### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 001, 12 pages      cond-mat/0701075      https://doi.org/10.3842/SIGMA.2007.001
Contribution to the Proceedings of the O'Raifeartaigh Symposium

### Non-Local Finite-Size Effects in the Dimer Model

Nickolay Sh. Izmailian a, b, c, Vyatcheslav B. Priezzhev d and Philippe Ruelle e
a) Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan
b) Yerevan Physics Institute, Alikhanian Brothers 2, 375036 Yerevan, Armenia
c) National Center of Theoretical Sciences at Taipei, Physics Division, National Taiwan University, Taipei 10617, Taiwan
d) Bogolyubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia
e) Institut de Physique Théorique, Université catholique de Louvain, 1348 Louvain-La-Neuve, Belgium

Received September 29, 2006, in final form December 12, 2006; Published online January 04, 2007

Abstract
We study the finite-size corrections of the dimer model on ∞ × N square lattice with two different boundary conditions: free and periodic. We find that the finite-size corrections depend in a crucial way on the parity of N, and show that, because of certain non-local features present in the model, a change of parity of N induces a change of boundary condition. Taking a careful account of this, these unusual finite-size behaviours can be fully explained in the framework of the c = -2 logarithmic conformal field theory.

Key words: dimer model; finite-size corrections; conformal field theory.

pdf (351 kb)   ps (235 kb)   tex (190 kb)

References

1. Fowler R.H., Rushbrooke G.S., Statistical theory of perfect solutions, Trans. Faraday Soc. 33 (1937), 1272-1294.
2. Kasteleyn P.W., The statistics of dimers on a lattice. I. The number of dimer arrangements on a quadratic lattice, Physica 27 (1961), 1209-1225.
3. Kasteleyn P.W., Dimer statistics and phase transitions, J. Math. Phys. 4 (1963), 287-293.
4. Fisher M.E., Statistical mechanics of dimers on a plane lattice, Phys. Rev. 124 (1961), 1664-1672.
5. Temperley H.N.V., Fisher M.E., Dimer problem in statistical mechanics - an exact result, Philos. Mag. (8) 6 (1961), 1061-1063.
6. Fisher M.E., Stephenson J., Statistical mechanics of dimers on a plane lattice. II. Dimer correlations and monomers, Phys. Rev. 132 (1963), 1411-1431.
7. Hartwig R.E., Monomer pair correlations, J. Math. Phys. 7 (1966), 286-299.
8. Fendley P., Moessner R., Sondhi S.L., Classical dimers on the triangular lattice, Phys. Rev. B 66 (2002), 214513, 14 pages, cond-mat/0206159.
9. Basor E.L., Ehrhardt T., Asymptotics of block Toeplitz determinants and the classical dimer model, math-ph/0607065.
10. Tseng W.J., Wu F.Y., Dimers on a simple-quartic net with a vacancy, J. Statist. Phys. 110 (2003), 671-689.
11. Kong Y., Logarithmic corrections in the free energy of monomer-dimer model on plane lattices with free boundaries, Phys. Rev. E 74 (2006), 011102, 8 pages.
12. Wu F.Y., Pfaffian solution of a dimer-monomer problem: single monomer on the boundary, Phys. Rev. E 74 (2006), 020140(R), 4 pages, Erratum, Phys. Rev. E 74 (2006), 020104(E), cond-mat/0607647.
13. Ferdinand A.E., Statistical mechanics of dimers on a quadratic lattice, J. Math. Phys. 8 (1967), 2332-2339.
14. McCoy B.W., Wu T.T., The two-dimensional Ising model, Harvard University Press, Cambridge, MA, 1973.
15. Bhattacharjee S.M., Nagle F.F., Finite-size effect for the critical point of an anisotropic dimer model of domain walls, Phys. Rev. A 31 (1985), 3199-3213.
16. Brankov J.G., Priezzhev V.B., Critical free energy of a Möbius strip, Nuclear Phys. B 400 (1993), 633-652.
17. Lu W.T., Wu F.Y., Dimer statistics on the Möbius strip and the Klein bottle, Phys. Lett. A 259 (1999), 108-114, cond-mat/9906154.
18. Lu W.T., Wu F.Y., Close-packed dimers on nonorientable surfaces, Phys. Lett. A 293 (2002), 235-246, Erratum, Phys. Lett. A 298 (2002), 293, cond-mat/0110035.
19. Ivashkevich E., Izmailian N.Sh., Hu C.-K., Kronecker's double series and exact asymptotic expansion for free models of statistical mechanics on torus, J. Phys. A: Math. Gen. 35 (2002), 5543-5561.
20. Izmailian N.Sh., Oganesyan K.B., Hu C.-K., Exact finite-size corrections of the free energy for the square lattice dimer model under different boundary conditions, Phys. Rev. E 67 (2003), 066114, 14 pages.
21. Izmailian N.Sh., Priezzhev V.B., Ruelle P., Hu C.-K., Logarithmic conformal field theory and boundary effects in the dimer model, Phys. Rev. Lett. 95 (2005), 260602, 4 pages, cond-mat/0512703.
22. Itzykson C., Saleur H., Zuber J.-B., Conformal invariance of nonunitary 2d-models, Europhys. Lett. 2 (1986), 91-96.
23. Blote H.W.J., Hilhorst H.J., Roughening transitions and the zero-temperature triangular Ising antiferromagent, J. Phys. A: Math. Gen. 15 (1982), L631-L637.
24. Kenyon R., Dominos and the Gaussian free field, Ann. Probab. 29 (2001), 1128-1137.
25. Baxter R.J., Exactly solved models in statistical mechanics, Academic Press, New York, 1982.
26. Temperley H.N.V., Combinatorics, in Proceedings of the British Combinatorial Conference, London Math. Soc. Lecture Notes Series 13 (1974), 202-204.
27. Priezzhev V.B., The dimer problem and the Kirchhoff theorem, Sov. Phys. Usp. 28 (1985), 1125-1135.
28. Blöte H.W., Cardy J.L., Nightingale M.P., Conformal invariance, the central charge, and universal finite-size amplitudes at criticality, Phys. Rev. Lett. 56 (1986), 742-745.
29. Affleck I., Universal term in the free energy at a critical point and the conformal anomaly, Phys. Rev. Lett. 56 (1986), 746-748.
30. Cardy J.L., Effect of boundary conditions on the operator content of two-dimensional conformally invariant theories, Nuclear Phys. B 275 (1986), 200-218.
31. Brankov J.G., Isomorphism of dimer configurations and spanning trees on finite square lattices, J. Math. Phys. 36 (1995), 5071-5083.
32. Majumdar S.N., Dhar D., Equivalence between the Abelian sandpile model and the q ® 0 limit of the Potts model, Phys. A 185 (1992), 129-145.
33. Ruelle P., A c = -2 boundary changing operator for the Abelian sandpile model, Phys. Lett. B 539 (2002), 172-177, hep-th/0203105.
34. Piroux G., Ruelle P., Pre-logarithmic and logarithmic fields in a sandpile model, J. Stat. Mech. Theory Exp. (2004), P10005, 24 pages, hep-th/0407143.
35. Piroux G., Ruelle P., Logarithmic scaling for height variables in the Abelian sandpile model, Phys. Lett. B 607 (2005), 188-196, cond-mat/0410253.
36. Jeng M., Piroux G., Ruelle P., Height variables in the Abelian sandpile model: scaling fields and correlations, J. Stat. Mech. Theory Exp. (2006), P10015, 63 pages, cond-mat/0609284.
37. Ghosh A., Dhar D., Jacobsen J.L., Random trimer tilings, cond-mat/0609322.
38. Priezzhev V.B., Ruelle P., Boundary monomers in the dimer model, in preparation.