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arXiv:1811.11338 · Journal of Physics A Mathematical and Theoretical

Inferring the smoothness of the autocorrelation function from that of the initial state

K L Yang, J M Zhang

Published Oct 2, 2019• 0 citations
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Abstract

Abstract We point out that by the ‘smoothness means fast decay’ principle in Fourier analysis, it is possible to infer the smoothness (or nonsmoothness) of the autocorrelation function from a mere glimpse of the initial state. Specifically, for a generic system with smooth eigenstates, the smoother an initial state is, the faster its decomposition coefficients with respect to the eigenstates of the system decay, and in turn the smoother the autocorrelation function is. The idea is illustrated with three increasingly smooth functions in an infinite square well. By using the Mellin transform, we also find that the nonsmoothness or singularity of the initial state affects the short-time behavior of its autocorrelation function. In particular, a sufficiently nonsmooth or singular initial state could decay in a nonquadratic power law, with the exponent continuously tunable. We now understand the periodic cusps of the autocorrelation function in the quench dynamics of a Bloch state, which was observed previously (Zhang and Yang 2016 Europhys. Lett . 114 60001).

DOI
10.1088/1751-8121/ab4a2b

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Abstract We point out that by the ‘smoothness means fast decay’ principle in Fourier analysis, it is possible to infer the smoothness (or nonsmoothness) of the autocorrelation function from a mere glimpse of the initial state. Specifically, for a generic system with smooth eigenstates, the smoother an initial state is, the faster its decomposition coefficients with respect to the eigenstates of the system decay, and in turn the smoother the autocorrelation function is. The idea is illustrated with three increasingly smooth functions in an infinite square well. By using the Mellin transform, we also find that the nonsmoothness or singularity of the initial state affects the short-time behavior of its autocorrelation function. In particular, a sufficiently nonsmooth or singular initial state could decay in a nonquadratic power law, with the exponent continuously tunable. We now understand the periodic cusps of the autocorrelation function in the quench dynamics of a Bloch state, which was observed previously (Zhang and Yang 2016 Europhys. Lett . 114 60001).
We point out that by the ``smoothness means fast decay'' principle in Fourier analysis, it is possible to infer the smoothness (or nonsmoothness) of the autocorrelation function from a mere glimpse of the initial state. Specifically, for a generic system with smooth eigenstates, the smoother an initial state is, the faster its decomposition coefficients with respect to the eigenstates of the system decay, and in turn the smoother the autocorrelation function is. The idea is illustrated with three increasingly smooth functions in an infinite square well. By using the Mellin transform, we also find that the nonsmoothness or singularity of the initial state affects the short-time behavior of its autocorrelation function. In particular, a sufficiently nonsmooth or singular initial state could decay in a nonquadratic power law, with the exponent continuously tunable. We now understand the periodic cusps of the autocorrelation function in the quench dynamics of a Bloch state, which was observed previously [Zhang and Yang, EPL 114, 60001 (2016)].
We report some nonsmooth dynamics of a Bloch state in a one-dimensional tight binding model with the periodic boundary condition. After a sudden change of the potential of an arbitrary site, quantities like the survival probability of the particle in the initial Bloch state show cusps periodically, with the period being the Heisenberg time associated with the energy spectrum. This phenomenon is a nonperturbative counterpart of the nonsmooth dynamics observed previously (Z hang J. M. and H aque M., arXiv:1404.4280) in a periodically driven tight binding model. Underlying the cusps is an exactly solvable model, which consists of equally spaced levels extending from − ∞ to + ∞ , between which two arbitrary levels are coupled to each other by the same strength.
We take a one-dimensional tight-binding chain with periodic boundary condition and put a particle in an arbitrary Bloch state, then quench it by suddenly changing the potential of an arbitrary site. In the ensuing time evolution, the probability density of the wave function at an arbitrary site jumps indefinitely between plateaus . This phenomenon adds to a former one in which the survival probability of the particle in the initial Bloch state shows cusps periodically, which was found in the same scenario ( Zhang J. M. and Yang H.-T. , EPL , 114 (2016) 60001). The plateaus support the scattering wave picture of the quench dynamics of the Bloch state. Underlying the cusps and jumps is the exactly solvable, nonanalytic dynamics of a Luttinger-like model, based on which, the locations of the jumps and the heights of the plateaus are accurately predicted.
From the analysis of the relaxation process of isolated lattice many-body quantum systems quenched far from equilibrium, we deduce a criterion for predicting when they are certain to thermalize. It is based on the algebraic behavior $\ensuremath{\propto}{t}^{\ensuremath{-}\ensuremath{\gamma}}$ of the survival probability at long times. We show that the value of the power-law exponent $\ensuremath{\gamma}$ depends on the shape and filling of the weighted energy distribution of the initial state. Two scenarios are explored in detail: $\ensuremath{\gamma}\ensuremath{\ge}2$ and $\ensuremath{\gamma}<1$. Exponents $\ensuremath{\gamma}\ensuremath{\ge}2$ imply that the energy distribution of the initial state is ergodically filled and the eigenstates are uncorrelated, so thermalization is guaranteed to happen. In this case, the power-law behavior is caused by bounds in the energy spectrum. Decays with $\ensuremath{\gamma}<1$ emerge when the energy eigenstates are correlated and signal lack of ergodicity. They are typical of systems undergoing localization due to strong onsite disorder and are found also in clean integrable systems.
The rate and extent of the exploration of the available phase space of a bound quantum mechanical system are shown to depend on the repulsion of energy eigenstates. Central to the argument is the Fourier transform relating the survival probability (in time) of an initially prepared nonstationary state and the (frequency) autocorrelation function of the excitation spectrum. Strong repulsion of states, as in the Wigner surmise, leads to a rapid dephasing of the initially coherently prepared state. The rate and extent of sampling of phase space depend not only on the spacing distribution but also on the intensity fluctuations. The rate of dephasing is equal to that inferred from the width of the spectral autocorrelation function.
The eigenstate thermalization hypothesis (ETH) describes the properties of diagonal and off-diagonal matrix elements of local operators in the eigenenergy basis. In this work, we propose a relation between (i) the singular behavior of the off-diagonal part of ETH at small energy differences and (ii) the smooth profile of the diagonal part of ETH as a function of the energy density. We establish this connection from the decay of the autocorrelation functions of local operators, which is constrained by the presence of local conserved quantities whose evolution is described by hydrodynamics. We corroborate our predictions through numerical simulations of two distinct nonintegrable spin-1 Ising models, exhibiting diffusive and superdiffusive transport behaviors. The simulations are performed using dynamical quantum typicality up to 18 spins, for both infinite- and finite-temperature regimes.