Unbounded Growth of Entanglement in Models of Many-Body Localization

Bardarson, Jens H.; Pollmann, Frank; Moore, Joel E.

doi:10.1103/physrevlett.109.017202

Cited by 1,080 publications

(1,269 citation statements)

References 28 publications

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“…The existence of the MBL phase can be proved with minimal assumptions [20]; many of its properties are phenomenologically understood [10,11,16], and some cases can be explored using strong-randomness renormalization group methods [9,[21][22][23]. While the eigenstate properties of MBL systems are in some respects similar to those of noninteracting Anderson insulators, there are important differences in the dynamics, such as the logarithmic spreading of entanglement in the MBL phase [6,9,11,18,19,24,25].…”

Section: Introductionmentioning

confidence: 99%

Low-frequency conductivity in many-body localized systems

Gopalakrishnan¹,

Müller²,

Khemani³

et al. 2015

Phys. Rev. B

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: Introductionmentioning

confidence: 99%

Low-frequency conductivity in many-body localized systems

Gopalakrishnan¹,

Müller²,

Khemani³

et al. 2015

Phys. Rev. B

215

294

View full text Add to dashboard Cite

show abstract

“…This irreversible growth of entanglementquantified by the growth of the von Neumman entropyis important for several reasons. It is an essential part of thermalization, and as a result has been addressed in diverse contexts ranging from conformal field theory [1][2][3][4] and holography [5][6][7][8][9][10][11][12] to integrable [13][14][15][16][17][18][19], nonintegrable [20][21][22][23], and strongly disordered spin chains [24][25][26][27][28][29][30]. Entanglement growth is also of practical importance as the crucial obstacle to simulating quantum dynamics numerically, for example, using matrix product states or the density matrix renormalization group [31].…”

Section: Introductionmentioning

confidence: 99%

Quantum Entanglement Growth under Random Unitary Dynamics

et al. 2017

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Characterizing how entanglement grows with time in a many-body system, for example, after a quantum quench, is a key problem in nonequilibrium quantum physics. We study this problem for the case of random unitary dynamics, representing either Hamiltonian evolution with time-dependent noise or evolution by a random quantum circuit. Our results reveal a universal structure behind noisy entanglement growth, and also provide simple new heuristics for the "entanglement tsunami" in Hamiltonian systems without noise. In 1D, we show that noise causes the entanglement entropy across a cut to grow according to the celebrated KardarParisi-Zhang (KPZ) equation. The mean entanglement grows linearly in time, while fluctuations grow like ðtimeÞ 1=3 and are spatially correlated over a distance ∝ ðtimeÞ 2=3 . We derive KPZ universal behavior in three complementary ways, by mapping random entanglement growth to (i) a stochastic model of a growing surface, (ii) a "minimal cut" picture, reminiscent of the Ryu-Takayanagi formula in holography, and (iii) a hydrodynamic problem involving the dynamical spreading of operators. We demonstrate KPZ universality in 1D numerically using simulations of random unitary circuits. Importantly, the leading-order time dependence of the entropy is deterministic even in the presence of noise, allowing us to propose a simple coarse grained minimal cut picture for the entanglement growth of generic Hamiltonians, even without noise, in arbitrary dimensionality. We clarify the meaning of the "velocity" of entanglement growth in the 1D entanglement tsunami. We show that in higher dimensions, noisy entanglement evolution maps to the wellstudied problem of pinning of a membrane or domain wall by disorder.

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“…The MBL phase resembles noninteracting Anderson insulators in some ways (e.g., spatial correlations decay exponentially, and eigenstates have area-law entanglement [35]). However, there are also important distinctions in entanglement dynamics [36,37], dephasing [38][39][40], linear [41] and nonlinear [42][43][44][45][46][47][48] response, and the entanglement spectrum [49,50]. These developments (reviewed in Refs.…”

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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Unbounded Growth of Entanglement in Models of Many-Body Localization

Cited by 1,080 publications

References 28 publications

Low-frequency conductivity in many-body localized systems

Low-frequency conductivity in many-body localized systems

Quantum Entanglement Growth under Random Unitary Dynamics

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

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