Phase transitions in $XY$ models with randomly oriented crystal fields

We obtain a representation of the free energy of an $XY$ model on a fully connected graph with spins subjected to a random crystal field of strength $D$ and with random orientation $\alpha$. Results are obtained for an arbitrary probability distribution of the disorder using large deviation theory, for any $D$. We show that the critical temperature is insensitive to the nature and strength of the distribution $p\left(\alpha \right)$, for a large family of distributions which includes quadriperiodic distributions, with $p\left(\alpha \right)=p(\alpha +\frac{\pi }{2})$, which includes the uniform and symmetric bimodal distributions. The specific heat vanishes as temperature $T\to 0$ if $D$ is infinite, but approaches a constant if $D$ is finite. We also studied the effect of asymmetry on a bimodal distribution of the orientation of the random crystal field and obtained the phase diagram comprising four phases: a mixed phase (in which spins are canted at angles which depend on the degree of disorder), an $x$-Ising phase, a $y$-Ising phase, and a paramagnetic phase, all of which meet at a tetracritical point. The canted mixed phase is present for all finite $D$, but vanishes when $D\to \infty$.

Figure 1

Low energy states with (a) $J\to \infty$ and (b) $D\to \infty$ on a fully connected graph with $N=5$ (five spins). Spins are represented by blue arrows and the random anisotropy axes by dotted red lines. In (a) the spins align with each other while in (b) the spins align with the random anisotropy axes.

Figure 2

Truncated Landau functional for uniform distribution obtained by expanding until $10\text{th}$ ($12\text{th}$) power in $r$ shown as the dotted (solid) line, for (a) $D=0$ and $\beta =2.1$; (b) $D=2$ and $\beta =2.1$.

Figure 3

$p\text{−}T$ phase diagram for bimodal distribution for infinite $D$.

Figure 4

Phase diagram for the pure $XY$ model with crystal field in the $y$ direction. The solid lines are critical curves separating the Ising phases from the paramagnetic phase. Across the dotted line there is a first order spin flop transition between the two Ising phases. As $D\to \pm \infty$, the two critical lines approach $T=1$.

Figure 5

Phase diagram in the $p\text{−}T$ plane for (a) $D=0.2$ and (b) $D=1$ for the bimodal distribution. There is a tetracritical point at $(1/2,1/2)$ from which four critical curves emanate, separating the four phases.

Figure 6

Transverse (${\chi }_{11}$) and longitudinal (${\chi }_{22}$) susceptibility in the case of asymmetric bimodal distribution with $D=0.2$ and $p=0.4$.

Figure 7

Phase diagram for $w=1$ in the $p\text{−}T$ plane. Four critical curves meet at a tetracritical point at $(1/2,1/2)$ for all $w$. The phases represented by $|x_1|>0$ and $|x_2|>0$ are transverse and longitudinal Ising phases with $|x_2|=0$ and $|x_1|=0$, respectively.

Figure 8

The two red vectors represent the average magnetization vectors along the $x$ and $y$ directions for $D=\infty$ at $T=0$, which have magnitude $p$ and $1-p$, respectively. Blue vectors represent the canted average magnetic vectors for large finite $D$ at $T=0$. We have taken $p$ such that $p>1-p$ and hence ${\theta }_x<{\theta }_y$.

Figure 9

Possible phase diagrams from a two parameter Landau theory. Solid line represents the coordinate axes. Dashed and dotted lines represent the locus of continuous and first order transitions, respectively. For $u_1{u}_2<u_{12}^2$, the phase diagram consists of two lines of continuous transition and a line of first order transition meeting at a bicritical point as shown in (a). For $u_1{u}_2>u_{12}^2$ there are four phases, which meet at a tetracritical point as shown in (b), (c), and (d). The topology, though, can be different depending on the ratios of $\frac{u_{12}}{u_1}$ and $\frac{u_{12}}{u_2}$.