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Journal of Physics A Mathematical and General

Critical behaviour and logarithmic corrections of a quantum model with three-spin interaction

Carlo Vanderzande, Ferenc Iglói

Published Sep 11, 1987• 16 citations
Publisher

Abstract

Extended finite-size scaling and series expansion methods are used to determine the critical properties of the one-dimensional quantum Ising model with three-spin interaction. By supposing the presence of logarithmic corrections the universality class of the model is investigated. The results obtained by different methods support the conjecture that the model belongs to the same universality class as the four-state Potts model.

DOI
10.1088/0305-4470/20/13/052

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Extended finite-size scaling and series expansion methods are used to determine the critical properties of the one-dimensional quantum Ising model with three-spin interaction. By supposing the presence of logarithmic corrections the universality class of the model is investigated. The results obtained by different methods support the conjecture that the model belongs to the same universality class as the four-state Potts model.
A quantum spin model with three-spin coupling is studied by finite-lattice methods. The critical indices nu and alpha are obtained by standard finite-size scaling analysis. Conformal invariance is used to estimate the exponents nu and eta , the central charge or conformal anomaly c and the anomalous dimension of a spin 1/4 parafermion. The authors conclude that the model belongs to the same universality class as the four-state Potts and Baxter-Wu models, although the convergence of the various finite lattice estimators is extremely slow.
We investigate three Ising models on the simple cubic lattice by means of Monte Carlo methods and finite-size scaling. These models are the spin-1/2 Ising model with nearest-neighbour interactions, a spin-1/2 model with nearest-neighbour and third-neighbour interactions, and a spin-1 model with nearest-neighbour interactions. The results are in accurate agreement with the hypothesis of universality. Analysis of the finite-size scaling behaviour reveals corrections beyond those caused by the leading irrelevant scaling field. We find that the correction-to-scaling amplitudes are strongly dependent on the introduction of further-neighbour interactions or a third spin state. In a spin-1 Ising model, these corrections appear to be very small. This is very helpful for the determination of the universal constants of the Ising model. The renormalization exponents of the Ising model are determined as y t =1.587 (2), y h =2.4815 (15) and y i =-0.82 (6). The universal ratio Q=(m 2 ) 2 /(m 4 ) is equal to 0.6233 (4) for periodic systems with cubic symmetry. The critical point of the nearest-neighbour spin-1/2 model is K c =0.2216546 (10).
Strong and weak coupling series expansions are used to investigate the critical properties of the S=1 multispin coupling Ising chain in the presence of a transverse field. The order of the transition goes from second order to first order when more than three spins are coupled. According to the numerical results the two-spin and three-spin coupling models belong to the Ising and the four-state Potts universality classes, respectively. Thus the critical properties of the model do not depend on the value of the spin, even when three spins are coupled.
We perform high-statistics Monte Carlo simulations of three three-dimensional Ising spin glass models: the ± J Ising model for two values of the disorder parameter p , p = 1/2 and 0.7, and the bond-diluted ± J model for bond-occupation probability p b = 0.45. A finite-size scaling analysis of the quartic cumulants at the critical point shows conclusively that these models belong to the same universality class and allows us to estimate the scaling-correction exponent ω related to the leading irrelevant operator, ω = 1.0(1). We also determine the critical exponents ν and η. Taking into account the scaling corrections, we obtain ν = 2.53(8) and η = −0.384(9).
We compute an improved action for the Ising universality class in three dimensions that has suppressed leading corrections to scaling. It is obtained by tuning models with two coupling constants. We studied three different models: the $\ifmmode\pm\else\textpm\fi{}1$ Ising model with nearest-neighbor and body diagonal interaction, the spin-1 model with states $0,\ifmmode\pm\else\textpm\fi{}1$, and nearest-neighbor interaction, and ${\ensuremath{\varphi}}^{4}$ theory on the lattice (Landau-Ginzburg model). The remarkable finite-size scaling properties of the suitably tuned spin-1 model are compared in detail with those of the standard Ising model. Great care is taken to estimate the systematic errors from residual corrections to scaling. Our best estimates for the critical exponents are $\ensuremath{\nu}=0.6298(5)$ and $\ensuremath{\eta}=0.0366(8)$, where the given error estimates take into account the statistical and systematic uncertainties.