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arXiv:1712.05793 · Physical review. B./Physical review. B

Remnants of Anderson localization in prethermalization induced by white noise

Salvatore Lorenzo, Tony J. G. Apollaro, G. Massimo Palma, Rahul Nandkishore, Alessandro Silva, Jamir Marino

Published Aug 3, 2018• 20 citations
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Abstract

We study the nonequilibrium evolution of a one-dimensional quantum Ising chain with spatially disordered, time-dependent, transverse fields characterized by white noise correlation dynamics. We establish prethermalization in this model, showing that the quench dynamics of the on-site transverse magnetization first approaches a metastable state unaffected by noise fluctuations, and then relaxes exponentially fast toward an infinite temperature state as a result of the noise. We also consider energy transport in the model, starting from an inhomogeneous state with two domain walls which separate regions characterized by spins with opposite transverse magnetization. We observe at intermediate timescales a phenomenology akin to Anderson localization: energy remains localized within the two domain walls, until the Markovian noise destroys coherence and, accordingly, disorder-induced localization, allowing the system to relax toward the late stages of its nonequilibrium dynamics. We compare our results with the simpler case of a noisy quantum Ising chain without disorder, and we find that the prethermal plateau is a generic property of spin chains with weak noise, while the phenomenon of prethermal Anderson localization is a specific feature arising from the competition of noise and disorder in the real-time transport properties of the system.

DOI
10.1103/physrevb.98.054302

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We study the nonequilibrium evolution of a one-dimensional quantum Ising chain with spatially disordered, time-dependent, transverse fields characterized by white noise correlation dynamics. We establish prethermalization in this model, showing that the quench dynamics of the on-site transverse magnetization first approaches a metastable state unaffected by noise fluctuations, and then relaxes exponentially fast toward an infinite temperature state as a result of the noise. We also consider energy transport in the model, starting from an inhomogeneous state with two domain walls which separate regions characterized by spins with opposite transverse magnetization. We observe at intermediate timescales a phenomenology akin to Anderson localization: energy remains localized within the two domain walls, until the Markovian noise destroys coherence and, accordingly, disorder-induced localization, allowing the system to relax toward the late stages of its nonequilibrium dynamics. We compare our results with the simpler case of a noisy quantum Ising chain without disorder, and we find that the prethermal plateau is a generic property of spin chains with weak noise, while the phenomenon of prethermal Anderson localization is a specific feature arising from the competition of noise and disorder in the real-time transport properties of the system.
We study the dynamics of thermalization resulting from a time-dependent noise in a quantum Ising chain subject to a sudden quench of the transverse magnetic field. For weak noise the dynamics shows a prethermalized state at intermediate time scales, eventually drifting towards an asymptotic infinite temperature steady state characterized by diffusive behavior. By computing analytically the density of kinks, as well as the transverse and longitudinal magnetic field correlators, we characterize these two regimes, their observability, and their signatures in the various physical quantities.
We discuss the nonequilibrium dynamics of a quantum Ising chain following a quantum quench of the transverse field and in the presence of a Gaussian time-dependent noise. We discuss the probability distribution of the work done on the system both for static and dynamic noise. While the effect of static noise is to smooth the low energy threshold of the statistic of the work, appearing for sudden quenches, a dynamical noise protocol affects also the spectral weight of such features. We also provide a detailed derivation of the kinetic equation for the Green's functions on the Keldysh contour and the time evolution of observables of physical interest, extending previously reported results [Marino and Silva, Phys. Rev. B 86, 060408 (2012)], and discussing the extension of the concept of prethermalization which can be used to study noisy quantum many-body Hamiltonians driven out of equilibrium.
The results presented in this PhD Thesis aim at understanding thermalization dynamics and, strictly speaking, pre-thermalization in both open and closed quantum spin chains driven out-of-equilibrium. In order to make analytical progress, we consider perturbations of the Quantum Ising Chain (QIC), one of the simplest integrable models, whose non-equilibrium dynamics has been studied in great details in the last few years. In Chapter 2 we introduce for the non-technical reader the main recent ideas behind quantum thermalization and relaxation in integrable and non-integrable quantum many body systems, while Chapter 3 contains an introduction to the current understanding of pre-thermalization. In Chapter 4 we start our study considering a QIC perturbed by a time dependent noise in the direction of the transverse magnetic field [6] [7]. We generalize the concept of pre-thermalization to a noisy quantum many body system driven away from equilibrium simultaneously by a quantum quench of the transverse field and by the noise, highlighting the mechanism behind pre-thermalization and the subsequent relaxation towards the thermal state in a wide class of physically relevant observables. Our main achievement is to show that pre-thermalization occurs quite generally up to time scales where the noise is dominated by the non-equilibrium dynamics induced by the quench of the transverse field, while for longer times pre-thermal plateaux disappear and the noise leaves different signatures in different observables. In Chapter 5 we consider the more challenging problem of pre-thermalization in a closed non-integrable quantum spin chain [8]. We show using a mapping of a weakly non-integrable spin chain into a hard core boson model, that pre-thermalization is successfully captured by a low-density description of the theory, which results in a integrable, bosonic model, that constitutes an excellent description for the out-of-equilibrium dynamics up to intermediate time scales. On the other hand, in order to investigate the late time dynamics we resort to a diagrammatic approach, which serves for a two-fold scope. First, we focus on the effect of the leading inelastic processes, realizing that they constitute only a sub-leading correction to the pre-thermal plateau, found within the effective low-energy description; and secondly, we show from a Quantum Boltzmann equation perspective how inelastic processes are the fundamental ingredient in order to have thermalization in the late time dynamics. The main technical tool used to derive the results in Chapter 4 and 5 is the Keldysh contour technique which is reviewed in the Appendix. This PhD Thesis is based on the following works: 1) J. Marino, A. Silva, Relaxation, pre-thermalization and diffusion in a noisy Quantum Ising Chain, Phys. Rev. B 86, 060408 (Rapid Communications) (2012). 2) M. Marcuzzi, J. Marino, A. Gambassi, A. Silva, Pre-thermalization in a non-integrable quantum spin chain after a quench, arXiv:1307.3738 (2013) (submitted to Phys. Rev. Lett.). 3) J. Marino, A. Silva, Non-Equilibrium Dynamics of a Noisy Quantum Ising Chain: statistics of the work and prethermalization after a sudden quench of the transverse field, arXiv:1309.7595 (2013) (submitted to Phys. Rev. B).
Abstract Nonequilibrium dynamics in an isolated quantum spin chain with long-range Ising interactions that decay as ( ) with the distance r is studied. It turns out that long-range interactions give rise to a big timescale separation, which causes prethermalization for all . This conclusion is deduced by comparing two important timescales relevant for relaxation dynamics; one is the relaxation time of local permutation operators, which are quasi-conserved quantities in this system, and the other is the timescale of the initial relaxation of local quantities due to the growth of quantum fluctuations. We also explore the entire nonequilibrium dynamics by using the discrete truncated Wigner approximation, which is consistent with the result mentioned above.
We study the dynamics caused by transport of transverse magnetization in one dimensional transverse Ising chain at zero temperature. We observe that a class of initial states in fermionic momentum-space produce spatial variation in transverse magnetization. Starting from such a state, we obtain the transverse magnetization analytically and then observe its dynamics in presence of a homogeneous constant field Γ\GammaΓ. In contradiction with general expectation, whatever be the strength of the field, the magnetization of the system does not become homogeneous even after infinite time. In each site, the dynamics is associated with oscillations having two different timescales. The envelope of the larger timescale oscillation decays algebraically having an exponent which is universal for the entire class of such initial states. Signature of quantum criticality is also embedded in the dynamics in two aspects : Behavioral change shown by the frequency of large timescale oscillation at any site and the corresponding magnetization at infinity when they cross the quantum critical point.
We study the effect of spatially correlated classical noise on both Anderson and many-body localization of a disordered fermionic chain. By analyzing the evolution of the particle density imbalance following a quench from an initial charge density wave state, we find prominent signatures of localization also in the presence of the time-dependent noise, even though the system eventually relaxes to the infinite temperature state. In particular, for sufficiently strong static disorder, we observe the onset of metastability, which becomes more prominent the stronger the spatial correlations of the noise. In this regime, we find that the imbalance decays as a stretched-exponential---a behavior characteristic of glassy systems. We identify a simple scaling behavior of the relevant relaxation times in terms of the static disorder and of the noise correlation length. We discuss how our results could be exploited to extract information about the localization length in experimental setups.