(non) thermalization of 1d systems: numerical studies with mps
DESCRIPTION
(NON) THERMALIZATION OF 1D SYSTEMS: numerical studies with MPS. M. 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 - PowerPoint PPT PresentationTRANSCRIPT
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
different approximate contraction strategies
Heisenberg picture
OBSERVABLE AS TNti
me
space
evolved operator as MPO
different approximate contraction strategies
transverse contraction,
folding
MCB, Hastings, Verstraete, Cirac, PRL 2009Müller-Hermes, Cirac, MCB, NJP 2012
OBSERVABLE AS TNti
me
space
different approximate contraction strategies
transverse contraction,
folding
MCB, Hastings, Verstraete, Cirac, PRL 2009Müller-Hermes, Cirac, MCB, NJP 2012
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
MCB, Hastings, Verstraete, Cirac, PRL 2009Müller-Hermes, Cirac, MCB, NJP 2012
toy model tn
MCB, Hastings, Verstraete, Cirac, PRL 2009Müller-Hermes, Cirac, MCB, NJP 2012
toy model tn
MCB, Hastings, Verstraete, Cirac, PRL 2009Müller-Hermes, Cirac, MCB, NJP 2012
toy model tn
MCB, Hastings, Verstraete, Cirac, PRL 2009Müller-Hermes, Cirac, MCB, NJP 2012
toy model tn
MCB, Hastings, Verstraete, Cirac, PRL 2009Müller-Hermes, Cirac, MCB, NJP 2012
toy model tn
MCB, Hastings, Verstraete, Cirac, PRL 2009Müller-Hermes, Cirac, MCB, NJP 2012
entanglement also in the transverse eigenvector
folding can reduce the entanglement in this case
MCB, Hastings, Verstraete, Cirac, PRL 2009Müller-Hermes, Cirac, MCB, NJP 2012
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
MCB, Hastings, Verstraete, Cirac, PRL 2009Müller-Hermes, Cirac, MCB, NJP 2012
eigenstate of the evolution
no entanglement
created in space
a second case
toy model tn
MCB, Hastings, Verstraete, Cirac, PRL 2009Müller-Hermes, Cirac, MCB, NJP 2012
fast growing entanglement in transverse direction
folding worksIsing GS
folded
transv
ers
e
GS found via iTEBD
OBSERVABLE AS TN
minimal TN
MCB, Hastings, Verstraete, Cirac, PRL 2009Müller-Hermes, Cirac, MCB, NJP 2012
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?
Application: thermalization
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
Application: thermalization
instant distance
Application: thermalization
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!