Integrability and conformal data of the dimer model
(2016) Journal of Physics A: Mathematical and Theoretical — Vol. 49, p. 174002 (2016)
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- The central charge of the dimer model on the square lattice is still being debated in the literature. In this paper, we provide evidence supporting the consistency of a c=−2 description. Using Lieb's transfer matrix and its description in terms of the Temperley-Lieb algebra TL_n at β=0, we provide a new solution of the dimer model in terms of the model of critical dense polymers on a tilted lattice and offer an understanding of the lattice integrability of the dimer model. The dimer transfer matrix is analysed in the scaling limit and the result for L_0−c/24 is expressed in terms of fermions. Higher Virasoro modes are likewise constructed as limits of elements of TL_n and are found to yield a c=−2 realisation of the Virasoro algebra, familiar from fermionic bc ghost systems. In this realisation, the dimer Fock spaces are shown to decompose, as Virasoro modules, into direct sums of Feigin-Fuchs modules, themselves exhibiting reducible yet indecomposable structures. In the scaling limit, the eigenvalues of the lattice integrals of motion are found to agree exactly with those of the c=−2 conformal integrals of motion. Consistent with the expression for L_0−c/24 obtained from the transfer matrix, we also construct higher Virasoro modes with c=1 and find that the dimer Fock space is completely reducible under their action. However, the transfer matrix is found not to be a generating function for the c=1 integrals of motion. Although this indicates that Lieb's transfer matrix description is incompatible with the c=1 interpretation, it does not rule out the existence of an alternative, c=1 compatible, transfer matrix description of the dimer model.
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Morin Duchesne, A., Ruelle, P., & Rasmussen, J. (2016). Integrability and conformal data of the dimer model. Journal of Physics A: Mathematical and Theoretical, 49, 174002. https://doi.org/10.1088/1751-8113/49/17/174002 (Original work published 2016)