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A generalized Beraha conjecture for non-planar graphs

AuthorsJacobsen, Jesper Lykke; Salas, Jesús
KeywordsBerker–Kadanoff phase
Transfer matrix
Generalized Petersen graphs
Beraha conjecture
Non-planar graphs
Potts model
Issue Date2013
CitationNuclear Physics B 875: 678- 718 (2013)
AbstractWe study the partition function ZG(nk,k)(Q,v) of the Q-state Potts model on the family of (non-planar) generalized Petersen graphs G(nk, k). We study its zeros in the plane (Q,v) for 1≤k≤7. We also consider two specializations of ZG(nk,k), namely the chromatic polynomial PG(nk,k)(Q) (corresponding to v=-1), and the flow polynomial ΦG(nk,k)(Q) (corresponding to v=-Q). In these two cases, we study their zeros in the complex Q-plane for 1≤k≤7. We pay special attention to the accumulation loci of the corresponding zeros when n→∞. We observe that the Berker-Kadanoff phase that is present in two-dimensional Potts models, also exists for non-planar recursive graphs. Their qualitative features are the same; but the main difference is that the role played by the Beraha numbers for planar graphs is now played by the non-negative integers for non-planar graphs. At these integer values of Q, there are massive eigenvalue cancellations, in the same way as the eigenvalue cancellations that happen at the Beraha numbers for planar graphs. © 2013 Elsevier B.V.
Identifiersdoi: 10.1016/j.nuclphysb.2013.07.012
issn: 0550-3213
Appears in Collections:(CFMAC-IEM) Artículos
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