'}$.
  • Figure 5: Energy bands associated with the $\pi$-flux phase.
  • Figure 6: Graphical representation of the cut torus.
  • ...and 5 more figures

Theorems & Definitions (41)

  • Remark 2.1
  • Definition 2.2: Physical pure-gauge observables.
  • Example 2.3
  • Definition 2.4: Physical observables
  • Proposition 2.5: Gauge equivalent configurations are isospectral
  • proof
  • Remark 2.6
  • Proposition 2.7: Resolution of the gauge constraint
  • proof
  • Remark 2.8
  • ...and 31 more