Tensor networks and the numerical study of quantum and classical systems on infinite lattices

Tensor networks and the numerical study of quantum

and classical systems on infinite lattices

Román Orús School of Physical Sciences,

The University of Queensland, Brisbane (Australia)in collaboration with Guifré Vidal and Jacob Jordan

Trobada de Nadal 2006 ECM, December 21st 2006

Outline

0.- Introduction

1.- Entanglement renormalization of environment degrees of freedom

2.- Contraction of infinite 2-dimensional tensor networks

3.- Outlook

Matrix Product States (MPS)

Matrix Product Density Operators (MPDO)

Projected Entangled Pair States (PEPS)

Infinite 1-dimensional thermal states and disentanglers

Infinite 1-dimensional master equations

Critical correlators of the classical Ising model

€

ψ = c i1i2 ...ini{ }

∑ i1,i2,...,in

n2

State of a quantum system of n spins 1/2:

coefficients (very inneficient to handle classically) niiic ...21

Introduction

A natural ansatz for relevant states of quantum mechanical systems is given in terms of the contraction of an appropriate tensor network:

€

ˆ H = ˆ h i, j

<i, j>

∑

€

ψ0 = c i1i2 ...

i{ }

∑ i1,i2,...

Inspires classical techniques to compute properties of quantum systems which are free from the sign problem, and which can be implemented in the thermodynamic limit


[Afflek et al., 1987] [Fannes et al., 1992] [White, 1992] [Ostlund and Rommer, 1995] [Vidal, 2003]

Physical local system of dimension Bonds of dimension χ

€

d

For finite systems, the state is represented with parameters, instead of .

€

ndχ 2

€

dn

Any quantum state can be represented as an MPS, with large enough .

Physical observables (e.g. correlators) can be computed in time.

€

O( poly(χ ))€

χ

Great in 1 spatial dimension because of the logarithmic scaling of the entaglement entropy [Vidal et al., 2003]

DMRGDynamics

Imaginary-time evolutionThermal states

Master equations

… …


€

ˆ ρ = c i1 ,i2 ,...j1 , j2 ,... i1,i2,... j1, j2,...

{i}{ j}

∑

Physical local system of dimension Bonds of dimension χ

€

d

… …Purification of local dimension

€

p


€

2ndpχ 2

€

d2n

Any density operator can be represented as an MPDO, with large enough and


€

O( poly(χ )poly( p))€

χ

€

p

Useful in the computation of 1-dimensional thermal states.

[Verstraete, García-Ripoll, Cirac, 2004]


Physical local system of dimension

€

d

Bonds of dimension

€

D


€

ndD4

€

dn


€

O( poly(D))

Exact contraction of an arbitrary PEPS for a finite system is an #P-Complete problem [N. Schuch et al., 2006].

Successfully applied to variationally compute the ground state of finite quantum systemsin 2 spatial dimensions (up to 11 x 11 sites, [Murg, Verstraete and Cirac, 2006]).

… …

……

[Verstraete and Cirac, 2004]

Outline

0.- Introduction



3.- Outlook







On thermal states in 1 spatial dimension…

€

ˆ ρ = c i1 ,i2 ,...j1 , j2 ,... i1,i2,... j1, j2,...

{i}{ j}

∑

OR … …

MPDO

€

ˆ ρ ∝ e−β ˆ H / 2 ˆ I e−β ˆ H / 2Both ansatzs can be applied to compute thermal states. However, MPDOs can introduce unphysical correlations between the environment degrees of freedom

environment

swap

€

χ >1

“Unnecessary”entanglement!

… …

MPS-like

[Zwolak and Vidal, 2004]

Disentanglers on the environment of MPDOs

swap

€

χ >1

U

Disentangler (renormalization of correlations flowing across the environment)

This effect is not negligible in the computation of thermal states with MPDOs

€

χ =1

Less expensive representation

Quantum Ising spin chain,

€

β =20

€

h =1.1

Schmidt coefficientsof the MPS-like representation

BIG!!!

Simulating master equations with MPDOs

€

d ˆ ρ

dt= L[ ˆ ρ ] = −i H, ˆ ρ [ ] + 2Aμ

ˆ ρ Aμ+ − Aμ Aμ

+ ˆ ρ − Aμ+Aμ

ˆ ρ ( )μ >0

∑

€

L = Lr,r+1

r

∑

€

ˆ ρ (t + dt) ≅ ⊗r odd

edtLr ,r+1

( ) ⊗r even

edtLr ,r+1

( ) ˆ ρ (t)

€

(edtLr ,r+1 )[ ˆ ρ (t)] = Mμ r ,r+1ˆ ρ (t)

μ r ,r+1

∑ Mμ r ,r+1

+Kraus operators

W

M

It is possible to introduce “disentangling isometries” acting in the environment

subspace that truncate the proliferation of indices at

each step

BUT…M

M

M

M

M

M

M

M

M

M

M

M

M

M

Proliferation of indices makes “naive” simulation not feasible

… …

Quantum Ising spin chain with amplitude damping,

€

h =1.1

€

γ=0.1

€

ˆ ρ (t = 0) = + +( )∞⊗

Quantum Ising spin chain with amplitude damping,

€

h =1.1

€

γ=0.1

€

ˆ ρ (t = 0) ∝ e−β ˆ H

€

β =20 with and without partial disentanglement

Outline

0.- Introduction



3.- Outlook







The difficult problem of a PEPS…In order to compute expected values of observables, one must necessarily contract the PEPS

tensor network, and this is an #P-Complete problem in general. For finite systems, there is a variational technique to efficiently approximate such a contraction (up to lattices of 11 x 11 spins) due to Verstraete and Cirac (2004).

We have developed a technique to contract the whole PEPS tensor network in the

thermodynamic limit for translationally-invariant systems.

€

D

€

d

€

D

€

D

€

D

€

D

€

D

€

D

€

D

… …

……

€

D2

The difficult problem of a PEPS…

… ……

…

In order to compute expected values of observables, one must necessarily contract the PEPS

tensor network, and this is an NP-hard problem in general. For finite systems, there is a variational technique to efficiently approximate such a contraction (up to lattices of 11 x 11 spins) due to Verstraete and Cirac.



The difficult problem of a PEPS…

…

……

…

In order to compute expected values of observables, one must necessarily contract the PEPS








…

……

…

Boundary MPS with bond dimension

Action of non-unitary gates on an infinite

MPS

Can be efficiently computed, taking care of orthonormalization issues

€

χ





…

……

…

Iterate until a fixed point is found





…

……

…

Once there is convergence, contract it from the other side and compute e.g. correlators on the diagonal with the obtained MPS

… …





…

……

…

Once there is convergence, contract it from the other side and compute e.g. correlators on the diagonal with the obtained MPS

r

€

σ z

€

σ z

r

€

D

€

D

€

D

€

D

€

D

€

D

€

D

€

D

€

σ z

An example: classical Ising model at criticality

€

H = − σ i

<i, j>

∑ σ j

€

βC =1

2ln 1+ 2( )

€

σ iσ i+r β C≈

1

r1

4

It is possible to build a quantum PEPS such that the expected values correspond to those of

the classical ensemble

€

C(r) = ψ βˆ σ i

z ˆ σ i+rz ψ β = σ iσ i+r β

€

β =βC − 0.1

€

χ =20€

χ =30

exact

Very good agreement up to ~100 sites with modest computational effort!€

logC(r)

€

log(r)

Outline

0.- Introduction



3.- Outlook







Outlook

Question: why tensor networks are good for you?

Answer: because, potentially, you can apply them to study…

strongly-correlated quantum many-body systems in 1, 2, and more spatial dimensions, in the finite case and in the thermodynamic limit, Hubbard models, high-Tc superconductivity, frustrated lattices, topological effects, finite-temperature systems, systems away from equilibrium, master equations and dissipative systems, classical statistical models, quantum field theories on infinite lattices, at finite temperature and away from equilibrium, effects of boundary conditions, RG transformations, computational complexity of physical systems, etc

Soon application to compute the ground state properties and dynamics of infinite quantum many-

body systems in 2 spatial dimensions

in collaboration with G. Vidal, J. Jordan, F. Verstraete and I. Cirac

Tensor networks and the numerical study of quantum and classical systems on infinite lattices

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