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Survival Probability of the Neel State in Clean and Disordered Systems: An Overview

Eduardo Jonathan Torres-Herrera, Marco Távora, Léa Ferreira dos Santos

Published Jan 1, 2016• 12 citations
OpenAlex

Abstract

"In this work we provide an overview of our recent results about the quench dynamics of one-dimensional many-body quantum systems described by spin-1/2 models. To illustrate those general results, here we employ a particular and experimentally accessible initial state, namely the Neel state. Both cases are considered: clean chains with- ´ out any disorder and disordered systems with static random on-site magnetic fields. The quantity used for the analysis is the probability for finding the initial state later in time, the so-called survival probability. At short times, the survival probability may decay faster than exponentially, Gaussian behaviors and even the limit established by the energy-time uncertainty relation are displayed. The dynamics at long times slows down significantly and shows a powerlaw behavior. For both scenarios, we provide analytical expressions that agree very well with our numerical results."

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"In this work we provide an overview of our recent results about the quench dynamics of one-dimensional many-body quantum systems described by spin-1/2 models. To illustrate those general results, here we employ a particular and experimentally accessible initial state, namely the Neel state. Both cases are considered: clean chains with- ´ out any disorder and disordered systems with static random on-site magnetic fields. The quantity used for the analysis is the probability for finding the initial state later in time, the so-called survival probability. At short times, the survival probability may decay faster than exponentially, Gaussian behaviors and even the limit established by the energy-time uncertainty relation are displayed. The dynamics at long times slows down significantly and shows a powerlaw behavior. For both scenarios, we provide analytical expressions that agree very well with our numerical results."
The isolated one-dimensional Heisenberg model with static random magnetic fields has become paradigmatic for the analysis of many-body localization. Here, we study the dynamics of this system initially prepared in a highly-excited nonstationary state. Our focus is on the probability for finding the initial state later in time, the so-called survival probability. Two distinct behaviors are identified before equilibration. At short times, the decay is very fast and equivalent to that of clean systems. It subsequently slows down and develops a power-law behavior with an exponent that coincides with the multifractal dimension of the eigenstates.
Possible strategies to describe analytically the dynamics of many-body quantum systems out of equilibrium include the use of solvable models and of full random matrices. None of the two approaches represent actual realistic systems, but they serve as references for the studies of these ones. We take the second path and obtain analytical expressions for the survival probability, density imbalance, and out-of-time-ordered correlator. Using these findings, we then propose an approximate expression that matches very well numerical results for the evolution of realistic finite quantum systems that are strongly chaotic and quenched far from equilibrium. In the case of the survival probability, the expression proposed covers all different time scales, from the moment the system is taken out of equilibrium to the moment it reaches a new equilibrium. The realistic systems considered are described by one-dimensional spin-1/2 models.
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.
In a disordered system, a quantity is self-averaging when the ratio between its variance for disorder realizations and the square of its mean decreases as the system size increases. Here, we consider a chaotic disordered many-body quantum system and search for a relationship between self-averaging behavior and the properties of the distributions over disorder realizations of various quantities and at different timescales. An exponential distribution, as found for the survival probability at long times, explains its lack of self-averaging, since the mean and the dispersion are equal. Gaussian distributions, however, are obtained for both self-averaging and non-self-averaging quantities. Our studies show also that one can make conclusions about the self-averaging behavior of one quantity based on the distribution of another related quantity. This strategy allows for semianalytical results, and thus circumvents the limitations of numerical scaling analysis, which are restricted to few system sizes.
We present a theoretical study of the survival probability of a state initially prepared in the one-particle sector of a multi-qubit system. The motivation for our work is the ongoing laboratory development of multi-qubit platforms based on superconducting circuits. Using elementary concepts of random matrix theory, we obtain analytic expressions for the survival probability in mathematical models of platforms which, albeit stylized, have been previously shown to provide relevant benchmarks for experimental data. In particular, we show that the decay properties are sensitive to the property of the Hamilton operator to have extended states. The survival probability does not appear instead to depend on whether the interaction between qubits is described by a Gaussian orthogonal ensemble (often interpreted as a model of ''chaotic'' dynamics) or is modeled by an analytically solvable chain. We interpret this phenomenon as a manifestation of a general mechanism for the emergence of equilibration in purely unitary dynamics. Finally, under the same hypothesis of an initial preparation with projection on a large fraction of the extended eigenstates of the Hamilton operator, we show how to extend the classical Kac-Mazur-Montroll estimate of the return time to the quantum survival probability.
We study numerically and analytically the quench dynamics of isolated many-body quantum systems. Using full random matrices from the Gaussian orthogonal ensemble, we obtain analytical expressions for the evolution of the survival probability, density imbalance, and out-of-time-ordered correlator. They are compared with numerical results for a one-dimensional-disordered model with two-body interactions and shown to bound the decay rate of this realistic system. Power-law decays are seen at intermediate times, and dips below the infinite time averages (correlation holes) occur at long times for all three quantities when the system exhibits level repulsion. The fact that these features are shared by both the random matrix and the realistic disordered model indicates that they are generic to nonintegrable interacting quantum systems out of equilibrium. Assisted by the random matrix analytical results, we propose expressions that describe extremely well the dynamics of the realistic chaotic system at different time scales.