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Constraints on Beta Functions from Duality

P.H. Damgaard, Peter E. Haagensen

Published Jan 1, 1997• 12 citations
OpenAlex

Abstract

We analyze the way in which duality constrains the exact beta function and correlation length in single-coupling spin systems. A consistency condition we propose shows very concisely the relation between self-dual points and phase transitions, and implies that the correlation length must be duality invariant. These ideas are then tested on the 2-d Ising model, and used towards finding the exact beta function of the q-state Potts model. Finally, a generic procedure is given for identifying a duality symmetry in other single-coupling models with a continuous phase transition.

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Constraints on beta functions from duality

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We analyse the way in which duality constrains the exact beta function and correlation length in single-coupling spin systems. We propose a consistency condition which shows very concisely the relation between self-dual points and phase transitions, and implies that the correlation length must be duality invariant. These ideas are then tested on the two-dimensional Ising model, and used towards finding the exact beta function of the q-state Potts model. Finally, a generic procedure is given for identifying a duality symmetry in other single-coupling models with a continuous phase transition.

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We discover a Ising-type duality in the general N-state chiral Potts model, which is the Kramers-Wannier duality of planar Ising model when N = 2. This duality relates the spectrum and eigenvectors of one chiral Potts model at a low temperature (of small k ′) to those of another chiral Potts model at a high temperature (of k ′−1). The τ (2)-model and chiral Potts model on the dual lattice are established alongside of the dual chiral Potts models. With the aid of this duality relation, we exact a precise relationship between the Onsager-algebra symmetry of a homogeneous superintegrable chiral Potts model and the sl2-loop-algebra symmetry of its associated spin- N−1 2 XXZ chain through the identification of their eigenstates.

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We analyze the way in which duality constrains the exact beta function and correlation length in single-coupling spin systems. A consistency condition we propose shows very concisely the relation between self-dual points and phase transitions, and implies that the correlation length must be duality invariant. These ideas are then tested on the 2-d Ising model, and used towards finding the exact beta function of the q-state Potts model. Finally, a generic procedure is given for identifying a duality symmetry in other single-coupling models with a continuous phase transition.
We analyse the way in which duality constrains the exact beta function and correlation length in single-coupling spin systems. We propose a consistency condition which shows very concisely the relation between self-dual points and phase transitions, and implies that the correlation length must be duality invariant. These ideas are then tested on the two-dimensional Ising model, and used towards finding the exact beta function of the q-state Potts model. Finally, a generic procedure is given for identifying a duality symmetry in other single-coupling models with a continuous phase transition.
Inspired by previous work on the constraints that duality imposes on beta functions of spin models, we propose a consistency condition between those functions and RG flows at different points in coupling constant space. We show that this consistency holds for a non self-dual model which admits an exact RG flow, but that it is violated when the RG flow is only approximate. We discuss the use of this deviation as a test for the ``goodness'' of proposed RG flows in complicated models, and the use of the proposed consistency in suggesting RG equations.
We discover an Ising-type duality in the general N -state chiral Potts model, which is the Kramers–Wannier duality of a planar Ising model when N = 2. This duality relates the spectrum and eigenvectors of one chiral Potts model at a low temperature (of small k ') to those of another chiral Potts model at a high temperature (of k ' −1 ). The τ (2) model and chiral Potts model on the dual lattice are established alongside of the dual chiral Potts models. With the aid of this duality relation, we exact a precise relationship between the Onsager-algebra symmetry of a homogeneous superintegrable chiral Potts model and the sl 2 -loop-algebra symmetry of its associated spin- ( N −1)/2 XXZ chain through the identification of their eigenstates.
Abstract Exact solutions of model systems are the most reliable source of information. However, there are not many model systems that can be exactly solved and, consequently, many approximate schemes such as series expansions have been developed. It is sometimes possible, though, to extract exact information without directly solving those model systems. Arguments using duality transformations make it possible to derive the exact location of the phase transition point and the exact value of the energy at the transition point when the model is self dual. Duality not only determines the exact location of the transition point of the two-dimensional Ising model and related models but also is useful to rewrite the XYXYXY model into a different form, the Villain model and its dual, which reveals new physical aspects of the system. Classical and quantum dualities can be related by certain mappings and Fourier transformation is at the root of such deep results.
We have studied spin-spin correlation functions in the ordered phase of the two-dimensional q -state Potts model with q = 10, 15, and 20 at the first-order transition point β t . Through extensive Monte Carlo simulations we obtain strong numerical evidence that the correlation length in the ordered phase agrees with the exactly known and recently numerically confirmed correlation length in the disordered phase: ξ o (β t ) = ξ d (β t ). As a by-product we find the energy moments in the ordered phase at β t in very good agreement with a recent large q -expansion.
We discover a Ising-type duality in the general N-state chiral Potts model, which is the Kramers-Wannier duality of planar Ising model when N = 2. This duality relates the spectrum and eigenvectors of one chiral Potts model at a low temperature (of small k ′) to those of another chiral Potts model at a high temperature (of k ′−1). The τ (2)-model and chiral Potts model on the dual lattice are established alongside of the dual chiral Potts models. With the aid of this duality relation, we exact a precise relationship between the Onsager-algebra symmetry of a homogeneous superintegrable chiral Potts model and the sl2-loop-algebra symmetry of its associated spin- N−1 2 XXZ chain through the identification of their eigenstates.