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arXiv:1103.0787 · Physical Review E

Quantum quenches in disordered systems: Approach to thermal equilibrium without a typical relaxation time

Ehsan Khatami, Marcos Rigol, Armando Relaño, Antonio M. Garcı́a-Garcı́a

Published May 15, 2012• 79 citations
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Abstract

We study spectral properties and the dynamics after a quench of one-dimensional spinless fermions with short-range interactions and long-range random hopping. We show that a sufficiently fast decay of the hopping term promotes localization effects at finite temperature, which prevents thermalization even if the classical motion is chaotic. For slower decays, we find that thermalization does occur. However, within this model, the latter regime falls in an unexpected universality class, namely, observables exhibit a power-law (as opposed to an exponential) approach to their thermal expectation values.

DOI
10.1103/physreve.85.050102

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We study spectral properties and the dynamics after a quench of one-dimensional spinless fermions with short-range interactions and long-range random hopping. We show that a sufficiently fast decay of the hopping term promotes localization effects at finite temperature, which prevents thermalization even if the classical motion is chaotic. For slower decays, we find that thermalization does occur. However, within this model, the latter regime falls in an unexpected universality class, namely, observables exhibit a power-law (as opposed to an exponential) approach to their thermal expectation values.
In the present thesis we study the unitary dynamics and the thermalization properties of free-fermion-like Hamiltonians after a sudden quantum quench in presence of disorder. With analytical and numerical arguments, we show that the existence of a stationary state and its description with a generalized Gibbs ensemble (GGE) depend crucially on the observable considered (local versus extensive, one-body versus many-body) and on the localization properties of the final Hamiltonian. We then show an extension of the Wang-Landau algorithm which allows the computation of weighted distributions associated to quantum quenches, like the diagonal and the GGE ensemble expectation-value distributions. We present results on three one-dimensional models, the Anderson model, a disordered one-dimensional fermionic chain with long-range hopping, and the disordered Ising/XY spin chain.
It has been shown that a quantum quench of interactions in a one-dimensional fermion system at zero temperature induces a universal power law proportional to t^{-2} t − 2 in its long-time dynamics. In this paper we demonstrate that this behaviour is robust even in the presence of thermal effects. The system is initially prepared in a thermal state, then at a given time the bath is disconnected and the interaction strength is suddenly quenched. The corresponding effects on the long times dynamics of the non-equilibrium fermionic spectral function are considered. We show that the non-universal power laws, present at zero temperature, acquire an exponential decay due to thermal effects and are washed out at long times, while the universal behaviour proportional to t^{-2} t − 2 is always present. To verify our findings, we argue that these features are also visible in transport properties at finite temperature. The long-time dynamics of the current injected from a biased probe exhibits the same universal power law relaxation, in sharp contrast with the non-quenched case which features a fast exponential decay of the current towards its steady value, and thus represents a fingerprint of quench-induced dynamics. Finally, we show that a proper tuning of the probe temperature, compared to that of the one-dimensional channel, can enhance the visibility of the universal power-law behaviour.
Many-body localized (MBL) systems are characterized by the absence of transport and thermalization and, therefore, cannot be described by conventional statistical mechanics. In this paper, using analytic arguments and numerical simulations, we study the behavior of local observables in an isolated MBL system following a quantum quench. For the case of a global quench, we find that the local observables reach stationary, highly nonthermal values at long times as a result of slow dephasing characteristic of the MBL phase. These stationary values retain the local memory of the initial state due to the existence of local integrals of motion in the MBL phase. The temporal fluctuations around stationary values exhibit universal power-law decay in time, with an exponent set by the localization length and the diagonal entropy of the initial state. Such a power-law decay holds for any local observable and is related to the logarithmic in time growth of entanglement in the MBL phase. This behavior distinguishes the MBL phase from both the Anderson insulator (where no stationary state is reached) and from the ergodic phase (where relaxation is expected to be exponential). For the case of a local quench, we also find a power-law approach of local observables to their stationary values when the system is prepared in a mixed state. Quench protocols considered in this paper can be naturally implemented in systems of ultracold atoms in disordered optical lattices, and the behavior of local observables provides a direct experimental signature of many-body localization.
We study the unitary dynamics and the thermalization properties of free-fermion-like Hamiltonians after a sudden quantum quench, extending the results of S. Ziraldo et al. [Phys. Rev. Lett. 109, 247205 (2012)]. With analytical and numerical arguments, we show that the existence of a stationary state and its description with a generalized Gibbs ensemble (GGE) depend crucially on the observable considered (local versus extensive) and on the localization properties of the final Hamiltonian. We present results on two one-dimensional (1D) models, the disordered 1D fermionic chain with long-range hopping and the disordered Ising/XYXYXY spin chain. We analytically prove that, while time averages of one-body operators are perfectly reproduced by GGE (even for finite-size systems, if time integrals are extended beyond revivals), time averages of many-body operators might show clear deviations from the GGE prediction when disorder-induced localization of the eigenstates is at play.
Quench dynamics in a two-dimensional system of interacting fermions is analyzed within the semiclassical truncated Wigner approximation (TWA). The models with short-range and long-range interactions are considered. We show that in the latter case, the TWA is very accurate, becoming asymptotically exact in the infinite-range limit, provided that the semiclassical Hamiltonian is correctly identified. Within the TWA, different dynamical timescales of charges and spins can be clearly distinguished. Interestingly, for a weak and moderate disorder strength, we observe subdiffusive behavior of charges, while spins exhibit diffusive dynamics. At strong disorder, the quantum Fisher information shows logarithmic growth in time with a slower increase for charges than for spins. It is shown that in contrast to the short-range model, strong inhomogeneities such as domain walls in the initial state can significantly slow down thermalization dynamics, especially at weak disorder. This behavior can put additional challenges in designing cold-atom experimental protocols aimed to analyze possible many-body localization in such systems. While within this approach we cannot make any definite statements about the existence of a many-body localized phase, we see a very fast crossover as a function of disorder strength from rapidly thermalizing to a slow glassylike regime both for the short-range and long-range models.
We study the expectation values of observables and correlation functions at long times after a global quantum quench. Our focus is on metallic (`gapless') fermionic many-body models and small quenches. The system is prepared in an eigenstate of an initial Hamiltonian, and the time evolution is performed with a final Hamiltonian which differs from the initial one in the value of one global parameter. We first derive general relations between time-averaged expectation values of observables as well as correlation functions and those obtained in an eigenstate of the final Hamiltonian. Our results are valid to linear and quadratic order in the quench parameter g and generalize prior insights in several essential ways. This allows us to develop a phenomenology for the thermalization of local quantities up to a given order in g. Our phenomenology is put to a test in several case studies of one-dimensional models representative of four distinct classes of Hamiltonians: quadratic ones, effectively quadratic ones, those characterized by an extensive set of (quasi-) local integrals of motion, and those for which no such set is known (and believed to be nonexistent). We show that for each of these models, all observables and correlation functions thermalize to linear order in g. The more local a given quantity, the longer the linear behavior prevails when increasing g. Typical local correlation functions and observables for which the term O(g) vanishes thermalize even to order g^2. Our results show that lowest order thermalization of local observables is an ubiquitous phenomenon even in models with extensive sets of integrals of motion.