Max Planck Institut of Quantum Optics (Garching) New perspectives on Thermalization Aspen 17-21.3.2014 (NON) THERMALIZATION OF 1D SYSTEMS: numerical studies

Max Planck Institutof Quantum Optics(Garching)

New perspectives on Thermalization

Aspen 17-21.3.2014

(NON) THERMALIZATION OF 1D SYSTEMS: numerical studies

with MPSM. C. Bañuls, A. Müller-Hermes, J. I. Cirac

M. Hastings, D. Huse, H. Kim, N. Yao, M. Lukin

Using Tensor Network techniques (MPS) for numerical studies of

thermalization

Tensor Network States: MPS techniques for dynamics

Applications to out-of-equilibrium problems

In this talk...

What are TNS?

Context: quantum many body systems

• TNS = Tensor Network States

interacting with each other

Goal: describe equilibrium

statesground, thermal states

Goal: describe interesting

states

What are TNS?

A general state of the N-body Hilbert

space has exponentially many

coefficients

A TNS has only a polynomial number of parameters

N-legged tensor

• TNS = Tensor Network States

Which properties characterize physically interesting states?

Area law

finite range gapped

Hamiltonians

states withlittle entanglement

WHY SHOULD TNS BE USEFUL?

MPS and PEPS satisfy the area law by

construction

TNS parametrize the structure of

entanglementlots of theoretical progress going on

mps

Area law by construction

• MPS = Matrix Product States

number of parameters

Bounded entanglement

good approximation of ground states

Verstraete, Cirac, PRB 2006Hastings J. Stat. Phys 2007

gapped finite range Hamiltonian ⇒ area law (ground state)

extremely successful for GS, low energy

time evolution can be simulated too

MPS

Verstraete, Porras, Cirac, PRL 2004White, PRL 1992

Schollwöck, RMP 2005, Ann. Phys. 2011

Vidal, PRL 2003, PRL 2007White, Feiguin, PRL 2004Daley et al., 2004

Completar lo del TDVP!!!!!Completar lo del TDVP!!!!!

but entanglement can grow fast!

little entangled

alternatively...time dependent

observables as TN

TN describe observables,

not states

problem is contracting the

network

exact contraction not possible

#P complete

observable as tnti

me

space

OBSERVABLE AS TNti

me

space

different approximate contraction strategies

standard (TEBD, tDMRG)

OBSERVABLE AS TNti

me

space

evolved state approximated as

MPS


Heisenberg picture

OBSERVABLE AS TNti

me

space

evolved operator as MPO


transverse contraction,

folding

MCB, Hastings, Verstraete, Cirac, PRL 2009Müller-Hermes, Cirac, MCB, NJP 2012

OBSERVABLE AS TNti

me

space


transverse contraction,

folding


relevant: entanglement in the network

OBSERVABLE AS TN

in particular, for infinite system

transverse eigenvectors as

MPS

Toy Tensor Network model helps to understand entanglement in

the network

toy model tnintuition: model free propagating excitations


toy model tn


toy model tn


toy model tn


toy model tn


toy model tn


entanglement also in the transverse eigenvector

folding can reduce the entanglement in this case


observable as tnclosest real case: global quench in

free fermionic models

XY model

other problems may benefit

from combined strategies

foldedtransv

ers

e

toy model tn


eigenstate of the evolution

no entanglement

created in space

a second case

toy model tn


fast growing entanglement in transverse direction

folding worksIsing GS

folded

transv

ers

e

GS found via iTEBD

OBSERVABLE AS TN

minimal TN


combined techniques

XY model

TN-MPS tools can be used to study out-of-equilibrium problems, thermalization

questions

Application: thermalization

Closed quantum system initialized out of equilibrium

Does it thermalize?

Local observables: do they reach thermal equilibrium

values?


MCB, Cirac, Hastings, PRL 2011

compute for small number of sites

compare to the thermal state with the same energy

thermalization of infinite quantum spin chain

fix non-integrable Hamiltonian

varying initial state

can also be computed using TN

APPLICATION: THERMALIZATION

MCB, Cirac, Hastings, PRL 2011

non-integrable regime

We observed different regimes of thermalization for the same Hamiltonian parameters

strong instantaneous state relaxes

weak only after time average

for some initial states, none of them


instant distance


time averaged

Other problems showing absence of thermalization: MBL

ongoing collaboration with D. Huse, N. Yao,

M. Lukin

many body localization

Interactions and disorder more interesting scenario

Many-body localization

Anderson localization: single particle states localized due to disorder

Basko, Aleiner, Altshuler, Ann. Phys. 2006Gornyi, Mirlin, Polyakov, PRL 2005Oganesyan, Huse, PRB 2007

Rigol et al PRL 2007Znidaric, Prosen, Prelovsek, PRB 2008Pal, Huse, PRB 2010Gogolin, Müller, Eisert, PRL 2011Bardarsson, Pollmann, Moore, PRL 2012Bauer, Nayak, JStatMech 2013Serbyn, Papic, Abanin, PRL 2013

environment destroys localization

weak interactions ⇒ MBL phase

highly excited states localizedsystem will not thermalize

MPS and mbl

Allow to study larger sizes than ED

Discover TI models exhibiting MBL

Oganesyan, Huse, PRB 2007Pal, Huse, PRB 2010

states at high temperature

study spin transport to decide thermalization

plus small modulation of spin density

MPS description of mixed states

TN description of evolution

MBL transition

MPS AND MBLSimilar model with discrete valued fields

pola

riza

tion

time

10 random realization

s

moderate bond dimensions needed

many body localizationDiscover TI models exhibiting MBL

work in progress

with J=0 produces average over ALL realizations of single chain with discrete values of radom fields

Paredes, Verstraete, Cirac, PRL 2005

MBL in TI model!!

phase diagram being explored

MPS AND MBLp

ola

riza

tion

time

prelimina

ry

(40 spins)

moderate bond dimensions needed

MPS AND MBLp

ola

riza

tion

time

prelimina

ry

(40 spins)

things will change with interactions J

many body localizationDiscover TI models exhibiting MBL

work in progress

Open questionsphase diagram J, Bcharacterize MBL from results accessible to finite t simulations?limitations of the mixed state MPO description of time evolution?

conclusionsVersatile TNS tools can be used for time evolution

approximations involved

statemore general TN contractionunderstanding

entanglement in TN important

Applications to non-equilibriumthermalizationMBL in TI systemsevolution of operators

ongoing work with M. Hastings, H.

Kim, D. Huse

Max Planck Institutof Quantum Optics(Garching)

THANKS!

Documents

Max Planck Institut of Quantum Optics (Garching) New perspectives on Thermalization Aspen 17-21.3.2014 (NON) THERMALIZATION OF 1D SYSTEMS: numerical studies