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Plaquette Renormalized Tensor Network States:
Application to Frustrated Systems
Ying-Jer Kao 高英哲
Department of Physics andCenter for Quantum Science and Engineering
National Taiwan UniversityHsin-Chih Hsiao, Ji-Feng Yu, NTU
Anders W. Sandvik, Boston UniversityLing Wang, University of Vienna
NDNS: from DMRG to TNF Kyoto, Japan10/29/2010
Department of PhysicsNational Taiwan University
Frustration
• Large number of degenerate classical ground states
• Emergence of novel spin-disordered ground states due to quantum fluctuations
• Hard to study numerically: size limitation in Exact Diagonalization, sign problem in QMC (also fermion), dimension limit for DMRG
or
AF
Kagome Pyrochlore Hyperkagome Garnet
ZnCu3(OH)6Cl2 Tb2Ti2O7 Na4Ir3O8 Gd3Ga5O12
Department of PhysicsNational Taiwan University
Tensor product states• Tensor product states, projected entangled paired
state (PEPS)
|ψ� =�
{s}
tTr{A(s1)A(s2) · · ·A(sN )}|s1, s2, . . . , sN �
Aijkl: rank-4 tensor
As
ki
s =±1 j
l =1...D
1sA
8sA 9sA
5sA 6sA7sA
4sA
3sA2sA
F. Verstraete and J. I. Cirac, arXiv:cond-mat/0407066
Department of PhysicsNational Taiwan University
Tensor Contraction• Contracting the internal indices,
the four-leg tensor can be viewed as a single tensor.
• External link dimension becomes D2 after one contraction; exponential growth as we keep contracting D4,D8......
• Computationally intensive; Impossible to store intermediate results.
• Need some RG scheme.
Department of PhysicsNational Taiwan University
Plaquette renormalized trial wavefunction
• We treat the elements in the A and S tensors as variational parameters, and minimize the total energy.
• Optimization: Principal axis method / random optimization, derivative-free, computationally expensive.
Department of PhysicsNational Taiwan University
Plaquette-Renormalization
Wang, Kao, Sandvik, arXiv:0901.0214
S1
S1
S1
S1
T0 T0
T0 T0 T0 T0
T0 T0
a
b
cd
e
f
g h
i
j
k
l
ar
br
crdr
er
frgr hr
al
bl
cldlel
flgl hl
i
j
k
l
T1
a
b
cd
e
f
g h
i
j
k
l
ar
br
crdr
er
frgr hr
al
bl
cldlel
flgl hl
i
j
k
l
8-index tensor: D8 4-index tensor: Dcut4
a,b,...g,h=1...D i,j,k,l=1...Dcut
a
b
cd
e
f
g h
i
j
k
l
ar
br
crdr
er
frgr hr
al
bl
cldlel
flgl hl
i
j
k
l
a,b,...g,h=1...D i,j,k,l=1...Dcut2
8-index dbl-tensor: D16 4-index dbl-tensor: Dcut8
Renormalization of an 8-index plaquette tensor using auxiliary 3-index tensors S.
Department of PhysicsNational Taiwan University
Building block: plaquette
• 12 internal sums. Maximum computational effort: D8.
• Each free index and summation contributes D.
T (1),αβγδ =
�
α1,α2,...,δ1,δ2,ijkl
S(1),αα1α2
S(1),ββ1β2
S(1),γγ1γ2
S(1),δδ1δ2
T (0),α1lγ2i T (0),α2j
iδ1T (0),jβ1
kδ2T (0),lβ2
γ1k
α
β
δγδ1
δ2
β1
γ1
γ2
β2
α1 α2
i
j
kl
T0
S1
T0
T0T0
S1
S1
S1
D7 D7 D8 D7 D6
Department of PhysicsNational Taiwan University
Plaquette-Renormalization of TNS
• Effective reduced tensor network for a 8x8 lattice
• Summing over all unequivalent bonds and sites
• Method is variational
• Optimize T and R globally
• Size scaling: L2Log(L)
R1
R2
T1
T0
T2
Wang, Kao, Sandvik, arXiv:0901.0214
Department of PhysicsNational Taiwan University
Expectation values
Department of PhysicsNational Taiwan University
Transverse Ising Model
• Assume translational invariance: all initial T’s are the same.
• Globally optimized T and R.
• hc=3.33 (3.04,QMC)
• mz~ (h-hc)β, β~0.40• h~hc, β~0.50 mean-field like.
0.0
0.2
0.4
0.6
0.8
2.5 3.0 3.5
mz 4
0.0
0.2
0.4
0.6
0.8
2.5 3.0 3.5
mz 4
0.0
0.2
0.4
0.6
0.8
2.5 3.0 3.5
mz 4
8
0.0
0.2
0.4
0.6
0.8
2.5 3.0 3.5
mz 4
8
0.0
0.2
0.4
0.6
0.8
2.5 3.0 3.5
mz 4
816
0.0
0.2
0.4
0.6
0.8
2.5 3.0 3.5
mz 4
816
0.0
0.2
0.4
0.6
0.8
2.5 3.0 3.5
mz 4
81632
0.0
0.2
0.4
0.6
0.8
2.5 3.0 3.5
mz 4
81632
0.0
0.2
0.4
0.6
0.8
2.5 3.0 3.5
mz 4
8163264
0
0.2
0.4
3.2 3.3 0
0.2
0.4
3.2 3.3 0
0.2
0.4
3.2 3.3 0
0.2
0.4
3.2 3.3 0
0.2
0.4
3.2 3.3 0
0.2
0.4
3.2 3.3
0.6
0.7
0.8
0.9
2.5 3.0 3.5
mx
h/J
48
1632
L. Wang, Y.-J. Kao, A. W. Sandvik, arXiv:0901.0214
D=Dcut=2
Department of PhysicsNational Taiwan University
Transverse Ising Model
• Spins at different lattice sites inside the plaquette have different environments.
• We use different tensors inside a plaquette.
Department of PhysicsNational Taiwan University
Computational Costs
• Globally optimized T and R.
• Contraction calculation is highly parallelizable.
• IBM Blue Gene/L at BU. D=2, takes weeks to optimize.
• Bottleneck: plaquette contractions.
• Take advantage of GPU (Graphic Processing Unit)
• SIMT (Single-instruction, multiple-thread) device.
Department of PhysicsNational Taiwan University
J1-J2 Model
• Need large size and large D to see the real physics.
• Currently L=8, D=3 is very time consuming on CPU.
H. J. Schulz, T. A. Ziman and D. Poilblanc, J. Phys. I 6 (1996) 675-703A. W. Sandvik, Phys. Rev. B 56, 11678 (1997)
0 0.2 0.4 0.6 0.8 1J2/J1
0
0.05
0.1
|ΔE|/E
ex
L=4, D=2L=4, D=3L=4, D=4
0 0.2 0.4 0.6 0.8 1J2/J1
-0.8
-0.7
-0.6
-0.5
-0.4
E/N
L=8, D=2L=8, D=3L=4, D=2L=4, D=3
H = J1
�
�ij�
Si · Sj + J2
�
��ij��
Si · Sj
Department of PhysicsNational Taiwan University
J1-J2 Model
• Order parameters M2(qx, qy) =
�
1N
�
j
ei(qxjx+qyjy)Sj
2�
M1=M(π,π) M2=M(0,π))(a
)(b
)(c
Figure 4.10: A graphric representation of the f(j) in Eq. (4.12) associatedwith (a) Q = (!, !), (b) Q = (!, 0) , and (c) Q = (0, !).
36
)(a
)(b
)(c
Figure 4.10: A graphric representation of the f(j) in Eq. (4.12) associatedwith (a) Q = (!, !), (b) Q = (!, 0) , and (c) Q = (0, !).
36
H. J. Schulz, T. A. Ziman and D. Poilblanc, J. Phys. I 6 (1996) 675-703
Department of PhysicsNational Taiwan University
0 0.2 0.4 0.6 0.8 1J2/J1
-0.65
-0.6
-0.55
-0.5
-0.45
-0.4
E/N
L=8, D=2L=8, D=3L=8, D=4
D E Error
2 -0.6331 5.99%
3 -0.6401 4.94%
4 -0.6495 3.55%
QMCE=-0.673487
Department of PhysicsNational Taiwan University
0 0.2 0.4 0.6 0.8 1J2/J1
0
0.05
0.1
0.15
0.2M
2
D=2 M21(π,0)
D=2 M22(π,π)
D=3 M21(π,π)
D=3 M22(π,0)
D=4 M21(π,0)
D=4 M22(π,0)
QMC
L=8
QMCM2=0.17784
Department of PhysicsNational Taiwan University
Summary and Outlook
• Tensor network states are promising candidates to understand frustrated quantum spin systems.
• In plaquette renormalized tensor network representation, no approximations are made when contracting the effective renormalized tensor network.
• Non-MFT results even with the smallest possible non- trivial tensors and truncation (D = 2) in 2d transverse Ising model.
• GPU can potentially speed up the computationally intensive part of the calculation.
• GS for J2/J1~0.6? VBS order?