Numerical study of the $F$ model with domain-wall boundaries

We perform a numerical study of the $F$ model with domain-wall boundary conditions. Various exact results are known for this particular case of the six-vertex model, including closed expressions for the partition function for any system size as well as its asymptotics and leading finite-size corrections. To complement this picture we use a full lattice multicluster algorithm to study equilibrium properties of this model for systems of moderate size, up to $L=512$. We compare the energy to its exactly known large-$L$ asymptotics. We investigate the model's infinite-order phase transition by means of finite-size scaling for an observable derived from the staggered polarization in order to test the method put forward in our recent joint work with Duine and Barkema. In addition we analyze local properties of the model. Our data are perfectly consistent with analytical expressions for the arctic curves. We investigate the structure inside the temperate region of the lattice, confirming the oscillations in vertex densities that were first observed by Syljuåsen and Zvonarev and recently studied by Lyberg et al. We point out “(anti)ferroelectric” oscillations close to the corresponding frozen regions as well as “higher-order” oscillations forming an intricate pattern with saddle-point-like features.

Figure 1

The six vertices allowed by the ice rule and their weights for the six-vertex model. Often one assumes arrow-reversal symmetry: $a_{\pm }=a, b_{\pm }=b, c_{\pm }=c$. The $F$ model is defined by $a=b<c$.

Figure 2

The phase diagram of the six-vertex model, parametrized by the ratios $a/c$ and $b/c$ since common rescalings of the vertex weights yield only an overall factor for the partition function. The colors show contours for $\mathrm{\Delta }$ at steps of $1/2$ for $-4\le \mathrm{\Delta }\le 4$. The dashed arc is the so-called free-fermion line. The dotted line corresponds to the $F$ model, with an infinite-order phase transition between the antiferroelectric (AF) and disordered (D) phases. The thick dot is the ice point $a=b=c$, which can be interpreted as $\beta =0$. There are two ferroelectric (FE) phases.

Figure 3

The average energy per site $e\left(\beta \right)$ (top) and specific heat per site $c_v\left(\beta \right)$ (bottom) as functions of inverse temperature $\beta$. The critical points at ${\beta }_{\text{c}}$ are indicated by the gray lines. The black lines are the exact asymptotic expressions extracted from Eqs. (5) and (7). From the data points, indicating the temperatures at which the simulations are done, the colored solid lines are calculated using the multihistogram method. For both observables we see a convergence of the data to the analytically known expressions in the thermodynamic limit.

Figure 4

The difference between the energy per site (top) and total energy of the system (bottom), with $E_{\infty }\left({\beta }_{\text{c}}\right):=L^2{e}_{\infty }\left({\beta }_{\text{c}}\right)=2L^2/3$, is shown as a function of system size. The solid blue disks represent the exact known values for small system sizes obtained from Eq. (4). The open red squares denote best estimates obtained from our simulations. The error bars are estimates based on the fluctuations in the energy and the number of measurements taken. The expressions from Eq. (15) with only the leading correction ($C_1\equiv C_2\equiv 0$) and with first subleading correction ($C_1=0.669\pm 0.018$), as well as the expression from Eq. (16) ($C_2=1.6\pm 1.2, C_3=14\pm 12, \alpha =1.91\pm 0.36$ [47]) are shown as dotted, dashed, and solid curves, respectively.

Figure 5

The observable $dln{P}_0/d\beta$ is shown in (a) as a function of inverse temperature $\beta$ for various system sizes up to $L=256$. In the thermodynamic limit, under the assumption that $P_0$ is a valid order parameter, this function must have its peak at the critical point. In (b) the peak and the point where the curves attain $95\%$ of their peak height (at higher $\beta$) are scaled on top of each other, with the two points indicated by black circles. From this collapse we extract the peak position and typical width for further analysis; cf. Fig. 6.

Figure 6

Values of the inverse peak height $h_{\text{max}}^{-1}$ (upper panel), peak position ${\beta }_{\text{max}}$ (central panel), and peak width $w$ (lower panel) are shown for $dln{P}_0/d\beta$ up to system size $L=256$ as a scaled function of ${ln}^{-2}L$. Exact values for small system sizes obtained from Eq. (13) are shown as solid blue disks and best estimates obtained from our simulations as open red squares. The best unweighed fits of the form (17) are drawn as black dashed lines. For ${\beta }_{\text{max}}$ a fit through all data points results in a best estimate for the critical point $A_{{\beta }_{\text{max}}}=0.87\pm 0.03$ far from the analytically known value ${\beta }_{\text{c}}=ln2\approx 0.69$. For $h_{\text{max}}^{-1}$ ($A_{h_{\text{max}}^{-1}}=-0.01\pm 0.03, B_{h_{\text{max}}^{-1}}=5.4\pm 1.1, C_{h_{\text{max}}^{-1}}=-3\pm 3, D_{h_{\text{max}}^{-1}}=-2\pm 3$), and $w$ ($A_w=-0.009\pm 0.009, B_w=2.4\pm 0.4, C_w=-5\pm 1, D_w=3.0\pm 0.8$) the fits work well and are in agreement with $dln{P}_0/d\beta$ becoming a Dirac delta-like distribution as $L\to \infty$.

Figure 7

The thermally averaged density $\rho \left(c\right)$ of $c$ vertices at $L=512$ show phase separation at different temperatures. The density lies between zero (shown in purple) and one (light red). White solid contour lines are drawn at the values $1/3,1/2,2/3,\text{and}0.98$, as indicated, of $\rho \left(c\right)$. The outer white solid contours are drawn at temperature-dependent values of $\rho \left(c\right)$ (see Table 1) that give the best qualitative match with the arctic curves [31, 32, 33, 34, 35], which are shown as dashed black curves. At zero temperature (a) the AF region is a diamond. At slightly elevated temperatures (b) the AF and FE regions are separated by a temperate region. As the temperature increases past its critical value (c), at which the AF region disappears, the arctic curve deforms to a circle at the free-fermion point (d). The system at infinite temperature is shown in (e), in which the arctic curve is a sort of inflated circle, with the arcs deformed somewhat towards the corners of the domain.

Figure 8

The thermally averaged densities $\rho$ for all six vertices in a system of linear size $L=512$ at the critical point $\mathrm{\Delta }=-1$ are shown along the diagonal from the $b_{-}$-dominated FE region to the center at $r=0$. The grey vertical line marks the transition between the FE and temperate region [35]. Each vertex density oscillates in the temperate region. Note that $\rho \left(c_{\pm }\right)$ are in antiphase around their average.

Figure 9

The difference $\delta \rho \left(c\right)$ is shown as a function of distance $r$ along the diagonal to the center (at $r=0$) for systems up to size $L=512$ at $\mathrm{\Delta }=-1$. The colored lines are a guide to the eye, and the gray vertical line denotes the transition between the FE-frozen and temperate region as in Ref. [35]. The wavelength of the oscillations seems to increase sublinearly while the average wave amplitude decreases monotonically with system size, suggesting that these are finite-size effects.

Figure 10

The thermally averaged density difference $\delta \rho \left(c\right)$ is shown for a system of size $L=512$ at $\mathrm{\Delta }=-1$ (upper panel) and $\mathrm{\Delta }=-7$ (lower panel). Each pixel corresponds to a vertex. The FE-frozen region in the bottom left contains only $b_{-}$ vertices. Below the critical point the AF-frozen region appears (lower panel) in which $\delta \rho \left(c\right)=0$ due to the twofold degeneracy for even $L$. Inside the intermediate temperate region at least two types of oscillations are visible. There appear to be checkerboard-like patterns in the “AF oscillations” even after thermal averaging, with opposite checkerboards in neighboring oscillations. The “FE oscillations” are dominated by the vertices constituting the FE-frozen region (here $b_{-}$), with $\delta \rho \left(c\right)>0$ between them.

Figure 11

The thermally averaged $c_{\pm }$-density difference $\delta \rho \left(c\right)$ for size $L=512$ at $\mathrm{\Delta }=-1$ (upper panel), truncated at $10\%$ of the values from the upper panel in Fig. 10. Every pixel represents one vertex. This reveals weak “higher-order” oscillations in the temperate region with various saddle-point-like features; we can distinguish at least four of these along the diagonal, and more along the top and right. The lower panels show $\delta \rho \left(c\right)$ at $\mathrm{\Delta }=0$ (left) and $\mathrm{\Delta }=1/2$ (right), each again truncated at $10\%$ of its minimal and maximal value.