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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.

Superconductivity and Low Energy Excitations in an Attractive Hubbard Model

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.

Paper Structure

This paper contains 13 sections, 8 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Model and Properties
  3. Shiba Transformation and Interaction Hamiltonian
  4. Grand Canonical Formalism and Charge Neutrality
  5. Positivity and Bounds for the Correlation
  6. Positivity of Superconducting Correlation
  7. Bounds for Superconducting Correlation
  8. Restriction of Fermion Fock Space
  9. Long Range Order of Superconductivity
  10. Spectral Gap of Single-Fermion Excitations
  11. The Case of Hypercubic Lattice
  12. The Case of Lieb Lattice
  13. Low-Lying Excitations

Key Result

Lemma 4.1

The following bound is valid: for the order parameter $O_{\rm super}^{(\Lambda)}$ of superconductivity in (Os).

Theorems & Definitions (9)

  • Lemma 4.1
  • Theorem 5.1
  • Theorem 5.3
  • Theorem 6.1
  • Example 6.2
  • Lemma 6.3
  • Theorem 6.4
  • Theorem 7.1
  • Theorem 7.2