ii. rg of tensor network states · 2016. 8. 2. · larger simplex pess p. corboz et al, prb 86,...
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Renormalization of Tensor Network States
Tao Xiang
Institute of Physics
Chinese Academy of Sciences
II. RG of Tensor Network States
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Tensor-Network Ansatz
of the ground state wavefunction
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: 1D: Matrix Product State (MPS)
| =
𝑚1,…𝑚𝐿
(𝑚1, …𝑚𝐿)|𝑚1, …𝑚𝐿
m1
m2
m3mL-2
mL-1
mL
…
Parameter number grows
exponentially with L
𝑑𝐿 parameters
1
1 1[ ]... [ ] ...L
L L
m m
Tr A m A m m m
𝑑𝐷2𝐿 parameters
m1 m2 m3 … … mL-1 mL
A[m2 ]
D
d
Parameter number grows
linearly with L
Virtual basis state
MPS is the wave function generated by the DMRG
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1
1 1[ ]... [ ] ...
0 0 1 0 0 2[ 1] [0] [1]
0 12 0 0 0
L
L L
m m
Tr A m A m m m
A A A
2
1 1
1 1 2
2 3 3i i i i
i
H S S S S
Affleck, Kennedy, Lieb, Tasaki, PRL 59, 799 (1987)
Example:S=1 AKLT valence bond solid state
A[m]
m
virtual S=1/2 spin
A[m] :
To project two virtual
S=1/2 states, and ,
onto a S=1 state m
A[m]
m
=
1 =1
21
2
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m1 m2 m3 … … mL-1 mL
1
1 1[ ]... [ ] ...L
L L
m m
Tr A m A m m m A[m]
m
Matrix product state (MPS)
Gauge Invariance
MPS wavefunction is unchanged if one replaces A[m] by
𝐴 𝑚 → 𝐴′ 𝑚 = 𝑃𝐴 𝑚 𝑃-1
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MPS as a Projection of 2D Tensor-Network Model
Ai
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Matrix Product Operator
𝑚1′ 𝑚2
′ 𝑚3′ … … 𝑚𝐿−1
′ 𝑚𝐿′
M
𝑚1 𝑚2 𝑚3 … … 𝑚𝐿−1 𝑚𝐿
𝑂 =
𝑚
𝑇𝑟 𝑀 𝑚1, 𝑚1′ ⋯𝑀 𝑚𝐿 , 𝑚𝐿
′ | 𝑚1 ⋯𝑚𝐿 𝑚1′⋯𝑚𝐿′ |
=
Ground state eigen-operator: 𝑂=| |
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𝑥 𝑥′
2D: Projected Entangled Pair State (PEPS)
𝑇𝑥𝑥′𝑦𝑦′ [𝑚] =
𝑦
y'
𝑚
Physical
basis
Local
tensor
Virtual
basis
D
Key parameter: virtual basis dimension D
Virtual spins at each bond form a maximally entangled state
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Tensor product states
H. Niggemann and J. Zittarz, Z. Phys. B 101, 289 (1996)
G. Sierra and M. Martin-Delgado, 1998
Variational approach:
Nishino, Okunishi, Maeshima, Hieida, Akutsu, Gendiar (since 1998)
Projected entangled pair states (PEPS) : Area law obeys
F. Verstraete and J. Cirac, cond-mat/0407066
Tensor-Network State as a Variational Ansatz
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PEPS: exact representation of Valence Bond Solid
𝑇𝑎𝑏𝑐𝑑 [𝑚𝑖] =𝑎 𝑏
𝑐
d𝑚𝑖
Physical stateVirtual basis state
S = 2
𝐻 =
𝑖𝑗
𝑃4(𝑖, 𝑗)
To project two S=2 spins on sites
i and j onto a total spin S=4 state
1/2
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Bond dimension dependence of the ground state energy
and magnetization
Gapless: Power law dependent on D
Gapped: Exponential law dependent on D
Bond Dimension Dependence of Physical Quantities
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1. Determine the local tensors
2. Evaluate the physical observables using the TRG
or other tensor network RG methods
Two Problems Need to Be Solved
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Evaluation of Expectation Values
|
|
𝐷
𝑑 = 𝐷2
|
𝐷2
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How Large Is the Virtual Bond Dimension Needed?
|
𝐷2
𝑑 = 𝐷2
In the DMRG or all tensor-related
approached, small physical dimension d
is much earlier to study than a larger 𝑑
(accuracy drops quickly with 𝑑)
The virtual bond dimension needed to
obtain a converged result is roughly of
order 𝑑, i.e. 𝐷2, of course, the larger the
better
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Computational Cost
Method CPU Time Minimum Memory
TMRG/CTMRG 𝐷12𝐿 𝐷10
TEBD 𝐷12𝐿 𝐷10
TRG 𝐷12ln𝐿 𝐷8
SRG 𝐷12ln𝐿 𝐷8
HOTRG 𝐷14ln𝐿 𝐷8
HOSRG 𝐷16ln𝐿 𝐷12
TNR 𝐷14ln𝐿 𝐷10
Loop-TNR 𝐷12ln𝐿 𝐷8
𝐷: bond dimension of PEPS bond dimension kept ~ 𝐷2 𝐿: lattice size
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Tensor Network States
in the Frustrated Lattices
Z. Y. Xie et al, PRX 4, 011025 (2014)
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Geometric frustration
1
1
N
i i
i
H J S S
J
J
J
Quantum frustration
e.g. S=1 bilinear-biquadratic Heisenberg model
2
cos sini j i j
ij
H S S S S
Two Kinds of Frustrations
More than two-body correlations/entanglements are important
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PEPS on Kagome or Other Frustrated Lattices
There is a serious cancellation in the tensor elements if three
tensors on a simplex (triangle here) are contracted
3-body (or more-body) entanglement is important
Max ( ) ~ 1
Max ( ) < 10-6
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Cancellation in the PEPS
1
1
N
i i
i
H J S S
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Projected Entangled Simplex States (PESS)
Projection tensor
Simplex tensor
Virtual spins at each simplex form a maximally entangled state
Remove the geometry frustration: The PESS is defined on the
decorated honeycomb lattice
Only 3 virtual bonds, low cost
Z. Y. Xie et al, PRX 4, 011025 (2014)
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PESS as an exact representation of Simplex Solid States
D. P. Arovas, Phys. Rev. B 77, 104404 (2008)
Example: S = 2 spin model on the Kagome lattice
A S = 2 spin is a symmetric superposition of two virtual S = 1 spins
Three virtual spins at each triangle form a spin singlet
Projection tensor
Simplex tensor
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S=2 Simplex Solid State on the Kagome Lattice
𝐴𝑎𝑏[𝜎] =1 1 2𝑎 𝑏 𝜎
antisymmetric tensor
C-G coefficients
Local tensors
Pn : projection operator
Parent Hamiltonians
or
Projection tensor
Simplex tensor
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To enlarge each simplex so that it contains more physical spins
5-PESS: a decorated
square lattice
9-PESS: a honeycomb
lattice
Larger Simplex PESS
P. Corboz et al, PRB 86,
041106 (2012).
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PESS on Triangular Lattice
PEPS PESS
Order of local tensors: dD6 Simplex tensor: D3
Projection tensor: dD3
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PESS on Square Lattice
J1 only J1- J2 model
Vertex-sharing Edge-sharing
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How to Determine
the Tensor-Network Wave Function
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Determination of Tensor Network Wavefunction
1. Imaginary time evolution
Simple update (entanglement mean-field approach)
Jiang, Weng, Xiang, PRL 101, 090603 (2008)
the solution can be used as the initial input of local tensors in I or
in the full update calculation
Full update
Murg, Verstraete, Cirac, PRA 75, 033605 (2007)
2. Minimize the ground state energy
Nishino et al, Nuclear Physics B 575 [FS] 504 (2000)
F. Verstraete and J. Cirac, cond-mat/0407066
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Variational Minimization of Ground State Energy
Determine local tensors by minimizing the ground state energy
𝐸 = 𝐻
|
Accurate
Cost is high
D is generally less than 13 without using symmetries
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[ ] [ ] i i i i i i j j jx y z x y z i x y z j i j
i blackj white
Tr A m B m m m
Bond vectors: measure approximately the “entanglement”
on the corresponding bonds
Simple Update: Entanglement Mean-Field Approach
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Simple Update: Entanglement Mean-Field Approach
state groundlim
He
The local tensors are determined by projection
Converge fast
D as large as 100 can be calculated (more if symmetry is
considered)
Exact on the Bethe lattice
Li, von Delft, Xiang, PRB 86, 195137 (2012)
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Canonical Form on the Bethe Lattice
Li, von Delft, Xiang,
PRB 86, 195137 (2012)
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ij x y z
ij
ij i j
H H H H H
H JS S
Heisenberg model
Simple Update: Imaginary Time Evolution
state groundlim
state groundlim
MH
M
H
e
e
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1. One iteration
2. Repeat the above iteration until converged
20
12
01
~
z
y
x
H
H
H
e
e
e
Trotter-Suzuki
decomposition),,(
)(
,
2
zyxHH
oeeee
blacki
ii
HHHH xyz
Imaginary Time Evolution
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,
ˆ
[ ] [ ] i jx
i i i i i i j j j
HH
i j i j x y z x y y i x y y j i j
i blackj i x
e Tr m m e m m A m B m m m
Step I
Step II
Step III
SVD: singular value decomposition
Step I
Step II
Step III Truncate basis space
One Step of Evolution
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To use bond vector as effective
fields to take into account the
environment contribution
The projection is done locally. This
keeps the locality of wavefunction,
making the calculation efficient
Truncation error not accumulated
,
ˆ
[ ] [ ] i jx
i i i i i i j j j
HH
i j i j x y z x y y i x y y j i j
i blackj i x
e Tr m m e m m A m B m m m
One Step of Evolution
Step I
Step II
Step III
SVD: singular value decomposition
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1
1
N
i i
i
H J S S
Application: Heisenberg Model on the Husimi Lattice
Locally similar to the Kagome lattice, but
less frustrated
Helpful to understand the Kagome
Heisenberg model
Kagome lattice Husimi lattice
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Heisenberg Model on the Husimi Lattice
Husimi lattice
1
1
N
i i
i
H J S S
PESS is defined on a unfrustrated
lattice and can be easily studied
Simple update is rigorous
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Is the Ground State a Spin Liquid?
Kagome lattice
If yes, then the g.s of the Kagome Heisenberg
should also be a spin liquid since the Kagome
is more frustrated than the Husimi lattice
1
1
N
i i
i
H J S S
Husimi lattice
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E0 converges algebraically with D the excitation is gapless
Ground State Energy
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1200 Neel ordered at any finite 𝐷, vanishes in the limit 𝐷 → ∞
𝑀 ~ 𝐷𝛼 with 𝛼 = −0.588(2))
Magnetization: Ground State is a Gapless Spin Liquid
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Kagome Lattice
Z. Y. Xie et al, PRX 4, 011025 (2014)
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1
1
N
i i
i
H J S S
Issue under debate:
Is the ground state a spin liquid?
S=1/2 Kagome Heisenberg Model
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Possible Ground States
Valence bond crystal
R. R. P. Singh and D. A. Huse, PRB 77, 144415 (2008) series expansion
G. Evenbly and G. Vidal, PRL 104, 187203 (2010) MERA
Y. Iqbal, F. Becca, and D. Poilblanc, PRB 83, 100404 (2011) VMC
Z2 Gapped spin liquid
S. Yan, D. A. Huse, and S. R. White, Science 332, 1173 (2011) DMRG
Depenbrock,McCulloch,Schollwock, PRL 109, 067201 (2012) DMRG
………….
U(1) Gapless spin liquid
Y. Ran, M. Hermele, P. A. Lee, and X.-G. Wen, PRL 98, 117205 (2007) Gutzwiller
Y. Iqbal, F. Becca, S. Sorella, and D. Poilblanc, PRB 87, 060405 (2013) VMC+Lanczos
………….
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Difficulty: Lack of Good Numerical Methods
Mean field or variational approach:
need accurate guess of trial wavefunction
Quantum Monte Carlo:
suffers from the minus sign problem on frustrated systems
Density Matrix Renormalization Group (DMRG):
limited to small lattice systems (area law), the number of states
need to be retained grows exponentially with the circumference
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Upper bound: iDMRG (cylinder Ly=12), D=5000, E=-0.4332
Yan et al: -0.4379(3)
Depenbrock et al: -0.4386(5), D=16000, Ly=17
DMRG results
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Valence bond solid
Gapped spin liquid
Gapless spin liquid
Ground state energy of the Kagome Heisenberg model
DMRG
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Valence bond solid
Gapped spin liquid
Gapless spin liquid
3-PESS
Ground state energy of the Kagome Heisenberg model
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Ground state energy of the Kagome Heisenberg model
The ground state energy converges algebraically with D,
indicating that the system is gapless.
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Comparison between Kagome and Husimi Lattices
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Bond Dimension D
No long-range magnetic order in the infinite D limit
The ground state is more likely a gapless spin liquid
120 Degree Neel Magnetization
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Deconfined Quantum Point
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2
cos sini j i j
ij
H S S S S
Honeycomb lattice: each site has only 3 neighbors,
quantum fluctuation is strong
Quantum Monte Carlo has the minus sign problem
when sin > 0
S = 1 Bilinear-biquadratic Heisenberg model on Honeycomb Lattice
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2
cos sini j i j
ij
H S S S S
FQ
IVFerro-magnetic
Antiferro-magnetic
Ferro-quadrupolar
Antiferro-quadrupolar
( )i
iQ
iQ
AFQ
Classical phase diagram
Quantum fluctuation may
lead to a deconfined
quantum critical point
Ground State Phase Diagram in the Classical Limit
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2nd order
There are four phases, but the AFQ phase is killed by quantum fluctuation
d 0.19
Ground State Energy and Orders
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d
FQ
Magnetic Orders
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Continuous phase transition point:
deconfined critical point?
red bon
black o
d
b nd
12
i j
i j
S SP
S S
d
Plaquette Valence Bond Crystal
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Proper tensor-network wave function in treating a
frustrated quantum lattice system
The simple update an approximate and efficient
algorithm for determining the local tensors
Kagome Heisenberg model is likely to be a gapless
spin liquid, but more study needed
Summary