tensor networks and the numerical study of quantum and classical systems on infinite lattices
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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
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
Matrix Product States (MPS)
[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
… …
Matrix Product Density Operators (MPDO)
€
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
For finite systems, the state is represented with parameters, instead of .
€
2ndpχ 2
€
d2n
Any density operator can be represented as an MPDO, with large enough and
Physical observables (e.g. correlators) can be computed in time.
€
O( poly(χ )poly( p))€
χ
€
p
Useful in the computation of 1-dimensional thermal states.
[Verstraete, García-Ripoll, Cirac, 2004]
Projected Entangled Pair States (PEPS)
Physical local system of dimension
€
d
Bonds of dimension
€
D
For finite systems, the state is represented with parameters, instead of .
€
ndD4
€
dn
Physical observables (e.g. correlators) can be computed in time.
€
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
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
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
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
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
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
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.
We have developed a technique to contract the whole PEPS tensor network in the
thermodynamic limit for translationally-invariant systems.
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.
We have developed a technique to contract the whole PEPS tensor network in the
thermodynamic limit for translationally-invariant systems.
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.
We have developed a technique to contract the whole PEPS tensor network in the
thermodynamic limit for translationally-invariant systems.
…
……
…
Boundary MPS with bond dimension
Action of non-unitary gates on an infinite
MPS
Can be efficiently computed, taking care of orthonormalization issues
€
χ
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.
We have developed a technique to contract the whole PEPS tensor network in the
thermodynamic limit for translationally-invariant systems.
…
……
…
Iterate until a fixed point is found
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.
We have developed a technique to contract the whole PEPS tensor network in the
thermodynamic limit for translationally-invariant systems.
…
……
…
Iterate until a fixed point is found
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.
We have developed a technique to contract the whole PEPS tensor network in the
thermodynamic limit for translationally-invariant systems.
…
……
…
Iterate until a fixed point is found
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.
We have developed a technique to contract the whole PEPS tensor network in the
thermodynamic limit for translationally-invariant systems.
…
……
…
Once there is convergence, contract it from the other side and compute e.g. correlators on the diagonal with the obtained MPS
… …
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.
We have developed a technique to contract the whole PEPS tensor network in the
thermodynamic limit for translationally-invariant systems.
…
……
…
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
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
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
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