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The Journal of Chemical Physics

Comments on a model influence functional for quantum systems

David H. Wu, Benny Carmeli, David Chandler

Published Feb 15, 1988• 2 citations
Publisher

Abstract

We continue the study of a model non-Gaussian influence functional proposed by Allinger, Carmeli, and Chandler [J. Chem. Phys. 84, 1724 (1986)] to approximate the exact influence functional resulting from integrating out all quantum states but those of primary interest. The premise of this work is that the influence of many secondary states on a single primary state can be closely approximated by the influence of a degenerate level of states with equal coupling to the primary state. The degeneracy reflects the fluctuations possible among the secondary states and in fact can be associated with the partition function of the exact secondary states. The new calculations presented herein emphasize the importance of entropic-like effects properly described by this degeneracy, and for the models we examine, our basic premise is shown to be correct.

DOI
10.1063/1.453977

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We continue the study of a model non-Gaussian influence functional proposed by Allinger, Carmeli, and Chandler [J. Chem. Phys. 84, 1724 (1986)] to approximate the exact influence functional resulting from integrating out all quantum states but those of primary interest. The premise of this work is that the influence of many secondary states on a single primary state can be closely approximated by the influence of a degenerate level of states with equal coupling to the primary state. The degeneracy reflects the fluctuations possible among the secondary states and in fact can be associated with the partition function of the exact secondary states. The new calculations presented herein emphasize the importance of entropic-like effects properly described by this degeneracy, and for the models we examine, our basic premise is shown to be correct.
Abstract In reply to Kaldor's [Int. J. Quantum Chem. XX, XXX (1985)] criticism of our study of simple four‐electron models, in which the degree of quasi‐degeneracy can be continuously varied, by the finite‐order nondegenerate many‐body perturbation theory, we examine in more detail a simple two‐state model that the used to substantiate his claim that “the low order sum of the perturbation series is not very meaningful” in view of its divergence. It is shown that in contrast to Kaldor's claim, the partitioning used increases the radius of convergence of the considered perturbation series and is in principle capable to make it convergent. It is also shown that the convergence of the series is not very essential and that even divergent series can provide useful estimate of the exact result, particularly when the resummation techniques, such as Padè approximants or continued fractions, are employed. Finally, the shortcomings of the existing multi‐reference perturbation approaches, which Kaldor advocates, are pointed out.
Feynman path integrals offer a convenient and elegant tool for dealing with the statistical physics of quantum-mechanical systems. By using the Trotter product formula, one can evaluate directly the propagation of many-body quantum systems in imaginary time. The idea of an influence functional arises when one wishes to consider division of the physical system into a primary system and a bath. We present a general approach to the problem of constructing influence functionals, one that is capable of dealing both with Boltzmann baths (classical oscillators) and quantum baths of bosons or fermions. The fermion bath of special interest is generally a submanifold of the electronic state such as the bridge structure connecting chromophores or electron localization subunits which are common in problems of mixed valency and superexchange. The fermion bath is treated by a general rewriting of the influence phase S as a sum of an eigenvalue part SEV{S}_{\mathrm{EV}}SEV​ and a remainder, S0{S}^{0}S0, that describes transitions. Reduction of the original Hamiltonian leads to a reduced Hamiltonian with effective off-diagonal matrix elements and influence functionals which can include memory effects. We present a general formalism for construction of influence functionals, and discuss particular applications to systems of interest, especially in electronic-structure problems. The following paper presents a variational technique for finding effective Hamiltonians.
We consider the quantum paths (in Euclidean time) through a space of N multiparticle states, and we derive the influence functional for a subset of m primary states in this space. The influence functional results from integrating out the occupation of the N-m nonprimary or bath states holding fixed the time-dependent occupation path through the primary states. The division into bath states and primary states is completely general in principle, though in practice, physical considerations will guide the choice of partitioning. The exact influence functional is nonlocal in time and very complicated in form. We argue, however, by a variational calculation that a simplified approximate functional can be constructed which is very accurate. Unlike the standard Gaussian influence functionals which are inapplicable for processes involving the large amplitude fluctuations of charge transfer and fermionic exchange, the influence functionals described herein are appropriate for models of these processes.
We consider quantum dynamics of the order parameter in the discrete pairing model (Richardson model) in thermodynamic equilibrium. The integrable Richardson Hamiltonian is represented as a direct sum of Hamiltonians acting in different Hilbert spaces of single-particle and paired/empty states. This allows us to factorize the full thermodynamic partition function into a combination of simple terms associated with real spins on singly occupied states and the partition function of the quantum XYXYXY model for Anderson pseudospins associated with the paired/empty states. Using coherent-state path integral, we calculate the effects of superconducting phase fluctuations exactly. The contribution of superconducting amplitude fluctuations to the partition function in the broken-symmetry phase is shown to follow from the Bogoliubov-de Gennes equations in imaginary time. These equations in turn allow several interesting mappings, e.g., they are shown to be in a one-to-one correspondence with the one-dimensional Schr\"odinger equation in supersymmetric quantum mechanics. However, the most practically useful approach to calculate functional determinants is found to be via an analytical continuation of the quantum order parameter to real time, $\ensuremath{\Delta}(\ensuremath{\tau}\ensuremath{\rightarrow}it)$, such that the problem maps onto that of a driven two-level system. The contribution of a particular dynamic order parameter, $\ensuremath{\Delta}(\ensuremath{\tau})$, to the partition function is shown to correspond to the sum of the Berry phase and dynamic phase accumulated by the pseudospin. We also examine a family of exact solutions for two-level-system dynamics on a class of elliptic functions and suggest a compact expression to estimate the functional determinants on such trajectories. The possibility of having quantum soliton solutions coexisting with classical BCS mean field is discussed.