Transfer matrix formalism has been used to study the phase transition in a two-dimensional isotropic planar model where one dimension is taken to be spatial and the second dimension is temporal. Character expansion has been used to calculate the eigenvalues of the transfer matrix operator. This has ensured very rapid convergence around the critical point. Fluxes have been generated at each lattice site of the spatial dimension by Monte Carlo simulation. Mass gap and free energy have been found in both theoretical calculation and computer simulation separately for different values of temperature. From the results I infer an algebraic divergence of correlation length rather than a Kosterlitz-Thouless type. The value of critical temperature is found to be ${\mathrm{k}}_{\mathrm{B}}$${\mathrm{T}}_{\mathrm{c}}$/J=0.899.
Transfer matrix formalism has been used to study the phase transition in a two-dimensional isotropic planar model where one dimension is taken to be spatial and the second dimension is temporal. Character expansion has been used to calculate the eigenvalues of the transfer matrix operator. This has ensured very rapid convergence around the critical point. Fluxes have been generated at each lattice site of the spatial dimension by Monte Carlo simulation. Mass gap and free energy have been found in both theoretical calculation and computer simulation separately for different values of temperature. From the results I infer an algebraic divergence of correlation length rather than a Kosterlitz-Thouless type. The value of critical temperature is found to be {\mathrm{k}}_{\mathrm{B}}$${\mathrm{T}}_{\mathrm{c}}/J=0.899.
Based on the short-time dynamic scaling form, a dynamic approach is proposed to numerically tackle the Kosterlitz-Thouless phase transition. Taking the two-dimensional XY model as an example, the exponential divergence of the spatial correlation length, the transition temperature TKT, and all critical exponents are computed. Compared with Monte Carlo simulations in equilibrium, we obtain data at temperatures closer to TKT.
We study phase transition properties of the two-dimensional q-state clock model by an extensive Monte Carlo simulation. By analyzing the Binder ratio and its temperature derivative, we confirm that the two-dimensional q-state clock model exhibits two distinct Kosterlitz-Thouless phase transitions for q=5,6 but it has one second-order phase transition for q=4. The critical temperatures are estimated quite accurately from the crossing behavior of the Binder ratio (for q<5) and from negative divergent dips of the derivative of the Binder ratio (for q≥5) around these critical points. We also calculate the correlation length, the helicity modulus, and the derivative of the helicity modulus, and analyze their behaviors in different phases in detail.
Recently, critical exponents in the continuum Baxter model have been calculated using an equivalence of this model to the Luttinger model of a one-dimensional spinless fermion gas. We discuss here the extension of this method to the two-dimensional planar model, using the Stoeckly-Scalapino representation of the transfer matrix. This representation is shown to be equivalent to the spin-1/2 fermion gas in one dimension, which further separates into a Luttinger model and the quantum-mechanical sine-Gordon equation. We emphasize the role that soliton bound states play in determining the critical properties. We find that a critical temperature Tc exists below which the susceptibility is infinite but without long-range order. At Tc, the correlation-function exponent $\ensuremath{\eta}$ takes the values 18, and decreases as the temperature is lowered, consistent with low-temperature results. As in the Baxter-model calculations, we argue that these results are exact for the continuum limit of the transfer matrix, and therefore provide a solution for the asymptotic properties near Tc. We discuss the correlation length and susceptibility, and sugget that nonuniversality is responsible for the disagreement between different numerical calculations.
The d-dimensional Ising transfer matrix is expressed exactly in terms of Fermi-type operators. In two dimensions this leads to the standard solution of the model. In three or more dimensions the transfer matrix is no longer a quadratic form in the Fermi variables, but rather contains quartic and higher-order terms. However, it can be expressed as a well-defined expansion about a quadratic form in Fermi operators. The zeroth-order transfer-matrix-called herein the "fermion model"---can be exactly diagonalized, and is well approximated in the critical region by ${V}^{(0)}\ensuremath{\approx}{[2sinh(2K)]}^{\frac{N}{2}}\mathrm{exp}\left[\ensuremath{-}N\ensuremath{\int}d\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}{\left({\ensuremath{\kappa}}^{2}+\ensuremath{\Sigma}{n=1}^{d\ensuremath{-}1}\ensuremath{\Sigma}{m=1}^{d\ensuremath{-}1}{q}_{n}{q}_{m}\right)}^{\frac{1}{2}}t(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})\right]$where N is the number of sites per layer, $\ensuremath{\mu}$ is the inverse correlation length, $t(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$ takes values \ifmmode\pm\else\textpm\fi{}1/2, K is the reduced inverse temperature, and the integral extends over a Brillouin zone of the ($d\ensuremath{-}1$)-dimensional layer. The critical point of the model is located by the equation ${x}_{c}^{d}+{x}_{c}^{d\ensuremath{-}1}+{x}_{c}=1$, where ${x}_{c}={e}^{\ensuremath{-}2{K}_{c}}$. The critical exponent $\ensuremath{\nu}$ describing the divergence of the correlation length at Kc is found to take on its two-dimensional value, $\ensuremath{\nu}=1$, in all dimensions d. The possibility of using the fermion model as the basis for an expansion of the Ising critical exponents as a function of $\ensuremath{\delta}(=d\ensuremath{-}2)$ is pointed out.
The principle and the efficiency of the Monte Carlo transfer-matrix algorithm are discussed. Enhancements of this algorithm are illustrated by applications to several phase transitions in lattice spin models. We demonstrate how the statistical noise can be reduced considerably by a similarity transformation of the transfer matrix using a variational estimate of its leading eigenvector, in analogy with a common practice in various quantum Monte Carlo techniques. Here we take the two-dimensional coupled XY-Ising model as an example. Furthermore, we calculate interface free energies of finite three-dimensional O(n) models, for the three cases n=1, 2, and 3. Application of finite-size scaling to the numerical results yields estimates of the critical points of these three models. The statistical precision of the estimates is satisfactory for the modest amount of computer time spent. \textcopyright{} 1996 The American Physical Society.
The two-dimensional planar spin model in the presence of p-fold random anisotropy has been investigated using Monte Carlo simulation and finite size scaling. The system has been examined for p=2, 3, 4, 5, 6 and lattice sizes N=4 2 , 8 2 , 16 2 . The critical behaviour is found to depend on the value of p and is of two types. For p=3, 5, 6 the results indicate a transition to a quasiferromagnetic state at the same temperature as the pure system. At lower temperatures there is evidence for a second transition; however, the system remains quasiferromagnetic. For p=2, 4 the behaviour is different and results suggest a second-order transition to a spin glass phase and no quasiferromagnetism.