KITPC Max Planck Institut of 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 approximation Can still produce good results in some cases product state! TNS = Tensor Network States

Successful history regarding static properties Introduction

➡ In particular in 1D DMRG methods ‣ Matrix Product State (MPS) representation of physical states White, PRL 1992 Schollwöck, 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 2008 Jiang, Weng, Xiang, 2008 Zhao et al., 2010 Verstraete, Cirac, 2004 Jordan et al PRL 2008 Wang, Verstraete, 2011 Corboz et al 2011, 2012

Dynamics is a more difficult challenge Introduction even in 1D

with many potential applications 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 Matrix Product States

Works great for ground state properties... ➡ finite chains → ➡ infinite chains → Östlund, Rommer, PRL 1995 Vidal, PRL 2007 White, PRL 1992 Schollwöck, 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 2003 White, Feiguin, PRL 2004 Daley et al., 2004 Vidal, PRL 2007 TEBD t-DMRG

Under time evolution entanglement can grow fast !

Entropy of evolved state may grow linearly required bond for fixed precision Osborne, PRL 2006 Schuch 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 state Used 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 hopping interaction limit t-J model

Adiabatic Heisenberg AFM hopping exchange interaction Fermi-Hubbard model describing fermions in an optical lattice

Adiabatic Heisenberg AFM hopping exchange interaction 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 gap non-interacting second OL Product of singlets Lubasch, Murg, Schneider, Cirac, MCB PRL 107, (2011)165301

Adiabatic Heisenberg AFM Feasible proposal Product of singlets Lower barriers Trotzky et al., PRL 2010 Lubasch, Murg, Schneider, Cirac, MCB PRL 107, (2011)165301

Adiabatic Heisenberg AFM Feasible proposal Final Hamiltonian Lubasch, Murg, Schneider, Cirac, MCB PRL 107, (2011)165301

Adiabatic Heisenberg AFM We find: Feasible time scales Fraction of magnetization

Adiabatic Heisenberg AFM We find: Local adiabaticity Large system ⇒ longer time antiferromagnetic state on 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

Adiabatic Heisenberg AFM Holes destroy magnetic order 2 holes simplified picture: free particle

Control holes with harmonic trap

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 system M. Lubasch, V. Murg, U. Schneider, J.I. Cirac, MCB PRL 107, (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

Apply operator

Observables as a TN the problem is contracting the TN t-DMRG

Observables as a TN MCB, Hastings, Verstraete, Cirac, PRL 102, (2009) Müller-Hermes, Cirac, MCB, NJP 14, (2012) transverse contraction

Observables as a TN MCB, Hastings, Verstraete, Cirac, PRL 102, (2009) Müller-Hermes, Cirac, MCB, NJP 14, (2012) infinite TI system

Observables as a TN MCB, Hastings, Verstraete, Cirac, PRL 102, (2009) Müller-Hermes, Cirac, MCB, NJP 14, (2012) infinite TI system reduces to dominant eigenvectors are they well approximated by MPS?

a question of the entanglement in the network

intuition from free propagation MCB, Hastings, Verstraete, Cirac, PRL 102, (2009) Müller-Hermes, Cirac, MCB, NJP 14, (2012) Observables as a TN

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, (2009) Müller-Hermes, Cirac, MCB, NJP 14, (2012)

Folded transverse method Contract the resulting network in the transverse direction larger tensor dimensions smaller transverse entanglement MCB, Hastings, Verstraete, Cirac, PRL 102, (2009) Müller-Hermes, Cirac, MCB, NJP 14, (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 contraction and folding

other observables

dynamical correlators

Dynamical correlators Müller-Hermes, Cirac, MCB, NJP 14, (2012)

Dynamical correlators Müller-Hermes, Cirac, MCB, NJP 14, (2012)

Dynamical correlators Müller-Hermes, Cirac, MCB, NJP 14, (2012) special case correlators in the GS

Dynamical correlators special case correlators in the GS optimal: combination of techniques Müller-Hermes, Cirac, MCB, NJP 14, (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, (2011)

Thermalization of infinite quantum systems different initial product states integrable if g=0 or h=0 MCB, Cirac, Hastings, PRL 106, (2011) Different applications

Compute the reduced density matrix for several sites ➡ computing all Reduced density matrix MCB, Cirac, Hastings, PRL 106, (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, (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!