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Physical review. B, Condensed matter

Critical properties of two-dimensional models

James L. Black, Victor J. Emery

Published Jan 1, 1981• 236 citations
Publisher

Abstract

Critical properties of two-dimensional classical models and (1 + 1)-dimensional quantum models are obtained by solving a general dimerized spin chain. The staggered-electric exponent of the $F$ model is derived. The central role of umklapp scattering in controlling renormalization and determining Potts-model critical exponents is pointed out. The thermal exponent of the Potts model is evaluated, and the validity of den Nijs conjecture is established. Logarithmic singularities at the multicritical point are obtained.

DOI
10.1103/physrevb.23.429

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Critical properties of two-dimensional classical models and (1 + 1)-dimensional quantum models are obtained by solving a general dimerized spin chain. The staggered-electric exponent of the FFF model is derived. The central role of umklapp scattering in controlling renormalization and determining Potts-model critical exponents is pointed out. The thermal exponent of the Potts model is evaluated, and the validity of den Nijs conjecture is established. Logarithmic singularities at the multicritical point are obtained.
The two-dimensional four-state Potts model at finite temperature can be transformed, via the transfer matrix, into a one-dimensional quantum mechanical model at zero temperature. The duality invariant renormalisation group introduced by Fernandez-Pacheco (1979) is then employed to study the ground-state critical properties of this model. The fixed point is located at exactly the self-dual critical point K*=1. The thermal exponent is calculated to be y T =1.3219. It is in excellent agreement with the recent series value of Ditzian and Kadanoff (1979) (y T =1.33). Although it is not inconsistent with den Nijs's (1979) conjectured exact value of 3 / 2 , the difference is nevertheless substantial.
We use the density matrix renormalization group (DMRG) to perform a detailed study of the critical properties of the two-dimensional QQQ state Potts model, including the magnetization and energy-density profiles, bulk and surface critical exponents, and the Casimir amplitudes. We apply symmetry breaking boundary conditions to a $L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ strip and diagonalize the corresponding transfer matrix for a series of moderately large systems (L< 64)(L<~64)(L< 64) by the DMRG method. The numerically very accurate finite lattice results are then extrapolated by efficient sequence extrapolation techniques. The density profiles and the Casimir amplitudes at the critical point are found to follow precisely the conformal predictions for Q=2Q=2Q=2 and 333. Similarly, the bulk and surface critical exponents of the models are in very good agreement with the conformal and exact values: their accuracy has reached or even exceeded the accuracy of other available numerical methods. For the Q=4Q=4Q=4 model both the profiles and the critical exponents show strong logarithmic corrections, which are also studied.
A theory of the scaling behavior near the qc=4{q}_{c}=4qc​=4 state multicritical point of the two-dimensional Potts lattice gas model is developed. Proceeding from the assumption that a dilution field becomes marginal at the multicritical point while the thermal and ordering fields are relevant, a set of differential renormalization group (RG) equations for these fields are constructed. Keeping terms through second order we find that these equations are characterized by five universal parameters which we evaluate using exact as well as conjectured results. Based upon these RG equations, we investigate the physical properties of the two-dimensional Potts lattice gas for qqq near qc{q}_{c}qc​. For the pure Potts model with q=qcq={q}_{c}q=qc​ we find logarithmic temperature corrections to the specific heat and the spontaneous magnetization. At Tc{T}_{c}Tc​ we find ln(r)\mathrm{ln}(r)ln(r) corrections to the power law behavior of the spin-spin correlation function. For the dilute Potts model with q=qcq={q}_{c}q=qc​ we find that the latent heat, the discontinuity in the magnetization, and the discontinuity in the coexisting densities vanish with an essential singularity as TTT approaches the multicritical point from the first-order side. Results for q>qcq>{q}_{c}q>qc​ and q<qcq<{q}_{c}q<qc​ are also given.
A method is presented by which critical and multicritical points of spin-1 (three-state) vertex models and classical O(n) models on two-dimensional lattices are determined. It is a straightforward generalization of the ideas that earlier led to the determination of critical points and critical exponents of a honeycomb O(n) model. On the square lattice the methods leads to tricritical as well as critical loci. For n=2 a larger critical manifold is found than for other values of n. At the critical and multicritical points thus produced the models turn out to be soluble. The method is applicable to O(n) models and spin-1 vertex models on any planar lattice.
The thermal critical exponent of the four-state Potts model in two dimensions is evaluated as ${\ensuremath{\nu}}^{\ensuremath{-}1}=1.49\ifmmode\pm\else\textpm\fi{}0.01$, using the Monte Carlo renormalization-group method. The presence of a marginal scaling direction is also indicated. These results confirm previous conjectures, universality, and logarithmic corrections. A purely Potts Hamiltonian is considered. A four-point interaction is used to control the chemical potential of effective vacancies, without the introduction of explicit vacancy states.