Conformal Field Theory and the Exact Solution of the BCS Hamiltonian

TL;DR

This paper establishes a novel link between conformal field theory and the exact solvability of the reduced BCS superconductivity model, using a perturbed SU(2)_k-WZW model to derive wave functions and integrals of motion.

Contribution

It introduces a new CFT-based framework for solving the BCS Hamiltonian exactly, connecting integrability with conformal field theory techniques.

Findings

Derived Richardson's wave function using CFT methods

Identified integrals of motion for the BCS model

Linked BCS solution to perturbed WZW model

Abstract

We propose a connection between conformal field theory (CFT) and the exact solution and integrability of the reduced BCS model of superconductivity. The relevant CFT is given by the $SU(2)_k$-WZW model in the singular limit when the level k goes to -2. This theory has to be perturbed by an operator proportional to the inverse of the BCS coupling constant. Using the free field realization of this perturbed Wess-Zumino-Witten model, we derive the exact Richardson's wave function and the integrals of motion of the reduced BCS model in the saddle point approximation. The construction is reminiscent of the CFT approach to the Fractional Quantum Hall effect.