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arXiv:1701.04748 · Physical Review Letters

Disorder-Free Localization

A. Smith, Johannes Knolle, Dmitry L. Kovrizhin, Roderich Moessner

Published Jun 27, 2017• 237 citations
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Abstract

The venerable phenomena of Anderson localization, along with the much more recent many-body localization, both depend crucially on the presence of disorder. The latter enters either in the form of quenched disorder in the parameters of the Hamiltonian, or through a special choice of a disordered initial state. Here, we present a model with localization arising in a very simple, completely translationally invariant quantum model, with only local interactions between spins and fermions. By identifying an extensive set of conserved quantities, we show that the system generates purely dynamically its own disorder, which gives rise to localization of fermionic degrees of freedom. Our work gives an answer to a decades old question whether quenched disorder is a necessary condition for localization. It also offers new insights into the physics of many-body localization, lattice gauge theories, and quantum disentangled liquids.

DOI
10.1103/physrevlett.118.266601

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The venerable phenomena of Anderson localization, along with the much more recent many-body localization, both depend crucially on the presence of disorder. The latter enters either in the form of quenched disorder in the parameters of the Hamiltonian, or through a special choice of a disordered initial state. Here, we present a model with localization arising in a very simple, completely translationally invariant quantum model, with only local interactions between spins and fermions. By identifying an extensive set of conserved quantities, we show that the system generates purely dynamically its own disorder, which gives rise to localization of fermionic degrees of freedom. Our work gives an answer to a decades old question whether quenched disorder is a necessary condition for localization. It also offers new insights into the physics of many-body localization, lattice gauge theories, and quantum disentangled liquids.
Localization due to the presence of disorder has proven crucial for our current understanding of relaxation in isolated quantum systems. The many-body localized phase constitutes a robust alternative to the thermalization of complex interacting systems, but recently the importance of disorder has been brought into question. A number of disorder-free localization mechanisms have been put forward connected to local symmetries of lattice gauge theories. Here, starting from translationally invariant (1+1)(1+1)(1+1)-dimensional quantum electrodynamics, we modify the dynamics of the gauge field which allows us to construct a lattice model with a UUU(1) local gauge symmetry revealing a mechanism of disorder-free localization. We consider two different discretizations of the continuum model resulting in a free-fermion soluble model in one case and an interacting model in the other. We diagnose the localization of our translationally invariant model in the far-from-equilibrium dynamics following a global quantum quench.
Disorder-free localization is a paradigm of nonergodicity in translation-invariant quantum many-body systems hosting gauge symmetries. The quench dynamics starting from simple initial states, which correspond to extensive superpositions of gauge superselection sectors, exhibits many-body localization with the system dynamically inducing its own disorder. An open question concerns the stability of disorder-free localization in the presence of gauge-breaking errors, and whether processes due to the latter can be controllably suppressed. Here, we show that translation-invariant \textit{single-body gauge terms} induce a quantum Zeno effect that reliably protects disorder-free localization against errors up to times at least polynomial in the protection strength. Our experimentally feasible scheme not only shows that disorder-free localization can be reliably stabilized, but also opens promising prospects for its observation in quantum simulators.
We study the time evolution after a quantum quench in a family of models whose degrees of freedom are fermions coupled to spins, where quenched disorder appears neither in the Hamiltonian parameters nor in the initial state. Focusing on the behavior of entanglement, both spatial and between subsystems, we show that the model supports a state exhibiting combined area and volume-law entanglement, being characteristic of the quantum disentangled liquid. This behavior appears for one set of variables, which is related via a duality mapping to another set, where this structure is absent. Upon adding density interactions between the fermions, we identify an exact mapping to an XXZ spin chain in a random binary magnetic field, thereby establishing the existence of many-body localization with its logarithmic entanglement growth in a fully disorder-free system.
We study quantum quenches in two-dimensional lattice gauge theories with fermions coupled to dynamical Z2{\mathbb{Z}}_{2}Z2​ gauge fields. Through the identification of an extensive set of conserved quantities, we propose a generic mechanism of charge localization in the absence of quenched disorder both in the Hamiltonian and in the initial states. We provide diagnostics of this localization through a set of experimentally relevant dynamical measures, entanglement measures, as well as spectral properties of the model. One of the defining features of the models that we study is a binary nature of emergent disorder, related to Z2{\mathbb{Z}}_{2}Z2​ degrees of freedom. This results in a qualitatively different behavior in the strong disorder limit compared to typically studied models of localization. For example, it gives rise to a possibility of a delocalization transition via a mechanism of quantum percolation in dimensions higher than 1D. We highlight the importance of our general phenomenology to questions related to dynamics of defects in Kitaev's toric code, and to quantum quenches in Hubbard models. While the simplest models we consider are effectively noninteracting, we also include interactions leading to many-body localizationlike logarithmic entanglement growth. Finally, we consider effects of interactions that generate dynamics for conserved charges, which gives rise to only transient localization behavior, or quasi-many-body localization.
The phenomenon of localization is usually accompanied with the presence of quenched disorder. To what extent disorder is necessary for localization is a well-known open problem. In this paper, we prove the instability of localization in translation-invariant systems. For any translation-invariant local Hamiltonian exhibiting either Anderson or many-body localization, an arbitrarily small translation-invariant random local perturbation almost surely leads to the following manifestations of delocalization: (i) Transport: For any (inhomogeneous) initial state, the spatial distribution of energy or any other local conserved quantity becomes uniform at late times. (ii) Scrambling: The out-of-time-ordered correlator of any traceless local operators decays to zero at late times. (iii) Thermalization: Random product states locally thermalize to the infinite temperature state with overwhelming probability.
Anderson localization (AL) phenomena usually exist in systems with random potential. However, disorder-free quantum many-body systems with local conservation can also exhibit AL or even many-body localization transition. We show that the AL phase exists in a modified Kondo lattice without external random potential. The density of state, inverse participation ratio and temperature-dependent resistance are computed by classical Monte Carlo simulation, which uncovers an AL phase from the previously studied Fermi liquid and Mott insulator regimes. The occurrence of AL roots from quenched disorder formed by conservative localized moments. Interestingly, a many-body wavefunction is found, which captures elements in all three paramagnetic phases and is used to compute their quantum entanglement. In light of these findings, we expect that the disorder-free AL phenomena can exist in generic translation-invariant quantum many-body systems.