Tensor networks for dynamical observables in 1D systems

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


number of parameters

MPS good at states with small entanglement

controlled by parameter D



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


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


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


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


hoppingexchangeinteraction




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 protocol

• initial state with low S

• tune interactions to

how long does it take?what if there are defects?


Feasible proposal

Band insulator

big gapnon-interactingsecond OL

Product of singlets

Lubasch, Murg, Schneider, Cirac, MCB

PRL 107, 165301 (2011)

http://arxiv.org/ct?url=http%3A%2F%2Fdx.doi.org%2F10%252E1103%2FPhysRevLett%252E107%252E165301&v=1bbcc66f


Feasible proposal

Product of singletsLower barriers

Trotzky et al., PRL 2010


PRL 107, 165301 (2011)



Feasible proposal

Final Hamiltonian


PRL 107, 165301 (2011)



We find: Feasible time scalesFraction of

magnetization


We find: Local adiabaticity

Large system ⇒ longer time

antiferromagnetic stateon a sublattice


Fraction of magnetization

We find: Local adiabaticity


Experiments at finite T

➡ holes expected


Holes destroy magnetic order

simplified picture:free particle

2 holes


Hole dynamics

Control holes with harmonic trap




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


(2012)

infinite TI system

Observables as a TN


(2012)

infinite TI systemreduces to dominant

eigenvectors

are they well approximated by

MPS?

a question of the entanglement in the

network

intuition fromfree propagation


(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


(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)





special casecorrelators in the GS


special casecorrelators in the GS

optimal: combination of techniques


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)


different initial product states

integrable if g=0 or h=0



Compute the reduced density matrix for several sites

➡ computing all

Reduced density matrix


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)





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!

Documents

Tensor networks for dynamical observables in 1D systems