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Physical review. B, Condensed matter

Spinor exponents for the two-dimensional Potts model

Bernard Nienhuis, H. J. F. Knops

Published Aug 1, 1985• 30 citations
Publisher

Abstract

The spinor operator in the two-dimensional Ising model can be readily generalized to other self-dual models as the product of the order parameter and its dual image, the disorder operator. Recently, the exponent of this and other operators in the $q$-state Potts model received renewed attention, as the theory of conformal invariance produces complete lists of critical exponents for some of these models. In this paper we calculate the critical and tricritical indices of the spinor operator in the two-dimensional $q$-state Potts model, via a well-known mapping into a solid-on-solid model. Our results give a physical identification for the exponents predicted by the conformal theory.

DOI
10.1103/physrevb.32.1872

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The spinor operator in the two-dimensional Ising model can be readily generalized to other self-dual models as the product of the order parameter and its dual image, the disorder operator. Recently, the exponent of this and other operators in the qqq-state Potts model received renewed attention, as the theory of conformal invariance produces complete lists of critical exponents for some of these models. In this paper we calculate the critical and tricritical indices of the spinor operator in the two-dimensional qqq-state Potts model, via a well-known mapping into a solid-on-solid model. Our results give a physical identification for the exponents predicted by the conformal theory.
Making complex CFTs real: The two-dimensional Potts model for Q>4Q>4Q>4 and complex QQQ
The two-dimensional QQQ-state Potts model with real couplings has a first-order transition for Q>4Q>4Q>4. Starting from a triangular-lattice Potts model with two- and three-spin interactions, we study an equivalent loop model in which QQQ is a continuous parameter. By a combination of analytical and numerical arguments, we show that this loop model allows for the collision of a critical and a tricritical fixed point at Q=4Q=4Q=4. These then emerge as a pair of complex conformally invariant theories at Q>4Q>4Q>4, or even complex QQQ, for suitable complex coupling constants. We conjecture that all conformal data (such as the central charge, critical exponents, and three-point structure constants) can be obtained by analytic continuation of known exact results for the loop model with Q≤4Q \le 4Q≤4. This conjecture is checked, both for real Q>4Q>4Q>4 and for Q∈CQ \in \mathbb{C}Q∈C, by extensive transfer-matrix computations and comparison to previous studies for Q=5Q=5Q=5.
The two-dimensional Q-state Potts model with real couplings has a first-order transition for Q>4. We study a loop-model realization in which Q is a continuous parameter. This model allows for the collision of a critical and a tricritical fixed point at Q=4, which then emerge as complex conformally invariant theories at Q>4, or even complex Q, for suitable complex coupling constants. All critical exponents can be obtained as analytic continuation of known exact results for Q≤4. We verify this scenario in detail for Q=5 using transfer-matrix computations.
In this paper we review the critical Ising model by using the properties of conformal invariance. We use the known mapping of the Ising model to a theory of Majorana fermions and recognize that the self-duality property appears as a double degeneracy of the periodic ground state. The Ising model is doubled obtaining an Ashkin-Teller model (two Ising models coupled with a four-spin coupling) at the decoupling point. Using the bosonization technique this model is mapped onto the Gaussian model at a particular value of the temperature (K=1/\ensuremath{\pi}). This allows us to give the expressions for products of order, disorder, and energy operators of both Ising models in terms of operators in the Gaussian model. We compute several correlation functions of order, disorder, energy and spinor operators and show that they reproduce the operator product expansions predicted by conformal invariance. We explicitly discuss the physics by which the correlation functions obtained from the Gaussian model (a theory with conformal anomaly c=1) reproduce those of the Ising model (c=1/2). This provides a proof of Kadanoff and Brown's equivalences and conjectures. We provide a simple prescription to compute all n-point correlation functions at criticality, including mixed correlations of order, disorder, energy, and spinor operators. A bosonized form of the continuum limit of the transfer matrix for the critical Ising model is constructed. It is nonlocal and contains both spin-wave and vortex operators of the Gaussian model.
Using suitable Monte Carlo methods and finite-size scaling, we investigate critical and tricritical surface phenomena of two-dimensional Potts models. For the critical two- and three-state models, we determine a surface scaling dimension describing percolation properties of the so-called Potts clusters near the edges. On this basis, we propose an exact expression describing this exponent for the whole critical branch. For tricritical Potts models we find that varying the surface coupling constant or the surface magnetic field can induce a continuous phase transition. At bulk tricriticality and sufficiently strong surface couplings, spontaneous one-dimensional order occurs on the edges. We determine several critical exponents describing these edge transitions. On the basis of these results and conformal field theory, we conjecture exact expressions for these exponents.
We determine the backbone exponent X(b) of several critical and tricritical q-state Potts models in two dimensions. The critical systems include the bond percolation, the Ising, the q=2-sqrt[3], 3, and 4 state Potts, and the Baxter-Wu model, and the tricritical ones include the q=1 Potts model and the Blume-Capel model. For this purpose, we formulate several efficient Monte Carlo methods and sample the probability P2 of a pair of points connected via at least two independent paths. Finite-size-scaling analysis of P2 yields X(b) as 0.3566(2), 0.2696(3), 0.2105(3), and 0.127(4) for the critical q=2-sqrt[3], 1,2, 3, and 4 state Potts model, respectively. At tricriticality, we obtain X(b)=0.0520(3) and 0.0753(6) for the q=1 and 2 Potts model, respectively. For the critical q-->0 Potts model it is derived that X(b)=3/4. From a scaling argument, we find that, at tricriticality, X(b) reduces to the magnetic exponent, as confirmed by the numerical results.