Free and freer $XY$ models

We study two versions of the XY model where the spins but also the interaction topology is allowed to change. In the free XY model, the number of links is fixed, but their positions in the network are not. We also study a more relaxed version where even the number of links is allowed to vary, we call it the freer XY model. When the interaction networks are dense enough, both models have phase transitions visible both in spin configurations and the network structure. The low-temperature phase in the free XY model is characterized by tightly connected clusters of spins pointing in the same direction and isolated spins disconnected from the rest. For the freer XY model the low-temperature phase is almost completely connected. In both models, exponents describing the magnetic ordering are mostly consistent with values of the mean-field theory of the standard XY model.

Figure 1

Low (a and d), middle (b and e), and high (c and f) temperature configurations of the free XY model (a, b, c) with $N=32$ and $M=64$ and the freer XY model (d, e, f) with $N=32$. The colors represent the spin angles.

Figure 2

Binder's cumulant for the free [panels (a)–(d)] and freer [panel (e)] XY models as a function of temperature for various system sizes. For the free XY model, we plot four values of the average degree—$k=1$ in panel (a), $k=2$ in panel (b), $k=4$ in panel (c), and $k=8$ panel (d). The $T$ axes are logarithmic.

Figure 3

Determining the critical exponent $\nu$ by finite size effects on $U$ near the critical point. For the free XY model [panels (a) and (b)—the former for $k=4$, the latter for $k=8$] and the freer XY model [panel (c)]. Colors define the same system size as in Fig. 2. The axes are logarithmic.

Figure 4

Crossing plots for the magnetization to determine $T_c$ and the exponent $\nu$. For the free XY model [panels (a)–(c)] and the freer XY model [panel (d)]. The lines have the same color as in Fig. 2. The $T$ axes are logarithmic.

Figure 5

Collapse plots to determine $\nu$ and $\beta$. For the free XY model [panels (a), (b), and (c)] and the freer XY model [panel (d)].

Figure 6

The specific heat for the free [panel (a)–(c)] and freer [panel (d)] XY models as a function of temperature for various system sizes. For the free XY model, we plot the results for $k=2$ in panel (a), $k=4$ in panel (b), $k=8$ in panel (c), and the freer XY model panel (d). The lines symbolize the same as in Fig. 2. Note the axes are logarithmic.

Figure 7

The crossing plot for the susceptibility of the free [panels (a)–(c)] and freer [panel (d)] XY models as a function of temperature for various system sizes. For the free XY model, we plot the results for $k=1$ in panel (a), $k=2$ in panel (b), $k=4$ in panel (c), and $k=8$ panel (d). The lines symbolize the same as in Fig. 2. The axes are logarithmic.

Figure 8

The connectance [number of links $M$ divided by the maximal possible number of links $N(N-1)/2$] in the freer XY model as a function of temperature. The lines symbolize the same as in Fig. 2. The $T$ axis is logarithmic.

Figure 9

The size of the largest connected component of the free XY model as a function of temperature for various system sizes. We show results for $k=1$ in panel (a), $k=2$ in panel (b), $k=4$ in panel (c), and $k=8$ panel (d). The lines symbolize the same as in Fig. 2. The $T$ axes are logarithmic.

Figure 10

Panel (a) shows the logarithmic scaling of the diameter of an Erdős-Rényi model of the same $N$ and $M$ (i.e., the diameter for infinite temperature), $d_{\infty }$. Panels (b) and (c) show the diameter relative to $d_{\infty }$ for $k=2$ (b) and $k=4$ (c). The lines symbolize the same as in Fig. 2. The $T$ axes are logarithmic.