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Physical review. E

Universal Matsubara time decay of quantum autocorrelations for Boltzmann particles

F. Barocchi, Eleonora Guarini

Published Oct 20, 2022• 1 citation
Publisher

Abstract

The general properties of time dependent autocorrelations in many-body quantum systems are here analyzed at thermodynamic equilibrium in the Boltzmann canonical ensemble at temperature T, by means of the exponential expansion theory (EET). It is shown that the Kubo-Martin-Schwinger (KMS) symmetry applied to the exponential expansion of the correlation leads to the existence of two different sets of decay modes (channels) here indicated as "Matsubara modes" and "system modes," respectively. The Matsubara modes are a series of pure decay channels with time constants representing a direct action of the thermostat upon the correlation, with a characteristic principal decay time τ_{1}=ℏ/(2πk_{B}T), where ℏ and k_{B} are the Planck and Boltzmann constants, and T is the temperature. Moreover, the KMS condition implies that the amplitudes pertaining to the even and odd contribution of the system modes to the quantum correlation are not independent. These two properties are quantum mechanical in nature and "universal," in the sense that they are present for any autocorrelation of a quantum system at equilibrium at a temperature T. The Matsubara modes' contribution to the time behavior of a quantum correlation is limited to times of the order of τ_{1}, which however can be comparable with some of the characteristic decay times of the system modes. In addition, since the parameters representing the overall time behavior of the quantum correlation can be given in terms of the parameters of its Kubo transform, the EET representation turns out to be useful in calculations exploiting the outputs of some widespread quantum simulation methods. A discussion of the properties of these relations is described in detail with numerical examples. The case of the velocity autocorrelation function of para hydrogen at low temperature is also reported as a final example for a real system.

DOI
10.1103/physreve.106.044128

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The general properties of time dependent autocorrelations in many-body quantum systems are here analyzed at thermodynamic equilibrium in the Boltzmann canonical ensemble at temperature T, by means of the exponential expansion theory (EET). It is shown that the Kubo-Martin-Schwinger (KMS) symmetry applied to the exponential expansion of the correlation leads to the existence of two different sets of decay modes (channels) here indicated as "Matsubara modes" and "system modes," respectively. The Matsubara modes are a series of pure decay channels with time constants representing a direct action of the thermostat upon the correlation, with a characteristic principal decay time τ_{1}=ℏ/(2πk_{B}T), where ℏ and k_{B} are the Planck and Boltzmann constants, and T is the temperature. Moreover, the KMS condition implies that the amplitudes pertaining to the even and odd contribution of the system modes to the quantum correlation are not independent. These two properties are quantum mechanical in nature and "universal," in the sense that they are present for any autocorrelation of a quantum system at equilibrium at a temperature T. The Matsubara modes' contribution to the time behavior of a quantum correlation is limited to times of the order of τ_{1}, which however can be comparable with some of the characteristic decay times of the system modes. In addition, since the parameters representing the overall time behavior of the quantum correlation can be given in terms of the parameters of its Kubo transform, the EET representation turns out to be useful in calculations exploiting the outputs of some widespread quantum simulation methods. A discussion of the properties of these relations is described in detail with numerical examples. The case of the velocity autocorrelation function of para hydrogen at low temperature is also reported as a final example for a real system.
In the past few years, the exponential expansion analysis of time autocorrelation functions has provided profound insight into the leading microscopic processes driving the atomic-scale dynamics and has made it possible to highlight the presence and the role of various relaxation channels through which the fundamental correlation functions decay with time. Here we apply this method to the determination of the full time dependence of a correlation function c(t) in a quantum system at nonzero temperature, by making explicit its relationship with its Kubo transform c_{K}(t), which in some cases can be approximately computed with the presently available quantum simulation techniques. We obtain an exact expression for c(t) in terms of the exponential modes that describe the time behavior of c_{K}(t). The relative importance of the various modes in determining the overall shape of c(t) can then be studied in detail. This work extends to the full time domain the results of a previous paper [Guarini et al., Phys. Rev. Lett. 123, 135301 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.135301], in which we employed the same method to calculate the zero time value of the velocity autocorrelation function, to obtain a microscopic description of the quantum mean kinetic energy in a fluid. In particular, we show that the decay constants and the frequencies of the dominant microscopic modes of c(t) are the same as those of c_{K}(t), but the dynamics of the quantum system also contains an additional term decaying on a time scale determined solely by temperature of the system.
Matsubara dynamics has recently emerged as the most general form of a quantum-Boltzmann-conserving classical dynamics theory for the calculation of single-time correlation functions. Here, we present a generalization of Matsubara dynamics for the evaluation of multitime correlation functions. We first show that the Matsubara approximation can also be used to approximate the two-time symmetrized double Kubo transformed correlation function. By a straightforward extension of these ideas to the multitime realm, a multitime Matsubara dynamics approximation can be obtained for the multitime fully symmetrized Kubo transformed correlation function. Although not a practical method, due to the presence of a phase-term, this multitime formulation of Matsubara dynamics represents a benchmark theory for future development of Boltzmann preserving semiclassical approximations to general higher order multitime correlation functions. It also reveals a connection between imaginary time-ordering in the path integral and the classical dynamics of multitime correlation functions.
We develop an exact quantum thermostat that generates the quantum Boltzmann distribution in path integral phase space and so correctly captures the imaginary position–momentum correlations. This extends beyond the capability of existing quantum thermostats such as the PILE thermostat, which can only achieve the marginal of this distribution in position, with the momentum acting purely as a sampling tool and satisfying classical Maxwell–Boltzmann statistics. The dynamics correspond to the treatment of the Caldeira–Leggett system–bath model with Matsubara dynamics, a theory for approximating nuclear quantum dynamics by propagating smooth imaginary-time Feynman paths in real time. A Matsubara generalised Langevin equation that obeys certain quantum fluctuation–dissipation relations is mapped onto a Markovian stochastic process by introducing auxiliary variables; this allows the construction of a corresponding Fokker–Planck equation from which the stationary distribution in the extended variable space is determined. We show that this corresponds to the joint quantum Boltzmann distribution in position and momentum, a finding that holds for any bath spectral density and for anharmonic systems. The sole approximation in Matsubara dynamics is the neglect of real-time quantum coherence; consequently, provided there is sufficient quantum decoherence, the dynamics of the exact quantum thermostat also well approximates the exact quantum dynamics of the system–bath. In the limit of infinite system–bath coupling, the level of real-time coherence becomes negligible and so the Matsubara dynamics of the thermostat tends to the exact dynamics. The quantum thermostat can therefore be used to study this limit exactly and, in so doing, we demonstrate the existence of a quantum mechanical overdamped regime analogous to its classical counterpart that is characterised by a Matsubara Smoluchowski equation. The drawback of the quantum thermostat is that one of its associated quantum fluctuation–dissipation relations is imaginary. This results in complex noise and dynamical instabilities that often preclude the numerical implementation of the thermostat. Instead, it primarily serves as a theoretical framework that can be used as a starting point for introducing approximations that yield more practical methods. One such approximation that results in real noise is considered here and leads to a practical approach for calculating linear time-correlation functions of bound open quantum systems. These exhibit good agreement with exact quantum mechanical calculations, particularly at strong system–bath coupling, where the real-time quantum coherence is increasingly suppressed.
In recent years, there has been intense attention on the constraints posed by quantum mechanics on the dynamics of the correlation at low temperatures, triggered by the postulation and derivation of quantum bounds on the transport coefficients or on the chaos rate. However, the physical meaning and the mechanism enforcing such bounds is still an open question. Here, we discuss the quantum fluctuation-dissipation theorem (the KMS conditions) as the principle underlying bounds on correlation time scales. By restating the problem in a replicated space, we show that the quantum bound to chaos is a direct consequence of the KMS condition, as applied to a particular pair of two-time correlation and response functions. Encouraged by this, we describe how quantum fluctuation-dissipation relations act in general as a blurring of the time-dependence of correlations, which can imply bounds on their decay rates. Thinking in terms of fluctuation-dissipation opens a direct connection between bounds and other thermodynamic properties.
Lee [Phys. Rev. B 47, 8293 (1993)] has shown that the odd-numbered derivatives of the Kubo autocorrelation function vanish at t=0. We show that this condition is based on a more general property of nondiagonal Kubo correlation functions. This general property provides that certain functional forms (e.g., simple exponential decay) are not admissible for any symmetric or antisymmetric Kubo correlation function in a Hermitian many-body system. Lee's result emerges as a special case of this result. Applications to translationally invariant systems and systems with rotational symmetries are also demonstrated.
This is the first of a series of papers which describe the functional-integral approach to the study of the statistical and kinetic properties of nonequilibrium quantum fields in flat and curved spacetimes. In this paper we treat a system of self-interacting bosons described by \ensuremath{\lambda}${\ensuremath{\varphi}}^{4}$ scalar fields in flat space. We adopt the closed-time-path (CTP or ``in-in'') functional formalism and use a two-particle irreducible (2PI) representation for the effective action. These formalisms allow for a full account of the dynamics of quantum fields, and put the correlation functions on an equal footing with the mean fields. By assuming a thermal distribution we recover the real-time finite-temperature theory as a special case. By requiring the CTP effective action to be stationary with respect to variations of the correlation functions we obtain an infinite set of coupled equations which is the quantum-field-theoretical generalization of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. Truncation of this series leads to dissipative characteristics in the subsystem. In this context we discuss the nature of dissipation in interacting quantum fields.To one-loop order in a perturbative expansion of the CTP effective action, the 2PI formalism yields results equivalent to the leading 1/N expansion for an O(N)-symmetric scalar field. To higher-loop order we introduce a two-time approximation to separate the quantum-field effects of radiative correction and renormalization from the statistical-kinetic effects of collisions and relaxation. In the weak-coupling quasiuniform limit, the system of nonequilibrium quantum fields can subscribe to a kinetic theory description wherein the propagators are represented in terms of relativistic Wigner distribution functions. From a two-loop calculation we derive the Boltzmann equation for the distribution function and the gap equation for the effective mass of the quasiparticles. One can define an entropy function for the quantum gas of quasiparticles which satisfies the H theorem. We also calculate the limits to the validity of the binary collision approximation from a three-loop analysis. The theoretical framework established here can be generalized to nonconstant background fields and for curved spacetimes.