plaquette renormalized tensor network states: application ... · plaquette renormalized tensor...

17
Plaquette Renormalized Tensor Network States: Application to Frustrated Systems Ying-Jer Kao 高英哲 Department of Physics and Center for Quantum Science and Engineering National Taiwan University Hsin-Chih Hsiao, Ji-Feng Yu, NTU Anders W. Sandvik, Boston University Ling Wang, University of Vienna NDNS: from DMRG to TNF Kyoto, Japan10/29/2010

Upload: others

Post on 09-Jun-2020

6 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Plaquette Renormalized Tensor Network States: Application ... · Plaquette Renormalized Tensor Network States: Application to Frustrated Systems Ying-Jer Kao 高英哲 Department

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

Page 2: Plaquette Renormalized Tensor Network States: Application ... · Plaquette Renormalized Tensor Network States: Application to Frustrated Systems Ying-Jer Kao 高英哲 Department

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

Page 3: Plaquette Renormalized Tensor Network States: Application ... · Plaquette Renormalized Tensor Network States: Application to Frustrated Systems Ying-Jer Kao 高英哲 Department

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

Page 4: Plaquette Renormalized Tensor Network States: Application ... · Plaquette Renormalized Tensor Network States: Application to Frustrated Systems Ying-Jer Kao 高英哲 Department

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.

Page 5: Plaquette Renormalized Tensor Network States: Application ... · Plaquette Renormalized Tensor Network States: Application to Frustrated Systems Ying-Jer Kao 高英哲 Department

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.

Page 6: Plaquette Renormalized Tensor Network States: Application ... · Plaquette Renormalized Tensor Network States: Application to Frustrated Systems Ying-Jer Kao 高英哲 Department

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.

Page 7: Plaquette Renormalized Tensor Network States: Application ... · Plaquette Renormalized Tensor Network States: Application to Frustrated Systems Ying-Jer Kao 高英哲 Department

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

Page 8: Plaquette Renormalized Tensor Network States: Application ... · Plaquette Renormalized Tensor Network States: Application to Frustrated Systems Ying-Jer Kao 高英哲 Department

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

Page 9: Plaquette Renormalized Tensor Network States: Application ... · Plaquette Renormalized Tensor Network States: Application to Frustrated Systems Ying-Jer Kao 高英哲 Department

Department of PhysicsNational Taiwan University

Expectation values

Page 10: Plaquette Renormalized Tensor Network States: Application ... · Plaquette Renormalized Tensor Network States: Application to Frustrated Systems Ying-Jer Kao 高英哲 Department

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

Page 11: Plaquette Renormalized Tensor Network States: Application ... · Plaquette Renormalized Tensor Network States: Application to Frustrated Systems Ying-Jer Kao 高英哲 Department

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.

Page 12: Plaquette Renormalized Tensor Network States: Application ... · Plaquette Renormalized Tensor Network States: Application to Frustrated Systems Ying-Jer Kao 高英哲 Department

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.

Page 13: Plaquette Renormalized Tensor Network States: Application ... · Plaquette Renormalized Tensor Network States: Application to Frustrated Systems Ying-Jer Kao 高英哲 Department

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

Page 14: Plaquette Renormalized Tensor Network States: Application ... · Plaquette Renormalized Tensor Network States: Application to Frustrated Systems Ying-Jer Kao 高英哲 Department

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

Page 15: Plaquette Renormalized Tensor Network States: Application ... · Plaquette Renormalized Tensor Network States: Application to Frustrated Systems Ying-Jer Kao 高英哲 Department

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

Page 16: Plaquette Renormalized Tensor Network States: Application ... · Plaquette Renormalized Tensor Network States: Application to Frustrated Systems Ying-Jer Kao 高英哲 Department

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

Page 17: Plaquette Renormalized Tensor Network States: Application ... · Plaquette Renormalized Tensor Network States: Application to Frustrated Systems Ying-Jer Kao 高英哲 Department

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?