Author

# E. Jonathan Torres-Herrera

Bio: E. Jonathan Torres-Herrera is an academic researcher from Benemérita Universidad Autónoma de Puebla. The author has contributed to research in topics: Random matrix & Quantum. The author has an hindex of 7, co-authored 17 publications receiving 203 citations.

Topics: Random matrix, Quantum, Quantum system, Observable, Correlation function

##### Papers

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TL;DR: In this paper, the authors identify the time scales involved in the relaxation process of isolated quantum systems that have many interacting particles by analyzing dynamical manifestations of spectral correlations and show that the Thouless time and the relaxation time increase exponentially with system size.

Abstract: A major open question in studies of nonequilibrium quantum dynamics is the identification of the time scales involved in the relaxation process of isolated quantum systems that have many interacting particles. We demonstrate that long time scales can be analytically found by analyzing dynamical manifestations of spectral correlations. Using this approach, we show that the Thouless time ${t}_{\text{Th}}$ and the relaxation time ${t}_{\text{R}}$ increase exponentially with system size. We define ${t}_{\text{Th}}$ as the time at which the spread of the initial state in the many-body Hilbert space is complete and verify that it agrees with the inverse of the Thouless energy. ${t}_{\text{Th}}$ marks the point beyond which the dynamics acquire universal features, while relaxation happens later when the evolution reaches a stationary state. In chaotic systems, ${t}_{\text{Th}}\ensuremath{\ll}{t}_{\text{R}}$, while for systems approaching a many-body localized phase, ${t}_{\text{Th}}\ensuremath{\rightarrow}{t}_{\text{R}}$. Our analytical results for ${t}_{\text{Th}}$ and ${t}_{\text{R}}$ are obtained for the survival probability, which is a global quantity. We show numerically that the same time scales appear also in the evolution of the spin autocorrelation function, which is an experimental local observable. Our studies are carried out for realistic many-body quantum models. The results are compared with those for random matrices.

94 citations

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58 citations

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TL;DR: In this article, a single defect site brings integrable spin-1/2 and spin 1 models to the chaotic domain and the correlation hole and the off-diagonal elements of observables detect this transition despite the presence of symmetries.

Abstract: This work shows that a single defect site brings integrable spin-1/2 and spin-1 models to the chaotic domain. The correlation hole and the off-diagonal elements of observables detect this transition despite the presence of symmetries.

37 citations

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TL;DR: In this paper, the authors show that the survival probability in chaotic systems is not self-averaging at any time scale, even when evolved under full random matrices, and they also analyze the participation ratio, Renyi entropies, the spin autocorrelation function from experiments with cold atoms, and the connected spin-spin correlation function from experiment with ion traps.

Abstract: Despite its importance to experiments, numerical simulations, and the development of theoretical models, self-averaging in many-body quantum systems out of equilibrium remains underinvestigated. Usually, in the chaotic regime, self-averaging is taken for granted. The numerical and analytical results presented here force us to rethink these expectations. They demonstrate that self-averaging properties depend on the quantity and also on the time scale considered. We show analytically that the survival probability in chaotic systems is not self-averaging at any time scale, even when evolved under full random matrices. We also analyze the participation ratio, Renyi entropies, the spin autocorrelation function from experiments with cold atoms, and the connected spin-spin correlation function from experiments with ion traps. We find that self-averaging holds at short times for the quantities that are local in space, while at long times, self-averaging applies for quantities that are local in time. Various behaviors are revealed at intermediate time scales.

29 citations

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TL;DR: In this article, the authors analyzed the degree of energy-level repulsion, the structure of the eigenstates, and the time evolution of the finite 1D Anderson model as a function of the parameter W ∈ {W ∈ √ 2 L √ (W √ √ 1/L) √ W √ 3/L √ N √ n/L}, and showed that the probability in time to find a particle initially placed on the first site of an open chain decays as fast as in full random matrices and much

Abstract: This work shows that dynamical features typical of full random matrices can be observed also in the simple finite one-dimensional (1D) noninteracting Anderson model with nearest-neighbor couplings. In the thermodynamic limit, all eigenstates of this model are exponentially localized in configuration space for any infinitesimal on-site disorder strength $W$. But this is not the case when the model is finite and the localization length is larger than the system size $L$, which is a picture that can be experimentally investigated. We analyze the degree of energy-level repulsion, the structure of the eigenstates, and the time evolution of the finite 1D Anderson model as a function of the parameter $\ensuremath{\xi}\ensuremath{\propto}{({W}^{2}L)}^{\ensuremath{-}1}$. As $\ensuremath{\xi}$ increases, all energy-level statistics typical of random matrix theory are observed. The statistics are reflected in the corresponding eigenstates and also in the dynamics. We show that the probability in time to find a particle initially placed on the first site of an open chain decays as fast as in full random matrices and much faster than when the particle is initially placed far from the edges. We also see that at long times, the presence of energy-level repulsion manifests in the form of the correlation hole. In addition, our results demonstrate that the hole is not exclusive to random matrix statistics, but emerges also for $W=0$, when it is in fact deeper.

16 citations

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TL;DR: In this article, the authors present a basic introduction to the topic of many-body localization, using the simple example of a quantum spin chain that allows us to illustrate several of the properties of this phase.

547 citations

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TL;DR: It is argued that the ergodicity breaking transition in interacting spin chains occurs when both time scales are of the same order, t_{Th}≈t_{H}, and g becomes a system-size independent constant, and carries certain analogies with the Anderson localization transition.

Abstract: Characterizing states of matter through the lens of their ergodic properties is a fascinating new direction of research. In the quantum realm, the many-body localization (MBL) was proposed to be the paradigmatic ergodicity breaking phenomenon, which extends the concept of Anderson localization to interacting systems. At the same time, random matrix theory has established a powerful framework for characterizing the onset of quantum chaos and ergodicity (or the absence thereof) in quantum many-body systems. Here we numerically study the spectral statistics of disordered interacting spin chains, which represent prototype models expected to exhibit MBL. We study the ergodicity indicator $g={log}_{10}({t}_{\mathrm{H}}/{t}_{\mathrm{Th}})$, which is defined through the ratio of two characteristic many-body time scales, the Thouless time ${t}_{\mathrm{Th}}$ and the Heisenberg time ${t}_{\mathrm{H}}$, and hence resembles the logarithm of the dimensionless conductance introduced in the context of Anderson localization. We argue that the ergodicity breaking transition in interacting spin chains occurs when both time scales are of the same order, ${t}_{\mathrm{Th}}\ensuremath{\approx}{t}_{\mathrm{H}}$, and $g$ becomes a system-size independent constant. Hence, the ergodicity breaking transition in many-body systems carries certain analogies with the Anderson localization transition. Intriguingly, using a Berezinskii-Kosterlitz-Thouless correlation length we observe a scaling solution of $g$ across the transition, which allows for detection of the crossing point in finite systems. We discuss the observation that scaled results in finite systems by increasing the system size exhibit a flow towards the quantum chaotic regime.

232 citations

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TL;DR: For weak interactions, the entanglement entropy grows as ξln(Vt/ℏ), where V is the interaction strength, and ξ is the single-particle localization length.

Abstract: Recent numerical work by Bardarson, Pollmann, and Moore revealed a slow, logarithmic in time, growth of the entanglement entropy for initial product states in a putative many-body localized phase. We show that this surprising phenomenon results from the dephasing due to exponentially small interaction-induced corrections to the eigenenergies of different states. For weak interactions, we find that the entanglement entropy grows as ξln(Vt/ℏ), where V is the interaction strength, and ξ is the single-particle localization length. The saturated value of the entanglement entropy at long times is determined by the participation ratios of the initial state over the eigenstates of the subsystem. Our work shows that the logarithmic entanglement growth is a universal phenomenon characteristic of the many-body localized phase in any number of spatial dimensions, and reveals a broad hierarchy of dephasing time scales present in such a phase.

231 citations

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TL;DR: In this article, the ergodic phase of many-body localization (MBL) is considered and the available numerically exact and approximate methods for its study are discussed and a phenomenological explanation of its dynamical properties is presented.

Abstract: Recent studies point towards nontriviality of the ergodic phase in systems exhibiting many-body localization (MBL), which shows subexponential relaxation of local observables, subdiffusive transport and sublinear spreading of the entanglement entropy Here we review the dynamical properties of this phase and the available numerically exact and approximate methods for its study We discuss in which sense this phase could be considered ergodic and present possible phenomenological explanations of its dynamical properties We close by analyzing to which extent the proposed explanations were verified by numerical studies and present the open questions in this field

228 citations

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TL;DR: It is argued that the two parameter scaling breaks down in the vicinity of the transition to the localized phase, signaling a slowing-down of dynamics.

Abstract: Spectral statistics of disordered systems encode Thouless and Heisenberg timescales, whose ratio determines whether the system is chaotic or localized. We show that the scaling of the Thouless time with the system size and disorder strength is very similar in one-body Anderson models and in disordered quantum many-body systems. We argue that the two parameter scaling breaks down in the vicinity of the transition to the localized phase, signaling a slowing-down of dynamics.

159 citations