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Journal of Physics A Mathematical and General

Extended scaling relations for the chiral and cubic crossover exponents

Marcel den Nijs

Published Apr 1, 1984• 37 citations
Publisher

Abstract

The critical dimensions x CH (q) of the chiral and x CB (q) of the cubic operator in the two-dimensional q-state Potts model satisfy the extended scaling relations x CH =(n 2 -y 2 )/(4x)+x and x CB =(m 2 -y 2 )/(4x)+x with x+y=2, 2 cos( pi y/2)= square root q, n an odd and m an even integer; n=1 and m=0 give the leading exponents. At q=3 x CH is relevant, but takes the special value x T +1=9/5. The crossover exponent at the percolation point in the bond-diluted random Ising model, determined by x CB at q=1, is equal to one. Along the Baxter line in the Ashkin-Teller, eight-vertex and ANNNI model x CH =x T /4+1/x T .

DOI
10.1088/0305-4470/17/5/015

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The critical dimensions x CH (q) of the chiral and x CB (q) of the cubic operator in the two-dimensional q-state Potts model satisfy the extended scaling relations x CH =(n 2 -y 2 )/(4x)+x and x CB =(m 2 -y 2 )/(4x)+x with x+y=2, 2 cos( pi y/2)= square root q, n an odd and m an even integer; n=1 and m=0 give the leading exponents. At q=3 x CH is relevant, but takes the special value x T +1=9/5. The crossover exponent at the percolation point in the bond-diluted random Ising model, determined by x CB at q=1, is equal to one. Along the Baxter line in the Ashkin-Teller, eight-vertex and ANNNI model x CH =x T /4+1/x T .
By using the replica trick and a duality transformation, the bond-diluted random Ising model is mapped onto a new Hamiltonian. This demonstrates the higher order critical nature of the percolation point and identifies the appropriate crossover scaling variables. Taking the n to 0 limit of the replica method near the percolation point is shown to be equivalent to the q to 1 limit of the Potts model. The critical line near p c is calculated, yielding for a square lattice exp(-2K c )=2ln2(p- 1 / 2 )+0(p- 1 / 2 ) 2 .
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.
We present results of a numerical simulation of the qqq-state random bond Potts model in two dimensions and for large qqq. In particular, care is taken to study the crossover from the pure model to the random model, as well as the crossover from the percolation to the random model. We show how to determine precisely the random fixed point and measure critical exponents at this point.
The relation between critical exponents and the amplitude of the correlation length divergence at the critical point of two-dimensional systems as a function of finite size is investigated and generalised in two ways. Correlations more general than those of order-order type are included. A form appropriate for anisotropic systems is proposed. The authors present (a) exact results for the Ising and Gaussian models and (b) numerical results for the symmetric eight-vertex (Baxter), continuous q-state Potts, and continuous N-component cubic models.
The Q-state Potts model can be extended to noninteger and even complex Q by expressing the partition function in the Fortuin-Kasteleyn (F-K) representation. In the F-K representation the partition function Z(Q,a) is a polynomial in Q and v=a-1 (a=e(betaJ)) and the coefficients of this polynomial, Phi(b,c), are the number of graphs on the lattice consisting of b bonds and c connected clusters. We introduce the random-cluster transfer matrix to compute Phi(b,c) exactly on finite square lattices with several types of boundary conditions. Given the F-K representation of the partition function we begin by studying the critical Potts model Z(CP)=Z(Q,a(c)(Q)), where a(c)(Q)=1+square root[Q]. We find a set of zeros in the complex w=square root[Q] plane that map to (or close to) the Beraha numbers for real positive Q. We also identify Q(c)(L), the value of Q for a lattice of width L above which the locus of zeros in the complex p=v/square root[Q] plane lies on the unit circle. By finite-size scaling we find that 1/Q(c)(L)-->0 as L-->infinity. We then study zeros of the antiferromagnetic (AF) Potts model in the complex Q plane and determine Q(c)(a), the largest value of Q for a fixed value of a below which there is AF order. We find excellent agreement with Baxter's conjecture Q(AF)(c)(a)=(1-a)(a+3). We also investigate the locus of zeros of the ferromagnetic Potts model in the complex Q plane and confirm that Q(FM)(c)(a)=(a-1)(2). We show that the edge singularity in the complex Q plane approaches Q(c) as Q(c)(L) approximately Q(c)+AL(-y(q)), and determine the scaling exponent y(q) for several values of Q. Finally, by finite-size scaling of the Fisher zeros near the antiferromagnetic critical point we determine the thermal exponent y(t) as a function of Q in the range 2</=Q</=3. Using data for lattices of size 3</=L</=8 we find that y(t) is a smooth function of Q and is well fitted by y(t)=(1+Au+Bu2)/(C+Du) where u=-(2/pi)cos(-1)(squareroot[Q]/2). For Q=3 we find y(t) approximately 0.6; however if we include lattices up to L=12 we find y(t) approximately 0.50(8) in rough agreement with a recent result of Ferreira and Sokal [J. Stat. Phys. 96, 461 (1999)].