Canonical Transformation of the Three-Band Hubbard Model and Hole Pairing

TL;DR

This paper develops a canonical transformation method for the three-band Hubbard model to derive an effective interaction between holes, revealing a Cooper-like instability and generalizing previous cluster calculation results.

Contribution

It introduces an analytic expression for the effective interaction including 4-body virtual states, enabling a comprehensive analysis of hole pairing in the model.

Findings

Derives a closed-form expression for the effective interaction.

Identifies a Cooper-like instability in the Fermi liquid.

Generalizes two-hole bound states beyond cluster calculations.

Abstract

We propose a canonical tranformation approach to the effective interaction $W_{eff}$ between two holes, based on the three-band Hubbard model but ready to include extra interactions as well. An effective two-body Hamiltonian can in principle be obtained including any kind of virtual intermediate states. We derive the closed-form analytic expression of the effective interaction including 4-body virtual states, describing the exchange of an electron-hole pair to all orders. The resulting integral equation, valid for the full plane, leads to a Cooper-like instability of the Fermi liquid. The two-hole bound states generalize those reported earlier in cluster calculations by exact diagonalisation methods.