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Journal of Physics A Mathematical and General

Transfer matrix study of the chiral clock model in the Hamiltonian limit

H. U. Everts, Heinrich Röder

Published Jul 7, 1989• 12 citations

Abstract

The logarithm of the transfer matrix of the two-dimensional, three state chiral clock model is shown to be equivalent to a one-dimensional non-Hermitian quantum Hamiltonian. In the region of small chirality the critical behaviour of the quantum model is investigated by a series expansion and by finite-size scaling. A single phase boundary between a modulated paramagnetic and a ferromagnetic phase is located with good accuracy. Both methods yield comparable results for the critical exponent beta of the wavevector of the modulated high temperature phase. The critical exponents nu x , nu tau for the correlation lengths obtained from finite-size scaling point towards the existence of a Lifshitz point at finite chirality. The results of the series expansion for the exponent nu x , however, are inconsistent with finite-size scaling. For larger chirality the authors' finite-size results clearly reveal the occurrence of an incommensurate phase with algebraically decaying correlations between the ferromagnetic and the modulated paramagnetic phase. They analyse the critical behaviour of this incommensurate phase at its melting line and determine the exponent eta that controls the algebraic decay of correlations.

DOI: 10.1088/0305-4470/22/13/040

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The logarithm of the transfer matrix of the two-dimensional, three state chiral clock model is shown to be equivalent to a one-dimensional non-Hermitian quantum Hamiltonian. In the region of small chirality the critical behaviour of the quantum model is investigated by a series expansion and by finite-size scaling. A single phase boundary between a modulated paramagnetic and a ferromagnetic phase is located with good accuracy. Both methods yield comparable results for the critical exponent beta of the wavevector of the modulated high temperature phase. The critical exponents nu x , nu tau for the correlation lengths obtained from finite-size scaling point towards the existence of a Lifshitz point at finite chirality. The results of the series expansion for the exponent nu x , however, are inconsistent with finite-size scaling. For larger chirality the authors' finite-size results clearly reveal the occurrence of an incommensurate phase with algebraically decaying correlations between the ferromagnetic and the modulated paramagnetic phase. They analyse the critical behaviour of this incommensurate phase at its melting line and determine the exponent eta that controls the algebraic decay of correlations.

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.

The magnetic critical exponent of the two-dimensional axial next-nearest-neighbor Ising model is estimated by means of the finite-size scaling of the order parameter. For a finite system without ordering field, this magnitude is computed from the two leading eigenvectors of the transfer matrix. The ferromagnetic-paramagnetic transition temperatures estimated with this method exhibit a small lattice-size dependence. Moreover, they are in good agreement with those obtained using phenomenological renormalization, and also with analytical approximations. The correlation-length critical exponent is calculated by means of phenomenological renormalization. The results found with these numerical techniques indicate that the ferromagnetic-paramagnetic transition of the model seems to be continuous, in contradiction with a previous result obtained using the persistence-length criterion, which indicated that the transition is always first order. We found that the phase transition is first order only when the ratio of the next-nearest-neighbor and nearest-neighbor interactions equals one-half.

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).

The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems of square geometry with periodic boundaries oriented either along <10> or along <11> direction have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction proportional to 1/L^0.25 of a very small magnitude also has been detected in agreement with predictions of our recently developed GFD (grouping of Feynman diagrams) theory. A refined analysis of the recent MC data for 3D Ising, phi^4, and XY lattice models has been performed. It includes an analysis of the partition function zeros of 3D Ising model, an estimation of the correction-to-scaling exponent omega from the Binder cumulant data near criticality, as well as a study of the effective critical exponent eta and the effective amplitudes in the asymptotic expansion of susceptibility at the critical point. In all cases a refined analysis is consistent with our (GFD) asymptotic values of the critical exponents (nu=2/3, omega=1/2, eta=1/8 for 3D Ising model and omega=5/9 for 3D XY model), while the actually accepted "conventional" exponents are, in fact, effective exponents which are valid for approximation of the finite--size scaling behavior of not too large systems.

Based on Novotny's transfer-matrix method, we simulated the (stacked) triangular Ising anti-ferromagnet embedded in the space with the dimensions variable in the range 2 < or = d < or = 3. Our aim is to investigate the criticality of the XY universality class for 2 < or = d < or = 3. For that purpose, we employed an extended version of the finite-size-scaling analysis developed by Novotny, who utilized this scheme to survey the Ising criticality (ferromagnet) for 1 < or = d < or = 3. Diagonalizing the transfer matrix for the system sizes N up to N = 17 , we calculated the d -dependent correlation-length critical exponent nu(d). Our simulation result nu(d) appears to interpolate smoothly the known two limiting cases, namely, the Kosterlitz-Thouless (KT) and d = 3 XY universality classes, and the intermediate behavior bears close resemblance to that of the analytical formula via the 1/N-expansion technique. Methodological details including the modifications specific to the present model are reported.