Multicritical behavior of the ferromagnetic Blume-Emery-Griffiths model with repulsive biquadratic couplings

The ferromagnetic ${\displaystyle (J>0)}$ version of the Blume-Emery-Griffiths model in the region of repulsive biquadratic couplings ${\displaystyle (K<0)}$ is considered on a Cayley tree of coordination ${\displaystyle z}$, reducing the statistical problem to the analysis of a two-dimensional nonlinear discrete map. In order to investigate the effect of the coordination ${\displaystyle z}$ on the system multicritical behavior, we study the particular case ${\displaystyle K/J=-3.5}$ with the inclusion of crystal fields ${\displaystyle (D\ne 0)}$, but vanishing external magnetic fields ${\displaystyle (H=0)}$, for two distinct lattice coordinations (${\displaystyle z=4}$ and ${\displaystyle z=6}$). The thermodynamic solutions on the Bethe lattice (the central region of a large Cayley tree) are associated with the attractors of the two-dimensional map. The phase diagrams display several thermodynamic phases (paramagnetic, ferromagnetic, ferrimagnetic, and staggered quadrupolar). In some cases, there are regions of numerical costability of two different attractors of the map, associated with discontinuous phase transitions between the corresponding phases. To verify the thermodynamic stability of the phases and to locate the first-order boundaries, the analytical expression of the Gibbs free energy was obtained by the method proposed by Gujrati [Phys. Rev. Lett. 74, 809 (1995)]. For lower coordinations ${\displaystyle (z=4)}$ the transition between the ferrimagnetic and the staggered quadrupolar phases is always continuous, while the transition between the ferromagnetic and the ferrimagnetic phases is discontinuous at low temperatures, turning into continuous for temperatures above a tricritical point. On the other hand, for higher coordinations ${\displaystyle (z=6)}$, the transition between the ferromagnetic and the ferrimagnetic phases is always continuous. However, the transition between the ferrimagnetic and the staggered quadrupolar phases is continuous for higher temperatures and discontinuous for temperatures below a tricritical point, in agreement with previous results obtained in the mean-field approximation (infinity-coordination limit). In both cases, the occurrence and the thermodynamic stability of the ferrimagnetic phase is confirmed.

Figure 1

Cayley tree with coordination number ${\displaystyle z=3}$ and ${\displaystyle N=5}$ generations of sites. The generations are labeled from the surface ${\displaystyle (n=0)}$ to the central site ${\displaystyle (n=N=5)}$.

Figure 2

Bifurcation diagrams of the fixed points ${\displaystyle (\overline{m},\overline{q})}$ of the auxiliary variables as a function of the parameter ${\displaystyle D/Jz}$, with ${\displaystyle z=4}, {\displaystyle K/J=-3.5}$, and ${\displaystyle t=0.3}$. The thermodynamic phase labels (F, Fi, SQ) follow the main text. For the alternate Fi and SQ phases, the two branches shown correspond to the ${\displaystyle ({\overline{m}}_a,{\overline{m}}_b,{\overline{q}}_a,{\overline{q}}_b)}$ values on the two distinct sublattices ${\displaystyle (a,b)}$.

Figure 3

Bifurcation diagrams of the fixed points ${\displaystyle (\overline{m},\overline{q})}$ of the auxiliary variables as a function of the parameter ${\displaystyle D/Jz}$, with ${\displaystyle z=4}, {\displaystyle K/J=-3.5}$, and ${\displaystyle t=0.15}$. The dashed lines are associated with an unstable branch of the recurrence relations. Note the costability of two different attractors in the abscissa range ${\displaystyle -2.503⪅D/Jz⪅-2.501}$, which coincides with the region of existence of the unstable branch.

Figure 4

Absolute value of the largest eigenvalue ${\displaystyle \left|{\mathrm{\Lambda }}_1\right|}$ of the Jacobian matrix as a function of the parameter ${\displaystyle D/Jz}$, with ${\displaystyle z=4}, {\displaystyle K/J=-3.5}$, and ${\displaystyle t=0.15}$. The solid lines represent the numerically stable solutions, while the dashed line represents a numerically unstable branch that indicates the occurrence of a discontinuous (first-order) transition between the coexisting F and Fi phases. Note the costability of both attractors in the abscissa range ${\displaystyle -2.503⪅D/Jz⪅-2.501}$. A horizontal dotted line delimits ${\displaystyle \left|{\mathrm{\Lambda }}_1\right|\le 1}$ and the point with ${\displaystyle \left|{\mathrm{\Lambda }}_1\right|=1}$ denotes a continuous (second-order) transition between the Fi and the SQ phases.

Figure 5

Bifurcation diagrams of the fixed points ${\displaystyle (\overline{m},\overline{q})}$ of the auxiliary variables as a function of the parameter ${\displaystyle D/Jz}$, with ${\displaystyle z=6}, {\displaystyle K/J=-3.5}$, and ${\displaystyle t=0.4}$. The phase labels and the meaning of the distinct branches are the same as in Fig. 2.

Figure 6

Bifurcation diagrams of the fixed points ${\displaystyle (\overline{m},\overline{q})}$ of the auxiliary variables as a function of the parameter ${\displaystyle D/Jz}$, with ${\displaystyle z=6}, {\displaystyle K/J=-3.5}$, and ${\displaystyle t=0.1}$. The dashed lines are associated with an unstable branch of the recurrence relations. A logarithmic scale for ${\displaystyle \overline{q}}$ is used for a better visualization of the branches profile, since they are close to ${\displaystyle \overline{q}=1}$. Note the costability of two different attractors in the abscissa range ${\displaystyle -2.523⪅D/Jz⪅-2.493}$, which coincides with the region of existence of the unstable branch.

Figure 7

Absolute value of the largest eigenvalue ${\displaystyle \left|{\mathrm{\Lambda }}_1\right|}$ of the Jacobian matrix as a function of the parameter ${\displaystyle D/Jz}$, with ${\displaystyle z=6}, {\displaystyle K/J=-3.5}$, and ${\displaystyle t=0.1}$. The solid lines represent the numerically stable solutions, while the dashed line represents a numerically unstable branch that indicates the occurrence of a discontinuous (first-order) transition between the coexisting Fi and SQ phases. Note the costability of both attractors in the abscissa range ${\displaystyle -2.523⪅D/Jz⪅-2.493}$. A horizontal dotted line delimits ${\displaystyle \left|{\mathrm{\Lambda }}_1\right|\le 1}$ and the point with ${\displaystyle \left|{\mathrm{\Lambda }}_1\right|=1}$ denotes a continuous (second-order) transition between the F and the Fi phases.

Figure 8

Gibbs free-energy profiles (solid lines) associated with the unique numerically stable solutions, obtained for ${\displaystyle K/J=-3.5}$ and temperatures above the tricritical temperature. The dashed line is only a reference straight line. Only continuous phase transitions (indicated by the arrows) between the different labeled phases are observed, associated with the absence of numerical costability of distinct attractors. These continuous transitions are given by the criterion ${\displaystyle \left|{\mathrm{\Lambda }}_1\right|=1}$ on the absolute value of the largest eigenvalue of the Jacobian matrix.

Figure 9

Gibbs free-energy profiles associated with the numerically stable (solid lines) and unstable (dashed lines) solutions, obtained for ${\displaystyle K/J=-3.5}$ and temperatures below the tricritical temperature. The numerical costability of two distinct attractors indicates the occurrence of a discontinuous (first-order) phase transition between the two coexisting labeled phases. The point where the Gibbs free energies of the stable branches intersect (indicated by the arrows) locates the discontinuous phase transition.

Figure 10

Global phase diagram of the ferromagnetic BEG model with repulsive biquadratic interations ${\displaystyle (K/J=-3.5)}$ on a BL with coordination ${\displaystyle z=4}$. The numerical costability region is shaded in gray. The dashed line represents the discontinuous (first-order) phase boundary, while the solid lines correspond to continuous (second-order) transitions. The labels of the thermodynamic phases (P, F, Fi, SQ) and the classification of the special points (M, T, S) follow the main text.

Figure 11

Global phase diagram of the ferromagnetic BEG model with repulsive biquadratic interations ${\displaystyle (K/J=-3.5)}$ on a BL with coordination ${\displaystyle z=6}$. The numerical costability region is shaded in gray. The dashed line represents the discontinuous (first-order) phase boundary, while the solid lines correspond to continuous (second-order) transitions. The labels are the same as those used in Fig. 10.