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The Journal of Chemical Physics

Non-Gaussian influence functional for quantum systems

Kurt Allinger, Benny Carmeli, David Chandler

Published Feb 1, 1986• 8 citations
Publisher

Abstract

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.

DOI
10.1063/1.450470

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Influence functionals: General methodology for subsystem calculations

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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 ${S}_{\mathrm{EV}}$ and a remainder, ${S}^{0}$, 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.

Physical Review A•Jan 1, 1989•8 citations

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A powerful numerical procedure is developed for calculating temperature-dependent quantities, such as partition functions for quantum systems coupled to thermal or other baths. A variational procedure is worked out to construct model Hamiltonians starting from the general results of the previous paper for influence functionals. Explicit, though intricate, formulas for all relevant quantities are presented. The methods developed are applied to a generalized model for a boson-bath interaction.

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Comments on a model influence functional for quantum systems

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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.

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Constructing tensor network influence functionals for general quantum dynamics

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We describe an iterative formalism to compute influence functionals that describe the general quantum dynamics of a subsystem beyond the assumption of linear coupling to a quadratic bath. We use a space-time tensor network representation of the influence functional and investigate its approximability in terms of its bond dimension and time-like entanglement in the tensor network description. We study two numerical models, the spin-boson model and a model of interacting hard-core bosons in a 1D harmonic trap. We find that the influence functional and the intermediates involved in its construction can be efficiently approximated by low bond dimension tensor networks in certain dynamical regimes, which allows the quantum dynamics to be accurately computed for longer times than with direct time evolution methods. However, as one iteratively integrates out the bath, the correlations in the influence functional can first increase before decreasing, indicating that the final compressibility of the influence functional is achieved via non-trivial cancellation.

arXiv:2101.05466•The Journal of Chemical Physics•Jul 26, 2021•56 citations
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.
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 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.
The recently developed (energy) sudden representation of the Schrödinger propagator is examined in conjunction with the path integral method. The perturbation expansion of the propagator in this representation (with the sudden approximation as the zeroth order term) is easily understood using Feynman diagrams. The zeroth order term is a form involving only sudden paths (i.e., paths resulting from the sudden approximation). Each higher order term is a series of sudden paths which are connected to one another due to the nonsudden effects of the dynamics. The sudden representation, and in particular its perturbation expansion, is also used to study the bath effects in many body systems which are expressed in the form of influence functionals in the path integral formalism. In our treatment, the bath coordinates are assumed to be the slow coordinates. The zeroth order (sudden approximation) influence functionals are expressed in a very simple form; they involve only an ordinary integral over the bath coordinates. Finally, to illustrate the approach, a harmonic oscillator coupled linearly to an unspecified primary system is used as a simple model bath. The reduction of the exact influence functional to the sudden approximation form is also considered.
An exact hierarchical equations of motion formalism for a quantum dissipation system in a grand canonical bath ensemble surrounding is constructed, on the basis of the calculus-on-path-integral algorithm, together with the parametrization of arbitrary non-Markovin bath that satisfies fluctuation-dissipation theorem. The influence functionals for both the Fermion or Boson bath interaction are found to be of the same path-integral expression as the canonical bath, assuming they all satisfy the Gaussian statistics. However, the equation of motion formalism are different, due to the fluctuation-dissipation theories that are distinct and used explicitly. The implications of the present work to quantum transport through molecular wires and electron transfer in complex molecular systems are discussed.
We describe an iterative formalism to compute influence functionals that describe the general quantum dynamics of a subsystem beyond the assumption of linear coupling to a quadratic bath. We use a space-time tensor network representation of the influence functional and investigate its approximability in terms of its bond dimension and time-like entanglement in the tensor network description. We study two numerical models, the spin-boson model and a model of interacting hard-core bosons in a 1D harmonic trap. We find that the influence functional and the intermediates involved in its construction can be efficiently approximated by low bond dimension tensor networks in certain dynamical regimes, which allows the quantum dynamics to be accurately computed for longer times than with direct time evolution methods. However, as one iteratively integrates out the bath, the correlations in the influence functional can first increase before decreasing, indicating that the final compressibility of the influence functional is achieved via non-trivial cancellation.