Post on 19-Jan-2016
description
KITPC 9.8.2012
Max Planck Institutof Quantum Optics(Garching)
Tensor networks for dynamical
observables in 1D systems
Mari-Carmen Bañuls
Tensor network techniques and dynamics
An application to experimental situation
Limitations, advances
Introduction
Approximate methods are fundamental for the numerical study of many body problems
Introduction
Efficient representations of many body systems
• Tensor Network States
• states with little entanglement are easy to describe
We also need efficient ways of computing with them
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
What are TNS?
A particular example
Mean field approximatio
n
Can still produce good
results in some cases
product state!
• TNS = Tensor Network States
Successful history regarding static properties
Introduction
Introduction
In particular in 1D
DMRG methods
‣ Matrix Product State (MPS) representation of physical states
White, PRL 1992Schollwöck, RMP 2005
Verstraete et al, PRL 2004
Introduction
In 2D
MPS generalized by PEPS
‣ much higher computational cost
‣ recent developments
‣ Tensor Renormalization
‣ iPEPS
Gu, Levin, Wen 2008Jiang, Weng, Xiang, 2008Zhao et al., 2010
Verstraete, Cirac, 2004
Jordan et al PRL 2008Wang, Verstraete, 2011Corboz et al 2011, 2012
Dynamics is a more difficult challenge
Introduction
even in 1D
with many potential applications
Introduction
Introduction
with many potential applications
theoretical
applied
non-equilibrium dynamics
transport problems
predict experiments
What can we say about dynamics with
MPS?
The tool: MPS
Matrix Product States
Matrix Product States
number of parameters
MPS good at states with small entanglement
controlled by parameter D
Matrix Product States
Matrix Product States
Works great for ground state properties...
➡ finite chains →
➡ infinite chains →Östlund, Rommer, PRL 1995Vidal, PRL 2007
White, PRL 1992Schollwöck, RMP 2005
MPS are a good ansatz!
(Most) ground states satisfy an area law
...because of entanglement
Matrix Product States
Can also do time evolution
• finite chains
• infinite (TI) chains ⇒ iTEBD
• but...
Vidal, PRL 2003White, Feiguin, PRL 2004Daley et al., 2004
Vidal, PRL 2007
TEBDt-DMRG
Under time evolution entanglement can grow fast !
Entropy of evolved state may grow linearly
required bond for fixed precision
Osborne, PRL 2006Schuch et al., NJP 2008
bond dim
time
Matrix Product States
But not completely hopeless...
Will work for short times
For states close to the ground stateUsed to simulate adiabatic processes
Predictions at short times
Imaginary time (Euclidean) evolution → ground states
Matrix Product States
Simulating adiabatic dynamics for the experiment
A particular application
Adiabatic preparation of Heisenberg
antiferromagnet with ultracold fermions
Adiabatic Heisenberg AFM
Fermi-Hubbard model describing fermions in an optical lattice
hoppinginteraction
limit t-J model
Adiabatic Heisenberg AFM
hoppingexchangeinteraction
Fermi-Hubbard model describing fermions in an optical lattice
Fermi-Hubbard model describing fermions in an optical lattice
Adiabatic Heisenberg AFM
hoppingexchangeinteraction
Heisenberg model
Simulation of dynamics in OL experiments
Fermionic Hubbard model realized in OL Jördens et al., Nature 2008
Schneider et al., Science 2008
Observed Mott insulator, band insulating phases
Simulation of dynamics in OL experiments
Challenge: prepare long-range antiferromagnetic order
Problem: low entropy required beyond direct preparation
Jördens et al., PRL 2010
e.g. t-J at half filling
Adiabatic Heisenberg AFM
Adiabatic protocol
• initial state with low S
• tune interactions to
how long does it take?what if there are defects?
Adiabatic Heisenberg AFM
Feasible proposal
Band insulator
big gapnon-interactingsecond OL
Product of singlets
Lubasch, Murg, Schneider, Cirac, MCB
PRL 107, 165301 (2011)
Adiabatic Heisenberg AFM
Feasible proposal
Product of singletsLower barriers
Trotzky et al., PRL 2010
Lubasch, Murg, Schneider, Cirac, MCB
PRL 107, 165301 (2011)
Adiabatic Heisenberg AFM
Feasible proposal
Final Hamiltonian
Lubasch, Murg, Schneider, Cirac, MCB
PRL 107, 165301 (2011)
Adiabatic Heisenberg AFM
We find: Feasible time scalesFraction of
magnetization
Adiabatic Heisenberg AFM
We find: Local adiabaticity
Large system ⇒ longer time
antiferromagnetic stateon a sublattice
Adiabatic Heisenberg AFM
Fraction of magnetization
We find: Local adiabaticity
Adiabatic Heisenberg AFM
Experiments at finite T
➡ holes expected
Adiabatic Heisenberg AFM
Holes destroy magnetic order
simplified picture:free particle
2 holes
Adiabatic Heisenberg AFM
Hole dynamics
Control holes with harmonic trap
Adiabatic Heisenberg AFM
Adiabatic Heisenberg AFM
Adiabatic Heisenberg AFM
Harmonic trap can control the effect of holes
feasible proposal for adiabatic preparation (time scales)
local adiabaticity:
• AFM in a sublattice faster
holes can be controlled by harmonic trap
generalize to 2D systemM. Lubasch, V. Murg, U. Schneider, J.I.
Cirac, MCBPRL 107, 165301 (2011)
We found
Is this all we can do with MPS techniques?
Not really
In some cases, longer times attainable with new tricks
Key: observables as contracted tensor network
entanglement in network ⇒ MPS tools
Observables as a TN
Apply evolution operator
non local! discretize time
still non local
Observables as a TN
Observables as a TN
Observables as a TN
Apply operator
Observables as a TN
the problem is contracting
the TN
t-DMRG
Observables as a TN
MCB, Hastings, Verstraete, Cirac, PRL 102, 240603 (2009)Müller-Hermes, Cirac, MCB, NJP 14, 075003
(2012)
transverse contractio
n
Observables as a TN
MCB, Hastings, Verstraete, Cirac, PRL 102, 240603 (2009)Müller-Hermes, Cirac, MCB, NJP 14, 075003
(2012)
infinite TI system
Observables as a TN
MCB, Hastings, Verstraete, Cirac, PRL 102, 240603 (2009)Müller-Hermes, Cirac, MCB, NJP 14, 075003
(2012)
infinite TI systemreduces to dominant
eigenvectors
are they well approximated by
MPS?
a question of the entanglement in the
network
intuition fromfree propagation
MCB, Hastings, Verstraete, Cirac, PRL 102, 240603 (2009)Müller-Hermes, Cirac, MCB, NJP 14, 075003
(2012)
Observables as a TN
A toy TN model
A toy TN model
A toy TN model
A toy TN model
A toy TN model
A toy TN model
A toy TN model
A toy TN model
eigenvector
more efficient description of entanglement is
possible!
• Bring together sites corresponding to the same time step
Folded transverse method
MCB, Hastings, Verstraete, Cirac, PRL 102, 240603 (2009)Müller-Hermes, Cirac, MCB, NJP 14, 075003
(2012)
Folded transverse method
Contract the resulting network in the transverse direction
• larger tensor dimensions
• smaller transverse entanglementMCB, Hastings, Verstraete, Cirac, PRL 102,
240603 (2009)Müller-Hermes, Cirac, MCB, NJP 14, 075003 (2012)
toy model, checked with real time evolution under
free fermion models
Ising model
Maximum entropy in the transverse
eigenvector
unfolded
folded
Longer times than standard approach
Smooth effect of error: qualitative description possible
Results
• Try Ising chain
Real time evolution
without folding
Real time evolution
transverse contractionand folding
other observables
dynamical correlators
Dynamical correlators
Müller-Hermes, Cirac, MCB, NJP 14, 075003 (2012)
Dynamical correlators
Müller-Hermes, Cirac, MCB, NJP 14, 075003 (2012)
Dynamical correlators
Müller-Hermes, Cirac, MCB, NJP 14, 075003 (2012)
special casecorrelators in the GS
Dynamical correlators
special casecorrelators in the GS
optimal: combination of techniques
Müller-Hermes, Cirac, MCB, NJP 14, 075003 (2012)
GS MPS can be found by iTEBD
XY model
transverse
minimal TN
Fix non-integrable Hamiltonian
vary initial state
Compute for small number of sites
Compare to the thermal state with the same energy
Thermalization of infinite quantum systems
Different applications
MCB, Cirac, Hastings, PRL 106, 050405 (2011)
Thermalization of infinite quantum systems
different initial product states
integrable if g=0 or h=0
MCB, Cirac, Hastings, PRL 106, 050405 (2011)
Different applications
Compute the reduced density matrix for several sites
➡ computing all
Reduced density matrix
MCB, Cirac, Hastings, PRL 106, 050405 (2011)
Application: thermalization
Compute the reduced density matrix for several sites
➡ computing all
Compare to thermal state
➡ corresponding to the same energy
Measure non-thermalization as distance MCB, Cirac, Hastings, PRL 106, 050405 (2011)
Thermalization of infinite quantum systems
Different applications
Thermalization of infinite quantum systems
Different applications
Mixed states
thermal states
open systems
Long range interactions
Schwinger model
Ongoing work
with K. Jansen and K. Cichy (DESY)
Take home message:
Tensor network techniques can be useful for the study of dynamics
Conclusions
Tensor network techniques can be useful for the study of dynamics
TEBD, t-DMRG‣ short times and close to equilibrium‣ don’t forget imaginary time evolution!
Transverse contraction + Folding + ...‣ longer times than other methods‣ qualitative description at very long
times
Applications...
Thanks!