PEPS, matrix product operators and the algebraic Bethe ansatz

Frank VerstraeteUniversity of Vienna

Valentin Murg, Ignacio Cirac (MPQ) B. Pirvu (Vienna)

Matrix Product States and Projected Entangled Pair States as variational states for simulating strongly correlated

quantum systems

• Why?

– History of Quantum Mechanics is one in which we try to find approximate solutions to Schrödinger equation

• Actually, this is the reason why quantum computers would be so exciting

– Most relevant breakthroughs in context of many-body physics: guess the right wavefunction (BCS, Laughlin, …)

– Is there a way to come up with a systematic way of parameterizing the wavefunctions arising in relevant Hamiltonians?

• In case of 1-D quantum spin chains: NRG / DMRG : MPS• In case of 2-D quantum spin systems: PEPS / MERA / ….

Many-body Hilbert space is a convenient illusion

• Size of Hilbert space of system of N particles / modes / … scales exponentially with N.

– What is the fraction of states that are physical, i.e. can be created as low-energy states of local Hamiltonians or by a quantum computer in poly time? Exponentially small !!!

– Ground states (and low-energy states …) have very special properties

• Amount of entanglement is very small: can be formalized using so-called area laws• Ground states have extremal local correlations: all (quasi-)long range correlations are a

consequence of the fact that those local correlations must be made compatible with translational invariance

– If we want to simulate a many-body system, we should be smarter

Matrix Product States

• Valence bond state: translational invariant by construction• Has extremal local correlations• Obeys area law by construction

• Theorem: if an area law is satisfied, then the state can be well approximated by a MPS:

• In case of local gapped 1-D Hamiltonians: area law is guaranteed• Conclusion: all states in finite 1-D chains can be represented by

MPS: breakdown of exponential wall !

S.R. White’s DMRG et al.

• DMRG can be understood as variational method within this class of VBS/MPS states: alternating least squares– Extension to periodic BC: trivial once formulated like this

• A local Hamiltonian (e.g. Heisenberg model) is a special case of a more general type of operators: matrix product operators

H

Matrix product operators

• Slight modification of DMRG allows to approximate extremal eigenvectors (and eigenvalues) of hermitean MPO by MPS:

• Where else do MPO appear?– Transfer matrices of classical spin systems!

DMRG is therefore basically a method for finding leading eigenvector and eigenvalue of transfer matrix (cfr. Nishino, Baxter)

• We can also solve another optimization problem involving MPO: given a MPS and an MPO O, find the MPS that minimizes

– It turns out that this is also a multiquadratic optimization problem that is very well conditioned and can be solved using DMRG-like sweeping!

– Core method for simulating PEPS

– The error in the approximation is known exactly!

– Leads to a massively parallel time evolution algorithm that does not break translational invariance:

Matrix Product Operators and the Bethe ansatz:

• Algebraic Bethe ansatz is all about MPO:

• Crucial Property of this family of MPO: they all commute (==Yang-Baxter equation):

– Gauge transformation of MPS/MPO leave it invariant!

• What has this to do with the Heisenberg model?

– This can easily be seen because is the shift operator (shifts qubits 1,2,3,…N to 2,3,4,…1); taking the derivative replaces one of those “swaps” with the idenity; logarithmic derivative undoes all the other swaps, leaving the Heisenberg Hamiltonian!

– It follows that and hence they have the same eigenvectors

• Let’s now define new operators similar to but with OBC:

– These will play the role of creation operators and commute for all

• All eigenstates of the Heisenberg model are of the form

– The parameters are found by imposing that these are eigenstates of = Bethe equations (follows simply from working out commutation relations; this leads to coupled equations between the )

• In terms of MPS/MPO: all eigenstates can exactly be represented as

– Can therefore easily be simulated using MPS algorithms: correlation functions, … with absolute error bars!

– How to extend to higher dimensions? PEPS!

Generalizing MPS to higher dimensions: PEPS

• Area law is satisfied by construction : scalable!• Precursors: AKLT, Nishino; PEPS introduced in context of

measurement-based quantum computation

How to calculate expectation values?

• Equivalent to contracting tensor network consisting of MPS and MPO!

– Obvious way of doing this: recursively use

– Optimization: alternating least squares as in DMRG• Alternatively: imaginary time evolution ; infinite algorithm ; renormalization (Gu et al.)

arXiv:cond-mat/0407066

Holographic principle: dimensional reduction

• Crucial property of MPS/PEPS: dimensional reduction

– Start from quantum system in 2 dimensions (2+1)

– The PEPS ansatz maps the quantum Hamiltonian to a state corresponding to a partition function in 2 dimensions (2+0)

– The properties of such a state are described by a (1+1) dimensional theory (eigenvectors of transfer matrices)

– Those eigenvectors are well described by MPS

– Properties of MPS are trivial to calculate: reduction to a partition function of a 1-D system (1+0)

Goal:

Simulation of the Operation

| PEPS |iH te PEPS

Procedure:

For , the state remains a PEP-state, but with increased virtual Dimension:

1t |iH te PEPS

D D

Problem: Virtual Dimension increases exponentially with the number of steps.

Workaround:

Approximate at each step the PEP-state by a PEP-state with reduced virtual Dimension.

|iH te PEPS

Dimension

Dimension D

Dimension D

The Method Time Evolution

From here on: all work and slides by Valentin Murg

The Method Time Evolution

8

| PEPS | PEPS Dimension D Dimension D

Algorithm:

Optimization of the distance site by site:

( )ijA M w

( )0

ijk

K

A

Goal:

2 10

8

# ~

# ~steps

memory

N D

D

Optimal scaling of the Algorithm:

Dimension D

Dimension D

| PEPS

| PEPS

2| | Min.K PEPS PEPS

J1-J2-J3 Heisenberg model

• Frustrated system• We did calculation for systems with open boundary conditions of

size up to 14x14 and D=4

J3

J1

J2

arXiv:0901.2019

Classical Phase Diagram

( , ) I. Néel

J1

J2

II. Independent Sublattices

J1

J2

( , )q IV. Helicoidal

J1

J2

( , )q qIII. Helicoidal

J1

J2

J1J2J3-Model

Quantum Phase Diagram

( ,0) (0, )II. Collinear

J1

J2

J1

J2

(Order-by-Disorder)

J1J2J3-Model

Long Range Order

3 1/ 0J J 3 1/ 0.5J J 3 1/ 1J J

( , ) Q ( / 2, / 2) Q

Néel Order: Néel Order onfour Sublattices:

No long range order!

Structure Factor

J1J3-Model

Plaquette order parameter, 8x8 lattice, J3/J1=1/2

Pure Plaquette state: in Pl Q=1, in between Pl Q=1/4

Intermediate Phase

Comparison with short range resonating valence bond ground state

24

J1J3-Model

J1J2-Model Long Range Order

Structure Factor

34

( , ) Q ( ,0), (0, ) Q

Néel Order: Columnar Order:No long range order!

Long Range Order

Structure Factor

2 1/ 0J J 2 1/ 0.6J J 2 1/ 1J J

J1J2-Model

Long Range Order

Local Spin Directions

2 1/ 0.1J J 2 1/ 1J J

J1J2-Model

General features of PEPS

• Pretty reliable and well-conditioned method with absolute error bars for expectation values of variables

– In principle unbiased, like DMRG: we can make the system completely translational invariant

– Especially suited for describing spin liquids et al., gapped systems (cfr. MERA for critical systems!)

– In case of fermions: make use of classical gauge theories to get right statistics

• Everything that can be done with MPS can be done with PEPS (but at a much higher cost)

• Cost of algorithm still scales as ND10 , which is very good as a function of systems size, but bad with respect to the bond dimension

– Not too bad if compared with DMRG: D5 versus D3

• Note also: way less entanglement in 2-D than in 1-D: frustration!

• Work in progress for finding alternative methods for contracting tensor networks– Renormalization methods of Gu et al.– Calculating expectation values by Monte Carlo sampling (see talk of Schuch)– Infinite methods (cfr. Talk of Orus)

Conclusion

• MPS/MPO/PEPS might be a smarter tool to study new states of quantum matter

• MPS/MPO/PEPS formalism is very natural way of representing wave functions of strongly correlated quantum systems

• How does it compare to MERA (Cfr. Guifre)???

• Workshop and long-term programme in Erwin Schrodinger Institute for Mathematical Physics in Vienna on topic of “Entanglement and Correlations in many-body quantum physics” from Aug. 10 – Oct. 17

• http://qit.univie.ac.at/conference

• PhD and Postdoc positions available in Vienna