Bethe Ansatz approach to the pairing fluctuations in the mesoscopic regime

Abstract

Dedicated to Ulrich Eckern on the occasion of his 60th birthday.We review the exact treatment of the pairing correlation functions in the canonical ensemble. The key for the calculations has been provided by relating the discrete BCS model to known integrable theories corresponding to the so called Gaudin magnets with suitable boundary terms. In the present case the correlation functions can be accessed beyond the formal level, allowing the description of the cross-over from few electrons to the thermodynamic limit. In particular, we summarize the results on the finite size scaling behavior of the canonical pairing clarifying some puzzles emerged in the past. Some recent developments and applications are outlined.

Integrability of an extended -wave pairing Hamiltonian

“…The Bethe ansatz equations for D = 0 would admit an electrostatic analogy as has been used in other studies. 4,7,12,13,16,25,26 In that setting the solutions y j are equilibrium positions of M mobile +1 charges subject to electrostatic forces from −1/2 charges associated to the L fixed positions ε i , and a charge −C/2 at the origin. Even though the electrostatic analogy no longer holds for D = 0, the equation can still be solved by use of complex analysis techniques.…”

“…Some time later it was shown that this system is integrable, 5 and following on from this the integrability and exact solution were unified through approaches involving the quantum inverse scattering method based on the Yang-Baxter equation, 6 and Gaudin algebra methods. 7 From those developments up to the present day, the s-wave pairing model has continued to be the subject of many investigations including the implementation of numerical techniques for solving the associated Bethe ansatz equations, 8-10 studies using analytic methods, [11][12][13] the computation of correlation functions, [14][15][16] and quantum dynamics, 17,18 and relations to conformal field theory. 19 There have also been several searches for examples of integrable, exactly solvable BCS Hamiltonians which go beyond the s-wave case.…”

Relative phase and Josephson dynamics between weakly coupled Richardson models

“…On the other hand, in the strong coupling regime states group into eigenstates of the total angular momentum, as seen from the spin representation (27). Since the distance among the energies of these subspaces is of order g, in this regime, even a tunneling term of several times the gap cannot mix the different subspaces among them.…”

“…The RM is particularly relevant for the study of finite-size scaling effects in the BCS theory of superconductivity [19][20][21][22][23][24][25][26][27]. The reason is that the classic BCS approach to superconductivity [28] in the presence of a pairing interaction violates particle number conservation [3]: number fluctuations are negligible in the thermodynamic limit, but important for small number of particles [5].…”

Жорданова деформация открытой $s\ell(2)$-модели Годена

“…Модели Годена применялись во многих областях современной физики, от квантовой оптики [1], [2] до физики металлических нанозерен (см. статью [3] и приведенную в ней библиографию). Модель цепочки взаимодействующих спинов впервые была рассмотрена в работах [4], [5], где такие модели вводились как квазиклассический предел интегрируемых квантовых цепочек.…”