### The multi-scale Entanglement Renormalization Ansatz

```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
⊗
Ψ ∈ (ℂ )

complex numbers
Multi-scale entanglement
renormalization ansatz
(MERA)
Matrix product state
(MPS)
MERA
also MERA !
Efficiency
⊗
Ψ ∈ (ℂ )
+

1 1
+ + … = 1+ + +⋯
2
4
2 4

≤ 2
complex numbers
Multi-scale entanglement
renormalization ansatz
(MERA)
Matrix product state
(MPS)
spins ⇒  tensors
⇒ () parameters
log()

spins
⇒  log() tensors ?
2 tensors ⇒ () parameters
Q1
efficiency
Matrix product state
(MPS)
Q2
〈Ψ Ψ
cost ()
〈Ψ| Ψ
cost ()

efficiency
|Ψ

isometric tensors!

†
〈Ψ|Ψ =

†

†
=1
=
=
Q3
cost = 0 !!!
efficiency
〈Ψ|Ψ =
isometric tensors!

†

†

†
=1
=
=
efficiency
〈Ψ|Ψ =
isometric tensors!

†

†

†
=1
=
=
efficiency
〈Ψ|Ψ =
isometric tensors!

†

†

†
=1
=
=
efficiency
〈Ψ|Ψ =
isometric tensors!

†

†

†
=1
=
=
efficiency
〈Ψ|Ψ =
isometric tensors!

†

†

†
=1
=
=
efficiency
〈Ψ|Ψ = 1
isometric tensors!

†

†

†
=1
=
=

〈Ψ| Ψ =
isometric tensors!

†

†

†
=1
=
=
=
(1)
=
〈Ψ| Ψ =
(2)
=
(3)
cost O(log  )
= ΨΨ
Structural properties
⊗
Ψ ∈ (ℂ )

• Decay of correlations
• Scaling of entanglement
complex numbers
⟨Ψ| 0 ()|Ψ
MPS
=
=
−1
=
≈
−1 =  −/
≡−
⇒ Exponential decay of correlations
1
log
Ψ 0   Ψ
⋯
⋯
⋯
=
MERA
⋯
⋯
⋯
Ψ 0   Ψ
⋯
MERA
⋯
=
≈
=
log3

log3 ()
= 2 log3() = 2 log3() = −
log3() =  log3()
≡ −2 log 3 ()
⇒ Polynomial decay of correlations
⋯
⋯
Correlations: summary and interpretation
matrix product state
(MPS)
multi-scale entanglement renormalization ansatz
(MERA)

log()
structure of geodesics:
⟨ 0

≈  −/
exponential
structure of geodesics:
⟨ 0
≈ −
power-law
Entanglement entropy
matrix product state
multi-scale entanglement renormalization ansatz
(MPS)
(MERA)

log()
connectivity:

() ≈
boundary law!
Q4

connectivity:
() ≈ log
logarithmic correction!
Q5
⋮
⋯
⋯
() ≈ log
Example: operator content of quantum Ising model

⊗ +1
+ℎ
=

for ℎ = ℎ = 1

scaling
dimension
(exact )
scaling operators/dimensions:
identity
spin
energy
disorder
fermions

0
scaling
dimension
(MERA)
0
error
----
0.125
0.124997
0.003%

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
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,+1
on  sites
(Hilbert space dimension   )

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
→  ′ →  ′′ → ⋯ →  (⋆)
Ψ     |Ψ〉 for  −  → ∞
2) Low energy, short distance, detailed MICROSCOPIC properties
e.g. 〈Ψ|  |Ψ〉,
Ψ     |Ψ〉, for all ,
The Renormalization Group: goals
Example: given the
(transverse field) Ising Hamiltonian

⊗ +1
+ℎ
=

(ℎ)
spontaneous
magnetization
0
ℎ
ℎ
magnetic field
1) Low energy, large distance, UNIVERSAL behavior?
Is the spontaneous magnetization
m(h) ≡ Ψ   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|  |Ψ〉 as a function on ℎ?
The Renormalization Group
on Hamiltonians:
on ground state
wave-functions:
on classical
partition functions:
→
′
→
′′ →
⋯ →  (⋆)
|Ψ〉 → |Ψ′〉 → |Ψ′′〉 → ⋯ → |Ψ (⋆) 〉
→  ′ →  ′′ → ⋯ →  (∗)
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
→  ′ →  ′′ → ⋯ →  (⋆)
such that
(for instance,  =
in quantum Ising model)
Ψ  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?
→  ′ →  ′′ → ⋯ →  (⋆)
(for instance,  =
in quantum Ising model)
such that
Ψ  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
′ = /3 sites

ℒ′

†
ℒ
=
sites
coarse-graining transformation
=  ⊗  ⊗ ⋯⊗
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
=  ⊗  ⊗ ⋯⊗
Ψ → Ψ ′ =  † |Ψ〉
=
Ψ
†
A’
B’
C’
D’
F’
E’
Ψ′
A’ B’ C’ D’ E’ F’
→ ′ =  †

=
A’
B’
C’

D’
E’
F’
†
=
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
=  ⊗  ⊗ ⋯⊗
Ψ → Ψ ′ =  † |Ψ〉
Ψ′
Ψ
=
†
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
=  ⊗  ⊗ ⋯⊗
|Ψ ′ 〉 retains short-range entanglement = MICROSCOPIC details
Ψ  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!
(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
=  ⊗  ⊗ ⋯⊗
|Ψ ′ 〉 retains short-range entanglement = MICROSCOPIC details
Ψ  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
′ = /3 sites

ℒ′
†
ℒ

sites
coarse-graining transformation
= (⋯  ⊗  ⊗  ⊗ ⋯ )(⋯  ⊗  ⊗  ⊗ ⋯)
Q7
=
A’
B’
C’
D’
E’
F’
A’ B’ C’ D’ E’ F’
†
=
=
The Renormalization Group
Type 2B:
example: MERA
Proper RG transformation:
|Ψ〉 → |Ψ′〉 → |Ψ′′〉 → ⋯ → |Ψ (⋆) 〉
fixed point:
only UNIVERSAL
properties
coarse-graining transformation
= (⋯  ⊗  ⊗  ⊗ ⋯ )(⋯  ⊗  ⊗  ⊗ ⋯)
Ψ → Ψ ′ =  † |Ψ〉
=
Ψ
†
Ψ′
A’ B’ C’ D’ E’ F’
→ ′ =  †
=
A’
B’
C’

D’
E’
F’
=
A’ B’ C’
D’ E’ F’
A’ B’ C’ D’ E’ F’
′
The Renormalization Group
Type 2B:
example: MERA
Proper RG transformation:
|Ψ〉 → |Ψ′〉 → |Ψ′′〉 → ⋯ → |Ψ (⋆) 〉
fixed point:
only UNIVERSAL
properties
coarse-graining transformation
= (⋯  ⊗  ⊗  ⊗ ⋯ )(⋯  ⊗  ⊗  ⊗ ⋯)
Ψ → Ψ ′ =  † |Ψ〉
Ψ
†
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
= (⋯  ⊗  ⊗  ⊗ ⋯ )(⋯  ⊗  ⊗  ⊗ ⋯)
Same fixed-point wave-function
for the same phase!
|Ψ 〉 → |Ψ′〉 → |Ψ′′〉 → ⋯ →
|Ψ
(⋆)
|Ψ
ℎ=0
(⋆)
|Ψ
ℎ
for ℎ < ℎ_
〉
for ℎ = ℎ_
〉
(⋆)
ℎ=∞
for ℎ > ℎ_
〉
(ℎ)
magnetic
field h
spontaneous
magnetization
ℎ=0
ℎ′
|Ψ
(⋆)
ℎ=0
〉
ℎ
ℎ
|Ψ
(⋆ )
ℎ
〉
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
= (⋯  ⊗  ⊗  ⊗ ⋯ )(⋯  ⊗  ⊗  ⊗ ⋯)
Same fixed-point wave-function
for the same phase!
|Ψ 〉 → |Ψ′〉 → |Ψ′′〉 → ⋯ →
|Ψ
(⋆)
|Ψ
ℎ=0
(⋆)
|Ψ
ℎ
for ℎ < ℎ_
〉
for ℎ = ℎ_
〉
(⋆)
ℎ=∞
〉
for ℎ > ℎ_
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
⟨ 0
≈ −
() ≈ log
The Renormalization Group
• Goals
ground state
wave-function
|Ψ〉
classical partition
function
• RG by isometries: why is it “wrong”?
TTN
TRG
• RG by isometries and disentanglers
MERA
TNR
(Zhiyuan’s lecture)
(Glen’s talk)
```