$XY$ model with competing higher-order interactions

We study effects of competing pairwise higher-order interactions (HOI) with alternating signs and exponentially decreasing intensity on critical behavior of the $XY$ model. It is found that critical properties of such a generalized model can be very different from the standard $XY$ model and can strongly depend on whether the number of HOI terms is odd or even. Inclusion of any odd number of HOI terms results in two consecutive phase transitions to distinct ferromagnetic quasi-long-range order phases. Even number of HOI terms leads to two phase transitions only if the decay of the HOI intensities is relatively slow. Then the high-temperature transition to the ferromagnetic phase is followed by another transition to a peculiar competition-induced canted ferromagnetic phase. In the limit of an infinite number of HOI terms only one phase transition is confirmed, and under the conditions of fierce competition between the even and odd terms the transition temperature can be suppressed practically to zero.

Figure 1

Potential functions for the cases of (a) $p\to \infty$ for several values of $\alpha$, (b) the fixed $\alpha =1.01$ and various odd values of $p$, (c) the fixed $\alpha =1.2$ and various even values of $p$, and (d) the fixed $p=2$ and various values of $\alpha$. In panels (c) and (d), $p^{*}=20$ and ${\alpha }^{*}=2$ denote transition values from single- to double-well shape.

Figure 2

(a) SW approximation of the reduced correlation function exponent ${\eta }_{\mathrm{eff}}/{\eta }_{XY}$, shown as a function of (a) $\alpha$ for different $p$ and (b) $p$ for different $\alpha$. The solid and dashed curves represent the cases of even and odd values of $p$, respectively. In panel (a) the vertical broken lines denote the respective limiting values ${\alpha }^{*}\left(p\right)$ and the inset shows for even $p$ the area of $J_{\mathrm{eff}}>0$ in which ${\eta }_{\mathrm{eff}}$ is defined.

Figure 3

Temperature dependencies of (a), (b) the internal energy per spin and (c), (d) the specific heat for (a), (c) even and (b), (d) odd values of $p$, with $\alpha =1.01$ and $L=48$. The black symbols corresponding to the internal energy at $T=0$ shown in panels (a) and (b) are the GS values obtained by the global minimization of the Hamiltonian on the entire lattice.

Figure 4

Temperature dependencies of (a), (c) the generalized magnetizations $m_i$ and (b), (d) the generalized susceptibilities ${\chi }_i$, $i=1,\cdots ,p$, for (a), (b) $p=2$, (c), (d) $p=3$ and the values of $\alpha =1.01$ and $L=48$.

Figure 5

(a) Mean value of the standard deviation of spin angle distributions as a function of temperature, for $p=3$ with $\alpha =1.01$ and $L=48$. The inset in panel (a) shows the angle distribution in the phase ${\mathrm{F}}_0$ (at $T=0.35$) and the snapshot in panel (b) their spatial distribution on the lattice. Panels (c) and (d) show the angle distribution and the snapshot in the phase ${\mathrm{F}}_1$ (at $T=0.10$).

Figure 6

Phase diagrams in $\left(\alpha \text{−}T\right)$ planes for (a) even and (b) odd values of $p$, estimated from maxima of the specific heat curves for $L=48$. ${\mathrm{F}}_0$, ${\mathrm{F}}_1$, and CF denote two different ferromagnetic and canted ferromagnetic phases, respectively. The arrows in panel (a) show the values of ${\alpha }^{*}$ at which CF phase terminates for different $p$.

Figure 7

Temperature dependencies of (a) the specific heat and (b) the magnetization $m_1$, for $p\to \infty$, several values of $\alpha$ and $L=48$.

Figure 8

(a) FSS behavior of $m_1$ at very low temperatures. The inset shows the temperature dependence of the correlation function critical exponent $\eta$ and the adjusted coefficient of determination $R^2$ [27]. (b) The blue circles show the phase boundary for $p\to \infty$ as a function of the parameter $\alpha$, separating the ${\mathrm{F}}_0$ and paramagnetic phases. The (pseudo)transition temperatures are obtained from maxima of the specific heat curves shown in Fig. 7. The red star represents the FSS estimation of the true value at $\alpha =1.01$. For reference, the black squares show the phase boundary for the case of only positive coupling constants $J_k$. The filled squares represent the first-order transition points and the dashed line the transition temperature of the standard $XY$ model.