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Physical Review Letters

Finite-size-scaling amplitudes of the incommensurate phase

Hyunggyu Park, Mike Widom

Published Mar 5, 1990• 14 citations
Publisher

Abstract

We examine the finite-size-scaling amplitudes of the free energy in incommensurate phases on a torus with periodic and twisted boundary conditions. We show that these amplitudes are equivalent to those of the six-vertex model with electric and magnetic defect lines. The twist angle generates magnetic defect lines, while the electric defect lines are generated by competition between the domain-wall separation and the finite system size. We calculate the amplitudes exactly for the free-fermion model and the spin-(1/2 XXZ chain, and conjectire the form of these amplitudes for a more general model. Numerical calculations employing the Bethe ansatz confirm our conjecture.

DOI
10.1103/physrevlett.64.1076

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We examine the finite-size-scaling amplitudes of the free energy in incommensurate phases on a torus with periodic and twisted boundary conditions. We show that these amplitudes are equivalent to those of the six-vertex model with electric and magnetic defect lines. The twist angle generates magnetic defect lines, while the electric defect lines are generated by competition between the domain-wall separation and the finite system size. We calculate the amplitudes exactly for the free-fermion model and the spin-(1/2 XXZ chain, and conjectire the form of these amplitudes for a more general model. Numerical calculations employing the Bethe ansatz confirm our conjecture.
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The critical Ising model in two dimensions with a specific defect line is analyzed to deliver the first exact solution with twisted boundary conditions. We derive exact expressions for the eigenvalues of the transfer matrix and obtain analytically the partition function and the asymptotic expansions of the free energy and inverse correlation lengths for an infinitely long cylinder of circumference L x . We find that finite-size corrections to scaling are of the form a k / L x 2 k − 1 for the free energy f and b k ( p ) / L x 2 k − 1 and c k ( p ) / L x 2 k − 1 for inverse correlation lengths ξ p − 1 and ξ L − p − 1 , respectively, with integer values of k . By exact evaluation we find that the amplitude ratios b k ( p ) / a k and c k ( p ) / a k are universal and verify this universal behavior using a perturbative conformal approach.
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A new and general analytic method for calculating finite-size corrections and central charges is applied to the 6- and 19-vertex models and their related spin-1/2 and spin-1 XXZ chains with twisted boundary conditions. Nonlinear integral equations are derived from which the central charge c can be extracted in terms of Rogers dilogarithms. For twist angle phi , the central charge is c=3S/S+1 (1-4(S-1) phi 2 / pi ( pi -2S gamma )) where gamma is the crossing parameter or chain anisotropy and spin S=1/2 or 1. For periodic boundary conditions ( phi =0) this reduces to the known results c=1 and c=3/2, respectively.