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Physical review. B, Condensed matter

Critical properties of the Z(p)-symmetric spin models from an exact finite-lattice approach

B. Bonnier, M. Hontebeyrie, C. Meyers

Published Mar 1, 1989• 16 citations
Publisher

Abstract

The Hamiltonian formulation of the two-dimensional spin systems with Z(p) symmetry, ranging from the Ising to the XY-plane rotor model, is used together with the Lanczos scheme to compute exactly the mass gap of finite systems. Critical properties are then deduced from a Callan-Symanzik \ensuremath{\beta} function. We find in particular a Kosterlitz-Thouless transition for p\ensuremath{\ge}5, with a p-dependent exponent which coincides with its renormalization-group estimate only in the p=\ensuremath{\infty} limit.

DOI
10.1103/physrevb.39.4079

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Physical review. B, Condensed matter•Feb 1, 1981•45 citations
The Hamiltonian formulation of the two-dimensional spin systems with Z(p) symmetry, ranging from the Ising to the XY-plane rotor model, is used together with the Lanczos scheme to compute exactly the mass gap of finite systems. Critical properties are then deduced from a Callan-Symanzik \ensuremath{\beta} function. We find in particular a Kosterlitz-Thouless transition for p\ensuremath{\ge}5, with a p-dependent exponent which coincides with its renormalization-group estimate only in the p=\ensuremath{\infty} limit.
We study the isotropic, ferromagnetic two-dimensional spin systems with Z(5) symmetry, which, in the general case, depend upon two couplings, a temperature and a parameter \ensuremath{\Omega}, used to interpolate continuously between the clock (\ensuremath{\Omega}=0) and the Potts (\ensuremath{\Omega}=1) models, passing through the integrable Fateev-Zamolodchikov case (\ensuremath{\Omega}=${\mathrm{\ensuremath{\Omega}}}_{\mathit{c}}$\ensuremath{\approxeq}0.618). Two methods, the high-temperature expansion and the exact \ensuremath{\beta} function on finite lattices, are applied within the Hamiltonian formulation to study critical properties. The transitions found are first order (${\mathrm{\ensuremath{\Omega}}}_{\mathit{c}}$\ensuremath{\le}1), second order of the Ising type (\ensuremath{\Omega}\ensuremath{\simeq}${\mathrm{\ensuremath{\Omega}}}_{\mathit{c}}$), or of the Kosterlitz-Thouless type (0\ensuremath{\le}\ensuremath{\Omega}${\mathrm{\ensuremath{\Omega}}}_{\mathit{c}}$) with a varying exponent.
The thermodynamics of the two-dimensional (2D) XY model is formulated by a transfer-matrix method and analyzed by a density-matrix renormalization group. The finite-size scaling and the beta function of the model are studied by the Roomany-Wyld renormalization-group theory. It is found that the 2D XY model has an essential finite-size effect and the Berezinskii-Kosterlitz-Thouless transition with the critical temperature TBKT=0.892{T}_{\mathrm{BKT}}=0.892TBKT​=0.892 appears in a finite system of 2000--3000 spins as a massless to massive transition with the effective critical temperature ${T}_{c}=1.07\ifmmode\pm\else\textpm\fi{}0.01.$
The critical behavior of the general isotropic, ferromagnetic two-dimensional spin system with openZ(5) symmetry is studied with use of high-temperature expansions of its mass gap. On the basis of these expansions we propose a simple analytic representation of the mass gap which naturally reproduces all the different phase transitions exhibited by this model (first order and second order of the Ising and of the Kosterlitz-Thouless types). In addition, the bifurcation point where the soft phases originate is clearly identified with the Fateev-Zamolodchikov value.
The critical behavior of 2D magnetic easy-plane systems has mainly been studied by the classical XY model. However, the z components of the spins have to be considered in order to describe real systems, and their fluctuations cannot positively be neglected when quantum effects are to be included, quantum spins being intrinsically three-component objects. Therefore, Monte Carlo simulations are performed for the Heisenberg ferromagnet with easy-plane anisotropy (XXZ model) on a two-dimensional square lattice with a twofold aim: first, to obtain accurate quantitative results about the critical behavior of the classical model, showing the relevant role played by the out-of-plane fluctuations; second, to open the way for approaching the quantum thermodynamics by means of the effective Hamiltonian method that reduces the quantum thermodynamics of the XXZ ferromagnet to the investigation of an effective classical model with temperature-dependent renormalized interaction parameters. Specific heat, magnetic susceptibility, and correlation length are calculated in the critical region for lattice sizes up to 128×128. These quantities preserve the Kosterlitz–Thouless behavior of the XY model. For the transition temperature of the classical XX0 model we obtain the estimate kBTc/(JS̃2)=0.69±0.01.
We propose an implementation of the nonperturbative renormalization group (NPRG) which applies to lattice models. Contrary to the usual NPRG approach where the initial condition of the RG flow is the mean-field solution, the lattice NPRG uses the (local) limit of decoupled sites as the (initial) reference system. In the long-distance limit, it is equivalent to the usual NPRG formulation and therefore yields identical results for the critical properties. We discuss both a lattice field theory defined on a d-dimensional hypercubic lattice and classical spin models. The simplest approximation, the local potential approximation, is sufficient to obtain the critical temperature and the magnetization of the three-dimensional Ising, XY, and Heisenberg models to an accuracy on the order of 1%. We show how the local potential approximation can be improved to include a nonzero anomalous dimension η and discuss the Berezinskii-Kosterlitz-Thouless transition of the two-dimensional XY model on a square lattice.
The phase diagrams of general qqq-state planar spin models on a square lattice are studied for $q=4\ensuremath{-}7$. The phase diagrams are calculated by a variational method and the infinitesimal Migdal renormalization-group (RG) transformations. Since the Migdal RG commutes with the self-dual transformations of Z(q)Z(q)Z(q) models, one expects reliable phase diagrams (but not accurate critical exponents). In fact all exactly known critical points are reproduced by our calculations together with a fair numerical simulation of lines of critical points. Our results clearly indicate the existence of massless phases for q=6and7q=6 \mathrm{and} 7q=6and7 but are inconclusive for q=5q=5q=5. More accurate methods are needed to determine the phase structure of the Z(5)Z(5)Z(5) model; see, for example, the finite-lattice approximation of Roomany and Wyld (preceding paper).