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Physical review. B, Condensed matter

Three-component model and tricritical points: A renormalization-group study. II. General dimensions and the three-phase monohedron

Julia M. Yeomans, Michael Ellis Fisher

Published Sep 1, 1981• 43 citations
Publisher

Abstract

In Part I the global phase diagram of the general spin-1, nearest-neighbor, ferromagnetic Ising model was studied for $d=2$ dimensions using an approximate renormalization-group approach of the Migdal-Kadanoff type. Here general $d$ is considered with emphasis on $\ensuremath{\epsilon}=d\ensuremath{-}1$ expansions and on the variation of the tricritical exponents with $d<~4$. A dilution field is introduced which can be chosen to yield tricritical exponents for $d=2$, in better agreement with more reliable estimates and conjectured exact results. The three-phase monohedra, which describe the densities of coexisting phases near a tricritical point, are calculated quantitatively for $d=2$, and their shape contrasted with that predicted by classical theory: they are found, in particular, to be significantly flattened.

DOI
10.1103/physrevb.24.2825

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Physical review. B, Condensed matter•Jan 1, 1980•56 citations
In Part I the global phase diagram of the general spin-1, nearest-neighbor, ferromagnetic Ising model was studied for d=2d=2d=2 dimensions using an approximate renormalization-group approach of the Migdal-Kadanoff type. Here general ddd is considered with emphasis on \ensuremathϵ=d\ensuremath−1\ensuremath{\epsilon}=d\ensuremath{-}1\ensuremathϵ=d\ensuremath−1 expansions and on the variation of the tricritical exponents with d< 4d<~4d< 4. A dilution field is introduced which can be chosen to yield tricritical exponents for d=2d=2d=2, in better agreement with more reliable estimates and conjectured exact results. The three-phase monohedra, which describe the densities of coexisting phases near a tricritical point, are calculated quantitatively for d=2d=2d=2, and their shape contrasted with that predicted by classical theory: they are found, in particular, to be significantly flattened.
The global phase diagram for a three-component lattice gas or spin-one Ising model with general single-site and nearest-neighbor "ferromagnetic" interactions is worked out for two-dimensional lattices using a Migdal-Kadanoff recursion relation. It differs in important qualitative respects from the corresponding mean-field phase diagram. The set of fixed points and flows provides the characteristic phase diagrams of the three-state Potts multicritical point and the ordinary (n=1n=1n=1) tricritical point in a complete set of symmetry-breaking fields. The latter is associated, in this renormalization-group scheme, with seven distinct critical fixed points, a number which is surprisingly large.
The spin-1 Ising model on the square lattice with nearest-neighbor ferromagnetic exchange interactions [both bilinear (JJJ) and biquadratic (KKK) and crystal-field interaction ($\ensuremath{\Delta}$) is studied via a renormalization-group transformation in position space. The phase diagram in JJJ, KKK, $\ensuremath{\Delta}$ space is found to have one surface of critical phase transitions and two surfaces of first-order phase transitions. These surfaces are variously bounded by an ordinary tricritical line, an isolated critical line, and a line of critical end points. These three lines joint at a special tricritical point corresponding to the transition of the three-state Potts model. The over-all phase diagram is qualitatively similar to that obtained with the mean-field approximation, except in the vicinity of the Potts transition where a four-phase coexistence line in mean-field theory shrinks into a special tricritical point in renormalization-group theory. Symmetry considerations guide the construction of our truncated renormalization-group transformation. The global connectivity and local exponents of the thirteen separate fixed points underlying this quite complicated structure are determined. Local analysis with respect to magnetic field (HHH) and another odd interaction (LLL) is performed. A one-adjusted-parameter version of our transformation yields remarkably quantitative results, predicting the Potts transition temperature, for example, within 0.3% of the exact value.
The spin-3/2 Ising model, with nearest-neighbor interactions only, is the prototypical system with two different ordering species, with concentrations regulated by a chemical potential. Its global phase diagram, obtained in d=3 by renormalization-group theory in the Migdal-Kadanoff approximation or equivalently as an exact solution of a d=3 hierarchical lattice, with flows subtended by 40 different fixed points, presents a very rich structure containing eight different ordered and disordered phases, with more than 14 different types of phase diagrams in temperature and chemical potential. It exhibits phases with orientational and/or positional order. It also exhibits quintuple phase transition reentrances. Universality of critical exponents is conserved across different renormalization-group flow basins via redundant fixed points. One of the phase diagrams contains a plastic crystal sequence, with positional and orientational ordering encountered consecutively as temperature is lowered. The global phase diagram also contains double critical points, first-order and critical lines between two ordered phases, critical end points, usual and unusual (inverted) bicritical points, tricritical points, multiple tetracritical points, and zero-temperature criticality and bicriticality. The four-state Potts permutation-symmetric subspace is contained in this model.
The tricirical exponents ${\ensuremath{\eta}}_{t}$ and ${\ensuremath{\varphi}}_{t}$ are calculated through order ${\ensuremath{\epsilon}}^{3}$ and ${\ensuremath{\epsilon}}^{2}$, respectively, in the isotropic nnn-component model, where $\ensuremath{\epsilon}=3\ensuremath{-}d$. The estimate for ${\ensuremath{\eta}}_{t}$, in two dimensions for the Ising case, is 0.027; the series for ${\ensuremath{\varphi}}_{t}$ is quite ill behaved, producing a negative estimate at this order. Beginning at this order in $\ensuremath{\epsilon}$, the spherical-model limit fails to exist. A scaling function for the spin-spin correlation function, appropriate for nonexceptional paths of approach to the tricritical point in the disordered phase, is calculated through ${\ensuremath{\epsilon}}^{2}$; its large momentum expansion is shown to involve the crossover exponent. These results are generalized to points where $\ensuremath{\theta}$ coexisting phases become critical.
We have performed large-scale numerical simulations for the ± J Ising model on the simple cubic and the square lattices with asymmetric weight of ferromagnetic and antiferromagnetic bonds. The Monte Carlo renormalization group method is used to estimate T c , γ/ν, β/ν and ν at various points along the phase boundary between ferromagnetic and paramagnetic phases including the tricritical point. The obtained values of T c and the resulting phase diagrams of the simple cubic and the square lattices are much more accurate than those estimated so far. In the simple cubic lattice, we find evidence for the possibility that universality of the ferromagnetic critical exponents holds along the critical line except at the tricritical point, while weak universality holds even at the tricritical point. Weak universality seems to hold also in the square lattice.
The phase diagrams and critical properties of a quenched random alloy of a ferromagnet and an antiferromagnet (or of two antiferromagnets with different periodicities) are studied in the mean-field approximation and by renormalization-group techniques in $d=6\ensuremath{-}\ensuremath{\epsilon}$ dimensions, using the $n\ensuremath{\rightarrow}0$ replica method. If only nearest-neighbor interactions are assumed, then one finds ferromagnetic (FFF), antiferromagnetic (AF), and spin-glass (SG) phases. As the average strength of the next-nearest-neighbor interactions is increased (e.g., by an additional component in the alloy), the FFF and the AF phases approach each other, and one may find regions in parameter space where a mixed FFF-AF phase exists, as discussed in Paper II of this series. The phase diagram exhibits lines of multicritical points where the paramagnetic (PPP), FFF (or AF) and SG phases coexist (PPP-SG-FFF or PPP-SG-AF), or where the PPP, FFF, and AF phases coexist ($P\ensuremath{-}F$-AF). These lines meet at a new multicritical point, where all four phases coexist (PPP-SG-FFF-AF). Recent renormalization-group analysis of PPP-SG-FFF, by Chen and Lubensky, yields complex exponents for the XY\mathrm{XY}XY and Heisenberg cases. We show that the phase diagram predicted by that analysis is difficult to understand even for the Ising case. The same method of analysis yields similar difficulties for PPP-SG-FFF-AF. A modified way to take the limit $n\ensuremath{\rightarrow}0$, which resolves these difficulties, is presented. This modified way still lacks rigorous justification.