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Conformal Field Theory and the Exact Solution of the BCS Hamiltonian

Sierra, G

Published Jan 1, 2000• 36 citations
OpenAlex

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.

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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)kSU(2)_kSU(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.
In this paper, we recall Richardson's solution of the reduced BCS model, its relationship with the Gaudin model, and the known implementation of these models in conformal field theory. The CFT techniques applied here are based on the use of the free field realization, or more precisely, on the calculation of saddle-point values of Coulomb gas integrals representing certain (perturbed) WZW conformal blocks. We identify the saddle-point limit as the classical limit of conformal blocks. We show that this observation implies a new method for calculating classical conformal blocks and can be further used in the study of quantum integrable models.
Abstract In this paper, we recall Richardson’s solution of the reduced BCS model, its relationship with the Gaudin model, and the known implementation of these models in conformal field theory. The CFT techniques applied here are based on the use of the free field realization, or more precisely, on the calculation of saddle-point values of Coulomb gas integrals representing certain (perturbed) WZW conformal blocks. We identify the saddle-point limit as the classical limit of conformal blocks. We show that this observation implies a new method for calculating classical conformal blocks and can be further used in the study of quantum integrable models.
A model is introduced for two reduced BCS systems which are coupled through the transfer of Cooper pairs between the systems. The model may thus be used in the analysis of the Josephson effect arising from pair tunneling between two strongly coupled small metallic grains. At a particular coupling strength the model is integrable and explicit results are derived for the energy spectrum, conserved operators, integrals of motion, and wave function scalar products. It is also shown that form factors can be obtained for the calculation of correlation functions. Furthermore, a connection with perturbed conformal field theory is made.
Correlation functions in the Wess-Zumino-Novikov-Witten (WZNW) theory satisfy a system of Knizhnik-Zamolodchikov (KZ) equations, which involve constants of motion of an exactly solvable model, known as Gaudin magnet. We show that modified KZ equations, where the Gaudin operators are replaced by constants of motion of the closely related pairing Hamiltonian, give rise to a deformed WZNW model that contains terms breaking translational symmetry. This boundary WZNW model is identified and solved. The solution establishes a connection between the WZNW model and the pairing Hamiltonian in the theory of superconductivity. We also argue and demonstrate on an explicit example that our general approach can be used to derive exact solutions to a variety of dynamical systems.
The QHE is studied in the context of a CFT. An effective field of NNN ``spins" associated with the cyclotron motion of particles is taken as an order parameter from which an effective Hamiltonian may be defined. This effective Hamiltonian describes the COM motion of the NNN particles (with coupling κ0κ_0κ0​) together with a current-current interaction (of strength κ1κ_1κ1​). Such a system gives rise to a CFT in the large NNN limit when κ0=κ1κ_0 = κ_1κ0​=κ1​. The Laughlin wavefunction is derived from this CFT as an N′N'N′-point correlation function of winding state vertex operators.