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Quantum Thermalization Dynamics: From Information Scrambling to Emergent Hydrodynamics

Michael Knap

Collaborators: A. Bohrdt, Ch. Mendl, M. Endres

MainzJuly 7, 2018

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What is quantum about quantum evolution?

many-body dynamics erases memory on initial state

quantum information stored in local objects is quickly lost

at late times: reduction to dynamics of slow modes (i.e., diffusion) da Vinci (~1510)

time

local thermalization (quantum)

hydrodynamics (classical)Is that all there is?

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Route to equilibrium

Relaxation is characterized by time scale :

→ diagnosed by time-ordered correlations

Fermi-Liquid:

strange metals (no of quasi-particles)

Halperin, Hohenberg (1977)

e.g. Sachdev, QPT (2011)

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Route to equilibrium

Characterized by out-of-time ordered correlators (OTOCs)

Probes operator spreading

New time-scale of quantum dynamics: information scrambling

Larkin, Ovchinnikov, JETP Lett. 28, 1200 (1969),Shenker, Stanford (2014), Kitaev (2015).

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Outline

Emergent hydrodynamics and informationscrambling of incoherent lattice bosons

Bohrdt, Mendl, Endres, MK, NJP (2017) MK, arXiv (2018)

Entanglement production and operatorspreading in noisy quantum dynamics

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Interacting lattice bosons

consider the 1D Bose-Hubbard model

in the incoherent high-temperature regime

solve finite temperature dynamics of quantum system exactly using MPO

see e.g. Verstraete, Murg, Cirac (2008), Schollwoeck (2010), Barthel (2013)

Bohrdt, Mendl, Endres, MK, NJP (2017)

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OTOCs in the Bose-Hubbard model

high temperature limit

single-particle GF decays quickly

OTOCs exhibit a pronounced light cone!

Operators spread ballistically even in incoherent regime

Bohrdt, Mendl, Endres, MK, NJP (2017)

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Characterizing information propagation

wave front of OTOCs broadens diffusively in 1D

von Keyserlingk, et al. (2017), Nahum, et al. (2017) Bohrdt, Mendl, Endres, MK, NJP (2017)

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Dynamics of conserved quantities

How do time-ordered correlators relax?

Conserved particle number:

hydrodynamic regime:

Chaikin, Lubensky (2000)

cf. Leviatan et al (2017) → TDVP for

Bohrdt, Mendl, Endres, MK, NJP (2017)

Emergent hydrodynamics

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A noisy spin system

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The stochastic Heisenberg model

white noise:

non-integrable already without noise

Q: How does quantum information spread in noisy systems?

MK, arXiv:1806.04686

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Entanglement production

von Neumann/Rényi entanglement entropy

Simple rules for random unitary circuits

More macroscopically:

BA

L

Entanglement production is ofKardar–Parisi–Zhang (KPZ)

universality class

A. Nahum et al, PRX (2017)

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Entanglement production

von Neumann entanglement entropy

subtle finite-size and finite-time scaling

sub-leading corrections (consistent with KPZ):

MK, arXiv:1806.04686

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Entanglement fluctuations

fluctuations of the entropy obey KPZ scaling

MK, arXiv:1806.04686

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The strong noise limit

Stochastic white-noise dynamics can be mapped onto a Lindblad equation

Second order perturbation theory in superoperator space

Spin-diffusion constant

Perturbation

MK, arXiv:1806.04686

Han, Hartnoll, arXiv:1806.01859Znidaric, NJP 12, 043001 (2010)

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What about entanglement?

Express a Lindblad master equation for

Purity obtained from judiciously contracting the indices

Perturbation theory → emergent time scale

MK, arXiv:1806.04686

cf. Rowlands, Lamacraft, arXiv:1806.01723

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Operator spreading

Compute OTOC in the presence of noise ( ) using Lanczos

linear spreading of OTOC in noisy environment

→ no light-cone velocity

MK, arXiv:1806.04686

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Diffusive wave front

wave front spreads as biased random walk = Gaussian

→ integrate Gaussian = OTOC:

MK, arXiv:1806.04686

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Velocity dependent Lyapunov exponent

Deissler, Phys. Lett. A (1984)Khemani, Huse, Nahum, (2018)

MK, arXiv:1806.04686

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Entanglement vs. butterfly velocity

hierarchy of velocities: entanglement velocity isslower than butterfly velocity

MK, arXiv:1806.04686

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Non-Markovian noise

Ornstein-Uhlenbeck process

Fermi’s golden rule type calculation

Incoherent hopping time scale for fast and slow noise should dominate dynamics

MK, arXiv:1806.04686

Gopalakrishnan, Islam, MK, PRL 119 046601 (2017)Amir et al. PRE 050105 (2009)

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Summary and Outlook

relaxation dynamics of the Bose-Hubbard model

emergent hydrodynamics at late times

Noisy dynamics in Heisenberg model

KPZ describes entanglement production

Strong noise-limit analytically treatable

Bohrdt, Mendl, Endres, MK, NJP, (2017)

MK, arXiv:1806.04686

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Financial support

Technical University of Munich - Institute for Advanced Study, funded by the German Excellence Initiative and the European Union FP7 under grant agreement 291763, DFG

grant KN 1254/1-1, Max Planck Society, and DFG TRR80

Collaborators

A. Bohrdt, C. Mendl, M.Endres

Thank you!