Ground-state energy of a Richardson-Gaudin integrable BCS model

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.

The Generalized Lipkin-Meshkov-Glick Model and the Modified Algebraic Bethe Ansatz

“…In the context of the "shifted" trigonometric r-matrix and the corresponding Richardson-type models interacting with the environment the mABA has appeared in the papers [3,11]. We also remark that integrable models associated with certain limits of the non-skew-symmetric elliptic r-matrix have been considered also in the recent papers [4,15].…”

Quantum nonequilibrium dynamics from Knizhnik-Zamolodchikov equations

“…Gaudin model Hamiltonian ĤG is integrable and is linear combination of integrals of motion ĤG = 2 N l=1 w l ĤG l . As we pointed in the introduction, a class of models including the celebrated Richardon's pairing Hamiltonian 5,[20][21][22][23][24][25][26][27][28][29][30][31][32][33] , whose conserved integrals of motion can be regarded as a generalization of those of the Gaudin magnet. [62][63][64]…”

“…where the matrix S µ 1 = (1, σ 1 , σ 2 , σ 3 ), µ = 0, 1, 2, 3 is given by the unity and the set of three Pauli matrices (the presence of the unity operator in the set of spin operators implied that we have algebra u(2) instead of su( 2)). In (33) vectors b k are found to be…”

“…The b µ 1 S µ 1 term corresponds to λS 3 modification in MKZ equations (14). Now it is natural to ask whether there is a sufficient number of integrals of motion for the Hamiltonian in the Schrödinger equation (33), namely…”

“…There has been considerable interest in nonequilibrium dynamics of quantum systems (see e.g., Refs 1,2) over the last couple of decades that surged again recently in connection with quantum information problems. [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] These iclude the dynamical discrete-state Bardeen-Cooper-Shriffer (BCS) pairing models 5,[20][21][22][23][24][25][26][27][28][29][30][31][32][33] , where for example the interaction strength can be made time dependent, and various multi-level Landau-Zenner tunneling models and their many body generalizations. [34][35][36][37][38][39][40][41][42][43][44][45] The cornerstone in the integrable properties of the discrete-state BCS hamiltonian is Richardson's exact solution 20,21 and its extensions to a class of mod...…”