Many-Body Localization and Thermalization in Quantum Statistical Mechanics

Nandkishore, Rahul; Huse, David A.

doi:10.1146/annurev-conmatphys-031214-014726

Cited by 2,417 publications

(2,491 citation statements)

References 99 publications

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“…It has recently been shown [19] by one of the present authors that localized systems subject to a nonzero-amplitude drive go nonlinear and display a highly non-local response at low enough frequencies. Further, an MBL system subject to a finite-frequency drive for a long enough duration will eventually leave the linear response regime and enter instead a Floquet MBL steady state or even thermalize due to the a.c. drive [8,19,32].…”

Section: Assumptionsmentioning

confidence: 99%

Low-frequency conductivity in many-body localized systems

Gopalakrishnan¹,

Müller²,

Khemani³

et al. 2015

Phys. Rev. B

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215

294

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We argue that the a.c. conductivity σ(ω) in the many-body localized phase is a power law of frequency ω at low frequency: specifically, σ(ω) ∼ ω α with the exponent α approaching 1 at the phase transition to the thermal phase, and asymptoting to 2 deep in the localized phase. We identify two separate mechanisms giving rise to this power law: deep in the localized phase, the conductivity is dominated by rare resonant pairs of configurations; close to the transition, the dominant contributions are rare regions that are locally critical or in the thermal phase. We present numerical evidence supporting these claims, and discuss how these power laws can also be seen through polarization-decay measurements in ultracold atomic systems.

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Section: Assumptionsmentioning

confidence: 99%

Low-frequency conductivity in many-body localized systems

Gopalakrishnan¹,

Müller²,

Khemani³

et al. 2015

Phys. Rev. B

Self Cite

215

294

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“…Such special cases, corresponding to integrable (and thus non-chaotic) dynamics, were thought to be fine-tuned points rather than stable dynamical phases. However, in recent years, a stable, nonthermalizing specifically quantum phase known as the 'many-body-localized' (MBL) phase, has been predicted to exist in certain systems [7][8][9][10][11]. The MBL phase is not known to have any direct classical equivalent [12,13], and unlike traditional integrable systems is, moreover, robust against generic local perturbations to the system's Hamiltonian, so is not fine-tuned.…”

Section: Introductionmentioning

confidence: 99%

Rare‐region effects and dynamics near the many‐body localization transition

et al. 2017

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The low-frequency response of systems near the many-body localization phase transition, on either side of the transition, is dominated by contributions from rare regions that are locally "in the other phase", i.e., rare localized regions in a system that is typically thermal, or rare thermal regions in a system that is typically localized. Rare localized regions affect the properties of the thermal phase, especially in one dimension, by acting as bottlenecks for transport and the growth of entanglement, whereas rare thermal regions in the localized phase act as local "baths" and dominate the low-frequency response of the MBL phase. We review recent progress in understanding these rare-region effects, and discuss some of the open questions associated with them: in particular, whether and in what circumstances a single rare thermal region can destabilize the many-body localized phase.

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“…The robust subharmonic synchronization of the many-body Floquet system is the essence of the discrete time crystal phase [7][8][9][10] . In a DTC, the underlying Floquet drive should generally be accompanied by strong disorder, leading to manybody localization 13 and thereby preventing the quantum system from absorbing the drive energy and heating to infinite temperatures [14][15][16][17] .…”

mentioning

confidence: 99%

Observation of a discrete time crystal

Zhang

Hess

Kyprianidis

et al. 2017

Nature

1,112

972

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Spontaneous symmetry breaking is a fundamental concept in many areas of physics, ranging from cosmology and particle physics to condensed matter 1 . A prime example is the breaking of spatial translation symmetry, which underlies the formation of crystals and the phase transition from liquid to solid. Analogous to crystals in space, the breaking of translation symmetry in time and the emergence of a "time crystal" was recently proposed 2,3 , but later shown to be forbidden in thermal equilibrium [4][5][6] . However, nonequilibrium Floquet systems subject to a periodic drive can exhibit persistent time-correlations at an emergent sub-harmonic frequency [7][8][9][10] . This new phase of matter has been dubbed a "discrete time crystal" (DTC) 10,11 . Here, we present the first experimental observation of a discrete time crystal, in an interacting spin chain of trapped atomic ions. We apply a periodic Hamiltonian to the system under many-body localization (MBL) conditions, and observe a sub-harmonic temporal response that is robust to external perturbations. Such a time crystal opens the door for studying systems with long-range spatial-temporal correlations and novel phases of matter that emerge under intrinsically non-equilibrium conditions 7 .For any symmetry in a Hamiltonian system, its spontaneous breaking in the ground state leads to a phase transition 12 . The broken symmetry itself can assume many different forms. For example, the breaking of spinrotational symmetry leads to a phase transition from paramagnetism to ferromagnetism when the temperature is brought below the Curie point. The breaking of spatial symmetry leads to the formation of crystals, where the continuous translation symmetry of space is replaced by a discrete one.We now pose an analogous question: can the translation symmetry of time be broken? The proposal of such a "time crystal" 2 for time-independent Hamiltonians has led to much discussion, with the conclusion that such structures cannot exist in the ground state or any thermal equilibrium state of a quantum mechanical system 4-6 . A simple intuitive explanation is that quantum equilibrium states have time-independent observables by construction; thus, time translation symmetry can only be spontaneously broken in non-equilibrium systems 7-10 . In particular, the dynamics of periodically-driven Floquet systems possesses a discrete time translation symmetry governed by the drive period. This symmetry can be further broken into "super-lattice" structures where physical observables exhibit a period larger than that of the drive. Such a response is analogous to commensurate charge density waves that break the discrete translation symmetry of their underlying lattice 1 . The robust subharmonic synchronization of the many-body Floquet system is the essence of the discrete time crystal phase 7-10 . In a DTC, the underlying Floquet drive should generally be accompanied by strong disorder, leading to manybody localization 13 and thereby preventing the quantum system from absorbing the drive energy...

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Many-Body Localization and Thermalization in Quantum Statistical Mechanics

Cited by 2,417 publications

References 99 publications

Low-frequency conductivity in many-body localized systems

Low-frequency conductivity in many-body localized systems

Rare‐region effects and dynamics near the many‐body localization transition

Observation of a discrete time crystal

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