Tensor Network States: Algorithms and Applications,
Beijing, December 2014
Tensor Network Renormalization
(Evenbly, Vidal, arXiv:1412.0732)
Glen Evenbly
Guifre Vidal
Tensor Renormalization Methods
What is the usefulness of renormalization (or
coarse-graining) in many-body physics???
description in terms of
very many microscopic
degrees of freedom
Iterative RG
transformations
each transformation removes
short-range (high energy)
degrees of freedom
description in terms of a
few effective (low-energy)
degrees of freedom
effective theory should contain
only universal information (i.e. no
microscopic details remaining)
Previous methods based upon tensor renormalization
can be very powerful and useful….
…but they don’t give you this!
Previous tensor RG methods do not address all short-ranged degrees
of freedom at each RG step. Some (unwanted) short ranged detail
always remains in the coarse-grained system
Tensor Renormalization Methods
Previous tensor RG methods do not address all short-ranged degrees
of freedom at each RG step. Some (unwanted) short ranged detail
always remains in the coarse-grained system
Consequences:
•
they do not give a proper RG flow (i.e. give wrong RG fixed points)
•
accumulation of short-ranged degrees of freedom can cause
computational breakdown (at or near criticality)
Tensor Network Renormalization (TNR)
(Evenbly, Vidal, arXiv:1412.0732)
A method of coarse-graining tensor networks that addresses
all short-ranged degrees of freedom at each RG step
Consequences:
•
gives a proper RG flow (i.e. correct RG fixed points)
•
gives a sustainable RG transformation (even at or near criticality)
Outline: Tensor Network Renormalization
Introduction: Tensor networks and methods based upon
tensor renormalization
Comparison: Tensor Renormalization Group (TRG) vs
Tensor Network Renormalization (TNR)
Discussion: Failure of previous tensor RG methods to
address all short-ranged degrees of freedom
Reformulation: A different prescription for implementing
tensor RG methods.
Resolution: how to build a tensor RG scheme that
addresses all short-ranged degrees of freedom
Can express the exact ground
state of quantum system (in terms
of a Euclidean path integral) as a
tensor network:
evolution in
imaginary time
Tensor Renormalization Methods
1D lattice in space
Tensor Renormalization Methods
Can express the exact ground
state of quantum system (in terms
of a Euclidean path integral) as a
tensor network:
Ground
state
|𝜓GS
lim 𝑒 −𝛽𝐻
1D quantum system
with Hamiltonian H:
𝛽→∞
1D lattice in space
Tensor Renormalization Methods
Can express the exact ground
state of quantum system (in terms
of a Euclidean path integral) as a
tensor network:
1D quantum system
with Hamiltonian H:
There are many different approaches
to evaluate a network of this form
(e.g. Monte-carlo, transfer matrix
methods, tensor RG…)
Tensor RG: evaluate expectation
value through a sequence of
controlled (quasi-exact) coarsegraining transformations of the
network
Expectation
value
= 𝜓𝐺𝑆 |𝑜|𝜓𝐺𝑆
Tensor Renormalization Methods
Sequence of coarse graining transformations applied to the tensor network:
𝐴
0
𝐴
1
Could represent:
•
Euclidean path integral of 1D quantum system
•
Partition function of 2D classical statistical system
𝐴
2
𝐴
3
Expectation value of local
observable, two-point
correlator etc
Tensor Renormalization Methods
Sequence of coarse graining transformations applied to the tensor network:
𝐴
𝐴
0
1
𝐴
2
𝐴
width N
log 𝑁
RG flow in the
space of tensors:
𝐴
0
→𝐴
1
→𝐴
2
steps
→⋯→𝐴
𝑆
scalar
→𝐶
3
Tensor Renormalization Methods
log 𝑁
RG flow in the
space of tensors:
𝐴
0
→𝐴
1
→𝐴
2
steps
→⋯→𝐴
𝑆
scalar
→𝐶
Tensor RG approaches borrow from earlier ideas (e.g. Kadanoff spin blocking)
First tensor RG approach:
•
Tensor Renormalization Group (TRG)
(Levin, Nave, 2006)
based upon truncated singular value decomposition (SVD):
≈
𝐴
dim
𝜒
𝑈
𝑆
dim
𝑉
=
𝑈 𝑆
𝑆𝑉
𝜒 ≤ 𝜒2
Incurs a truncation error that is related to the size of the
discarded singular values
Tensor Renormalization Group (TRG)
𝐴
(Levin, Nave, 2006)
0
𝐴
truncated
SVD
0.5
contract
coarser network
initial network
truncated SVD
contract
Tensor Renormalization Methods
log 𝑁
RG flow in the
space of tensors:
𝐴
0
→𝐴
1
→𝐴
2
steps
→⋯→𝐴
𝑆
scalar
→𝑍
Tensor RG approaches borrow from earlier ideas (e.g. Kadanoff spin blocking)
First tensor RG approach:
•
Tensor Renormalization Group (TRG)
(Levin, Nave, 2006)
Many improvements and generalizations:
•
Second Renormalization Group (SRG) (Xie, Jiang, Weng, Xiang, 2008)
•
Tensor Entanglement Filtering Renormalization (TEFR) (Gu, Wen, 2009)
•
Higher Order Tensor Renormalization Group (HOTRG)
+ many more…
(Xie, Chen, Qin, Zhu,
Yang, Xiang, 2012)
Give improvement in accuracy (e.g. by taking more of the local environment
into account when truncating) or allow application to higher dimensional
systems etc.
Tensor Renormalization Methods
But previous tensor RG approaches do not address all shortranged degrees of freedom at each RG step
Consequences:
•
they do not give a proper RG flow (i.e. gives wrong RG fixed points)
•
do not give a sustainable RG flow (at or near criticality)
Tensor Network Renormalization (TNR)
(Evenbly, Vidal, arXiv:1412.0732)
A method of coarse-graining tensor networks that addresses
all short-ranged degrees of freedom at each RG step
Consequences:
•
gives a proper RG flow (i.e. correct RG fixed points)
•
gives a sustainable RG flow (even at or near criticality)
Outline: Tensor Network Renormalization
Introduction: Tensor networks and methods based upon
tensor renormalization
Comparison: Tensor Renormalization Group (TRG) vs
Tensor Network Renormalization (TNR)
Discussion: Failure of previous tensor RG methods to
address all short-ranged degrees of freedom
Reformulation: A different prescription for implementing
tensor RG methods.
Resolution: how to build a tensor RG scheme that
addresses all short-ranged degrees of freedom
Tensor Renormalization Methods
RG flow in the
space of tensors:
𝐴
0
→𝐴
1
→𝐴
2
→⋯→𝐴
𝑠
→⋯
Consider 2D classical Ising ferromagnet at temperature T:
Phases:
Proper RG flow:
𝑇 < 𝑇𝐶
ordered phase
𝑇 = 𝑇𝐶
critical point (correlations
at all length scales)
𝑇 > 𝑇𝐶
disordered phase
0
𝐴 𝑇<𝑇𝐶
0
𝐴 𝑇=𝑇𝐶
𝐴(1)
𝐴order
𝑇=0
𝐴(3)
Encode partition
function (temp T) as
a tensor network:
0
𝐴𝑇
0
𝐴 𝑇>𝑇𝐶
𝐴(1)
𝐴(2)
𝐴(2)
𝐴crit
𝑇 = 𝑇𝐶
𝐴(3)
𝐴disorder
𝑇=∞
Proper RG flow: 2D classical Ising
Numerical results, Tensor renormalization group (TRG):
disordered
phase
|𝐴
1
|
|𝐴
2
|
|𝐴
3
|
|𝐴
4
|
𝑇 = 1.1 𝑇𝑐
𝑇 = 2.0 𝑇𝑐
should converge to
the same (trivial)
fixed point, but don’t!
Numerical results, Tensor Network Renormalization (TNR):
𝑇 = 1.1 𝑇𝑐
𝑇 = 2.0 𝑇𝑐
converge to the same
fixed point (containing
only information on the
universal properties of
the phase
Proper RG flow: 2D classical Ising
Numerical results for 2D classical Ising, Tensor Network Renormalization (TNR):
•
Converges to one of three RG fixed points, consistent with a proper RG flow
|𝐴
sub-critical
𝑇 = 0.9 𝑇𝑐
critical
𝑇 = 𝑇𝑐
super-critical
𝑇 = 1.1 𝑇𝑐
1
|
|𝐴
2
|
|𝐴
3
|
|𝐴
4
|
ordered (Z2)
fixed point
critical (scaleinvariant) fixed
point
disordered
(trivial) fixed
point
Tensor Renormalization Methods
RG flow in the
space of tensors:
𝐴
0
→𝐴
1
→𝐴
2
→⋯→𝐴
𝑠
→⋯
Consider 2D classical Ising ferromagnet at temperature T:
Phases:
𝑇 < 𝑇𝐶
Proper RG flow:
ordered phase
𝑇 = 𝑇𝐶
critical point (correlations
at all length scales)
𝑇 > 𝑇𝐶
disordered phase
0
𝐴 𝑇<𝑇𝐶
0
𝐴 𝑇=𝑇𝐶
𝐴(1)
𝐴order
𝑇=0
𝐴(3)
Encode partition
function (temp T) as
a tensor network:
0
𝐴𝑇
0
𝐴 𝑇>𝑇𝐶
𝐴(1)
𝐴(2)
𝐴(2)
𝐴crit
𝑇 = 𝑇𝐶
𝐴(3)
𝐴disorder
𝑇=∞
Sustainable RG flow: 2D classical Ising
RG flow in the
space of tensors:
𝐴
0
→𝐴
1
→𝐴
2
→⋯→𝐴
𝑠
What is a sustainable RG flow???
Key step of TRG:
truncated singular value decomposition (SVD):
𝐴
dim
𝑆𝑉
𝑈 𝑆
dim
𝜒
𝜒 ≤ 𝜒2
𝑠
Let 𝜒
be the number of singular values (or bond dimension)
needed to maintain fixed truncation error ε at RG step s
The RG is sustainable if 𝜒
Is TRG sustainable???
𝑠
is upper bounded by a constant.
→⋯
Sustainable RG flow: 2D classical Ising
Does TRG give a sustainable RG flow?
Spectrum of 𝐴(𝑠)
10
RG flow at criticality
0
s=0
s=1
s=6
s=2
10 -1
10 -2
10 -3
10 -4
10 0
10 1
10 2 10 0
10 1
10 2 10 0
Bond dimension 𝜒 required to
maintain fixed truncation error (~10-3):
~20
TRG:
~10
10 1
10 2 10 0
10 1
~40
>100
4 × 109
> 1012
Computational cost:
TRG,
𝑂(𝜒 6 ) : 1 × 106
6 × 107
Cost of TRG scales exponentially with RG iteration!
10 2
Sustainable RG flow: 2D classical Ising
Does TRG give a sustainable RG flow?
Spectrum of 𝐴(𝑠)
10
RG flow at criticality
0
s=0
s=1
s=6
s=2
10 -1
TRG
10 -2
10 -3
10 -4
10 0
TNR
10 1
10 2 10 0
10 1
10 2 10 0
10 1
Bond dimension 𝜒 required to
maintain fixed truncation error (~10-3):
~20
TRG:
~10
~40
~10
~10
TNR:
~10
10 2 10 0
10 1
Sustainable
>100
~10
Computational costs:
TRG,
TNR
𝑂(𝜒 6 ) : 1 × 106
𝑂(𝑘𝜒 6 ): 5 × 107
6 × 107
5 × 107
10 2
4 × 109
5 × 107
> 1012
5 × 107
Tensor Renormalization Methods
Previous RG methods for
contracting tensor networks
•
•
•
Tensor Network
Renormalization (TNR)
vs
•
do not address all short-ranged
degrees of freedom
do not give a proper RG flow
(wrong RG fixed points)
•
•
unsustainable RG flow (at or
near criticality)
can address all short-ranged
degrees of freedom
gives a proper RG flow
(correct RG fixed points)
can give a sustainable RG
flow
Analogous to:
Tree tensor network (TTN)
vs
Multi-scale entanglement
renormalization ansatz (MERA)
Tensor Renormalization Methods
Previous RG methods for
contracting tensor networks
•
•
•
do not address all short-ranged
degrees of freedom
do not give a proper RG flow
(wrong RG fixed points)
unsustainable RG flow (at or
near criticality)
Tensor Network
Renormalization (TNR)
vs
•
•
•
can address all short-ranged
degrees of freedom
gives a proper RG flow
(correct RG fixed points)
can give a sustainable RG
flow
Can we see how TRG fails to address all
short-ranged degrees of freedom?
… consider the fixed points of TRG
Outline: Tensor Network Renormalization
Introduction: Tensor networks and methods based upon
tensor renormalization
Comparison: Tensor Renormalization Group (TRG) vs
Tensor Network Renormalization (TNR)
Discussion: Failure of previous tensor RG methods to
address all short-ranged degrees of freedom
Reformulation: A different prescription for implementing
tensor RG methods.
Resolution: how to build a tensor RG scheme that
addresses all short-ranged degrees of freedom
Fixed points of TRG
𝑗
𝑖
𝐴
𝑙
𝑘
=
𝑗1 𝑗2
𝑖1
𝑖2
Partition function built from CDL
tensors represents a state with
short-ranged correlations
𝑘2
𝑘1
𝑙2 𝑙1
Imagine “A” is a special tensor such that each
index can be decomposed as a product of
smaller indices,
Aijkl  A( i1i2 )( j1 j2 )( k1k 2 )( l1l 2 )
such that certain pairs of indices are perfectly
correlated:
A( i1i2 )( j1 j2 )( k1k 2 )( l1l 2 )   i1 j1 
j2 k 2
k l l i
1 1
2 2
These are called corner double line (CDL)
tensors. CDL tensors are fixed points of TRG.
Fixed points of TRG
Singular value
decomposition
Contraction
=
=
new CDL tensor!
Some short-ranged always
correlations remain under TRG!
Fixed points of TRG
short-range correlated
short-range correlated
TRG
TRG removes some short ranged correlations, but…
others are artificially promoted to the next length scale
•
Coarse-grained networks always retain some
dependence on the microscopic (short-ranged) details
•
Accumulation of short-ranged correlations causes
computational breakdown when at / near criticality
Is there some way to ‘fix’ tensor renormalization such that all
short-ranged correlations are addressed?
Outline: Tensor Network Renormalization
Introduction: Tensor networks and methods based upon
tensor renormalization
Comparison: Tensor Renormalization Group (TRG) vs
Tensor Network Renormalization (TNR)
Discussion: Failure of previous tensor RG methods to
address all short-ranged degrees of freedom
Reformulation: A different prescription for implementing
tensor RG methods.
Resolution: how to build a tensor RG scheme that
addresses all short-ranged degrees of freedom
Reformulation of Tensor RG
Change in
formalism:
RG scheme based on insertion
of projectors into network
RG scheme based on
SVD decompositions
truncated singular value decomposition (SVD):
dim
dim
𝜒
𝐴
𝐴
=
dim
𝐴
=
𝑈 𝑆
𝜒
𝑊†
𝑊
rank
𝐴
𝜒 projector
𝑆𝑉
𝜒 ≤ 𝜒2
Reformulation of Tensor RG
Change in
formalism:
RG scheme based on insertion
of projectors into network
RG scheme based on
SVD decompositions
truncated singular value decomposition (SVD):
dim
dim
𝜒
𝐴
𝐴
=
𝑈 𝑆
dim
𝐴
=
set:
𝑊
𝜒 ≤ 𝜒2
𝑆𝑉
𝜒
Want to choose ‘w’
as to minimize error:
𝑊†
𝐴
𝑈†
𝐴
𝜀 = ‖𝐴 − 𝐴‖
𝑊=𝑈
𝑈
=
𝑈
𝑆𝑉
Reformulation of Tensor RG
Change in
formalism:
RG scheme based on insertion
of projectors into network
RG scheme based on
SVD decompositions
𝜒 dim
𝑈 𝑆
approximate decomposition
into pair of 3-index tensors
𝑆𝑉
𝜒 dim
𝐴
𝑊
𝑊†
if isometry w is optimised to act as an
approximate resolution of the identity,
then these are equivalent!
𝐴
=
𝑊
𝑊 †𝐴
Reformulation of Tensor RG
Change in
formalism:
RG scheme based on insertion
of projectors into network
RG scheme based on
SVD decompositions
𝑈
𝑆
𝑊
𝑊†
could be equivalent
decompositions
𝑉
𝐴
HOTRG can also be
done with insertion of
projectors
𝑊
𝐴
𝑊 †𝐴 𝑄
𝑄†
𝑄
𝑄†
Reformulation of Tensor RG
TRG
truncated
SVD
Equivalent scheme
contract
Can reduce cost of TRG
𝑂 𝜒6 → 𝑂 𝜒5
insert
projectors
𝑤†
𝑤
contract
Reformulation of Tensor RG
𝐴
≈
𝑤†
𝑤
=
𝐴
𝑤
𝑤 †𝐴
Insertion of projectors can mimic a matrix decomposition (e.g. SVD)…
…but can also do things that cannot be done using a matrix decomposition
dim 𝜒
𝐴
dim
≈
𝑢
𝜒≤𝜒
𝑢†
𝐴
=
projector that is
decomposed as a
product of fourindex isometries
Restrict to the case
𝜒=𝜒
such that u is a unitary
𝑢
𝑢†𝐴
Reformulation of Tensor RG
𝐴
≈
𝑤†
𝑤
𝐴
=
𝑤
𝑤 †𝐴
Insertion of projectors can mimic a matrix decomposition (e.g. SVD)…
…but can also do things that cannot be done using a matrix decomposition
dim 𝜒
𝐴
dim
=
𝜒=𝜒
𝑢†
𝑢
𝐴
exact resolution of
the identity
𝑢
𝑢†
=
=
𝑢
𝑢†𝐴
Reformulation of Tensor RG
𝐴
≈
𝑤†
𝑤
𝐴
=
𝑤
𝑤 †𝐴
Insertion of projectors can mimic a matrix decomposition (e.g. SVD)…
…but can also do things that cannot be done using a matrix decomposition
dim 𝜒
𝐴
dim
=
𝑢
𝜒=𝜒
𝑢†
𝐴
𝐴
=
𝑢
𝑢†
𝐴
=
𝑢
=
𝑢
𝐴
𝐴
𝑢†𝐴
Tensor network renormalization (TNR) approach follows from composition
of these insertions…
Outline: Tensor Network Renormalization
Introduction: Tensor networks and methods based upon
tensor renormalization
Comparison: Tensor Renormalization Group (TRG) vs
Tensor Network Renormalization (TNR)
Discussion: Failure of previous tensor RG methods to
address all short-ranged degrees of freedom
Reformulation: A different prescription for implementing
tensor RG methods.
Resolution: how to build a tensor RG scheme that
addresses all short-ranged degrees of freedom
Tensor Network Renormalization
𝑢†
𝑢
=
𝑤† 𝑤
≈
𝐴(s)
Insert exact resolutions
of the identity
Insert approximate
resolutions of the identity
Tensor Network Renormalization
𝑢†
𝑢
𝑤† 𝑤
=
≈
𝐴(s)
=
Contractions
=
=
Contract
Tensor Network Renormalization
𝑢†
𝑢
=
𝐴(s)
𝑤† 𝑤
≈
disentanglers
=
𝐴(s+1)
=
Contract
≈
Singular value decomposition
Contract
Tensor Network Renormalization
Equivalent to TRG
Tensor network
Renormalization (TNR)
Tensor Network Renormalization (TNR):
𝑢†
𝑢
=
𝑤†
𝑤
≈
Insert exact resolutions
of the identity
Insert approximate
resolutions of the identity
If the disentanglers ‘u’ are removed then the
TNR approach becomes equivalent to TRG
I will not here discuss the algorithm required to
optimize disentanglers ‘u’ and isometries ‘w’
Does TNR address all short-ranged degrees of freedom?
Tensor Network Renormalization (TNR):
short-range correlated
short-range correlated
What is the effect of
disentangling?
TRG
TNR can address all
short range degrees
of freedom!
TNR
Key step of TNR algorithm:
Insert unitary disentanglers:
𝑢†
=
𝑢
=
trivial (product) state
Tensor Network Renormalization (TNR):
short-range correlated
trivial (product) state
TEFR
trivial (product) state
TNR
An earlier attempt at resolving the problem of accumulation of shortranged degrees of freedom:
Tensor Entanglement Filtering Renormalization (TEFR) (Gu, Wen, 2009)
TEFR can transform the network of CDL tensors to a trivial network
Tensor Network Renormalization (TNR):
short-range correlated
Removing correlations
from CDL fixed point is
necessary, but not
sufficient, to generate a
proper RG flow
trivial (product) state
TNR
also
TEFR
More difficult case: can short-ranged correlations still be removed when
correlations at many length scales are present?
TEFR? No, does not appear so…
Tensor entanglement filtering renormalization only works removes
correlations from CDL tensors (i.e. systems far from criticality)
TNR? Yes…
Tensor network renormalization can remove short-ranged correlations
near or at criticality, and potentially in higher dimension D
Outline: Tensor Network Renormalization
Introduction: Tensor networks and methods based upon
tensor renormalization
Comparison: Tensor Renormalization Group (TRG) vs
Tensor Network Renormalization (TNR)
Discussion: Failure of previous tensor RG methods to
address all short-ranged degrees of freedom
Reformulation: A different prescription for implementing
tensor RG methods.
Resolution: how to build a tensor RG scheme that
addresses all short-ranged degrees of freedom
Benchmark numerics:
212 × 212
2D classical Ising model on lattice of size:
10
-3
1
Error in Free Energy, 𝛿𝑓
10
10
10
10
𝜒 = 4 TRG
-4
𝜒 = 8 TRG
-5
𝜒=4
TNR
-6
𝜒 = 8 TNR
-7
TC
0.8
0.7
0.6
0.5
0.4
0.3
Exact
𝜒=4
TNR
0.2
0.1
TC
10
Spontaneous Magnetization, 𝑀
0.9
0
-8
2
2.1
2.2
2.3
Temperature, T
2.4
2
2.1
2.2
2.3
Temperature, T
2.4
Tensor Renormalization Methods
RG flow in the
space of tensors:
𝐴
0
→𝐴
1
→𝐴
2
→⋯→𝐴
𝑠
→⋯
Consider 2D classical Ising ferromagnet at temperature T:
Phases:
𝑇 < 𝑇𝐶
Proper RG flow:
ordered phase
𝑇 = 𝑇𝐶
critical point (correlations
at all length scales)
𝑇 > 𝑇𝐶
disordered phase
0
𝐴 𝑇<𝑇𝐶
0
𝐴 𝑇=𝑇𝐶
𝐴(1)
𝐴order
𝑇=0
𝐴(3)
Encode partition
function (temp T) as
a tensor network:
0
𝐴𝑇
0
𝐴 𝑇>𝑇𝐶
𝐴(1)
𝐴(2)
𝐴(2)
𝐴crit
𝑇 = 𝑇𝐶
𝐴(3)
𝐴disorder
𝑇=∞
Proper RG flow: 2D classical Ising
Numerical results, Tensor renormalization group (TRG):
disordered
phase
|𝐴
1
|
|𝐴
2
|
|𝐴
3
|
|𝐴
4
|
𝑇 = 1.1 𝑇𝑐
𝑇 = 2.0 𝑇𝑐
Numerical results, Tensor Network Renormalization (TNR):
𝑇 = 1.1 𝑇𝑐
𝑇 = 2.0 𝑇𝑐
CDL tensor
fixed points
Proper RG flow: 2D classical Ising
more difficult!
|𝐴
1
|
|𝐴
2
|
|𝐴
3
|
|𝐴
8
|
|𝐴
7
|
|𝐴
6
|
|𝐴
9
10
|
11
|
|𝐴
4
|𝐴
5
|
|𝐴
12
|
𝑇 = 1.002 𝑇𝑐
TNR bond
dimension:
𝜒=4
|
|𝐴
|𝐴
|
Proper RG flow: 2D classical Ising
critical point:
|𝐴
1
|
|𝐴
2
|
|𝐴
3
|
|𝐴
8
|
|𝐴
7
|
|𝐴
6
|
|𝐴
9
10
|
11
|
|𝐴
4
|𝐴
5
|
|𝐴
12
|
𝑇 = 𝑇𝑐
TNR bond
dimension:
𝜒=4
|
|𝐴
|𝐴
|
Proper RG flow: 2D classical Ising
more difficult!
|𝐴
1
|
|𝐴
2
|
|𝐴
3
|
|𝐴
8
|
|𝐴
7
|
|𝐴
6
|
|𝐴
9
10
|
11
|
|𝐴
4
|𝐴
5
|
|𝐴
12
|
𝑇 = 1.002 𝑇𝑐
bond dimension:
𝜒=4
|
|𝐴
|𝐴
|
Sustainable RG flow: 2D classical Ising
Does TRG give a sustainable RG flow?
Spectrum of 𝐴(𝑠)
10
0
RG flow at criticality
(a)
s=0
s=1
s=6
s=2
10 -1
TRG
10 -2
10 -3
10 -4
10 0
TNR
10 1
10 2 10 0
10 1
10 2 10 0
10 1
Bond dimension 𝜒 required to
maintain fixed truncation error (~10-3):
~20
TRG:
~10
~40
~10
~10
TNR:
~10
10 2 10 0
10 1
>100
~10
Computational costs:
TRG,
TNR
𝑂(𝜒 6 ) : 1 × 106
𝑂(𝑘𝜒 6 ): 5 × 107
6 × 107
5 × 107
4 × 109
5 × 107
> 1012
5 × 107
10 2
Summary
We have introduced an RG based method for contracting tensor networks:
Tensor Network Renormalization (TNR)
key idea: proper arrangement of
isometric ‘w’ and unitary ‘u’ tensors
address all short-ranged degrees of
freedom at each RG step
𝑤
𝑤†
𝑢†
𝑢
key features of TNR:
•
Address all short-ranged degrees of freedom
•
Proper RG flow (gives correct RG fixed points)
•
Sustainable RG flow (can iterate without increase in cost)
Direct applications to study of 2D classical and 1D quantum many-body
systems, and for contraction of PEPS
In 2D quantum (or 3D classical) the accumulation of short-ranged degrees
of freedom in HOTRG is much worse (due to entanglement area law)
…but higher dimensional generalization of TNR could still generate a
sustainable RG flow