Quantum Monte Carlo study of molecular polarization and antiferroelectric ordering in squaric acid crystals

Effects of geometrical frustration and quantum fluctuation are theoretically investigated for the proton ordering in a quasi-two-dimensional hydrogen-bonded system, namely a squaric acid crystal. We elucidate the phase diagram for an effective model, the transverse-field Ising model on a frustrated checkerboard lattice, by using quantum Monte Carlo simulation. A crossover to a liquidlike paraelectric state with well-developed molecular polarizations is identified, distinguishably from long-range ordering. The emergence of long-range order from the liquidlike state exhibits peculiar aspects originating from the lifting of quasimacroscopic degeneracy, such as colossal enhancement of the transition temperature and a vanishingly small anomaly in the specific heat.

Figure 1

(a) Schematic picture of a layer of H${}_2$SQ molecules. The dashed lines represent hydrogen bonds connecting C${}_4$O${}_4$ units, and small circles on the bonds show protons. (b) Pseudospin representation of the proton displacement in (a). $J_1$, $J_2$, and $J_3$ denote the interactions in Eq. (1). (c) An example of intermediate liquidlike states in which every molecule bears a polarization but the system is globally disordered. (d) A ferroelectrically ordered state. In (c) and (d), the bold arrows in the center of the plaquettes represent the molecular polarizations. See the text for details.

Figure 2

QMC results of (a),(e) $P$ and ${\chi }_P$ (${\chi }_P$ is divided by 1000 and 500, respectively); (b),(f) $\rho$ and ${\chi }_{\rho }$; and (c),(g) $C/T$ for the system sizes $N=4L^2$ ranging from $L=24$ to 36. The data are at (a)–(c) $J_3=0.002$ and (e)–(g) $J_3=0.004$. (d) and (h) show the Binder parameter of $P$ in the vicinity of $T_c$ for $J_3=0.002$ and $0.004$, respectively. Insets of (c),(g) show the low-$T$ behaviors of $C/T$. All the results are calculated at $J_2=2$ along the axis with $\Gamma /T=\tan (\pi /6)$. See the text for details.

Figure 3

$T$ dependences of (a) $P$ and $\rho$, and (b) ${\chi }_P/2000$ and ${\chi }_{\rho }$, measured along various $\Gamma /T$. $\Gamma /T$ is parametrized by $\theta$ (degree) with $\Gamma /T=\tan \theta$. All the results are calculated at $J_2=2$ and $J_3=0.002$ for $N=4\times {36}^2$ sites.

Figure 4

Phase diagrams obtained for varying $J_3$. Each diagram for (a)–(d) corresponds to $J_3=0$, 0.002, 0.004, and 0.008, respectively. The solid circles show $T_c$ obtained by the Binder parameter of $P$ and triangular (square) points are $T^{*}$ evaluated from $C/T$ (${\chi }_{\rho }$). The dotted line shows the phase boundary of the ordered state and the paraelectric state, and the intermediate state is assigned as the region between $T_{{\chi }_{\rho }}^{*}$ and $T_c$. The gradation shows the intensity of $\rho$.