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Journal of the Korean Physical Society

Study of the antiferromagnetic Blume-Capel model by using the partition function zeros in the complex temperature plane

Seung‐Yeon Kim, Wooseop Kwak

Published Aug 1, 2014• 11 citations
Publisher

Abstract

The critical temperature and the thermal scaling exponent of the square-lattice Blume-Capel model are obtained as functions of fugacity by calculating the partition function zeros in the complex temperature plane of the square-lattice Blume-Capel antiferromagnet for the first time. The thermal scaling exponent y t determines the specific-heat critical exponent and the correlation-length critical exponent. The value of y t = 2 indicates a strong first-order phase transition in two dimensions whereas y t = 1 corresponds to an Ising-class second-order phase transition. The full spectrum (from y t = 2 to y t = 1) of the thermal scaling exponent for the square-lattice Blume-Capel model is evaluated for the first time. The phase diagram of the square-lattice Blume-Capel model, including the tricritical point, is also obtained, and it is compared with other results in the literature.

DOI
10.3938/jkps.65.436

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The critical temperature and the thermal scaling exponent of the square-lattice Blume-Capel model are obtained as functions of fugacity by calculating the partition function zeros in the complex temperature plane of the square-lattice Blume-Capel antiferromagnet for the first time. The thermal scaling exponent y t determines the specific-heat critical exponent and the correlation-length critical exponent. The value of y t = 2 indicates a strong first-order phase transition in two dimensions whereas y t = 1 corresponds to an Ising-class second-order phase transition. The full spectrum (from y t = 2 to y t = 1) of the thermal scaling exponent for the square-lattice Blume-Capel model is evaluated for the first time. The phase diagram of the square-lattice Blume-Capel model, including the tricritical point, is also obtained, and it is compared with other results in the literature.
Abstract Since the landmark work of Lee and Yang, locating the zeros of the partition function in the complex magnetic-field plane has become a powerful method for studying phase transitions. Fisher later extended this approach to complex temperatures, and subsequent generalizations introduced other control parameters, such as the crystal field. In previous works (Moueddene et al 2024 J. Stat. Mech. 023206; 2024 Phys. Rev. E 110 064144) we applied this framework to the two- and three-dimensional Blume–Capel model–a system with a rich phase structure where a second-order critical line meets a first-order line at a tricritical point. We showed that the scaling of Lee-Yang, Fisher, and crystal-field zeros yields accurate critical exponents even for modest lattice sizes. In the present study, we extend this analysis and demonstrate that simulations need not be performed exactly at the nominal transition point to obtain reliable exponent estimates. Strikingly, small system sizes are sufficient, which not only improves methodological efficiency but also advances the broader goal of reducing the carbon footprint of large-scale computational studies.
We study the square-lattice Ising model with nearest-neighbor ( J 1 ) and next nearest-neighbor ( J 2 ) interactions, using the Wang–Landau Monte Carlo algorithm and the partition function zeros in the complex temperature plane (Fisher zeros). Here, the competition between J 1 and J 2 interactions is defined by the coupling ratio R = J 2 / J 1 . For J 2 > 0, the critical behavior is trivial and is the same as the square-lattice Ising model with only J 1 interaction. On the other hand, the new phase of superantiferromagnet appears for J 2 < 0. However, the nature of the transition into the superantiferromagnetic phase is still controversial in the literature. To resolve the controversy, we estimate the critical temperature T c ( R ) and thermal scaling exponent y t ( R ) from the Fisher zeros, as a function of R . The estimated values of T c ( R ) agree well with the results in the literature. We obtain the simple critical behavior with a fixed value of the thermal scaling exponent, y t ( R ) = 1, for and J 2 < 0 (Ising antiferromagnet). For and J 2 < 0 (Ising superantiferromagnet), however, it is shown that y t ( R ) continuously varies depending on R values, and the estimated values of y t ( R ) are clearly less than that of the first-order phase transition, y t = 2.
We report on numerical simulations of the two-dimensional spin-1 Blume-Capel ferromagnet embedded in a triangular lattice. Utilizing a range of Monte Carlo and finite-size scaling techniques, we explore several critical aspects along the crystal field–temperature ( Δ , T ) transition line. Wang-Landau simulations measuring the joint density of states in combination with the method of field mixing allow us to probe the phase coexistence curve in high resolution, determining the tricritical point ( Δ t , T t ) with improved accuracy and verifying the tricritical exponents. Extensive multicanonical simulations identifying transition points across the phase diagram characterize the Ising universality class for Δ < Δ t with precise determination of thermal and magnetic critical exponents expected in the second-order regime. On the other hand, for Δ > Δ t , a finite-size scaling analysis is dedicated to revealing the first-order signature in the surface tension that linearly increases upon lowering the temperature deeper into the first-order transition regime. Finally, a comprehensive picture of the phase diagram for the model is presented, collecting transition points obtained from the combined numerical approach in this study and previous estimates in the literature.
The spin-5/2 Blume–Capel model was studied using the mean-field approximation and the Migdal–Kadanoff renormalization group method for two- and three-dimensional systems. We determined the phase diagrams in the (crystal field, temperature) plane where the system exhibited first- and second-order phase transitions as well as isolated critical, bicritical and triple points. In order to show first-order transitions at low temperature, we presented the total magnetization per site and the derivative of the free energy as a function of the crystal field. Moreover, the critical exponents of the system were calculated by linearizing the renormalization transformation at the vicinity of the second-order fixed points.
Abstract We show that accurate insights into the critical properties of the Blume–Capel model at two dimensions can be deduced from Monte Carlo simulations, even for small system sizes, when one analyses the behaviour of the zeros of the partition function. The phase diagram of the model displays a line of second-order phase transitions ending at a tricritical point, then a line of first-order transitions. We concentrate on critical and tricritical properties and compare the accuracy achieved via standard finite-size scaling of thermodynamic quantities with that from the zeros analysis. This latter analysis showcases spectacular precision, even for systems as small as 64 spins. We also show that the zeros are very sensitive to subtle crossover effects.