Lattice Statistics on Kagome-Type Lattices

F. Y. Wu

Northeastern University

Kagome-type lattices

Syozi

Physics Today, 56 (Feb) 12 (2003).

(a) (b) (c)

Kagome Triangular kagomeKagome lattice with an internal structure

Kagome lattice 3-12 lattice

Kagome lattice with 3-site interactions 3-12 lattice with 3-site interactions

There has been a surge of recent interest in considering the “triangular kagome” lattice such as

Closed-packed dimers on the triangular kagome latticeY. L. Loh, D.-X. Yao, C. L. Carlson, PRB 78, 224410 (2008)

Bond percolation on the triangular kagome latticeA. Haji-Ankabari and R. M. Ziff, PRE 79 021118 (2009)

Close-packed dimers on the kagome lattice(and the triangular kagome lattice)

6/)(constant NxyzZ

The constant can be determined by a simple mapping

6/4constant N

F. Y. Wu and F. Wang, Physica A 387 (2008) 4148

Dimer-dimer correlation vanishes identically at distances greater than 2 lattice spacing

F. Wang and F. Y. Wu, Physica A 387 (2008) 4157

Potts model on the kagome lattice

Triangular lattice with 3-site interactions

),(),(),(),(,,,

kjKrkkji

jKrjiKrjji

iKr MK

Exact duality relation: Baxter, Temperley and Ashley (1978) using algebraic analysis Wu and Lin (1979) using graphical method

kjiijk

jiij MK

,,,

1 Kev 1* * Kev,

233 KMK eey 23* ***3 KMK eey,

The duality relation reads

qvyv *2* qyy ,

The self-dual point is v=v*, y = y* = q, or

qee KMK 233

Using a continuity argument, Wu and Zia (1981) established that this is indeed the critical point of the ferromagnetic triangular Potts model.

Exact duality relation

Define

123312312

123312312

)(1

])1(1)[1)(1)(1(

yv

evvv M

Generally, for

123312312 )( CBA

qAC

the critical point is C/A=q, or

for the lattice

1 Kevthe critical point is y=q.

Generalization

Example 1: 2x2 subnet

=

)1)(1)(1(

)1)(1)(1)(1)(1)(1(

465645

35342624166,5,4

15

vvv

vvvvvv

123312312 )( CBA

1 Kev

)]61518(3)[3(

)72230()5(2

)32150()433(9

3254

324322

23223

vvvvqvvC

vvvvvqvvqB

vvvvqvvqqA

Bond percolation:

0124572453931 98765432 pppppppp

C=qA gives

cp 0.471 628 788 268… (exact)

Kep 1q=1,

In agreement withHaji-Akbari and Ziff (Phys. Rev. E 79 020102(R) (2009))who obtained the result using a different consideration

q=2,Ising2KK eeu

C=qA gives

and the Ising critical point

5IsingKe 2.236 067 977 500…

Ising model:

0)1)(5( 2223 uuu

For the q=3 Potts model, C=qA gives

Ke 2.493 123 120 701 …

Example 2: The martini lattice (Wu. PRL 96 (2006) 090602)

=

0)3()316()15(6 532224 vvvqvvvqvqq

In agreement with Ziff and Scullard, JPA 39 (2006) 15083

For bond percolation with q=1, v=p/(1-p), this gives

12)(2 65432 ppppp

qee KMK 233

Triangular lattice with alternate 3-site interactions M and 0:

Conjecture (Wu, 1979)

Triangular lattice with alternate 3-site interactions M and N:

qee KNMK 233MMM

M

N

M

N N N

M

Exact expression

qee KMK 2323

Triangular lattice with 3-site interactions M in every face:

M M MM M M M

Star-triangle transformation: Diced lattice

T

Duality transformation:

Diced lattice kagome lattice

This gives the kagome critical threshold

0692126 6543223 vvvqvqvvqq

1 cKev

(Wu, 1979)

N’

N

L’L

K

M’ K

K

M’M’ M’

M’M

M

M M

M

N’

N

K’

Duality relation for Potts model with multi-site interactions(Essam, 1979; Wu, 1982)

22' )1)(1( ,)1)(1( ,)1)(1(''

qeeqeeqee NMNMKK

Kagome lattice with 2- and 3-site interactions K and M

3-12 lattice with 2- and 3-site interactions K* and M*

=

Using the exact duality relation

2*

*

)1)(1(

)1)(1(

qee

qeeMM

KK

General formulation for the kagome-type lattices:

dualityM

MM

M*

M*

K

K*K

K*

K*

=K* .L L

123312312 )( CBA =L

LLM*

)1(

)1(

)]1)(3()[(

*3

*2

*3

M

M

M

eFvC

eFvB

evqvqFA

1 Lev

Solve F, , , hence , in terms of A, B, C Le *Me Ke Me

qee KMK 2323

The conjectured threshold

2222 )2()(3)3( CqCqBCqBAq

gives the threshold for the general problem in terms of A, B. C

=

=

123312312 )( CBA

qee KNMK 233

More generally, the conjectured threshold

')2()'')((3)''3')(3( 22 CCqCqBCqBCqBAqCqBAq

gives the threshold for the general A, B. C, A’, B’, C’ problem

=

=

123312312 )( CBA

123312312 ')('' CBA

The 3-12 lattice

For the 3-12 lattice with uniform interactions K, the threshold is:

23 ,1 vvrev K

632422233 )2())((3)]()[()( rvqrqvqrvqrvqrqvq

For bond percolation, q=1, v=p/(1-p), this gives

For the Ising model, q=2, this gives

08862 234 vvvv

]13[]3341[ cv = 4.073 446 135 … (Soyzi, 1972)

... 317 423 0.740 cp

0471 5432 ppppp

(Scullard and Ziff (PRE 73 (2006) 045102)

1 Kev

For the lattice with uniform interactions K:

)]61518(3)[3(

)72230()5(2

)32150()433(9

3254

324322

23223

vvvvqvvC

vvvvvqvvqB

vvvvqvvqqA

For percolation, q=1 and v=p/(1-p), this gives

0737863708994816359162368232

12617852642733039183118171615141312

1110967654

ppppppp

pppppppp

cp

cp

= 0.600 870 248 238 …

This is compared to the Ziff-Gu (2009) numerical result

= 0.600 862 4

For Ising model, q=2 and v=p/(1-p), this gives the exact result

0863283 8642 uuuu

3.024 382 957 092 ,,,, Ising2Keu

The End