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Physical review. B, Condensed matter

Correlation functions for Hermitian many-body systems: Necessary conditions

Edward B. Brown

Published Feb 1, 1994• 5 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.49.4305

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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.
We give rigorous analytical results on the temporal behavior of two-point correlation functions-also known as dynamical response functions or Green's functions-in closed many-body quantum systems. We show that in a large class of translation-invariant models the correlation functions factorize at late times ⟨A(t)B⟩_{β}→⟨A⟩_{β}⟨B⟩_{β}, thus proving that dissipation emerges out of the unitary dynamics of the system. We also show that for systems with a generic spectrum the fluctuations around this late-time value are bounded by the purity of the thermal ensemble, which generally decays exponentially with system size. For autocorrelation functions we provide an upper bound on the timescale at which they reach the factorized late time value. Remarkably, this bound is only a function of local expectation values and does not increase with system size. We give numerical examples that show that this bound is a good estimate in nonintegrable models, and argue that the timescale that appears can be understood in terms of an emergent fluctuation-dissipation theorem. Our study extends to further classes of two point functions such as the symmetrized ones and the Kubo function that appears in linear response theory, for which we give analogous results.
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.
We study the relation between the spectral gap above the ground state and the decay of the correlations in the ground state in quantum spin and fermion systems with short-range interactions on a wide class of lattices. We prove that, if two observables anticommute with each other at large distance, then the nonvanishing spectral gap implies exponential decay of the corresponding correlation. When two observables commute with each other at large distance, the connected correlation function decays exponentially under the gap assumption. If the observables behave as a vector under the U(1) rotation of a global symmetry of the system, we use previous results on the large distance decay of the correlation function to show the stronger statement that the correlation function itself, rather than just the connected correlation function, decays exponentially under the gap assumption on a lattice with a certain self-similarity in (fractal) dimensions D < 2. In particular, if the system is translationally invariant in one of the spatial directions, then this self-similarity condition is automatically satisfied. We also treat systems with long-range, power-law decaying interactions. Spectral Gap and Exponential Decay of Correlations 2 1
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.