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BCS-Spin-Model, its Thermodynamic Representations and Automorphisms

Received April

Published Jan 1, 1968• 0 citations
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Abstract

In this paper two equivalent explicit realizations of the thermo- dynamic representations of the BCS-spin-model are constructed and since they are of well known form their type is easily determined. Time evolution and vacuum properties of these representations are studied and it is shown that in general the time development of the system is representation-dependent.

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In this paper two equivalent explicit realizations of the thermo- dynamic representations of the BCS-spin-model are constructed and since they are of well known form their type is easily determined. Time evolution and vacuum properties of these representations are studied and it is shown that in general the time development of the system is representation-dependent.
We develop an explicit description of a time-dependent response of fermionic condensates to perturbations. The dynamics of Cooper pairs at times shorter than the energy relaxation time can be described by the BCS model. We obtain a general explicit solution for the dynamics of the BCS model. We also solve a closely related dynamical problem - the central spin model, which describes a localized spin coupled to a "spin bath". Here, we focus on presenting the solution and describing its general properties, but also mention some applications, e.g. to nonstationary pairing in cold Fermi gases and to the issue of electron spin decoherence in quantum dots. A typical dynamics of the BCS and central spin models is quasi-periodic with a large number of frequencies and stable under small perturbations. We show that for certain special initial conditions the number of frequencies decreases and the solution simplifies. In particular, periodic solutions correspond to the ground state and excitations of the BCS model.
The existence of inequivalent representations in quantum field theory with {\it finitely} many degrees of freedom is shown. Their properties are exemplified and analysed for concrete and simple models. In particular the relations to Bogoliubov--Valatin quasi-particles, to thermo field dynamics, and to qqq--deformed quantum theories are put foreward. The thermal properties of the non-trivial vacuum are given and it is shown that the thermodynamic equilibrium state is uniquely obtained by an irreversible vacuum dynamics. Finally, the theory is applied to a realistic model: the BCS--theory of superconductivity. An exact solution in order O(N−1)O(N^{-1})O(N−1) for the full particle number conserving BCS--Hamiltonian with particle number symmetric ground state is given.
We study a number of strongly correlated many-body quantum systems. Most of the thesis is devoted to effects of pairing between fermions. In particular, we develop an explicit description of a time-dependent response of fermionic condensates to external perturbations. The dynamics of Cooper pairs at times shorter than the energy relaxation time can be described by the BCS model. We obtain a general explicit solution for the dynamics of the BCS model. We also solve a closely related dynamical problem – the central spin model, which describes a localized spin coupled to a ‘spin bath’. This model has a range of interesting applications, in particular to the issue of electron spin coherence in semiconductor quantum dots. We evaluate finite size corrections for the BCS model, which is relevant for small systems such as metallic grains. In the reminder of the thesis, we study the connection between symmetry and degeneracy in quantum systems. We propose a method for extracting hidden symmetry from the energy spectrum and clarify the origin of crossings and degeneracies in the 1d Hubbard model.
By making use of time-dependent Bogoliubov transformations, we develop a calculation technique for time-dependent non-equilibrium systems of quantum fields in a time-representation (t-representation). The corrected one-body propagator in the t-representation turns out to have the form B−1 (diagonal matrix) B (B being a thermal Bogoliubov matrix). Applying the usual on-shell concept to the diagonal matrix part of the self-energy, we formulate a self-consistent renormalization scheme. This renormalization determines the vacuum and leads to a kinetic equation for the number density parameter, which reduces to the Boltzmann equation in the lowest approximation. This gives us the increasing entropy in time (the second law of thermodynamics).