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arXiv:1205.0026 · Physical Review Letters

Many-Body Localization in One Dimension as a Dynamical Renormalization Group Fixed Point

Ronen Vosk, Ehud Altman

Published Feb 7, 2013• 363 citations
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Abstract

We formulate a dynamical real space renormalization group (RG) approach to describe the time evolution of a random spin-1/2 chain, or interacting fermions, initialized in a state with fixed particle positions. Within this approach we identify a many-body localized state of the chain as a dynamical infinite randomness fixed point. Near this fixed point our method becomes asymptotically exact, allowing analytic calculation of time dependent quantities. In particular, we explain the striking universal features in the growth of the entanglement seen in recent numerical simulations: unbounded logarithmic growth delayed by a time inversely proportional to the interaction strength. This is in striking contrast to the much slower entropy growth as loglogt found for noninteracting fermions with bond disorder. Nonetheless, even the interacting system does not thermalize in the long time limit. We attribute this to an infinite set of approximate integrals of motion revealed in the course of the RG flow, which become asymptotically exact conservation laws at the fixed point. Hence we identify the many-body localized state with an emergent generalized Gibbs ensemble.

DOI
10.1103/physrevlett.110.067204

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We formulate a dynamical real space renormalization group (RG) approach to describe the time evolution of a random spin-1/2 chain, or interacting fermions, initialized in a state with fixed particle positions. Within this approach we identify a many-body localized state of the chain as a dynamical infinite randomness fixed point. Near this fixed point our method becomes asymptotically exact, allowing analytic calculation of time dependent quantities. In particular, we explain the striking universal features in the growth of the entanglement seen in recent numerical simulations: unbounded logarithmic growth delayed by a time inversely proportional to the interaction strength. This is in striking contrast to the much slower entropy growth as loglogt found for noninteracting fermions with bond disorder. Nonetheless, even the interacting system does not thermalize in the long time limit. We attribute this to an infinite set of approximate integrals of motion revealed in the course of the RG flow, which become asymptotically exact conservation laws at the fixed point. Hence we identify the many-body localized state with an emergent generalized Gibbs ensemble.
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An important and incompletely answered question is whether a closed quantum system of many interacting particles can be localized by disorder. The time evolution of simple (unentangled) initial states is studied numerically for a system of interacting spinless fermions in one dimension described by the random-field XXZ Hamiltonian. Interactions induce a dramatic change in the propagation of entanglement and a smaller change in the propagation of particles. For even weak interactions, when the system is thought to be in a many-body localized phase, entanglement shows neither localized nor diffusive behavior but grows without limit in an infinite system: interactions act as a singular perturbation on the localized state with no interactions. The significance for proposed atomic experiments is that local measurements will show a large but nonthermal entropy in the many-body localized state. This entropy develops slowly (approximately logarithmically) over a diverging time scale as in glassy systems.
We study a simple and tractable model of many-body localization. The main idea is to take a renormalization group perspective in which local entanglement is removed to reach a product state. The model is built from a random local unitary which implements a real space renormalization procedure and a fixed point Hamiltonian with random exponentially decaying interactions. We prove that every energy eigenstate is localized, that energy is not transported, and argue that despite being fine tuned, the model is stable to perturbations. We also show that every energy eigenstate obeys an area law for entanglement entropy and we consider the dynamics of entanglement entropy under perturbations. In the case of extensive pertubations we recover a logarithmic growth of entanglement observed in recent numerical simulations.
The nonequilibrium dynamics of disordered many-body quantum systems after a quantum quench unveils important insights about the competition between interactions and disorder, yielding, in particular, an interesting perspective toward the understanding of many-body localization. Still, the experimentally relevant effect of bond randomness in long-range interacting spin chains on their dynamical properties have so far not been investigated. In this Letter, we examine the entanglement entropy growth after a global quench in a quantum spin chain with randomly placed spins and long-range tunable interactions decaying with distance with power $\ensuremath{\alpha}$. Using a dynamical version of the strong disorder renormalization group we find for $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{c}$ that the entanglement entropy grows logarithmically with time and becomes smaller with larger $\ensuremath{\alpha}$ as $S(t)={S}_{p}ln(t)/(2\ensuremath{\alpha})$. Here, ${S}_{p}=2ln2\ensuremath{-}1$. We present results of numerical exact diagonalization calculations for system sizes up to $N\ensuremath{\sim}16$ spins, in good agreement with the analytical results for sufficiently large $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{c}\ensuremath{\approx}1.8$. For $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{c}$, we find that the entanglement entropy grows as a power law with time, $S(t)\ensuremath{\sim}{t}^{\ensuremath{\gamma}(\ensuremath{\alpha})}$ with $0<\ensuremath{\gamma}(\ensuremath{\alpha})<1$ a decaying function of the interaction exponent $\ensuremath{\alpha}$.
We introduce and implement a reformulation of the strong-disorder renormalization group method in real space, well suited to study bond-disordered antiferromagnetic power-law coupled quantum spin chains. We apply it to a chain of randomly placed spins coupled by power-law long-range interaction with power $\ensuremath{\alpha}$. First, keeping only interactions between adjacent spins, we derive the Master equation and confirm that it is solved by the infinite randomness fixed-point distribution. Then, we derive the master equation for power-law long-range interaction between all spins for any anisotropy $\ensuremath{\gamma}$ ranging from the XX limit, $\ensuremath{\gamma}=0$, to the isotropic Heisenberg limit, $\ensuremath{\gamma}=1,$ which corresponds to a tight-binding chain of disordered long-range interacting fermions with long-range hopping. We thereby show that the distribution function of couplings JJJ smaller than renormalization scale $\mathrm{\ensuremath{\Omega}}$ flows to the strong-disorder fixed-point distribution of finite width $2\ensuremath{\alpha}.$ We find only small corrections to that distribution, which depend on power exponent $\ensuremath{\alpha}$ and coupling anisotropy $\ensuremath{\gamma}.$ As a consequence, the low-temperature magnetic susceptibility diverges with an anomalous power law. The distribution of singlet lengths lll is found to decay as ${l}^{\ensuremath{-}2}$. The entanglement entropy of a subsystem of length nnn increases in the ground state logarithmically for all $\ensuremath{\alpha}$ and $\ensuremath{\gamma}$. After a global quantum quench, the entanglement entropy increases with time logarithmically as $S(t)\ensuremath{\sim}ln(t)/(2\ensuremath{\alpha})$.
Many-body localization (MBL) hinders the thermalization of quantum many-body systems in the presence of strong disorder. In this Letter, we study the MBL regime in bond-disordered spin-1/2 XXZ spin chain, finding the multimodal distribution of entanglement entropy in eigenstates, sub-Poissonian level statistics, and revealing a relation between operators and initial states required for examining the breakdown of thermalization in the time evolution of the system. We employ a real space renormalization group scheme to identify these phenomenological features of the MBL regime that extend beyond the standard picture of local integrals of motion relevant for systems with disorder coupled to on-site operators. Our results pave the way for experimental probing of MBL in bond-disordered spin chains.