Tensor Network States: Algorithms and Applications
December 1-5, 2014
Beijing, China
The multi-scale Entanglement
Renormalization Ansatz
MERA
-- a pedagogical introduction--
Guifre Vidal
outline
MERA
• Definition
• Efficiency
• Structural properties:
correlations and entropy
The Renormalization Group
• Goals
• RG by isometries (TTN): why is it “wrong”? TRG
• RG by isometries and disentanglers (MERA)
(Zhiyuan’s lecture)
TNR (Glen’s talk)
MERA: definition
𝑑 ⊗𝑁
Ψ ∈ (ℂ )
𝑑𝑁
complex numbers
Multi-scale entanglement
renormalization ansatz
(MERA)
Matrix product state
(MPS)
MERA
also MERA !
Efficiency
𝑑 ⊗𝑁
Ψ ∈ (ℂ )
𝑁 +
𝑁
𝑁
1 1
+ + … =𝑁 1+ + +⋯
2
4
2 4
𝑑𝑁
≤ 2𝑁
complex numbers
Multi-scale entanglement
renormalization ansatz
(MERA)
Matrix product state
(MPS)
𝑁 spins ⇒ 𝑁 tensors
⇒ 𝑂(𝑁) parameters
log(𝑁)
𝑁
𝑁 spins
⇒ 𝑁 log(𝑁) tensors ?
2𝑁 tensors ⇒ 𝑂(𝑁) parameters
Q1
efficiency
Matrix product state
(MPS)
Q2
⟨Ψ Ψ
cost 𝑂(𝑁)
⟨Ψ|𝑜 Ψ
cost 𝑂(𝑁)
𝑡
efficiency
|Ψ
𝑤
𝑢
isometric tensors!
𝑢
𝑤
𝑡
𝑡†
⟨Ψ|Ψ =
𝑤
𝑤†
𝑢
𝑢†
=1
=
=
Q3
cost = 0 !!!
efficiency
⟨Ψ|Ψ =
isometric tensors!
𝑡
𝑡†
𝑤
𝑤†
𝑢
𝑢
†
=1
=
=
efficiency
⟨Ψ|Ψ =
isometric tensors!
𝑡
𝑡†
𝑤
𝑤†
𝑢
𝑢
†
=1
=
=
efficiency
⟨Ψ|Ψ =
isometric tensors!
𝑡
𝑡†
𝑤
𝑤†
𝑢
𝑢
†
=1
=
=
efficiency
⟨Ψ|Ψ =
isometric tensors!
𝑡
𝑡†
𝑤
𝑤†
𝑢
𝑢
†
=1
=
=
efficiency
⟨Ψ|Ψ =
isometric tensors!
𝑡
𝑡†
𝑤
𝑤†
𝑢
𝑢
†
=1
=
=
efficiency
⟨Ψ|Ψ = 1
isometric tensors!
𝑡
𝑡†
𝑤
𝑤†
𝑢
𝑢
†
=1
=
=
𝑜
⟨Ψ|𝑜 Ψ =
isometric tensors!
𝑡
𝑡†
𝑤
𝑤†
𝑢
𝑢†
=1
=
=
=
(1)
=
⟨Ψ|𝑜 Ψ =
(2)
=
(3)
cost O(log 𝑁 )
= Ψ𝑜Ψ
Structural properties
𝑑 ⊗𝑁
Ψ ∈ (ℂ )
𝑑𝑁
• Decay of correlations
• Scaling of entanglement
complex numbers
⟨Ψ|𝑜 0 𝑜(𝐿)|Ψ
MPS
=
=
𝐿−1
=
≈
𝜆𝐿−1 = 𝐴𝑒 −𝐿/𝜉
𝜉≡−
⇒ Exponential decay of correlations
1
log 𝜆
Ψ𝑜 0 𝑜 𝐿 Ψ
⋯
⋯
⋯
=
MERA
⋯
⋯
⋯
Ψ𝑜 0 𝑜 𝐿 Ψ
⋯
MERA
⋯
=
≈ 𝜆
=
log3 𝐿
𝜆
log3 (𝐿)
= 𝜆2 log3(𝐿) = 𝐿2 log3(𝜆) = 𝐿−𝑝
𝑥 log3(𝑦) = 𝑦 log3(𝑥)
𝑝 ≡ −2 log 3 (𝜆)
⇒ Polynomial decay of correlations
⋯
⋯
Correlations: summary and interpretation
matrix product state
(MPS)
multi-scale entanglement renormalization ansatz
(MERA)
𝐿
log(𝐿)
structure of geodesics:
⟨𝑜 0 𝑜 𝐿
𝐿
≈ 𝑒 −𝐿/𝜉
exponential
structure of geodesics:
⟨𝑜 0 𝑜 𝐿
≈ 𝐿−𝑝
power-law
Entanglement entropy
matrix product state
multi-scale entanglement renormalization ansatz
(MPS)
(MERA)
𝐴
𝐿
log(𝐿)
connectivity:
𝐴
𝑆(𝐴) ≈ 𝑐𝑜𝑛𝑠𝑡
boundary law!
Q4
𝐿
connectivity:
𝑆(𝐴) ≈ log 𝐿
logarithmic correction!
Q5
⋮
⋯
⋯
𝑆(𝐴) ≈ log 𝐿
Example: operator content of quantum Ising model
𝑥
𝜎𝑖𝑥 ⊗ 𝜎𝑖+1
+ℎ
𝐻=
𝑖
𝜎𝑖𝑧
for ℎ = ℎ𝑐 = 1
𝑖
scaling
dimension
(exact )
scaling operators/dimensions:
identity
spin
energy
disorder
fermions
𝕀
0
scaling
dimension
(MERA)
0
error
----
𝜎 0.125
0.124997
0.003%
𝜀
0.99993
0.007%
0.1250002
0.0002%
0.5
<10−8 %
0.5
<10−8 %
1
 0.125
 0.5

0.5
OPE for local & non-local primary fields
C   1 / 2
C    i
C    1 / 2
C     i
C   e
 i / 4
C  e
i / 4
/
fusion rules

2
4
(  6  10 )
/
2
   I
    I+ 
   
   I
   
   I
{ I,  ,  ,  , , }
local and
semi-local
subalgebras
{ I,  }
{ I,  ,  }
   I
   
  
  
{ I,  ,  }
{ I,  , , }
...
MERA and HOLOGRAPHY
t
t
s
x
CFT1+1
x
x
AdS2+1
outline
MERA
• Definition
• Efficiency
• Structural properties:
correlations and entropy
The Renormalization Group
• Goals
• RG by isometries: why is it wrong?
• RG by isometries and disentanglers
The Renormalization Group: goals
Given a local Hamiltonian
𝐻=
ℎ𝑖,𝑖+1
on 𝑁 sites
(Hilbert space dimension 𝑑 𝑁 )
𝑖
Two type of questions:
1) Low energy, large distance, UNIVERSAL behavior:
e.g. disordered/symmetry-breaking phase,
topological order (S,T modular matrices),
quantum criticality (scaling operators, CFT data)
fixed point
𝐻 → 𝐻 ′ → 𝐻 ′′ → ⋯ → 𝐻 (⋆)
Ψ 𝑜 𝑥 𝑜 𝑦 |Ψ⟩ for 𝑥 − 𝑦 → ∞
2) Low energy, short distance, detailed MICROSCOPIC properties
e.g. ⟨Ψ|𝑜 𝑥 |Ψ⟩,
Ψ 𝑜 𝑥 𝑜 𝑦 |Ψ⟩, for all 𝑥, 𝑦
The Renormalization Group: goals
Example: given the
(transverse field) Ising Hamiltonian
𝑥
𝜎𝑖𝑥 ⊗ 𝜎𝑖+1
+ℎ
𝐻=
𝑖
𝜎𝑖𝑧
𝑖
𝑚(ℎ)
spontaneous
magnetization
0
ℎ
ℎ𝑐
magnetic field
1) Low energy, large distance, UNIVERSAL behavior?
Is the spontaneous magnetization
m(h) ≡ Ψ 𝜎 𝑥 Ψ ≠ 0, or = 0?
ordered
phase
disordered
phase
2) Low energy, short distance, detailed MICROSCOPIC properties?
How much is the spontaneous magnetization
m(ℎ) ≡ ⟨Ψ|𝜎 𝑥 |Ψ⟩ as a function on ℎ?
The Renormalization Group
on Hamiltonians:
on ground state
wave-functions:
on classical
partition functions:
𝐻 →
𝐻′
→
𝐻′′ →
⋯ → 𝐻 (⋆)
|Ψ⟩ → |Ψ′⟩ → |Ψ′′⟩ → ⋯ → |Ψ (⋆) ⟩
𝒵 → 𝒵 ′ → 𝒵 ′′ → ⋯ → 𝒵 (∗)
fixed point
Hamiltonian
fixed point
ground state
fixed point
partition function
Two types of Renormalization Group transformations:
• Type 1 is only required to preserve UNIVERSAL properties
• Type 2 is also required to preserve MICROSCOPIC properties
𝑜 → 𝑜 ′ → 𝑜 ′′ → ⋯ → 𝑜 (⋆)
such that
(for instance, 𝑜 = 𝜎 𝑥
in quantum Ising model)
Ψ 𝑜 Ψ = Ψ ′ |o′|Ψ ′ = 𝛹 ′′ 𝑜 ′′ 𝛹 ′′ = ⋯ = 𝛹 (∗) 𝑜(∗) 𝛹 (∗)
For instance, TRG, TTN, MERA, are of type 2
The Renormalization Group
• Type 1 is only required to preserve UNIVERSAL properties
• Type 2 is also required to preserve MICROSCOPIC properties?
𝑜 → 𝑜 ′ → 𝑜 ′′ → ⋯ → 𝑜 (⋆)
(for instance, 𝑜 = 𝜎 𝑥
in quantum Ising model)
such that
Ψ 𝑜 Ψ = Ψ ′ |o′|Ψ ′ = 𝛹 ′′ 𝑜 ′′ 𝛹 ′′ = ⋯ = 𝛹 (∗) 𝑜(∗) 𝛹 (∗)
Types 2A and 2B!!
Type 2A:
|Ψ⟩ → |Ψ′⟩ → |Ψ′′⟩ → ⋯ → |Ψ (⋆) ⟩
e.g. TTN, Zhiyuan’s lecture on TRG
fixed point:
mixture of UNIVERSAL
and MICROSCOPIC
properties
Type 2B:
|Ψ⟩ → |Ψ′⟩ → |Ψ′′⟩ → ⋯ → |Ψ (⋆) ⟩
e.g. MERA, Glen’s talk on TNR
fixed point:
only UNIVERSAL
properties
The Renormalization Group
Type 2A:
|Ψ⟩ → |Ψ′⟩ → |Ψ′′⟩ → ⋯ → |Ψ (⋆) ⟩
fixed point:
mixture of UNIVERSAL
and MICROSCOPIC
properties
example: TTN
𝑁 ′ = 𝑁/3 sites
𝑤
𝑤
ℒ′
𝑊
𝑤†
ℒ
=
𝑁 sites
coarse-graining transformation
𝑊 = 𝑤 ⊗ 𝑤 ⊗ ⋯⊗ 𝑤
Q6
=
A’
B’
C’
D’
E’
F’
A’
B’
C’
D’
E’
F’
The Renormalization Group
Type 2A:
|Ψ⟩ → |Ψ′⟩ → |Ψ′′⟩ → ⋯ → |Ψ (⋆) ⟩
example: TTN
fixed point:
mixture of UNIVERSAL
and MICROSCOPIC
properties
coarse-graining transformation
𝑊 = 𝑤 ⊗ 𝑤 ⊗ ⋯⊗ 𝑤
Ψ → Ψ ′ = 𝑊 † |Ψ⟩
=
Ψ
𝑊†
A’
B’
C’
D’
F’
E’
Ψ′
A’ B’ C’ D’ E’ F’
𝑜 → 𝑜′ = 𝑊 †𝑜 𝑊
𝑊
=
A’
B’
C’
𝑜
D’
E’
F’
𝑊†
=
A’ B’ C’
D’ E’ F’
A’ B’ C’ D’ E’ F’
𝑜′
The Renormalization Group
Type 2A:
|Ψ⟩ → |Ψ′⟩ → |Ψ′′⟩ → ⋯ → |Ψ (⋆) ⟩
example: TTN
fixed point:
mixture of UNIVERSAL
and MICROSCOPIC
properties
coarse-graining transformation
𝑊 = 𝑤 ⊗ 𝑤 ⊗ ⋯⊗ 𝑤
Ψ → Ψ ′ = 𝑊 † |Ψ⟩
Ψ′
Ψ
=
𝑊†
Already a fixed point wave-function!
It contains short-range entanglement = MICROSCOPIC details
The Renormalization Group
Type 2A:
|Ψ⟩ → |Ψ′⟩ → |Ψ′′⟩ → ⋯ → |Ψ (⋆) ⟩
example: TTN
fixed point:
mixture of UNIVERSAL
and MICROSCOPIC
properties
coarse-graining transformation
𝑊 = 𝑤 ⊗ 𝑤 ⊗ ⋯⊗ 𝑤
|Ψ ′ ⟩ retains short-range entanglement = MICROSCOPIC details
Ψ 𝑜 Ψ = Ψ ′ |o′|Ψ ′
⇒ results are still accurate
provided that we use a sufficiently large bond dimension
Different fixed-point wave-function
for the same phase!
𝑚(ℎ)
(⋆)
|Ψh1 ⟩ → |Ψh1 ′⟩ → |Ψℎ1 ′′⟩ → ⋯ → |Ψℎ1 ⟩
spontaneous
magnetization
(⋆)
(⋆)
|Ψℎ1 ⟩ |Ψℎ2 ⟩
0
(⋆)
|Ψh2 ⟩ → |Ψh2 ′⟩ → |Ψℎ2 ′′⟩ → ⋯ → |Ψℎ2 ⟩
(⋆)
(⋆)
|Ψℎ1 ⟩ and |Ψℎ2 ⟩ have the same UNIVERSAL information,
mixed with different MICROSCOPIC details
ℎ
ℎ1
ℎ2
magnetic
field
The Renormalization Group
Type 2A:
|Ψ⟩ → |Ψ′⟩ → |Ψ′′⟩ → ⋯ → |Ψ (⋆) ⟩
example: TTN
fixed point:
mixture of UNIVERSAL
and MICROSCOPIC
properties
coarse-graining transformation
𝑊 = 𝑤 ⊗ 𝑤 ⊗ ⋯⊗ 𝑤
|Ψ ′ ⟩ retains short-range entanglement = MICROSCOPIC details
Ψ 𝑜 Ψ = Ψ ′ |o′|Ψ ′
⇒ results are still accurate
provided that we use a sufficiently large bond dimension
(⋆)
(⋆)
|Ψℎ1 ⟩ and |Ψℎ2 ⟩ have the same UNIVERSAL information,
mixed with different MICROSCOPIC details
Two problems:
• Computational:
RG scheme is more expensive
(⋆)
• Conceptual: How do we separate UNIVERSAL from MICROSCOPIC in |Ψℎ ⟩ ?
The Renormalization Group
Type 2B:
Proper RG transformation:
fixed point:
only UNIVERSAL
properties
|Ψ⟩ → |Ψ′⟩ → |Ψ′′⟩ → ⋯ → |Ψ (⋆) ⟩
example: MERA
𝑁 ′ = 𝑁/3 sites
𝑤
𝑢
𝑤
ℒ′
𝑤†
ℒ
𝑢
𝑊
𝑁 sites
coarse-graining transformation
𝑊 = (⋯ 𝑢 ⊗ 𝑢 ⊗ 𝑢 ⊗ ⋯ )(⋯ 𝑤 ⊗ 𝑤 ⊗ 𝑤 ⊗ ⋯)
Q7
=
A’
B’
C’
D’
E’
F’
A’ B’ C’ D’ E’ F’
𝑢†
=
=
The Renormalization Group
Type 2B:
example: MERA
Proper RG transformation:
|Ψ⟩ → |Ψ′⟩ → |Ψ′′⟩ → ⋯ → |Ψ (⋆) ⟩
fixed point:
only UNIVERSAL
properties
coarse-graining transformation
𝑊 = (⋯ 𝑤 ⊗ 𝑤 ⊗ 𝑤 ⊗ ⋯ )(⋯ 𝑢 ⊗ 𝑢 ⊗ 𝑢 ⊗ ⋯)
Ψ → Ψ ′ = 𝑊 † |Ψ⟩
=
Ψ
𝑊†
Ψ′
A’ B’ C’ D’ E’ F’
𝑜 → 𝑜′ = 𝑊 †𝑜 𝑊
=
A’
B’
C’
𝑜
D’
E’
F’
=
A’ B’ C’
D’ E’ F’
A’ B’ C’ D’ E’ F’
𝑜′
The Renormalization Group
Type 2B:
example: MERA
Proper RG transformation:
|Ψ⟩ → |Ψ′⟩ → |Ψ′′⟩ → ⋯ → |Ψ (⋆) ⟩
fixed point:
only UNIVERSAL
properties
coarse-graining transformation
𝑊 = (⋯ 𝑤 ⊗ 𝑤 ⊗ 𝑤 ⊗ ⋯ )(⋯ 𝑢 ⊗ 𝑢 ⊗ 𝑢 ⊗ ⋯)
Ψ → Ψ ′ = 𝑊 † |Ψ⟩
Ψ
𝑊†
Product state wave-function!
It contains no MICROSCOPIC details
=
=
Ψ′
A’ B’ C’ D’ E’ F’
=
=
The Renormalization Group
Type 2B:
example: MERA
Proper RG transformation:
fixed point:
only UNIVERSAL
properties
|Ψ⟩ → |Ψ′⟩ → |Ψ′′⟩ → ⋯ → |Ψ (⋆) ⟩
coarse-graining transformation
𝑊 = (⋯ 𝑤 ⊗ 𝑤 ⊗ 𝑤 ⊗ ⋯ )(⋯ 𝑢 ⊗ 𝑢 ⊗ 𝑢 ⊗ ⋯)
Same fixed-point wave-function
for the same phase!
|Ψ ⟩ → |Ψ′⟩ → |Ψ′′⟩ → ⋯ →
|Ψ
(⋆)
|Ψ
ℎ=0
(⋆)
|Ψ
ℎ𝑐
for ℎ < ℎ_𝑐
⟩
for ℎ = ℎ_𝑐
⟩
(⋆)
ℎ=∞
for ℎ > ℎ_𝑐
⟩
𝑚(ℎ)
magnetic
field h
spontaneous
magnetization
ℎ=0
ℎ′
|Ψ
(⋆)
ℎ=0
⟩
ℎ
ℎ
|Ψ
(⋆ )
ℎ𝑐
⟩
ℎ′
ℎ=∞
|Ψ
(⋆)
ℎ=∞
⟩
The Renormalization Group
Type 2B:
example: MERA
Proper RG transformation:
|Ψ⟩ → |Ψ′⟩ → |Ψ′′⟩ → ⋯ → |Ψ (⋆) ⟩
fixed point:
only UNIVERSAL
properties
coarse-graining transformation
𝑊 = (⋯ 𝑤 ⊗ 𝑤 ⊗ 𝑤 ⊗ ⋯ )(⋯ 𝑢 ⊗ 𝑢 ⊗ 𝑢 ⊗ ⋯)
Same fixed-point wave-function
for the same phase!
|Ψ ⟩ → |Ψ′⟩ → |Ψ′′⟩ → ⋯ →
|Ψ
(⋆)
|Ψ
ℎ=0
(⋆)
|Ψ
ℎ𝑐
for ℎ < ℎ_𝑐
⟩
for ℎ = ℎ_𝑐
⟩
(⋆)
ℎ=∞
⟩
for ℎ > ℎ_𝑐
With MERA, we have solved the two problems of TTN:
• Computational:
RG scheme is now scalable
(⋆)
• Conceptual: |Ψ ⟩ only contains UNIVERSAL information
(it can be more easily extracted)
summary
MERA
• Definition
• Efficiency
• Structural properties:
correlations and entropy
⟨𝑜 0 𝑜 𝐿
≈ 𝐿−𝑝
𝑆(𝐴) ≈ log 𝐿
The Renormalization Group
• Goals
ground state
wave-function
|Ψ⟩
classical partition
function 𝒵
• RG by isometries: why is it “wrong”?
TTN
TRG
• RG by isometries and disentanglers
MERA
TNR
(Zhiyuan’s lecture)
(Glen’s talk)