"Probabilistic" approach to Richardson equations

Abstract

It is known that solutions of Richardson equations can be represented as stationary points of the "energy" of classical free charges on the plane. We suggest to consider "probabilities" of the system of charges to occupy certain states in the configurational space at the effective temperature given by the interaction constant, which goes to zero in the thermodynamical limit. It is quite remarkable that the expression of "probability" has similarities with the square of Laughlin wave function. Next, we introduce the "partition function", from which the ground state energy of the initial quantum-mechanical system can be determined. The "partition function" is given by a multidimensional integral, which is similar to Selberg integrals appearing in conformal field theory and random-matrix models. As a first application of this approach, we consider a system with the constant density of energy states at arbitrary filling of the energy interval, where potential acts. In this case, the "partition function" is rather easily evaluated using properties of the Vandermonde matrix. Our approach thus yields quite simple and short way to find the ground state energy, which is shown to be described by a single expression all over from the dilute to the dense regime of pairs. It also provides additional insights into the physics of Cooper-paired states.

20 pages, 1 figure; final version, accepted for publication in Journal of Physics: Condensed Matter

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Cited by in corpus (10)

• An eigenvalue-based method and determinant representations for general integrable XXZ Richardson-Gaudin models
• Density Matrices of Seniority-Zero Geminal Wavefunctions
• Generalised Heine-Stieltjes and Van Vleck polynomials associated with degenerate, integrable BCS models
• Reduced Density Matrices / Static Correlation Functions of Richardson-Gaudin States Without Rapidities
• Integrability of an extended d+id-wave pairing Hamiltonian
• Exact solution for the inhomogeneous Dicke model in the canonical ensemble: thermodynamical limit and finite-size corrections
• Electron-hole symmetry and solutions of Richardson pairing model
• Particle-hole duality, integrability, and Russian doll BCS model
• Near-exact treatment of seniority-zero ground and excited states with a Richardson-Gaudin mean-field
• Transition density matrices of Richardson-Gaudin states