Classical lattice spin models involving singular interactions isotropic in spin space

We address here a few classical lattice spin models, involving $n$-component unit vectors $\left(n=2,3\right)$, associated with a $D$-dimensional lattice ${\mathbb{Z}}^D,D=1,2$, and interacting via a pair potential restricted to nearest neighbors and being isotropic in spin space, i.e., defined by a function of the scalar product between the interacting spins. When the potential involves a continuous function of the scalar product, the Mermin-Wagner theorem and its generalizations exclude orientational order at all finite temperatures in the thermodynamic limit, and exclude phase transitions at finite temperatures when $D=1$; on the other hand, we have considered here some comparatively simple functions of the scalar product which are bounded from below, diverge to $+\infty$ for certain mutual orientations, and are continuous almost everywhere with integrable singularities. Exact solutions are presented for $D=1$, showing an absence of phase transitions and an absence of orientational order at all finite temperatures in the thermodynamic limit; for $D=2$, and in the absence of more stringent mathematical results, extensive simulations carried out on some of them point to the absence of orientational order at all finite temperatures and suggest the existence of a Berezinskiĭ-Kosterlitz-Thouless transition.

Figure 1

Comparison between the singular model $V$, its regular counterpart, and some truncated expansions [Eqs. (3a), (4a), (5a)] as functions of the angle $\mathrm{\Delta }$ between the two spins. Meaning of symbols: red solid line, regular counterpart; blue dashed line, model $V$; magenta dotted line, ${\mathcal{V}}_4$; brown dash-dotted line, ${\mathcal{V}}_{10}$.

Figure 2

Comparison between the singular model $W$, its regular counterpart, and some truncated expansions [Eqs. (3b), (4b), (5b)] as functions of the scalar product $\tau$ between the two spins. Meaning of symbols: red solid line, regular counterpart; blue dashed line, model $W$; magenta dotted line, ${\mathcal{W}}_2$; brown dash-dotted line, ${\mathcal{W}}_4$.

Figure 3

Comparison between the singular model $X$, its regular counterpart, and some truncated expansions [Eqs. (3c), (4c), (5c)] as functions of the scalar product $\tau$ between the two spins. Meaning of symbols: red solid line, regular counterpart; blue dashed line, model $X$; magenta dotted line, ${\mathcal{X}}_2$; brown dash-dotted line, ${\mathcal{X}}_4$. Notice that the quantity $-1$ has been added to $X$ as well as to the two truncated expansions in order to ease comparison.

Figure 4

The specific heat of model $V$ for different sample sizes against temperature; statistical errors (not shown here) range between $1\%$ and $5\%$. Meaning of symbols: red circles, $L=40$; green squares, $L=60$; blue triangles, $L=80$; magenta diamonds, $L=100$; cyan crosses, $L=160$; red asterisks, $L=160$.

Figure 5

The specific heat of model $W$ for different sample sizes against temperature; statistical errors (not shown here) range between $1\%$ and $5\%$; symbols are the same as in Fig. 4.

Figure 6

Simulation results for the magnetization of model $V$ obtained with different sample sizes; symbols are the same as in Fig. 4.

Figure 7

Simulation results for the susceptibility ${\chi }_1$ of model $V$ obtained with different sample sizes; symbols are the same as in Fig. 4. Assuming a BKT transition and fitting the largest sample size $L=160$ (upper solid orange curve) to the bulk behavior of the susceptibility [Eq. (18)] leads to a transition at $T_{\text{BKT}}=0.883\pm 0.007$.

Figure 8

Simulation results for the susceptibility ${\chi }_1$ of model $W$ obtained with different sample sizes; symbols are the same as in Fig. 4.

Figure 9

Simulation results for the short-range order parameters (16) of model $V$ obtained with the largest sample size. Meaning of symbols: red diamonds (upper curve), ${\sigma }_1$; green circles (lower curve), ${\sigma }_2$.

Figure 10

Simulation results for the short-range order parameters (16) of model $X$ obtained with the largest sample size. Meaning of symbols: red diamonds (upper curve), ${\sigma }_2$; green circles (lower curve), ${\sigma }_4$.

Figure 11

The specific heat of model $X$ for different sample sizes against temperature; statistical errors (not shown here) range between $1\%$ and $5\%$. Symbols are the same as in Fig. 4.

Figure 12

Simulation results for the second-rank order parameter ${\overline{P}}_2$ of model $X$ obtained with different sample sizes. Symbols are the same as in Fig. 4.

Figure 13

Simulation results for the fourth-rank order parameter ${\overline{P}}_4$ of model $X$ obtained with different sample sizes. Symbols are the same as in Fig. 4.

Figure 14

Simulation results for the susceptibility ${\chi }_2$ of model $X$ obtained with different sample sizes. Symbols are the same as in Fig. 4.