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Abstract: We consider quantum dynamics of the order parameter in the discrete pairingmodel Richardson model in thermodynamic equilibrium. The integrableRichardson Hamiltonian is represented as a direct sum of Hamiltonians acting indifferent Hilbert spaces of single-particle and paired-empty states. Thisallows us to factorize the full thermodynamic partition function into acombination of simple terms associated with real spins on singly-occupiedstates and the partition function of the quantum XY-model for Andersonpseudospins associated with the paired-empty states. Using coherent-statepath-integral, we calculate the effects of superconducting phase fluctuationsexactly. The contribution of superconducting amplitude fluctuations to thepartition function in the broken-symmetry phase is shown to follow from theBogoliubov-de Gennes equations in imaginary time. These equations in turn allowseveral interesting mappings, e.g., they are shown to be in a one-to-onecorrespondence with the one-dimensional Schr\-odinger equation insupersymmetric Quantum Mechanics. However, the most practically useful approachto calculate functional determinants is found to be via an analyticalcontinuation of the quantum order parameter to real time, \Delta\tau -> it,such that the problem maps onto that of a driven two-level system. Thecontribution of a particular dynamic order parameter to the partition functionis shown to correspond to the sum of the Berry phase and dynamic phaseaccumulated by the pseudospin. We also examine a family of exact solutions fortwo-level-system dynamics on a class of elliptic functions and suggest acompact expression to estimate the functional determinants on suchtrajectories. The possibility of having quantum soliton solutions co-existingwith classical BCS mean-field is discussed.

Author: Victor Galitski


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