$XY$ model with antinematic interaction

We consider the $XY$ model with ferromagnetic (FM) and antinematic (AN) nearest-neighbor interactions on a square lattice for a varying interaction strength ratio. Besides the expected FM and AN quasi-long-range order (QLRO) phases we identify at low temperatures another peculiar canted ferromagnetic (CFM) QLRO phase, resulting from the competition between the collinear FM and noncollinear AN ordering tendencies. In the CFM phase neighboring spins that belong to different sublattices are canted by a nonuniversal (dependent on the interaction strength ratio) angle and the ordering is characterized by a fast-decaying power-law intrasublattice correlation function. Compared to the FM phase, in the CFM phase correlations are significantly diminished by the presence of zero-energy domain walls due to the inherent degeneracy caused by the AN interactions. We present the phase diagram as a function of the interaction strength ratio and discuss the character of the respective phases as well as the transitions between them.

Figure 1

(a) Ground-state spin angles and (b) the corresponding energies per spin pair, obtained analytically (black solid curves) and from numerical optimization (light blue symbols). The dashed lines in (b) correspond to unstable solutions for the FM and AN states, within $J\in (0,0.8)$, and the red crosses to the energy per spin pair obtained from MC simulations at the lowest considered temperature.

Figure 2

Examples of spin configurations (left column) and local energy distributions (right column) in the CFM phase at $T=0.01$ for [(a) and (b)] $J=0.1$, [(c) and (d)] $J=0.2$, [(e) and (f)] $J=0.6$, and [(g) and (h)] $J=0.8$.

Figure 3

Temperature dependencies of (a) the specific heat $c$, (b) the vortex density $\rho$, (c) the magnetization $m_1$, (d) the nematic order parameter $m_2$, (e) the magnetic susceptibility ${\chi }_1$, and (f) the nematic susceptibility ${\chi }_2$, for various values of $J$.

Figure 4

Temperature dependencies of [(a) and (b)] the specific heat $c$; [(c) and (d)] the susceptibility ${\chi }_1$, corresponding to $m_1$ (insets); and [(e) and (f)] the susceptibility ${\chi }_2$, corresponding to $m_2$ (insets), for various lattice sizes $L$, with $J=0.6$ (left column) and $J=0.7$ (right column).

Figure 5

Phase diagram in the $J\text{−}T$ plane. FM, AN, CFM, and P denote, respectively, the ferromagnetic, antinematic, canted ferromagnetic, and paramagnetic phases. Empty (filled) symbols represent transition points determined from the specific heat peaks (correlation function analysis).

Figure 6

Correlation functions (a) $g_1$ and (b) $g_2$, for $J=0.5$ and $T=0.01$. The inset in (a) schematically depicts the first five distance lags.

Figure 7

Correlation functions (a) $g_1^d$ and (b) $g_1^s$, obtained for different values of $J$ at $T=0.01$.

Figure 8

(a) Temperature dependencies of the critical exponents ${\eta }_1^d$ (empty symbols) and ${\eta }_1^s$ (filled symbols), obtained for different values of $J$, and (b) the corresponding adjusted coefficients of determination $R^2$. The dash-dotted curves in (a) for $J=0.9$ and 1 correspond to the spin-wave approximation.

Figure 9

(a) Normalized autocorrelation function, $A\left(k\right)$, and the integrated autocorrelation time of the magnetization, ${\tau }_{\mathrm{int},M_1}$, close to the transition point at $T=0.0648$ for $L=120$ and $J=0.4$. (b) The FSS of the magnetic susceptibility at the AN-CFM phase boundary for $J=0.4$. The cyan squares correspond to data from 10 to 15 independent standard MC (SMC) runs and the magenta circles from the reweighting (RMC) method.