Download PEPS, matrix product operators and the Bethe ansatz

PEPS, matrix product operators and the
algebraic Bethe ansatz
Frank Verstraete
University 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.
H
• 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
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 (==YangBaxter 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
eigenvectors
and hence they have the same
• 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
arXiv:cond-mat/0407066
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.)
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)
From here on: all work and
slides by Valentin Murg
The Method
Time Evolution
eiH t | PEPS 
Goal:
Simulation of the Operation
| PEPS 

eiH t | PEPS 
Procedure:
Dimension 
Dimension D
 iH  t
| PEPS 
For  t 1 , the state e
remains a PEP-state, but with
increased virtual Dimension:
D D
Problem: Virtual Dimension increases
exponentially with the
number of steps.
Workaround:
Dimension  D
Approximate at each step the
PEP-state by a PEP-state with
reduced virtual Dimension.
8
The Method
Time Evolution
| PEPS 
Goal:
| PEPS 
Dimension  D
| PEPS 

Dimension D
2
K  | PEPS   | PEPS   Min.
Dimension  D
Algorithm:
Optimization of the distance site by site:
| PEPS 
K
0
Ak( ij )

MA(ij )  w
Optimal scaling of the Algorithm:
Dimension D
#steps
#memory
~ N 2 D10
~ D8
arXiv:0901.2019
J1-J2-J3 Heisenberg model
J1
J2
J3
• Frustrated system
• We did calculation for systems with open boundary conditions of
size up to 14x14 and D=4
J1J2J3-Model
IV. Helicoidal
Classical Phase Diagram
( q,  )
J1
III. Helicoidal
( q, q )
J1
J2
I. Néel
J1
J2
( ,  )
II. Independent Sublattices
J1
J2
J2

J1J2J3-Model
Quantum Phase Diagram
II. Collinear
( ,0) (0,  )
J1
J2
J1
J2
(Order-by-Disorder)
J1J3-Model
Long Range Order
Structure Factor
J 3 / J1  0
Néel Order:
Q  ( ,  )
J 3 / J1  0.5
J 3 / J1  1
No long range order!
Néel Order on
four Sublattices:
Q  ( / 2,  / 2)
Plaquette order parameter, 8x8 lattice, J3/J1=1/2
Pure Plaquette
state: in Pl
Q=1, in
between Pl
Q=1/4
24
J1J3-Model
Intermediate Phase
Comparison with short range resonating valence bond ground state
34
J1J2-Model
Long Range Order
Structure Factor
J1J2-Model
Long Range Order
Structure Factor
J 2 / J1  0
Néel Order:
Q  ( ,  )
J 2 / J1  0.6
J 2 / J1  1
No long range order!
Columnar Order:
Q  ( ,0), (0,  )
J1J2-Model
Long Range Order
Local Spin Directions
J 2 / J1  0.1
J 2 / J1  1
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