From hard spheres to hard-core spins

A system of hard spheres exhibits physics that is controlled only by their density. This comes about because the interaction energy is either infinite or zero, so all allowed configurations have exactly the same energy. The low-density phase is liquid, while the high-density phase is crystalline, an example of “order by disorder” as it is driven purely by entropic considerations. Here we study a family of hard spin models, which we call hard-core spin models, where we replace the translational degrees of freedom of hard spheres with the orientational degrees of freedom of lattice spins. Their hard-core interaction serves analogously to divide configurations of the many spin system into allowed and disallowed sectors. We present detailed results on the square lattice in $d=2$ for a set of models with ${\mathbb{Z}}_n$ symmetry, which generalize Potts models, and their $\text{U}\left(1\right)$ limits, for ferromagnetic and antiferromagnetic senses of the interaction, which we refer to as exclusion and inclusion models. As the exclusion and inclusion angles are varied, we find a Kosterlitz-Thouless phase transition between a disordered phase and an ordered phase with quasi-long-ranged order, which is the form order by disorder takes in these systems. These results follow from a set of height representations, an ergodic cluster algorithm, and transfer matrix calculations.

Figure 1

The classes of hard-core spin models studied in this paper, in which the pairwise potential takes the form of Eq. (2) or (3), specialized to the case $M=2$ on the square lattice. (a) Valid configurations for (left) an inclusion model with inclusion angle $\mathrm{\Delta }=0.48\pi$ and (right) an exclusion model with exclusion angle $\pi -\mathrm{\Delta }=0.52\pi$. Red shading indicates orientations forbidden by the nearest-neighbor interaction. (b) Sketch of the phase diagram in the XY limit of the inclusion model. Increasing the inclusion angle, defined in Eq. (5), tunes the system from a KT phase, consisting of both a vortex-forbidden region and a quasi-long-ranged order by disorder region, to a paramagnetic phase.

Figure 2

Nearest-neighbor bonds with periodic boundary conditions on a patch of a torus of width $W$. The element $T_{AB}$ is the Boltzmann weight due to the energy within and between adjacent columns in spin configurations $A$ and $B$. The bottom row is a copy of the top row for periodic boundary conditions; for twisted periodic boundary conditions, used for exclusion models when $W$ is odd, the dashed bonds are used instead.

Figure 3

Phase diagram in the $(N,p^{\prime })$ plane for $Z_N$ inclusion and exclusion models. $p^{\prime }$ families are represented by gray horizontal lines, with open circles along the gray dashed line denoting $N=N_c\left(p^{\prime }\right)$. Dark gray shaded regions are forbidden, and their black dashed boundaries are the trivial limiting cases: $p^{\prime }=1$ is strictly ordered with zero macroscopic entropy density, while $p^{\prime }=N$ is disordered (unconstrained and noninteracting). Constant inclusion angle corresponds to functions of the form $p^{\prime }=N\mathrm{\Delta }/\pi$; the blue dashed line has $\mathrm{\Delta }={\mathrm{\Delta }}_c\approx 0.56\pi$ and marks the upper boundary ($\eta =\frac{1}{4}$) of the quasi-long-ranged order by disorder phase [Eq. (37)].

Figure 4

$\xi \left(W\right)/N^2$ for the $p^{\prime }=2$, as determined from the transfer matrix for small values of $N$. Different-colored $x$'s correspond to different members of the family, while the solid (dashed) lines show linear fits for even (odd) widths. Very small widths ($W<4$) were not included in the fits.

Figure 5

Power-law scaling of the susceptibility. The left panel shows $log(\chi /L^2)$ as a function of $logL$ for $L=8,16,24,40,60,80$. Data from $L=8$ and 16, marked with $x$'s, were not included in the linear fit. From top to bottom, the inclusion angles are $0.44\pi , 0.46\pi , 0.48\pi , 0.5\pi , 0.52\pi , 0.54\pi , 0.55\pi , 0.56\pi$, and $0.57\pi$. The right panel shows residuals with respect to the fits, amplified by a factor of ${10}^3$ and with data from different inclusion angles scattered in the horizontal direction to enhance visibility. Error bars are defined as the standard error across 1000 consecutive bunches of 100 MCS each.

Figure 6

Critical exponents approaching the KT transition. The gray line in the left panel marks the value of $\eta =\frac{1}{4}$ at vortex unbinding. The critical exponents $\eta$, determined from finite-size scaling of the susceptibility, and ${\eta }^{\prime }$, determined from finite-size scaling of the average cluster size, are marked with circles and $x$'s respectively. The plotted error bars show the statistical uncertainty of the log-log fits in Fig. 5. Different colors denote models in the XY limit (blue), the smallest members of $p^{\prime }$ families, with $p^{\prime }$ ranging from 5 to 63 (orange), and clock models just outside the $p^{\prime }$ families, i.e., $N=N_c$, with $p^{\prime }$ ranging from 15 to 63 (green).

Figure 7

Data collapse of dimensionless quantities, the rescaled second moment correlation length, and Binder cumulant, near the KT phase transition. System sizes from top (blue) to bottom (brown) are $L=8$, 16, 24, 40, 60, and 80. The gray line in the left panel marks the value of ${lim}_{L\to \infty }{\xi }_{2\text{nd}}/L=0.7506912$ at vortex unbinding [56]; gray line in the right panel marks the value of the Binder cumulant $U=0.660603$ at the KT transition [70].

Figure 8

Scaling of the critical exponent $\eta$ in the critical phase. (a) Power-law scaling as a function of $\mathrm{\Delta }$ in the XY limit. Error bars are derived from the statistical uncertainties in the power-law fits to the susceptibility. The right panel shows residuals with respect to the fit $\eta =0.79{\mathrm{\Delta }}^{1.999}$, amplified by a factor of ${10}^3$. The $\mathrm{\Delta }/\pi$ axis uses a logarithmic scale. (b) Scaling of $\eta N^2$ as a function of $p^{\prime }$ for $p^{\prime }$ families of $Z_N$ models. Error bars are statistical uncertainties from power-law fits to the susceptibility of one model within the $p^{\prime }$ family (except $p^{\prime }=2$ where the theoretical value of 3 is used). The $p^{\prime }$ axis uses a logarithmic scale. The right panel shows the residuals with respect to the fit $\eta N^2=0.784p^{\prime 2}$, starting at $p^{\prime }=6$.

Figure 9

Discrepancy between $\eta$ and ${\eta }^{\prime }$ as a function of inclusion angle. Vertical axis is the relative error $(\eta -{\eta }^{\prime })/\eta$, with error bars estimated from the statistical uncertainty of the power-law fit used in finite-size scaling. Data from clock models were plotted by defining $\mathrm{\Delta }=\pi p^{\prime }/N$.

Figure 10

Sketch of the relationship between the Ising variable $\sigma$ and the piecewise function $C\left(\vartheta \right)$ defined in Eq. (B8). For the Ising pseudospins, $\sigma =1$ for spins pointed above the $x$ axis, and $\sigma =-1$ for spins pointed below the $x$ axis. For the function $C\left(\vartheta \right)$, we define $\vartheta$ with respect to an external magnetic field $\mathbf{H}$ oriented along the positive $y$ axis. Then, given an inclusion angle $\mathrm{\Delta }$, spins pointing in the blue shaded region have $C\left(\vartheta \right)=1$ and spins pointing in the unshaded region have $C\left(\vartheta \right)=-1$. For the inclusion angle shown, $C\left(\vartheta \right)=\sigma$ everywhere except in the narrow shaded region below the $x$ axis.

Figure 11

Rescaled probability distribution $\tilde{n}(s/L^{d_F})$ of cluster sizes at $\mathrm{\Delta }=0.5\pi$. The horizontal axis is cluster size $s$ rescaled by $L^{d_F}$, where $d_F=1.901$ is the fractal dimension determined from Eq. (B15), using $\eta =0.198$. The vertical axis is $n^{*}\left(s\right)s^{\tau -1}$. Optimal data collapse was achieved using a cluster exponent of $\tau -1=1.089$, obtained from a power-law fit described in the text. The distribution was measured over $\approx 8\times {10}^5$ cluster moves for $L=8, \approx 1.1\times {10}^6$ cluster moves for $L=16, \approx 1.65\times {10}^6$ cluster moves for $L=32, \approx 2.3\times {10}^6$ cluster moves for $L=64, \approx 3.2\times {10}^6$ cluster moves for $L=128$, and $\approx 4\times {10}^6$ moves for $L=200$.

Figure 12

Finite-size scaling of the bond susceptibility, defined in Eq. (B17), for $L=8,16,24,32,48,64,80,100,128,200$. Only data with $L\ge 24$ were included in the fit. Error bars were determined from the standard error across 100 or 1000 chunks. Right panel shows the residuals with respect to the fit, amplified by a factor of ${10}^3$.