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Physical review. B, Condensed matter

Phase diagram of the three-state chiral clock model studied by Monte Carlo renormalization-group calculations

J. M. Houlrik, Svend J. Knak Jensen

Published Jul 1, 1986• 18 citations
Publisher

Abstract

A Monte Carlo renormalization-group scheme based on a ``floating sublattice'' blocking rule is constructed for the two-dimensional three-state chiral clock model. The presence of a critical phase is established by matching certain wave-vector-dependent correlation functions derived from lattices differing in linear size by a factor of 2. The wave vector, which in a function of temperature and the asymmetry parameter \ensuremath{\Delta}, is determined from the simulations. For \ensuremath{\Delta}=0.45, a critical phase is established in a temperature range. For \ensuremath{\Delta}=0.35, no critical phase is observed, suggesting that the phase diagram has a Lifschitz point in the interval 0.35<\ensuremath{\Delta}<0.45. Alternatively, the wave vector and the width of the critical phase at \ensuremath{\Delta}=0.35 are much smaller than for \ensuremath{\Delta}=0.45..AE

DOI
10.1103/physrevb.34.325

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A Monte Carlo renormalization-group scheme based on a ``floating sublattice'' blocking rule is constructed for the two-dimensional three-state chiral clock model. The presence of a critical phase is established by matching certain wave-vector-dependent correlation functions derived from lattices differing in linear size by a factor of 2. The wave vector, which in a function of temperature and the asymmetry parameter \ensuremath{\Delta}, is determined from the simulations. For \ensuremath{\Delta}=0.45, a critical phase is established in a temperature range. For \ensuremath{\Delta}=0.35, no critical phase is observed, suggesting that the phase diagram has a Lifschitz point in the interval 0.35<\ensuremath{\Delta}<0.45. Alternatively, the wave vector and the width of the critical phase at \ensuremath{\Delta}=0.35 are much smaller than for \ensuremath{\Delta}=0.45..AE
The phase transition of the two-dimensional, three-state chiral clock model is studied by using the density matrix renormalization group method and the finite-size renormalization group technique. The transition temperature and the critical exponent ν of the correlation length are calculated for the transition from the commensurate to paramagnetic phases. It is found that ν is anisotropic and that ν for the correlation along the chiral axis decreases with increasing the chiral field Δ while the one perpendicular to the axis increases with Δ.
Within a real space renormalisation group approach, the authors study the phase diagram and the universality classes of the three-state chiral clock model in a self-dual planar hierarchical lattice. They find that the chiral field Delta is relevant at the pure Potts critical point with a crossover exponent phi approximately=0.32. The critical line which separates the ferromagnetic phase from the non-ferromagnetic ones is characterised by a multicritical (Lifshitz) point located at (K, Delta ) approximately= (1.452, 0.382).
Using the tensor renormalization group method based on the higher-order singular value decomposition, we have studied the phase transitions of the five-state clock model on the square lattice. The temperature dependence of the specific heat indicates the system has two phase transitions, as verified clearly by the correlation function at three representative temperatures. By calculating the magnetic susceptibility, we obtained only the upper critical temperature as T c2 = 0.9565(7). Investigating the fixed-point tensor, we precisely locate the transition temperatures at T c1 = 0.9029(1) and T c2 = 0.9520(1), consistent well with the Monte Carlo and the density matrix renormalization group results.
Monte Carlo renormalization-group methods were applied to the classical three-component Heisenberg model on a two-dimensional lattice. Expectation values of local correlations of spins and various sized block spins were computed using traditional Monte Carlo methods. By matching quantities at different length scales generated by different Monte Carlo Hamiltonians we directly determined the renormalization of the nearest-neighbor coupling. Using these data and the results of high-temperature expansions and low-temperature renormalization-group calculations we have determined that this model does not have a phase transition. We have also obtained the amplitude for the low-temperature divergence of the susceptibility and the correlation length.