The critical ground state of the antiferromagnetic triangular Ising model is studied with finite-size scaling, with the use of exact techniques and numerical transfer-matrix methods. Exact solution of the model in zero field yields a conformal anomaly equal to 1, and values of critical exponents. For nonzero fields, dominant eigenvalues of the transfer matrix are computed for systems with linear dimensions up to 27 with the use of an efficient coding of spin states as a system of nonintersecting strings. The results are in perfect agreement with the Gaussian-model--Coulomb-gas scenario proposed by Nienhuis et al. for this model. Quantitative agreement is found with the predictions for exponents of spin-wave and vortex operators. Furthermore, we locate the field-induced Kosterlitz-Thouless transition to a long-range-ordered state at a field
Hc=0.266\ifmmode\pm\else\textpm\fi{}0.010.
We study three-state Potts spins on a square lattice, in which all bonds are ferromagnetic along one of the lattice directions, and antiferromagnetic along the other. Numerical transfer-matrix methods are used on infinite strips of width L sites, 4 ⩽ L ⩽ 14. Based on the analysis of the ratio of scaled mass gaps (inverse correlation lengths) and scaled domain-wall free energies, we provide strong evidence that a critical (Kosterlitz–Thouless) phase is present, whose upper limit is, in our best estimate, T c = 0.29 ± 0.01. From an analysis of the (extremely anisotropic) nature of excitations below T c , we argue that the critical phase extends all the way down to T = 0. While domain walls parallel to the ferromagnetic direction are soft for the whole extent of the critical phase, those along the antiferromagnetic direction seem to undergo a softening transition at a finite temperature. Assuming a bulk correlation length varying, for T > T c , as ξ( T ) = a ξ exp[ b ξ ( T − T c ) −σ ], σ ≃ 1/2, we attempt finite-size scaling plots of our finite-width correlation lengths. Our best results are for T c = 0.50 ± 0.01. We propose a scenario in which such inconsistency is attributed to the extreme narrowness of the critical region.
An antiferromagnetic Ising system on a triangular lattice is studied by means of a transfer matrix and the phenomenological scaling method. System consists of a semi-infinite strip and helical boundary conditions are imposed at the end of each row. At external magnetic fields h =0 and 6 J , where J is a nearest neighbor coupling constant, a unite residual entropy is obtained at the zero temperature. With a small field, specific heat is found to have two peaks; a high-temperature peak corresponding to the enhanced short-range order, and a low-temperature peak to the appearance of a long-range order. A phase diagram in h - T space is obtained by means of the phenomenological scaling, and agrees fairly well with the one obtained by Monte Carlo simulation. The main deficiency is the finite transition temperature at h =0, which may be due to a crossover to the XY-like phase transition.
We investigate the finite-temperature corrections to scaling in the three-state square-lattice Potts antiferromagnet, close to the critical point at T=0. Numerical diagonalization of the transfer matrix on semi-infinite strips of width L sites, 4 < or = L < or = 14, yields finite-size estimates of the corresponding scaled gaps, which are extrapolated to L --> infinity. Owing to the characteristics of the quantities under study, we argue that the natural variable to consider is x identical with L e(-2 beta). For the extrapolated scaled gaps we show that square-root corrections, in the variable x, are present, and provide estimates for the numerical values of the amplitudes of the first- and second-order correction terms, for both the first and second scaled gaps. We also calculate the third scaled gap of the transfer matrix spectrum at T=0, and find an extrapolated value of the decay-of-correlations exponent, eta(3)=2.00(1). This is at odds with earlier predictions, to the effect that the third relevant operator in the problem would give eta(P(stagg))=3, corresponding to the staggered polarization.
Transfer-matrix methods are used, in conjunction with finite-size scaling and conformal invariance concepts, to generate an accurate phase diagram for a two-dimensional square-lattice Ising spin-1/2 magnet, with couplings which are positive along one coordinate axis and negative along the other, in a uniform external field. Our results indicate that the critical line starts horizontally at the zero-temperature end of the phase boundary, at variance with the re-entrant behavior predicted in some earlier studies. Estimates of the thermal scaling exponent are very close to the Ising value y(T)=1 along the critical line, except near T=0 , where strong crossover effects prevent a reliable analysis.
We consider the thermal phase transition from a paramagnetic to a stripe-antiferromagnetic phase in the frustrated two-dimensional square-lattice Ising model with competing interactions
J1<0 (nearest neighbor, ferromagnetic) and
J2>0 (second neighbor, antiferromagnetic). The striped phase breaks a
Z4 symmetry and is stabilized at low temperatures for
g=J2/∣J1∣>1/2. Despite the simplicity of the model, it has proved difficult to precisely determine the order and the universality class of the phase transitions. This was done convincingly only recently by Jin et al. [Phys. Rev. Lett. 108, 045702 (2012)]. Here, we further elucidate the nature of these transitions and their anomalies by employing a combination of cluster mean-field theory, Monte Carlo simulations, and transfer-matrix calculations. The
J1-
J2 model has a line of very weak first-order phase transitions in the whole region
1/2<g<g∗, where ${g}^{*}=0.67\ifmmode\pm\else\textpm\fi{}0.01$. Thereafter, the transitions from
g=g∗ to $g\ensuremath{\rightarrow}\ensuremath{\infty}$ are continuous and can be fully mapped, using universality arguments, to the critical line of the well-known Ashkin-Teller model from its four-state Potts point to the decoupled Ising limit. We also comment on the pseudo-first-order behavior at the Potts point and its neighborhood in the Ashkin-Teller model on finite lattices, which in turn leads to the appearance of similar effects in the vicinity of the multicritical point
g∗ in the
J1-
J2 model. The continuous transitions near
g∗ can therefore be mistaken for first-order transitions, and this realization was the key to understanding the paramagnetic-striped transition for the full range of
g>1/2. Most of our conclusions are based on Monte Carlo calculations, while the cluster mean-field and transfer-matrix results provide useful methodological benchmarks for weakly first-order behaviors and Ashkin-Teller criticality.