Guifre Vidal
Walter Burke Institute for Theoretical Physics
Caltech, Feb 23-24, 2015
Tensor network
renormalization
INAUGURAL CELEBRATION AND SYMPOSIUM
Sherman Fairchild Prize
Postdoctoral Fellow
(2003-2005)
Tensor network renormalization (TNR)
TNR = Renormalization group + Tensor networks
Renormalization group Tensor networks
TNR = Renormalization group + Tensor Networks
Renormalization group Tensor networks
In collaboration with
GLEN EVENBLY IQIM Caltech UC Irvine
superconductor
superfluid
Emergent phenomena in many-body systems
quantum criticality
topological order
metal
insulator
spin liquid
fractional quantum Hall effect
The problem
low energy, collective excitations, โฆ
โจฮจ|๐ ๐ฅ, ๐ก ๐(0,0)|ฮจโฉ ๐ป local Hamiltonian
we have
|ฮจโฉ
ground state
we want
Euclidean path integral
๐ = ๐ก๐ ๐โ๐ฝ๐ป
๐ป๐1๐ = ๐๐
๐ฅ๐๐๐ฅ
๐
+ ๐ ๐๐๐ง
๐
Example:
๐ ๐ = ๐ก๐ ๐โ๐ฝ ๐ป๐1๐
๐ฝ
๐ฟ
1d quantum
1d quantum Ising model
๐
โจ๐๐ฅโฉ
๐๐๐๐๐ก
๐(๐) = ๐โ1๐ ๐ป๐๐2๐
๐
๐ป๐๐2๐= ๐๐๐๐
<๐,๐>
~ ๐ฟ๐ฅ
๐ฟ๐ฆ 2d
classical
2d classical Ising model
๐
โจ๐โฉ
๐๐๐๐๐ก
Other examples: material science, quantum chemistry, QCD, โฆ
The Renormalization Group
Kenneth Wilson
Leo Kadanoff
๐ป ๐
๐ป[๐ฝ, ๐ ] = ๐ฝ ๐๐๐ฅ๐๐๐ฅ
๐
+ ๐ ๐๐๐ง
๐
๐ป[๐1, ๐2, 0,0,โฏ ] = ๐1 ๐๐๐ฅ๐๐๐ฅ
๐
+ ๐2 ๐๐๐ง
๐
๐ = (๐1, ๐2, ๐3, โฏ )
Phase A Phase B
stable fixed point A
stable fixed point B
critical fixed point
๐4
๐5
RG flow in the space of Hamiltonians
โถ๐ป[๐(๐ )]
Change of scale?
Leo Kadanof + some rule: majority vote, etc
block spin
Kenneth Wilson
๐ = ๐๐๐โ๐ป ๐,๐
๐ โคฮ
ฮ
0
|p|
๐ = ๐๐๐โ๐ป ๐,๐โฒ
๐ โฅฮโฒ
exact renormalization group equation (ERGE)
coarse-graining transformation
๐โ๐ป ๐,๐โฒ= ๐๐๐โ๐ป ๐,๐
ฮโฒ โค ๐ โค ฮ
ฮโฒ
(but we have not yet solved QCD, sorryโฆ)
We would like (wish list)
โข Non-perturbative RG approach
โข Universal coarse-graining rules valid for a generic system
โข Solve QCD โฆ !
๐ป โ ๐ปโฒ โ ๐ปโฒโฒ โ โฏ
โข Key ingredient: removal of short-rage entanglement
How far did we get ? (over the last 10 years)
โข Reformulated the RG using quantum information tools/concepts (quantum circuits, entanglement)
โข Efficient representation of ground state wave-functions (MERA) ฮจ
universal, non-perturbative, real-space RG approach!
Cirac
Verstraete
Swingle
Wen
Gu Xiang
White
Levin
Nave Eisert
Nishino
Qi
Corboz
Evenbly
Schuch
Pollmann
Orus Tagliacozzo
Read
Singh
and many moreโฆ
Chen
Hastings
Material science
Quantum chemistry
Holography
Condensed matter (Frustrated magnets, interacting fermions, quantum criticality)
Classification of gapped phases
Renormalization group
Classical statistical mechanics
TENSOR NETWORKS
Many-body wave-function of ๐ spins
ฮจ = ฮจ๐1๐2โฏ๐๐๐1,๐2,โฏ,๐๐
|๐1๐2โฏ๐๐โฉ
๐ ๐ = ๐ ๐ ๐
๐๐๐ = ๐ ๐๐๐๐๐๐
=
๐ = ๐ฆ โ โ ๐ โ ๐ฅ ๐ก๐(๐ด๐ต๐ถ๐ท)
๐
๐ ๐ graphical notation ๐ผ1 ๐ผ2 โฏ ๐ผ๐
๐ ๐i ๐ij ๐๐ผ1๐ผ2โฏ๐ผ๐
why bother? ๐ด๐๐๐๐ต๐๐๐๐ถ๐๐๐๐ท๐๐๐๐ฅ๐๐ฆ๐๐ง๐๐ฃ๐๐๐๐๐๐๐๐๐
2๐ parameters
๐1 ๐2 ๐๐ โฆ
= ฮจ
๐1 ๐2 ๐๐ โฆ
tensor network
2๐ parameters
inefficient
โ(๐)
ground states of local Hamiltonians
๐ = 1 ๐ = 2
๐ = 3
โฎ
generic state
โ(๐)
tensor network states
๐1 ๐2 ๐๐ โฆ
= ฮจ
๐1 ๐2 ๐๐ โฆ
tensor network ๐ผ = 1,2,โฏ , ๐
๐(๐2๐2)
parameters
efficient
Many-body wave-function of ๐ spins
ฮจ = ฮจ๐1๐2โฏ๐๐๐1,๐2,โฏ,๐๐
|๐1๐2โฏ๐๐โฉ 2๐
parameters
Vidal, 2006 Example of tensor network
Multi-scale entanglement renormalization ansatz
(MERA)
โข Variational ansatz for 1d systems, which extends in space and scale
โข Variational parameters for different scales
โข It is secretly a quantum circuit and an RG transformation
Vidal, 2006 Example of tensor network
Multi-scale entanglement renormalization ansatz
(MERA)
isometry disentangler
two-body gate
|0โฉ
also a two-body gate
=
|0โฉ |0โฉ
|0โฉ |0โฉ |0โฉ |0โฉ |0โฉ |0โฉ |0โฉ |0โฉ |0โฉ |0โฉ |0โฉ |0โฉ |0โฉ |0โฉ |0โฉ
|0โฉ
|0โฉ
|0โฉ
|0โฉ |0โฉ |0โฉ |0โฉ
|0โฉ |0โฉ |0โฉ |0โฉ |0โฉ |0โฉ |0โฉ |0โฉ
Multi-scale entanglement renormalization ansatz
(MERA)
quantum circuit
โtimeโ
Entanglement is introduced by the gates at different times (=scales)
๐
ground state ฮจ = 0 โ 0 โโฏโ |0โฉ ๐
RG Transformation
โ
โโฒ
Kadanoff (1966) + White (1992)
blocking variational optimization
โ
โโฒ
Entanglement renormalization (2005)
RG Transformation
โ
โโฒ
โ
โโฒ
Kadanoff (1966) + White (1992)
blocking variational optimization
Entanglement renormalization (2005)
failure to remove some short-range entanglement !
removal of all short-range entanglement
MERA -> RG flow in the space of ground state wave-functions
ฮจ โ ฮจโฒ โ ฮจโฒโฒ โ โฏ โ |ฮจ๐๐โฉ
MERA -> RG flow in the space of Hamiltonians
๐ป โ ๐ปโฒ โ ๐ปโฒโฒ โ โฏ โ ๐ป๐๐
ฮจ โ ฮจโฒ โ ฮจโฒโฒ โ โฏ โ |ฮจ๐๐โฉ
โข topological order (2+1)
โข quantum criticality (1+1)
fixed-point wave-function
= local operators are mapped into local operators !
MERA -> RG flow in the space of ground state wave-functions
MERA
โข Variational parameters for different length scales
โข It is secretly a quantum circuit โentanglement at different length scalesโ
Summary so far
โข Optimize variational parameters by energy minimization
Does it work?
โข blah, blah, blahโฆ
โpreservation of localityโ
๐ป โ ๐ปโฒ ฮจ โ |ฮจโฒโฉ โremoves short-range entanglementโ
and an RG transformation
scaling dimension (exact )
scaling dimension (MERA)
error
0 0 ----
0.125 0.124997 0.003%
1 0.99993 0.007%
0.125 0.1250002 0.0002%
0.5 0.5 <10โ8 %
0.5 0.5 <10โ8 %
Operator product expansion (OPE) coefficients
๐ถ๐๐๐ =1
2 ๐ถ๐๐๐ =
โ1
2 ๐ถ๐๐๐ =
๐โ๐๐4
2 ๐ถ๐ ๐๐ =
๐๐๐4
2 ๐ถ๐๐๐ = ๐ ๐ถ๐๐ ๐ = โ๐
(ยฑ6 ร 10โ4)
identity
spin
energy density
disorder
fermions
๐
๐
๐
Scaling dimensions of primary fields
Example: Critical quantum Ising model Pfeifer, Evenbly, Vidal (2008)
scale-invariant MERA โ conformal data of a CFT:
central charge ๐ scaling dimensions ฮ๐ผโก โ๐ผ + โ ๐ผ conformal spin s๐ผโก โ๐ผ โ โ ๐ผ OPE ๐ถ๐ผ๐ฝ๐พ
MERA and HOLOGRAPHY
โข entanglement entropy
๐๐ฟ โ log (๐ฟ)
parallel to area of minimal surface in Ryu-Takayanagi
โข two-point correlations
geodesic distance ๐ท โ log (๐ฟ) as in hyperbolic space
๐ถ ๐ฟ โ ๐ฟโ2ฮ
๐ถ ๐ฟ โ ๐โ๐ท = ๐โ2ฮlog(๐ฟ) = ๐ฟโ2ฮ ๐ฟ
log(๐ฟ)
๐ฅ space
๐ scale
AdS2+1
๐ฅ
s
๐ก
๐ฅ
๐ก
๐ฅ
CFT1+1
Swingle 2009
Qi
Harlow, Yoshida, Pastawki, Preskill
Sully, Czech
Hartman, Maldacena
Ryu, Takayanagi
Haegeman, Osborne, Verschelde, Verstraete
So, MERA seems to work!
Great! However โข variational optimization is expensive; local minima.
โข do we get the correct ground state?
โข Euclidean path integrals / classical partition functions?
Tensor Network Renormalization
Evenbly, Vidal 2014-2015
Euclidean path integral
๐ ๐ = ๐ก๐ ๐โ๐ฝ๐ป๐1๐
๐(๐) = ๐โ1๐ ๐ป๐๐2๐
๐
๐ฝ
๐ฟ ๐ฟ๐ฅ
๐ฟ๐ฆ ~ 1d quantum
2d classical
Statistical partition function
as a tensor network
๐ = ๐ด
Tensor Renormalization Group (TRG) Levin, Nave 2006
๐ด ๐ดโฒ
isometry!
Fixed-point of TRG
some short-range entanglement has not been removed
Levin, Nave 2006
CDL Tensor (zero correlation length)
๐ด ๐ดโฒ
isometry!
โ =
Tensor Network Renormalization (TNR) Evenbly, Vidal 2014-2015
isometry
disentangler!
๐ด ๐ดโฒ
Tensor Network Renormalization (TNR)
CDL Tensor (zero correlation length)
[for CDL tensors, see also Gu and Wen 2009 Tensor Entanglement Filtering Renormalization (TEFR)]
โ =
removal of all short-range entanglement ๐ด ๐ดโฒ
isometry
disentangler!
๐ = ๐๐
critical critical (scale-invariant)
fixed point
๐ = 1.1 ๐๐
above critical disordered (trivial)
fixed point
๐ = 0.9 ๐๐
below critical ordered (Z2) fixed point
|๐ด 1 | |๐ด 2 | |๐ด 3 | |๐ด 4 |
TNR -> proper RG flow Example: 2D classical Ising
๐ด โ ๐ดโฒ โ ๐ดโฒโฒ โ โฏ โ ๐ด๐๐
TNR yields MERA Evenbly, Vidal, 2015
MERA = variational ansatz
energy optimization
MERA = by-product of TNR
transform tensor network into MERA
energy minimization
โข 1000s of iterations over scale
โข local minima
โข correct ground ?
TNR -> MERA
โข single iteration over scale
โข rewrite tensor network for ground state
โข certificate of accuracy
โข Reformulation of the RG using quantum information tools/concepts (quantum circuits, entanglement)
ฮจ โ ฮจโฒ โ ฮจโฒโฒ โ โฏ ๐ด โ ๐ดโฒ โ ๐ดโฒโฒ โ โฏ ๐ป โ ๐ปโฒ โ ๐ปโฒโฒ โ โฏ 3 RG flows
Tensors Ground states Hamiltonians
โข universal, non-perturbative, real-space RG approach
Summary
โข Efficient representation of ground states (MERA) -> toy model for holography
โข Very accurate in 1+1 dimensions (Ising model, etc)
โข Key ingredient: removal of short-rage entanglement
What about 2+1, 3+1? (and QCD?)
IQI, 2005
Entanglement renormalization MERA
Tensor network renormalization
IQIM, 2014
GLEN EVENBLY
Sherman Fairchild Prize
Postdoctoral Fellow
(2011-2014)
Sherman Fairchild Prize
Postdoctoral Fellow
(2003-2005)
THANK YOU!