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arXiv:cond-mat/9712111

Correlation duality relations for the ($N_α, N_β$) model

Fa Yueh Wu, Wentao Lu

Published Dec 10, 1997• 0 citations
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Abstract

Duality relations for the correlation functions of $n$ sites on the boundary of a planar lattice are derived for the $(N_α, N_β)$ model of Domany and Riedel for $n=2,3$. Our result holds for arbitrary lattices which can have nonuniform interactions.

DOI
10.48550/arxiv.cond-mat/9712111

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Correlation duality relations for the (Nα,NβN_α, N_βNα​,Nβ​) model
Duality relations for the correlation functions of nnn sites on the boundary of a planar lattice are derived for the (Nα,Nβ)(N_α, N_β)(Nα​,Nβ​) model of Domany and Riedel for n=2,3n=2,3n=2,3. Our result holds for arbitrary lattices which can have nonuniform interactions.
The Kramers-Wannier duality is shown to hold for all the even number spin correlation functions of the two dimensional square lattice Ising model in the sense that the high temperature (T>Tc)(T>T_{c})(T>Tc​) expressions for these correlation functions are transformed into the low temperature $(T
We relate two classical dualities in low-dimensional quantum field theory: Kramers-Wannier duality of the Ising and related lattice models in 222 dimensions, with electromagnetic duality for finite gauge theories in 333 dimensions. The relation is mediated by the notion of boundary field theory: Ising models are boundary theories for pure gauge theory in one dimension higher. Thus the Ising order/disorder operators are endpoints of Wilson/'t Hooft defects of gauge theory. Symmetry breaking on low-energy states reflects the multiplicity of topological boundary states. In the process we describe lattice theories as (extended) topological field theories with boundaries and domain walls. This allows us to generalize the duality to non-abelian groups; finite, semi-simple Hopf algebras; and, in a different direction, to finite homotopy theories in arbitrary dimension.
In this paper we present the solutions of the star-triangle equation for Boltzmann weights of the (Nα,Nβ) model. This model consists of two scalar Potts models (of Nα and Nβ states, respectively) coupled together by four-spin interaction terms and was introduced by Domany and Riedel to study the phase transitions in adsorbed thin films. If Nα,Nβ≠2, the star-triangle equations have nontrivial self-dual and non-self-dual solutions only for Nα=Nβ; whereas if Nα≠Nβ=2, we have other self-dual and non-self-dual solutions. After applying partial duality transformations several of our solutions can be identified with known solutions for Interaction-Round-a-Face (IRF) models. Our results also mean that our high-genus solutions found earlier are not the only solutions to the star-triangle equations in chiral Potts models with six or more states per site.
Abstract A one-dimensional (1D) q -state Potts model with N sites, m -site interaction K in a field H is studied for arbitrary values of m . Exact results for the partition function and the two-point correlation function are obtained at H = 0. The system in a field is shown to be self-dual. Using a change of Potts variables, it is mapped onto a standard 2D Potts model, with first-neighbour interactions K and H , on a cylinder with helical boundary conditions (BC). The 2D system has a length N / m and a transverse size m . Thus the Potts chain with multi-site interactions is expected to develop a 2D critical singularity along the self-duality line, ( e q K − 1 ) ( e q H − 1 ) = q , when N / m → ∞ and m → ∞ .