Solving the Richardson equations close to the critical points

Abstract

We study the Richardson equations close to the critical values of the paring strength gc where the occurrence of divergencies preclude numerical solutions. We derive a set of equations for determining the critical g values and the non-collapsing pair energies. Studying the behavior of the solutions close to the critical points, we develop a procedure to solve numerically the Richardson equations for arbitrary coupling strength.

Quantum phase diagram of the integrablepx+ipyfermionic superfluid

“…(27) and (28). For certain values of µ and ∆ 2 there exists another set of values µ and ∆ 2 that lead to the same equations,…”

“…Just like in the rational case, the RG equations become singular when two or more pairons approach the same level 27,28 . Analyzing the residues in the equation, one finds that the number of singular pairons has to be equal to 2s k + 1 for a singularity around level η k .…”

“…If singularities with three or more pairons occur, this trick no longer works. More complicated changes of variables have been proposed to remediate this problem for the rational model 27,28 , but in the hyperbolic model we are confronted with a situation where all pairons can converge to zero and where the change of variables can no longer be performed with sufficient accuracy. Standard gradient methods to solve the Eqs.…”

Eigenvalue-based method and form-factor determinant representations for integrable XXZ Richardson-Gaudin models

“…As an illustration, the set of RG equations for the doubly degenerate XXX model [Eq. (14)] is isomorphic to the set of quadratic equations [20] …”

“…Unfortunately, the Bethe ansatz or Richardson-Gaudin equations [6,11,12] that need to be solved are highly nonlinear and give rise to singularities, making a straightforward numerical solution challenging [13,14]. Several methods have been introduced as a way to resolve this difficulty, such as a change in variables [14,15], a (pseudo)deformation of the algebra [16,17], or a Heine-Stieltjes connection, reducing the problem to a differential equation [18]. The ground-state energy in the thermodynamic limit has also been obtained by treating the interaction as an effective temperature [19].…”