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Physical Review Letters

Can the Velocity Autocorrelation Function Decay Exponentially?

M. Howard Lee

Published Oct 3, 1983• 93 citations
Publisher

Abstract

For Hermitian many-body systems, admissible functions for the velocity autocorrelation function form a certain invariant of time. This property originates from a basic recurrence relation. The exponentially decaying function cannot form an invariant and therefore cannot be admissible.
DOI
10.1103/physrevlett.51.1227

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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.
Brown [Phys. Rev. B 45, 10 809 (1991)] points out an error made in an article on the dynamics of a Heisenberg model by Wysin and Bishop [Phys. Rev. B 42, 810 (1990)]. After correcting the error, he arrives at the conclusion that the dynamic structure in this model cannot be in the form of Lorentzians. We show that this conclusion represents an orthogonality condition of certain basis vectors that span Hilbert space. Hence, it is applicable to all Hermitian many-body models. Also, a simple model is provided to illustrate both the orthogonality and transformation ideas. They can be very useful in the dynamical studies of Hermitian many-particle systems.
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.
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