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Journal of the Physical Society of Japan

Exact Renormalization Analysis of a Self-Similar Phase Diagram with Upsilon Points for an Ising Model

Kazuo Sasaki

Published Nov 15, 1999• 2 citations
Publisher

Abstract

The ground-state phase diagram is analyzed for a one-dimensional Ising model with long range interaction in the presence of uniform and staggered magnetic fields; the interaction is antiferromagnetic and its strength decays exponentially with distance. The exact renormalization formulas for the interactions between interfaces are derived, which is used to calculate the phase diagram hierarchically. The phase diagram is self-similar, and it contains upsilon points and devil's staircases; this agrees with the result of Aubry, Axel and Vallet [J. Phys. C 18 (1985) 753] for an equivalent model obtained by a different method. A simple “phase rule” presenting what kind of phases appear and how they are located in the phase diagram is derived. The application of this rule to the phase diagram of betaine calcium chloride dihydrate is discussed.

DOI
10.1143/jpsj.68.3562

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The ground-state phase diagram is analyzed for a one-dimensional Ising model with long range interaction in the presence of uniform and staggered magnetic fields; the interaction is antiferromagnetic and its strength decays exponentially with distance. The exact renormalization formulas for the interactions between interfaces are derived, which is used to calculate the phase diagram hierarchically. The phase diagram is self-similar, and it contains upsilon points and devil's staircases; this agrees with the result of Aubry, Axel and Vallet [J. Phys. C 18 (1985) 753] for an equivalent model obtained by a different method. A simple “phase rule” presenting what kind of phases appear and how they are located in the phase diagram is derived. The application of this rule to the phase diagram of betaine calcium chloride dihydrate is discussed.
A one-dimensional Ising chain with spin-lattice coupling is considered. If the lattice is harmonic but has no translational symmetry, there appears effectively through the spin-lattice coupling a four-spin interaction which decays exponentially with distance. It is shown rigorously that this spin system in the ground state exhibits a devil's staircase as a function of the exchange parameter J because of the long range interaction. The ground state phase diagram of the spin system under an external magnetic field is speculated.
A one-dimensional antiferromagnetic Ising chain with spin-lattice coupling is considered. If the underlying lattice is harmonic but has no translational symmetry, there appears at zero temperature an effective interaction between down spins, which is repulsive and decays exponentially with distance. It is shown rigorotusly that this system exhibits in the ground state a devil's staircase as a function of an external magnetic field on account of the effective long range interaction. The ground state phase diagram is also given.
The one-dimensional Ising model in an external magnetic field with uniform long-range interactions and random short-range interactions satisfying bimodal annealed distributions is studied. This generalizes the random model discussed by Paladin et al. (J. Phys. I France 4, 1994, p. 1597). Exact results are obtained for the thermodynamic functions at arbitrary temperatures, and special attention is given to the induced and spontaneous magnetization. At low temperatures the system can exist in a “ferrimagnetic” phase with magnetization 0 < σ < 1, in addition to the usual paramagnetic, ferromagnetic and antiferromagnetic phases. For a fixed distribution of the random variables the system presents up to three tricritical points for different intensities of the long-range interactions. Field-temperature diagrams can present up to four critical points. 1
Models with competing nearest-neighbor and very long-range interactions are solved exactly for several one-dimensional cases, including the usual Ising chain (ICCI), the XY model (XYCI), the transverse Ising model (TICCI), and the spherical model (SCCI). For certain ratios of the competing interaction strengths, ICCI and XYCI display triple points and two critical points in a field. In addition TICCI has an apparently enclosed phase in an H-T phase diagram; however, this phase is really paramagnetic as can be seen when an extended phase diagram is used. The extended phase diagrams for ICCI, XYCI, and TICCI display tricritical points and the tricritical exponents have different values from the usual classical values. In contrast to the rich phase behavior of the preceding models, SCCI shows very simple phase behavior, which is directly related to the ground state. Finally, the introduction of a staggered field to ICCI and a simple transformation allows reinterpretation as a metamagnetic model. Using ICCI as a guide the observability of the tricritical exponents is discussed.
The phase diagram of a two-sited magnetic Ising chain, with a long-range interaction in the form of ${1/r}^{1+\ensuremath{\sigma}},$ is studied. In this investigation, the finite-range scaling technique is employed and a proper transfer matrix is developed. The critical temperature of the chain, as a function of each type of interaction, i.e., the interaction between the similar sites and different sites, in both the classical and nonclassical regions is calculated. The results indicate that the critical temperature is strongly affected by the interaction range parameter \ensuremath{\sigma} for both types of interaction, but its behavior is quite different for each of the interactions. The behavior of the system is explained by calculating the strength of each interaction using the mean-field approximation. The study of the critical exponent of the correlation length indicates that the exponent is, within a good approximation, independent of the interaction constants.
Abstract The Ising model on an one-dimensional monoatomic equidistant lattice with different exchange interactions between atomic spins at the sites of nearest neighbors and different interactions between atomic spins at the sites of second neighbors is investigated. The generalized Kramers–Wannier transfer matrix with translation on two periods of a lattice is derived. The phase diagrams representing all possible magnetic types of ordering in the ground state are constructed. Many triple points of intersection of phases and lines of pair coexistence of phases in strict accordance with the Gibbs phase rule are found and a property similar to supercooling and superheating is detected. It is shown that at triple points the phases are not individualized but significantly frustrated, which corresponds to the phenomenon of critical opalescence. Exact analytical expressions for the Gibbs free energy, as well as for such thermodynamic characteristics of the system as internal energy, heat capacity and entropy including zero-temperature entropy are derived. A variety of particular cases are analyzed and a comparison is made with all known results including on a two-dimensional lattice.