Abstract
We investigate the ground-state energy of a Richardson-Gaudin integrable BCS model, generalizing the closed and open p+ip models. The Hamiltonian supports a family of mutually commuting conserved operators satisfying quadratic relations. From the eigenvalues of the conserved operators we derive, in the continuum limit, an integral equation for which a solution corresponding to the ground state is established. The energy expression from this solution agrees with the BCS mean-field result.
Highlights
Submission the px+ipy BCS Hamiltonian supports a family of mutually commuting operators [7]
Originating in the work of Babelon and Talalaev [19], it was shown, through a change of variables, that the Bethe Ansatz Equations (BAE) for certain Richardson-Gaudin type systems can be recast into a set of coupled polynomial equations
The same form of polynomial equations were adopted in [20] as means for efficient numerical solution of the conserved operator spectrum, and in [21] to compute wavefunction overlaps. This technique was shown to be generalizable in [22, 23]. It was subsequently shown for some systems that the conserved operator eigenvalues (COE), satisfying quadratic relations, are inherited from the same relations at the operator level [15,24]
Summary
Consider a system of fermions and let cαj, c†α′k, α, α′ ∈ {+, −}, j, k ∈ 1, 2, . . . , L denote the annihilation and creation operators, satisfying. The Hamiltonian has the form of a standard BCS model, (the terms in the first line of (1)), with additional terms. By setting λ = 0 and βx = βy ⇐⇒ β = 0, we recover the conserved operators underlying the open p + ip model. These operators satisfy the following commutation relations [Skz, Sk±] = ±Sk±, [Sk+, Sk−] = 2Skz. The Hamiltonian HBCS can be rewritten (omitting the constant term) in the following form. H = fi+fi−Qi, i=1 where {Qi} is a set of mutually commuting conserved operators. These operators have been shown [17, 18] to satisfy the following quadratic relations: Q2i
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