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 Triangular kagome Kagome lattice with an internal structure

5. Kagome lattice3-12 lattice Kagome lattice with 3-site interactions3-12 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. ,, 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. Exact duality relation Define

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

14. Example 1: 2x2 subnet = 

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

16. q=2, C=qA gives and the Ising critical point 2.236 067 977 500… Ising model: 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) = In agreement with Ziff and Scullard, JPA 39 (2006) 15083 For bond percolation with q=1, v=p/(1-p), this gives

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

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

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

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

22. 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 General formulation for the kagome-type lattices: duality M M M M* K K* K =. LL

23. = L LL M* 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 For the Ising model, q=2, this gives = 4.073 446 135 … (Soyzi, 1972) (Scullard and Ziff (PRE 73 (2006) 045102)

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