Lattice Statistics on Kagome-Type Lattices

1. Lattice Statistics on Kagome-Type Lattices F. Y. Wu Northeastern University

2. Kagome-type lattices Syozi

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

4. (a) (b) (c) Kagome lattice with an internal structure Kagome Triangular kagome

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

6. There has been a surge of recent interest in considering the “triangular kagome” lattice such as Closed-packed dimers on the triangular kagome lattice Y. L. Loh, D.-X. Yao, C. L. Carlson, PRB 78, 224410 (2008) Bond percolation on the triangular kagome lattice A. Haji-Ankabari and R. M. Ziff, PRE 79 021118 (2009)

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

8. The constant can be determined by a simple mapping F. Y. Wu and F. Wang, Physica A 387 (2008) 4148

9. Dimer-dimer correlation vanishes identically at distances greater than 2 lattice spacing F. Wang and F. Y. Wu, Physica A 387 (2008) 4157

10. Potts model on the kagome lattice

11. Triangular lattice with 3-site interactions Exact duality relation: Baxter, Temperley and Ashley (1978) using algebraic analysis Wu and Lin (1979) using graphical method

12. Exact duality relation Define , , The duality relation reads , The self-dual point is v=v*,y = y* = q, or Using a continuity argument, Wu and Zia (1981) established that this is indeed the critical point of the ferromagnetic triangular Potts model.

13. Generalization the critical point is y=q. Generally, for the critical point is C/A=q, or for the lattice

14. Example 1: 2x2 subnet = 

15. Bond percolation: q=1, C=qA gives 0.471 628 788 268… (exact) In agreement with Haji-Akbari and Ziff (Phys. Rev. E 79 020102(R) (2009)) who obtained the result using a different consideration

16. Ising model: q=2, C=qA gives and the Ising critical point 2.236 067 977 500… For the q=3 Potts model, C=qA gives 2.493 123 120 701 …

17. Example 2: The martini lattice (Wu. PRL 96 (2006) 090602) = For bond percolation with q=1, v=p/(1-p), this gives In agreement with Ziff and Scullard, JPA 39 (2006) 15083

18. Triangular lattice with alternate 3-site interactions M and 0: M M M Exact expression Conjecture (Wu, 1979) Triangular lattice with alternate 3-site interactions M and N: N M N N M N M

19. Triangular lattice with 3-site interactions M in every face: M M M M M M M Star-triangle transformation:  Diced lattice

20. Duality transformation: Diced lattice kagome lattice This gives the kagome critical threshold T (Wu, 1979)

21. Duality relation for Potts model with multi-site interactions (Essam, 1979; Wu, 1982) N’ N’ K L’ M M K K’ L M’ M’ M’ N N M M M M’ M’ K

22. General formulation for the kagome-type lattices: Using the exact duality relation . K* L L = K* K M* K K* duality K* M M M* M = Kagome lattice with 2- and 3-site interactions K and M 3-12 lattice with 2- and 3-site interactions K* and M*

23. L = M* L L Solve F, , , hence , in terms of A, B, C

24. The conjectured threshold gives the threshold for the general problem in terms of A, B. C = =

25. More generally, the conjectured threshold gives the threshold for the general A, B. C, A’, B’, C’ problem = =

26. The 3-12 lattice

27. For the 3-12 lattice with uniform interactions K, the threshold is: For bond percolation, q=1, v=p/(1-p), this gives (Scullard and Ziff (PRE 73 (2006) 045102) For the Ising model, q=2, this gives = 4.073 446 135 … (Soyzi, 1972)

28. For the lattice with uniform interactions K:

29. For percolation, q=1 and v=p/(1-p), this gives = 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 3.024 382 957 092 ,,,,

40. The End