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arXiv:1702.03165 · Journal of Statistical Mechanics Theory and Experiment

Periodically driven random quantum spin chains: real-space renormalization for Floquet localized phases

Cécile Monthus

Published Jul 1, 2017• 17 citations
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Abstract

Abstract When random quantum spin chains are submitted to some periodic Floquet driving, the eigenstates of the time-evolution operator over one period can be localized in real space. For the case of periodic quenches between two Hamiltonians (or periodic kicks), where the time-evolution operator over one period reduces to the product of two simple transfer matrices, we propose a block-self-dual renormalization procedure to construct the localized eigenstates of the Floquet dynamics. We also discuss the corresponding strong disorder renormalization procedure, that generalizes the RSRG-X procedure to construct the localized eigenstates of time-independent Hamiltonians.

DOI
10.1088/1742-5468/aa75dd

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Abstract When random quantum spin chains are submitted to some periodic Floquet driving, the eigenstates of the time-evolution operator over one period can be localized in real space. For the case of periodic quenches between two Hamiltonians (or periodic kicks), where the time-evolution operator over one period reduces to the product of two simple transfer matrices, we propose a block-self-dual renormalization procedure to construct the localized eigenstates of the Floquet dynamics. We also discuss the corresponding strong disorder renormalization procedure, that generalizes the RSRG-X procedure to construct the localized eigenstates of time-independent Hamiltonians.
Abstract The Vosk–Altman strong disorder renormalization for the unitary dynamics of various random quantum spin chains is reformulated as follows: the local degree of freedom characterized by the highest eigenfrequency Ω can be considered as a high-frequency-Floquet-periodic-driving for the neighboring slower degrees of freedom. Then the two first orders of the high-frequency expansion for the effective Floquet Hamiltonian can be used to generate the emergent local integrals of motion (LIOMs) and to derive the renormalization rules for the effective dynamics of the remaining degrees of freedom. The flow for this effective Floquet Hamiltonian is equivalent to the RSRG-X procedure to construct the whole set of eigenstates that generalizes the Fisher RSRG procedure constructing the ground state. This general framework is applied to the random-transverse-field XXZ spin chain in its many-body-localized phase, in order to derive the renormalization rules associated to the elimination of the biggest transverse field and to the elimination of the biggest coupling respectively.
We generalize the recently introduced Density-Matrix Renormalization Group (DMRG-X) [Khemani et al, PRL 2016] algorithm to obtain Floquet eigenstates of one-dimensional, periodically driven many-body localized systems. This generalization is made possible by the fact that the time-evolution operator for a period can be efficiently represented using a matrix-product operator. We first benchmark the method by comparing to exact diagonalization for small systems. We then obtain Floquet eigenstates for larger systems and show unambiguously that the characteristic area-law scaling remains robust.
We study the energy absorption in real time of a disordered quantum spin chain subjected to coherent monochromatic periodic driving. We determine characteristic fingerprints of the well-known ergodic (Floquet-Eigenstate thermalization hypothesis for slow driving/weak disorder) and many-body localized (Floquet--many-body localization for fast driving/strong disorder) phases. In addition, we identify an intermediate regime, where the energy density of the system---unlike the entanglement entropy a local and bounded observable---grows logarithmically slowly over a very large time window.
Quenched randomness can lead to robust nonequilibrium phases of matter in periodically driven (Floquet) systems. Analyzing transitions between such dynamical phases requires a method capable of treating the twin complexities of disorder and discrete time-translation symmetry. We introduce a real-space renormalization group approach, asymptotically exact in the strong-disorder limit, and exemplify its use on the periodically driven interacting quantum Ising model. We analyze the universal physics near the critical lines and multicritical point of this model, and demonstrate the robustness of our results to the inclusion of weak interactions.
We generalize the recently introduced Density‐Matrix Renormalization Group (DMRG‐X) [Khemani et al, PRL 2016] algorithm to obtain Floquet eigenstates of one‐dimensional, periodically driven many‐body localized systems. This generalization is made possible by the fact that the time‐evolution operator for a period can be efficiently represented using a matrix‐product operator. We first benchmark the method by comparing to exact diagonalization for small systems. We then obtain Floquet eigenstates for larger systems and show unambiguously that the characteristic area‐law scaling remains robust. image
We consider a one-dimensional spin chain system with quenched disorder and in the presence of a local periodic drive. We study the time evolution of the system in the Floquet basis and evaluate the fidelity susceptibility, which is a measure of how a given state changes under a small perturbation, of states to a weak periodic drive. We demonstrate that the statistical properties of the fidelity susceptibility over different disorder realizations can be used to identify two phases of the system: (1) the many-body localized phase, in which the susceptibility exhibits long tails while its average value decreases rapidly as disorder increases, and (2) the ergodic phase, in which the susceptibility distribution is narrow and its average value weakly depends on disorder. This distinction in the average value of the susceptibility between the two phases develops readily for systems with ten or more spins. Therefore, recently built networks of qubits subject to a local drive can simulate dynamics of a system in the many-body localization regime. We also show that the spin accumulation speed is correlated with the fidelity susceptibility and can also be used to distinguish the two phases.