entanglement renormalization on the triangular lattice...entanglement renormalization on the...

Entanglement renormalization on the

triangular lattice

October 28, 2010

K.Harada1 and N.Kawashima2

1Graduate School of Informatics, Kyoto Univ., JAPAN2Institute for Solid State Physics, Univ. of Tokyo, JAPAN

1

2010年10月28日木曜日

Introduction


S=1/2 anti-ferromagnetic Heisenberg model on a triangular lattice with spatial anisotropy

●Spin hamiltonian

●Related materials

� Cs2CuCl4� k-(ET)2Cu2(CN)3� EtMe3Sb[Pd (dmit)2]2� k-(ET)2Cu2[N(CN)2]Cl

3

H = J

�

�ij�

Si · Sj + J�

�

��ij��

Si · SjJ

J �

0.5 1.00

J �

J + J �

ChainSquare Triangle


Numerical methods for the anisotropic triangular lattice model

●Exact diagonalization (up to 36 sites)

●Variational methods

� VMC: S.Yunoki, et al (2006), D.Heidarian, et al (2009)

◊ BCS (or extended) + Gutzwiller projector + Long-range spin Jastrow factor

◊ Up to 24x24 sites

�DMRG: M.Q.Weng, et al (2006)�up to 8x18 sites

4


Ground state phase diagram

5methods used, and the results derived from them will bepresented. The series coefficients are not given here, but areavailable upon request. Previous studies using these methodshave agreed quantitatively with other numerical studies �in-cluding quantum Monte Carlo and exact diagonalizationstudies� for unfrustrated spin models. For frustrated two-dimensional models, both the Monte Carlo and the exactdiagonalization methods face problems �due to the minussigns in case of the former and tremendous variations withsize and shapes of the clusters in case of the latter�. Thus, webelieve that it would be difficult to get a comparable numeri-cal treatment of this strongly frustrated model by other nu-merical methods.

A. Ising expansions

To construct a T�0 expansion about the classical groundstate for this system, one has to introduce an anisotropy pa-rameter � , and write the Hamiltonian for the Heisenberg-Ising model as

H�H0�xV , �5�

where

H0�H1�t�iSiz,

V�J1H2�J2H3�t�iSiz . �6�

The last term of strength t in both H0 and V is a local fieldterm, which can be included to improve convergence. The

limits x�0 and x�1 correspond to the Ising model and theisotropic Heisenberg model, respectively. The operator H0 istaken as the unperturbed Hamiltonian, with the unperturbedground state being the usual ferromagnetically ordered state.The operator V is treated as a perturbation.To determine whether the system is in the ordered phase,

we need to compute the order parameter M defined by

M�1N �

i�Si

z�, �7�

where the angular brackets denote ground-state expectationvalues. Actually, in this expansion, we can choose q differentfrom that given in Eq. �2�; the favored q should give a lowerenergy. If q�� , we are expanding about the usual Neelphase.Ising series have been calculated for the ground-state en-

ergy per site, E0 /N , the order parameter M, and the tripletexcitation spectrum �for the case of q�� only� for severalratios of couplings x and �simultaneously� for several valuesof t up to order x10. This calculation involves 116 137 linkedclusters of up to 10 sites. The series for the case of y�1have been computed previously,5 and our results agree withthese previous results. This is a highly nontrivial check onthe correctness of the expansion coefficients, as the �aniso-tropic square-lattice� graphs considered here are completelydifferent from the �triangular-lattice� graphs considered inRef. 5 and the representation of the Hamiltonian as aHeisenberg-Ising model is also superficially different.At the next stage of the analysis, we try to extrapolate the

series to the isotropic point (x�1) for those values of theexchange coupling parameters which lie within the orderedphase at x�1. For this purpose, we first transform the seriesto a new variable

��1��1�x �1/2, �8�

to remove the singularity at x�1 predicted by spin-wavetheory. This was first proposed by Huse21 and was also usedin earlier work on the square lattice case.22 We then use bothintegrated first-order inhomogeneous differentialapproximants23 and Pade approximants to extrapolate the se-ries to the isotropic point ��1(x�1). The results of theIsing expansions will be presented in later sections.

B. Dimer expansions

For this system, we have carried out two types of dimerexpansions corresponding to two types of dimer coverings.One is the columnar covering displayed in Fig. 3, and theother is the diagonal covering displayed in Fig. 4�a�.

1. Columnar dimer expansions

In this expansion, we take the columnar bonds �D� de-noted by the thick solid bonds in Fig. 3 as the unperturbedHamiltonian H0 and the rest of the bonds as perturbation;that is, we define the following ‘‘dimerized J1-J2 model’’:

H�J2 �(i , j)�D

Si•Sj�x�J2 �(i , j)�” D

Si•Sj�J1��in�

Si•Sj� .�9�

FIG. 2. Phase diagram of the model. The wave vector q is plot-ted as a function of J1 /(J1�J2). The solid curve is the classicalordering wave vector, while the points are estimates of the locationof the minimum triplet gap from both columnar dimer expansions�solid circles� and diagonal dimer expansions �open circles�. Thelong dashed lines are the fit given in Eq. �16�. The vertical shortdashed lines indicate the region of spontaneously dimerized phase,and vertical dotted lines indicate another possible transition point.

PRB 59 14 369PHASE DIAGRAM FOR A CLASS OF SPIN-12 . . .

W. Zheng, et al., PRB 59 (1999) 14367

classical

quantum

J’/(J+J’)

2Cs CuCl4

(ET)2Cu2(CN)3

/J

SL SL AFMLO

0.0 ~0.65 ~0.8 1.0

IsotropicTriangular

1D

J’2/3 S. Yunoki, et al., PRB 74 (2006) 014408

!"#$%&'%!&($)

*+,-./012- %012-

343 /43

/0 #56789':75;<=>5=?6@

ABCA

D&<%&'&:

E34FGE34HD. Heidarin, et al., PRB 80 (2009) 012404

High temperature series expansion

Spiral order remain in J ! J’ regime ?

J �/J

J �/J

J/J’

J/J’


Entanglement renormalization on a

triangular lattice


Tensor network wave function

●Probability amplitude is defined by the contraction of small tensors

7

! !!

!

"

#$%

#&% #'%

i1 i2 i3 i4 i5 i6 i7 i8

|Ψ� =�

{i1,··· ,in}

Ti1···in |i1, · · · , in�


Entanglement renormalization (Vidal, 2005)

● Unitary tensor for removing entanglements (Disentangler)

� For 2 sites,

● Multi-scale entanglement renormalization ansatz (MERA)

8

U

i j

k l

U

T

U

T

U

U

T

T

T

T

T

T TT

T

T

T

T MERAArea law of

entanglement entropy is satisfied

G. Vidal, Phys. Rev. Lett. 99, 220405 (2007) (arXiv:cond-mat/0512165v2)


ER on a triangular lattice (1)

●Tensors

●Entanglement renormalization of 19 sites

9

Disentangler+isometryDisentangler Isometry


ER on a triangular lattice (2)

● Entanglement renormalization of 19 sites

1) Triangular disentangler2) Parallelogram disentangler

(6 sites to 2 sites)3) Hexagonal isometry

● Features

� Three-body interaction Hamiltonian converts to three-body one

� Width of causal cone is 610

Memory size : O(χ12)Computational cost : O(χ14)

χ : dimension of a tensor’s leg


Numerical results


1-step ER calculations● ER : 19 sites to a top site

● Top sites : 2 x 3 parallelogram

● The number of sites : (2x3)x19 = 114

● Periodic boundary condition

● All tensors are different : 36 tensors

� The number of causal cones : 228

◊ Calculations on the super computer system B at ISSP, Univ. of Tokyo

12

χ1 : χ2 : χ3 : χ4 =

χ1

χ2

χ3

χ4


Isotropic case : J=J’


Spin-Spin correlations on bonds

14

2:2:2:2 2:2:4:4

2:2:8:4 2:2:8:8

< Si · Sj >-0.3 0.0


Spin-Spin auto-correlation

15

-0.2 0.20

2:2:2:2 2:2:4:4 2:2:8:4 2:2:8:8

Radius = 0.2 + |G(0,i)|

The finite-correlation MERA for an infinite lattice can be used

G(0, i) ≡ (�S0 · Si� − �S0� · �Si�)/(3/4− |S0|2)


Magnetization of 3 sublattice

●Definition

●For 2:2:8:8

16

3-sublattice 120º magnetization

M3 = 0.316, cos(θ3) = −0.497

0

180˚

θ0i

Angle between the origin and a site


Isotropic case : J’= J

17For 2:2:8:8, E/N= -0.5418, M=0.314

2:2:2:2

2:2:4:4

2:2:8:4

2:2:8:8


Anisotropic case : J ! J’


Energy

19


Magnetization and angles

20

θ1

θ2θ3

χ = 2 : 2 : 8 : 6


Effect of periodic boundary conditions

21

0

180˚

θ0i

Angle between the origin and a site

J/J � = 0.8, χ = 2 : 2 : 8 : 82010年10月28日木曜日

2-step ER calculations● ER : 19 sites to a top site

● Top sites : 2 x 3 parallelogram

● The number of sites : (2x3)x19x19 =2166

● Periodic boundary condition

● All tensors are different : 720 tensors

� The number of causal cones : 4332

◊ Calculations on the super computer system B at ISSP, Univ. of Tokyo

22χ

1:χ

2:χ

3:χ

4:χ

5:χ

6:χ

7=


Energy

23


Magnetization and angles

24

θ1

θ2θ3

2-steps : χ = 2 : 2 : 4 : 4 : 4 : 4 : 4


Summary

●MERA scheme for the triangular model

� Entanglement renormalizaion for 19 sites

● S=1/2 antiferromagnetic Heisenberg models on a triangular lattice

� Isotropic case : 120˚ magnetization order

� Anisotropic case

◊ 1 " J/J’ " 0.7 : spiral long-range order from 2-steps ER calculations, no spin liquid

(Note: the dimension is small now)

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