Phase transition properties of the Bell-Lavis model

Using Monte Carlo calculations we analyze the order and the universality class of phase transitions into a low-density (honeycomb) phase of a triangular antiferromagnetic three-state Bell-Lavis model. The results are obtained in a whole interval of chemical potential $\mu$ corresponding to the honeycomb phase. Our results demonstrate that the phase transitions might be attributed to the three-state Potts universality class for all $\mu$ values except for the edges of the honeycomb phase existence. At the honeycomb phase and the low-density gas phase boundary the transitions become of the first order. At another, honeycomb-to-frustrated phase boundary, we observe the approach to the crossover from the three-state Potts to the Ising model universality class. We also obtain the Schottky anomaly in the specific heat close to this edge. We show that the intermediate planar phase, found in a very similar antiferromagnetic triangular Blume-Capel model, does not occur in the Bell-Lavis model.

Figure 1

Schematic representation of interactions and ordering in the Bell-Lavis model using triangular TMA molecules as an example: (a) a fragment of the HON phase, (b) tip-to-tip double H-bond interaction $\left(-{\epsilon }_H-{\epsilon }_{\mathrm{vdW}}\right)$, and (c) side-to-side interaction $\left(-{\epsilon }_{\mathrm{vdW}}\right)$. The C, O, and H atoms of the TMA molecule are shown by the black, red (gray) and white circles, respectively. In (b) and (c) the sites of sublattices A, B, and C are shown by triangles, crosses and squares, respectively.

Figure 2

Temperature dependence of the specific heat at $\xi =0$ and $L=120$ and different values of chemical potential: (a) $\mu /{\epsilon }_H=-0.1,0.1$ and 0.3 and (b) 0.5, 0.7, 0.9, 1.15, and 1.4. Inset in (a): $C_v\left(T\right)$ dependence at $\mu /{\epsilon }_H=0.3$ for different values of $L$. The points are obtained by thermal averaging and the curves are obtained by the reweighted energy histogram method. The black curves used to fit the low-temperature peaks at $\mu /{\epsilon }_H=0.1$ and 0.3 in (a) are the results of the two-level model (see text).

Figure 3

The BL model at $\xi =0$ and $L=120$: (a) Occupancy of the A (B) sublattice by +1 $(-1$) states (higher branch) and $-1$ (+1) states (lower branch) for $\mu /{\epsilon }_H=0.9$, 0.7, 0.5, 0.3, 0.1, 0.02, 0, and $-0.02$. (b) Phase diagram obtained from the temperature dependence of the specific heat. The solid circles are $T_c$ points, red crosses denote points of the Schottky anomaly, and the triangle denotes the tricritical point. The denotation HON corresponds to sublattice occupancies ${\rho }_A={\rho }_B,{\rho }_C>0$, and ${\mathrm{HON}}_0$ (“perfect” HON phase) corresponds to ${\rho }_A={\rho }_B\approx 1,{\rho }_C\approx 0$.

Figure 4

Snapshots of the HON structure fragment at $\mu /{\epsilon }_H=0.3$ and below the $T_c$ point: (a) $k_{B}T/{\epsilon }_H=0.35$, (b) 0.2, and (c) 0.06. The sublattices A and B are almost occupied by the +1 (blue tripod-shaped symbol) and $-1$ (gray tripod-shaped symbol) states, respectively, and the system undergoes gradual emptying of sublattice C with a decrease of temperature.

Figure 5

Temperature dependencies of order parameters at $\xi =0$ and $L=120$: (a) $m_s$ at $\mu /{\epsilon }_H=-0.1-0.9$ and $m_{10}$ at $\mu /{\epsilon }_H=0.7$, (b) $m_s$ and (c) $m_{10}$ at $\mu /{\epsilon }_H=0.7$ and different values of $L$, and (d) temperature dependencies of exponents ${\eta }_s$ and ${\eta }_{10}$ obtained at $\mu /{\epsilon }_H=0.7$ using order parameters $m_s$ and $m_{10}$, respectively. In (a)–(c) the points are the results of thermal averaging and curves are obtained from reweighted magnetization histograms.

Figure 6

Temperature dependencies of susceptibility at $\xi =0$ and $L=120$: (a) ${\chi }_s$ at $\mu /{\epsilon }_H=0.9,0.7,0.5$, and 0.3 and ${\chi }_{10}$ at $\mu /{\epsilon }_H=0.7$ close to $T_c$, (b) ${\chi }_s\left(T\right)$ and (c) ${\chi }_{10}\left(T\right)$ at $\mu /{\epsilon }_H=0.7$ and different values of $L$, and (d) ${\chi }_{10}$ at $\mu /{\epsilon }_H=0.7,0.3$ and 0.1 close to Schottky anomaly point $T_S$ $[{\chi }_{10}\left(T_S\right)\ll {\chi }_{10}\left(T_c\right)]$. The points are the results of thermal averaging and the curves are obtained by reweighted magnetization histograms.

Figure 7

Finite-size scaling of (a) specific heat, (b) both susceptibilities, and (c) order parameter $m_s$ at $\mu /{\epsilon }_H=0.7$ The results are fitted using formulas (5), $t=(T_c-T)/T_c$, and the background of $C_v$ is assumed to be zero. Large symbols correspond to the results of thermal averaging; lines correspond to the results obtained close to $T_c$ by the reweighted histogram method.

Figure 8

(a) The log-log dependencies of $X={\chi }_s,{\chi }_{10},D_{1s},D_{2s},C_v$ vs $L$ at $T_c$; (b) temperature dependence of the Binder cumulant of magnetization $m_s$ at $T_c$. The parameters $\mu /{\epsilon }_H=0.7$ and $L=120$.

Figure 9

Energy histograms: (a) at $\mu /{\epsilon }_H=0.3$ and $L=120$ close to the phase transition $\left(k_B{T}_c/{\epsilon }_H=0.3728\right)$ temperature, (b) at $\mu /{\epsilon }_H=1.45$, and (c) 1.48 for different lattice sizes at $T_c$. Insets in (b): Lattice size dependence of interface tension and latent heat at $\mu /{\epsilon }_H=1.45$.

Figure 10

Finite-size scaling of (a) specific heat at $\mu /{\epsilon }_H=0.1$ and 0.05 and (b) susceptibility ${\chi }_s$ at $\mu /{\epsilon }_H=0.1$. The log-log dependencies of $C_v$ (c), ${\chi }_s$ (d), and $D_{1s},D_{2s}$ (e) vs $L$ at $T_c$. The results are fitted using formulas (5), $t=(T_c-T)/T_c$ and the background of $C_v$ is assumed to be zero. Large symbols correspond to the results of thermal averaging; lines correspond to the results obtained close to $T_c$ by the reweighted histogram method.

Figure 11

Temperature dependence of order parameters of the BL model (1 and 2), AFM BC model (3) and $p$-state planar rotator model for $p=128$ (4). The curves 1–3 are obtained for $\mu /{\epsilon }_H=0.7$ and $L=120$ and the curve 4 for $L=72$ [51].