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arXiv:1706.07352 · Journal of Physics A Mathematical and Theoretical

Strong disorder renormalization for the dynamics of many-body-localized systems: iterative elimination of the fastest degree of freedom via the Floquet expansion

Cécile Monthus

Published Jun 4, 2018• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8121/aac672

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arXiv:1906.00004•May 31, 2019•1 citation
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.
Abstract For the many-body-localized phase of random Majorana models, a general strong disorder real-space renormalization procedure known as RSRG-X (Pekker et al 2014 Phys. Rev . X 4 011052) is described to produce the whole set of excited states, via the iterative construction of the local integrals of motion (LIOMs). The RG rules are then explicitly derived for arbitrary quadratic Hamiltonians (free-fermions models) and for the Kitaev chain with local interactions involving even numbers of consecutive Majorana fermions. The emphasis is put on the advantages of the Majorana language over the usual quantum spin language to formulate unified RSRG-X rules.
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 A fully many-body localized (FMBL) quantum disordered system is characterized by the emergence of an extensive number of local conserved operators that prevents the relaxation towards thermal equilibrium. These local conserved operators can be seen as the building blocks of the whole set of eigenstates. In this paper, we propose to construct them explicitly via block real-space renormalization. The principle is that each renormalization group step diagonalizes the smallest remaining blocks and produces a conserved operator for each block. The final output for a chain of N spins is a hierarchical organization of the N conserved operators with layers. The system size nature of the conserved operators of the top layers is necessary to describe the possible long-range order of the excited eigenstates and the possible critical points between different FMBL phases. We discuss the similarities and the differences with the strong disorder RSRG-X method that generates the whole set of the 2 N eigenstates via a binary tree of N layers. The approach is applied to the long-range quantum spin-glass Ising model, where the constructed excited eigenstates are found to be exactly like ground states in another disorder realization, so that they can be either in the paramagnetic phase, in the spin-glass phase or critical.
This work presents a theoretical investigation of dynamic stabilization in interacting quantum many-body systems via high-frequency periodic driving. Using Floquet theory and the Magnus expansion, we derive the effective time-independent Hamiltonian governing the slow dynamics of a driven spin chain. We show that the external field induces a parametric renormalization of the transverse tunneling terms through a zeroth-order Bessel function dependence, leading to a critical drive amplitude at which transport is strongly suppressed. This suppression of the lowest-order hopping term gives rise to a long-lived prethermal regime in which many-body thermalization is significantly delayed. The resulting dynamics exhibit a sharp reduction in the short-to-intermediate time entanglement entropy growth rate. The framework is consistent with known results in Floquet engineering and coherent destruction of tunneling, and provides a unified interpretation of transport suppression and delayed scrambling in periodically driven systems. A falsifiable experimental protocol using ultracold atoms in optical lattices is proposed to validate the predicted threshold behavior and prethermal stabilization mechanism.
We study the non-equilibrium dynamics of a disordered periodically driven quantum spin chain, with the competition between the interaction, disorder, and Floquet driving being of particular interest. We study dynamics of entanglement entropy, energy absorption to characterize dynamical regimes of the system whether it stays in the Floquet-MBL(many-body localization) region or thermalized region. Starting with a product state in the computational basis, followed by reduced density matrix which in turn gives rise to the entanglement entropy density. With the strength of the interaction, the transverse field, the parallel field, the disorder strength W and the driving frequency, we discover the distinct behaviors of fast delocalization and logarithmic entanglement growth and long-lasting memory of the initial state, indicative of localized or prethermal Floquet regimes. We observe that strong disorder arrests transport and enables slow entanglement dynamics, whereas strong driving frequency arrests energy absorption and creates a long-lived non-equilibrium state. Conversely, weak disorder or low driving frequency leads to delocalization. The outcomes show strong support for non-equilibrium phases in driven many-body systems.
Driving a quantum system periodically in time can profoundly alter its long-time correlations and give rise to exotic quantum states of matter. The complexity of the combination of many-body correlations and dynamic manipulations has the potential to uncover a whole field of new phenomena, but the theoretical and numerical understanding becomes extremely difficult. We now propose a promising numerical method by generalizing the density matrix renormalization group to a superposition of Fourier components of periodically driven many-body systems using Floquet theory. With this method we can study the full time-dependent quantum solution in a large parameter range for all evolution times, beyond the commonly used high-frequency approximations. Numerical results are presented for the isotropic Heisenberg antiferromagnetic spin-1/2 chain under both local(edge) and global driving for spin-spin correlations and temporal fluctuations. As the frequency is lowered, we demonstrate that more and more Fourier components become relevant and determine strong length- and frequency-dependent changes of the quantum correlations that cannot be described by effective static models.