Guifre Vidal
Coogee'15 Sydney Quantum Information Theory Workshop
Tensor network
renormalization
In collaboration with GLEN EVENBLY
IQIM Caltech UC Irvine
Quantum Mechanics
Paul Dirac
Erwin Schrodinger
Niels Bohr
Werner Heisenberg
Wolfgang Pauli
Enrico Fermi
Albert Einstein
Richard Feynman
๐โ๐
๐๐กฮจ = ๐ป ฮจ
Schrodinger equation
1920-1930
Collective quantum many-body phenomena
Exotic phases of quantum matter:
superconductors
fractional quantum Hall effect
supersolids (?) superfluids
Bose-Einstein condensation
quantum criticality topological order
spin liquids
โThe fundamental laws necessary for the mathematical treatment of a large part of physics and the whole of chemistry are thus completely known,
Paul Dirac
and the difficulty lies only in the fact that application of these laws leads to equations that are too complex to be solved.โ
๐โ๐
๐๐กฮจ = ๐ป ฮจ
Schrodinger equation
? Emergence: long-distance physics is often radically different from short-distance physics
There is a problemโฆ
โฆ and there is a solution: the Renormalization Group
Kenneth Wilson
Leo Kadanoff
๐ปcrit
๐ป3๐=๐๐
๐ปdisorder
๐ป2๐>๐๐
๐ = 0
๐ปorder
๐ป1๐<๐๐
๐ โ 0
Renormalization group flow:
โฆ and there is a solution: the Renormalization Group
Nice! But how do we do this in practice?
Outline of this talk
โข Real space RG of Euclidean path integral
โข TNR MERA (+ thermal states!)
โข Theory of minimal updates
โข Conformal transformations
arXiv:1412.0732
arXiv:1501.xxx
arXiv:1501.yyy
arXiv:1501.zzz
โข tensor network renormalization (new stuff)
โข entanglement renormalization (old stuff)
Real space RG transformation for
ground state wave-functions
Hamiltonians
Outline
โข Real space RG of Euclidean path integral
โข TNR MERA (+ thermal states!)
โข Theory of minimal updates
โข Conformal transformations
arXiv:1412.0732
arXiv:1501.xxx
arXiv:1501.yyy
arXiv:1501.zzz
โข tensor network renormalization (new stuff)
โข entanglement renormalization (old stuff)
Real space RG transformation for
ground state wave-functions
Hamiltonians
Entanglement renormalization (old stuff)
๐ข disentangler
๐ค isometry
โ
โโฒ
โ
โโฒ
Entanglement renormalization (old stuff)
๐ข disentangler
๐ค isometry
โ
โโฒ
โ
โโฒ
โข removal of all short-range entanglement
โข some short-range entanglement is not removed
RG transformation on ground state wave-functions
ฮจ
ฮจโฒ =
RG transformation on ground state wave-functions
ฮจ
=
Multi-scale entanglement renormalization ansatz (MERA)
โ โโฒ
โโฒโฒ
โข topological order
โข quantum criticality
โข holography
(๐ข, ๐ค)3๐=๐๐
(๐ข, ๐ค)2๐>๐๐ (๐ข, ๐ค)1
๐<๐๐
(๐ข, ๐ค)crit (๐ข, ๐ค)disorder
๐ = 0
(๐ข, ๐ค)order
๐ โ 0
RG flow in the space of wave-functions,
as parameterized by tensors ๐ข, ๐ค
๐ข ๐ค
RG transformation on Hamiltonians
๐ป
=
๐ปโฒ
RG transformation on Hamiltonians ๐ป = โ๐
๐
Local Hamiltonian
โ =
=
๐ข
๐ขโ
๐ค
๐คโ
RG transformation on Hamiltonians ๐ป = โ๐
๐
Local Hamiltonian
โ =
=
๐ข
๐ขโ
๐ค
๐คโ
RG transformation on Hamiltonians ๐ป = โ๐
๐
Local Hamiltonian
โ =
=
๐ข
๐ขโ
๐ค
๐คโ
=
โโฒ
โ3๐=๐๐
โ2๐>๐๐ โ1
๐<๐๐
๐ โ 0
โcrit โdisorder
๐ = 0
โorder
RG flow in the space of local Hamiltonias,
as parameterized by a strictly local Hamiltonian term โ
๐ป = โ๐
๐
โข entanglement renormalization (old stuff)
Real space RG transformation for
ground state wave-functions
Hamiltonians
MANY OPEN QUESTIONS
When is MERA a good approximation?
Better algorithms? ( branching, 2D)
RG on Euclidean path integral (Hamiltonian Lagrangian)
Theory of minimal updates (impurities, boundaries, etc)
MERA for thermal states? Classical partition functions?
Outline
โข Real space RG of Euclidean path integral
โข TNR MERA (+ thermal states!)
โข Theory of minimal updates
โข Conformal transformations
arXiv:1412.0732
arXiv:1501.xxx
arXiv:1501.yyy
arXiv:1501.zzz
โข tensor network renormalization (new stuff)
โข entanglement renormalization (old stuff)
Real space RG transformation for
ground state wave-functions
Hamiltonians
Tensor network renormalization (new stuff)
|ฮจโฉ
local
Hamiltonian
๐ป
ground state
ฮจ = lim๐ฝโโ
๐โ๐ฝ๐ป|๐0โฉ
||๐โ๐ฝ๐ป|๐0โฉ||
๐โ๐ฝ๐ป โ ๐โ๐๐ป๐ด๐โ๐๐ป๐ต๐โ๐๐ป๐ด๐โ๐๐ป๐ต โฏ
๐ป๐ด = โ๐,๐+1
๐๐๐๐
๐ป๐ต = โ๐,๐+1
๐๐๐ฃ๐๐
Suzuki-Trotter
decomposition
๐โ๐โ ๐โ๐๐ป๐ด
๐โ๐๐ป๐ต
๐โ๐๐ป๐ด
๐โ๐๐ป๐ต
๐โ๐โ
๐โ๐๐ป๐ด
ฮจ|ฮจ โ
๐โ๐โ ๐โ๐๐ป๐ด
๐โ๐๐ป๐ต
๐โ๐๐ป๐ด
๐โ๐๐ป๐ต
๐โ๐โ
๐โ๐๐ป๐ด
โฏ โฏ
โฎ
|ฮจโฉ
๐โ๐โ ๐โ๐๐ป๐ด
๐โ๐๐ป๐ต
๐โ๐๐ป๐ด
๐โ๐๐ป๐ต
๐โ๐โ
๐โ๐๐ป๐ด
โฎ
โจฮจ|
ฮจ ๐ ฮจ โ
๐โ๐โ ๐โ๐๐ป๐ด
๐โ๐๐ป๐ต
๐โ๐๐ป๐ด
๐โ๐๐ป๐ต
๐โ๐โ
๐โ๐๐ป๐ด
โฏ โฏ
โฎ
|ฮจโฉ
๐โ๐โ ๐โ๐๐ป๐ด
๐โ๐๐ป๐ต
๐โ๐๐ป๐ด
๐โ๐๐ป๐ต
๐โ๐โ
๐โ๐๐ป๐ด
โฎ
โจฮจ|
๐
= โ ๐โ๐โ ๐ข ๐ ๐ฃ โ ๐ ๐ข ๐ ๐ฃ โ
โ = ๐ค
๐ค โ
๐
๐
๐
๐ค โ
๐ค
๐โ๐โ
๐ด =
๐โ๐โ
๐ด
ฮจ ๐ ฮจ
ฮจ|ฮจ=
โฏ โฏ
โฏ
โฏ
โฏ โฏ โฏ
โฏ
โฏ โฏ ๐ด
Tensor network
Euclidean path integral / Euclidean time evolution operator
Classical partition function
Overlap between two PEPS
Monte Carlo sampling
๐โ๐โ ๐โ๐๐ป๐ด
๐โ๐๐ป๐ต
๐โ๐๐ป๐ด
๐โ๐๐ป๐ต
๐โ๐โ
๐โ๐๐ป๐ด
โข sampling error
โข (sign problem!)
Plan: sample important amplitudes!
โฏ โฏ ๐ด
Tensor network
Euclidean path integral / Euclidean time evolution operator
Classical partition function
Overlap between two PEPS
Tensor network techniques (I)
boundary MPS (or CTM)
โข poor approximation at/near criticality
Plan: sum over all amplitudes
โฏ โฏ ๐ด
Tensor network
Euclidean path integral / Euclidean time evolution operator
Classical partition function
Overlap between two PEPS
Tensor network techniques (II) Plan: sum over all amplitudes by coarse-grainning the tensor network
๐ด ๐ดโฒโฒ
๐ดโฒโฒโฒ
๐ดโฒ
โฏ โฏ ๐ด
Tensor network
Euclidean path integral / Euclidean time evolution operator
Classical partition function
Overlap between two PEPS
Can we define an RG flow in the space of tensors ๐ด ?
๐ดcrit
๐ด3๐=๐๐
๐ดdisorder
๐ด2๐>๐๐
๐ = 0
๐ดorder
๐ด1๐<๐๐
๐ โ 0
๐ด โ ๐ดโฒ โ ๐ดโฒโฒ โ โฏ
Tensor Renormalization Group (TRG) (Levin, Nave, 2006)
truncated SVD
๐ด
initial tensor network
๐ดโฒ
coarse-grained tensor network
Tensor Renormalization Group (TRG) (Levin, Nave, 2006)
Accurate results for gapped 1D systems
โข Second Renormalization Group (SRG)
โข Tensor Entanglement Filtering Renormalization (TEFR)
โข Higher Order Tensor Renormalization Group (HOTRG)
+ many moreโฆ
(Xie, Jiang, Weng, Xiang, 2008)
(Xie, Chen, Qin, Zhu, Yang, Xiang, 2012)
(Gu, Wen, 2009)
Many improvements and generalizations:
Improved accuracy, extensions to 2D systems (!?), etc
However, TRG fails to remove some of the short-range correlations.
โข Not a proper RG flow (wrong structure of fixed points) โข Computationally not sustainable at quantum criticality
Tensor Network Renormalization (Evenbly, Vidal, 2014)
Introduce disentanglers to remove all short-range correlations.
โข Proper RG flow (correct structure of fixed points) โข Works near/at quantum criticality โข Can be generalized to ๐ท > 1 dimensions
๐ = 2.0๐๐
๐ = 1.1๐๐
Proper RG flow: 2D classical Ising
|๐ด 1 | |๐ด 2 | |๐ด 3 | |๐ด 4 |
should converge to the same (trivial) fixed point,
but they donโt!
TNR (new)
๐ = 1.1๐๐
๐ = 2.0๐๐
converge to the same fixed point (containing only universal information of the phase)
TRG (Levin, Nave 2006)
e.g. disordered phase ๐ > ๐๐
๐ = ๐๐
critical critical (scale-invariant) fixed
point
๐ = 1.1๐๐
above critical disordered (trivial) fixed point
๐ = 0.9๐๐
below critical ordered (Z2) fixed point
|๐ด 1 | |๐ด 2 | |๐ด 3 | |๐ด 4 |
Tensor Network Renormalization (TNR):
โข Converges to one of three RG fixed points, consistent with a proper RG flow
Proper RG flow: 2D classical Ising
10 -4
10 -3
10 -2
10 -1
10 0
s = 0 s = 2 s = 6 s = 1
Spe
ctru
m o
f ๐ด
(๐ )
10 2 10 1 10 0 10 2 10 1 10 0 10 2 10 1 10 0 10 2 10 1 10 0
Does TRG give a sustainable RG flow?
TRG: ~10 ~20 ~40 > 100
Bond dimension required to maintain fixed truncation error (~10โ3):
๐
Sustainable RG flow: 2D classical Ising
4 ร 109 6 ร 107 1 ร 106 > 1012
Computational cost:
TRG : ๐(๐6)
Cost of TRG scales exponentially with RG iteration!
RG flow at criticality
Does TRG give a sustainable RG flow?
10 -4
10 -3
10 -2
10 -1
10 0
s = 0 s = 2 s = 6 s = 1
Spe
ctru
m o
f ๐ด
(๐ )
10 2 10 1 10 0 10 2 10 1 10 0 10 2 10 1 10 0 10 2 10 1 10 0
RG flow at criticality
TRG
TNR
Sustainable RG flow: 2D classical Ising
TNR: ~10 ~10 ~10 ~10
TRG: ~10 ~20 ~40 > 100
4 ร 109 6 ร 107 1 ร 106 > 1012
Computational costs:
TNR :
TRG :
5 ร 107 5 ร 107 5 ร 107 5 ร 107
๐(๐6) ๐(๐๐6)
Bond dimension required to maintain fixed truncation error (~10โ3):
๐ Sustainable
Proper RG flow: 2D classical Ising
|๐ด 1 | |๐ด 2 | |๐ด 3 | |๐ด 4 |
|๐ด 5 | |๐ด 6 | |๐ด 7 | |๐ด 8 |
|๐ด 9 | |๐ด 10 | |๐ด 11 | |๐ด 12 |
๐ = ๐๐
critical point:
๐ = 4
TNR bond dimension:
Proper RG flow: 2D classical Ising
๐ = 1.002๐๐
more difficult!
|๐ด 1 | |๐ด 2 | |๐ด 3 | |๐ด 4 |
|๐ด 5 | |๐ด 6 | |๐ด 7 | |๐ด 8 |
|๐ด 9 | |๐ด 10 | |๐ด 11 | |๐ด 12 |
๐ = 4
TNR bond dimension:
Outline
โข Real space RG of Euclidean path integral
โข TNR MERA (+ thermal states!)
โข Theory of minimal updates
โข Conformal transformations
arXiv:1412.0732
arXiv:1501.xxx
arXiv:1501.yyy
arXiv:1501.zzz
โข tensor network renormalization (new stuff)
โข entanglement renormalization (old stuff)
Real space RG transformation for
ground state wave-functions
Hamiltonians
TNR yields MERA (Evenbly, Vidal, 2015, in prep)
TNR yields MERA (Evenbly, Vidal, 2015, in prep)
โข reshape an existing representation of the ground state
โข TNR truncation errors: accuracy certificate
energy minimization trace optimization
โข 1000โs of iterations on scale
โข 1 iteration on scale
โข non-local in scale โข local in scale
MERA as a variational class of many-body states
Traditional perspective
MERA as a byproduct of tensor network renormalization (TNR)
New perspective
Extra bonus: thermal MERA (Evenbly, Vidal, 2015, in prep)
๐โ๐ฝ๐ป
Outline
โข Real space RG of Euclidean path integral
โข TNR MERA (+ thermal states!)
โข Theory of minimal updates
โข Conformal transformations
arXiv:1412.0732
arXiv:1501.xxx
arXiv:1501.yyy
arXiv:1501.zzz
โข tensor network renormalization (new stuff)
โข entanglement renormalization (old stuff)
Real space RG transformation for
ground state wave-functions
Hamiltonians
Extra bonus: theory of minimal updates (Evenbly, Vidal, 2015, in prep)
๐ฅ
๐
homogeneous system:
๐ค
๐ข
Extra bonus: theory of minimal updates (Evenbly, Vidal, 2015, in prep)
๐ฅ
๐
same system with impurity:
๐ข
๐ค ๐ฟ ๐ค ๐
๐ข
๐ค
๐(โ)
โ
Example: critical Ising model with one modified bond
๐ป = (โ๐๐๐๐+1 + ๐๐)
๐
๐ป๐๐๐ = 1 โ ๐ผ ๐0๐1
homogeneous system:
Extra bonus: theory of minimal updates
Extra bonus: theory of minimal updates same system with impurity:
Outline
โข Real space RG of Euclidean path integral
โข TNR MERA (+ thermal states!)
โข Theory of minimal updates
โข Conformal transformations
arXiv:1412.0732
arXiv:1501.xxx
arXiv:1501.yyy
arXiv:1501.zzz
โข tensor network renormalization (new stuff)
โข entanglement renormalization (old stuff)
Real space RG transformation for
ground state wave-functions
Hamiltonians
Extra bonus: conformal transformations (Evenbly, Vidal, 2015, in prep)
Conformal transformation 1:
Upper half-plane to ๐ด๐๐2
Holographic description
๐๐ฅ2 + ๐๐2 = ๐2๐๐ฅ2
๐2+
๐๐2
๐2
โ๐๐ฅ2
๐2+
๐๐2
๐2=
๐๐ฅ2
22๐ + ๐๐ 2
๐ = log2(๐)
Extra bonus: conformal transformations (Evenbly, Vidal, 2015, in prep)
Conformal transformation 2:
(radial quantization in CFT)
โข Extraction of scaling dimensions
Plane to cylinder
๐ง = 2๐ค
๐ง โก ๐ฅ + ๐๐
๐ค โก ๐ + ๐๐
๐ โก log2(๐ฅ2 + ๐2)
plane
cylinder
๐ฅ
๐
๐
๐
Summary
โข Real space RG of Euclidean path integral
โข TNR MERA (+ thermal states!)
โข Theory of minimal updates
โข Conformal transformations
arXiv:1412.0732
arXiv:1501.xxx
arXiv:1501.yyy
arXiv:1501.zzz
โข tensor network renormalization (new stuff)
โข entanglement renormalization (old stuff)
Real space RG transformation for
ground state wave-functions
Hamiltonians
THANKS !