Phase diagram of a square lattice model of XY spins with direction-dependent interactions

We study a generalization of the well-known classical two-dimensional square lattice compass model of XY spins (sometimes referred to as the ${90}^{\circ }$ compass model), which interpolates between the XY model and the compass model. Our model possesses the combined $C_4$ lattice and spin rotation symmetry of the compass model but is free of its fine-tuned subsystem symmetries. Using both field theoretic arguments and Monte Carlo simulations, we find that our model possesses a line of critical points with continuously varying exponents of the Ashkin-Teller type terminating at the four-state Potts point. Further, our Monte Carlo study uncovers that beyond the four-state Potts point, the line of phase transition is connected to the lattice-nematic Ising phase transition in the square lattice compass model through a region of first-order transitions.

Figure 1

Illustration of the XY model with generic direction-dependent interactions, Eq. (2). On the sites of the square lattice is a two-component classical XY spin. The bond-direction-dependent interactions in this model are given by $J_1{S}_i^x{S}_{i+\widehat{x}}^x+J_2{S}_i^y{S}_{i+\widehat{x}}^y$ in the $x$ direction and $J_2{S}_i^x{S}_{i+\widehat{y}}^x+J_1{S}_i^y{S}_{i+\widehat{y}}^y$ in the $y$ direction. This form of interaction possesses a symmetry operation that combines both internal and spatial ${\mathcal{C}}_4$ transformations, where neither the spatial operation nor the internal operation alone are symmetries of the system.

Figure 2

The main result of this work: the phase diagram of the generalized compass model, Eq. (2). In panel (a), we show the phase diagram for our model. The critical region of the BKT phase is denoted by a solid red line, while the BKT phase transition point is indicated by a red star marked KT. The nematic phase of the compass model is represented by a solid blue line, with the nematic transition, which has been shown in previous works to be an Ising transition marked by a blue star (NI). The continuous order-disorder phase transition with continuously varying critical exponents of the AT type is depicted by a solid black line; the first-order phase transition is illustrated by a dashed black line. The decoupled Ising (DI) point is marked by a green star. Panel (b) is a closer examination of the phase diagram within the range $\alpha =0.9\sim 1$. The four-state Potts point (4P) is marked by an orange star.

Figure 3

Binder cumulants as functions of temperature for (a) magnetic order $U_m$ and (b) nematic order $U_D$ for various system sizes $L$ at $\alpha =0.5$. While $U_D$ clearly exhibits a crossing point, showing a distinct signature of a continuous phase transition at $T_c\approx 0.649\left(5\right)$, it is observed that $U_m$ has a phase transition at the same $T_c$, though a clear crossing point is absent. We interpret this as arising from crossovers from the KT physics. We note that a clear crossing point does emerge for both $U_m$ and $U_D$ at larger $\alpha$; see Fig. 6. The analysis in Fig. 4 presents alternate evidence for a direct transition in the magnetic order parameter at the same $T_c$ within errors at $\alpha =0.5$.

Figure 4

Finite-size scaling data showing evidence for a direct magnetic order-disorder transition. The data is presented as a flow diagram at (a) $\alpha =0.5$ and (b) $\alpha =0$ (2D XY Model), with $x$ axis representing the Binder cumulant $U_m$ and the $y$ axis representing the spin correlation $C_S$. Each set of connected circles represents a fixed $T$ and sizes $L=2,3,4,\cdots$, showing a numerical RG flow to fixed point values in the thermodynamic limit. (a) The flow at $T=T_c\approx 0.650$ is denoted by red solid circles. The maximum lattice size is $L_{\max }=704$ for temperatures ranging from $T=0.61$ to 0.65, while for other temperatures, $L_{\max }=448$ or smaller. The flows at temperatures lower than $T_c$ converge towards locations where $U_m=1$ and $C_S$ takes on a finite value. Conversely, flows at temperatures higher than $T_c$ converge towards locations where $U_m=0$ and $C_S=0$. This behavior indicates the presence of an order-disorder phase transition. The arrows indicate the direction of increasing $L$. Panel (b) illustrates the numerical RG flow diagram for the 2D XY model ($\alpha =0$). The flow at $T=T_{\text{KT}}$ is also marked with red solid circles. The maximum lattice size is $L_{\max }=224$. Flows at temperatures lower than $T_{\text{KT}}$ converge towards locations with finite $U_m$, while $C_S=0$, consistent with the power law KT phase.

Figure 5

Crossing data analysis of $U_D$ to obtain $T_C^I$ for $\alpha =0.5$. Since the crossing points first increase, then decrease with $L$, the scaling form Eq. (15) with two powers are used and $T_C^I=0.649\left(5\right)$ is estimated, with the fit windows $L=10\sim 128$.

Figure 6

Binder cumulants (a) $U_m$ and (b) $U_D$ as functions of temperature $T$ at $\alpha =0.9$. Both $U_m$ and $U_D$ exhibit distinct crossing points for different sizes $L$, indicating the presence of a continuous phase transition.

Figure 7

Estimate anomalous exponent $\eta$ using spin correlation. The longest distance spin correlation $C_S(L/2)$ at $T_C^{(-)}=0.644,T_C^I=0.649,T_C^{(+)}=0.654$ of $\alpha =0.5$ are shown on a log-log scale. The Monte Carlo simulation data are indicated by open circles, and the straight lines are the power law fits to the data. The power law fits are statistically sound for $T=0.649$, with $\eta =0.2427\left(2\right)$ and a reduced ${\chi }^2$ value of 1.06, over the fit window $L=24\sim 256$; for $T=0.644$, with $\eta =0.2336\left(2\right)$ and a reduced ${\chi }^2=0.955$, over the fit window of $L=24\sim 256$; for $T=0.654$ with $\eta =0.258\left(1\right)$ and a reduced ${\chi }^2$ value of 1.04, over the fit window $L=80\sim 256$.

Figure 8

The finite-size analysis of $T_C\left(L\right)$ and $1/\nu \left(L\right)$ to estimate $T_C^{III}$ and $1/\nu$ at $\alpha =0.5$. Panel (a) shows $T_C\left(L\right)$ and panel (b) shows $1/\nu \left(L\right)$ obtained by polynomial fit of the scaling function $f_1\left(t{L}^{1/\nu }\right)$ up to $P=3$ for pairs of sizes $L$ and $2L$. The solid lines are power-law fits to $T_C\left(L\right)$ and to $1/\nu \left(L\right)$, respectively.

Figure 9

Comparison of the dependence of ${\eta }_D$ on $1/\nu$ from Monte Carlo studies on the gCM model along the line of continuous phase transitions (shown as data with error bars), with the behavior expected from the Coulomb gas description of Ashkin-Teller criticality ${\eta }_D=1-\frac{1}{2\nu }$ (solid line). Note that the Coulomb gas prediction has no fit parameters! The good quantitative agreement (critical exponents at all $\alpha$ studied are at most within a few error bars from the Coulomb gas prediction) establishes the Ashkin-Teller criticality in our gCM model and the correct identification of the microscopic spin and nematic order parameters.

Figure 10

Peak value $C_{\max }$ of the specific heat (suitably scaled) for the gCM model for different $\alpha$ and a comparison of the same quantity for the four-state Potts model. For Potts criticality it is expected that asymptotically, $C_{\max }=aL{(lnL)}^{-3/2}$, where $a$ is a nonuniversal number. Therefore graphing $C_{\max }{(lnL)}^{3/2}/L$ (as done above) should result in a constant value for large $L$. From our data it is apparent that the four-state Potts point is observed around $\alpha =0.96\sim 0.97$.

Figure 11

Binder cumulant crossing points $U_m^{*}\left(L\right)$ comparison with the four-state Potts point. This figure illustrates the crossing points $(L,2L)$ of the Binder cumulant $U_m^{*}\left(L\right)$ as the size $L$ increases. Markers of different colors represent data for various values of $\alpha$. The black solid pentagon marker is the $U_{4\mathrm{P}}^{*}(L=\infty )=0.792\left(4\right)$ for the four-state Potts model [28, 29]. While reliable extrapolations to the thermodynamic limit are difficult, it is clear that in the region $\alpha \sim 0.96\pm 0.01$ our data extrapolates to values very close to the known four state Potts point, $U_{4\mathrm{P}}^{*}$, providing further evidence for the four state Potts point in this regime.

Figure 12

The dependence of $\nu$ on $\alpha$, and estimation of ${\alpha }_{\mathrm{DI}}$ and ${\alpha }_{4\mathrm{P}}$. The green points represent $\nu$ from Table 1. The dashed black horizontal lines indicate $\nu =2/3,1$ which are the universal values for the four-state Potts model and Ising model. The data are fitted using a linear function $f\left(x\right)=ax+b$ (shown as a black solid line) and a quadratic function $f\left(x\right)=a{x}^2+bx+c$ (shown as a red solid line) over the range $\alpha =0.8\sim 0.98$, which give consistent estimates. The linear fit gives $\alpha (\nu =1)={\alpha }_{\mathrm{DI}}=0.885\left(23\right)$, which gives an estimate for the point at which $H_{\mathrm{gCM}}$ can be described as a pair of decoupled Ising fixed points (marked in Fig. 2). We can locate ${\alpha }_{4\mathrm{P}}$ similarly, $\alpha (\nu =2/3)={\alpha }_{4\mathrm{P}}=0.963\left(22\right)$, in general agreement with the estimates from $C_{\max }$ and $U_m^{*}$.

Figure 13

Comparative study of Binder cumulants at (a) continuous transition at $\alpha =0.95$ and (b) first-order transition at $\alpha =0.97$. Panels (a) and (b) show the Binder cumulants $U_m$ as a function of temperature $T$ for various system sizes $L$ at $\alpha =0.95,0.97$, respectively. $U_m$ show negative peaks approaching larger negative values as the lattice size increases near the phase transition point for $\alpha =0.97$, indicating a first-order phase transition, while staying positive in the whole temperature region for $\alpha =0.95$, typical for a continuous transition.

Figure 14

2D histogram of $\vec{m}$ at $\alpha =0.97$ and $\alpha =0.5$. The $x$ axis represents $m_x$, while the $y$ axis represents $m_y$. The color in the figures represents the probability density, with brighter colors indicating higher probabilities. Panels (a) and (b) display the histograms at $\alpha =0.97$ for $L=64$ at $T=0.256$ and 128 at $T=0.255$, respectively. The histogram reveals a coexistence of ordered and disordered phases, suggesting the possibility of a first-order phase transition. Panels (c) and (d) depict the histograms at $\alpha =0.5$ for $L=64$ at $T=0.72$ and $T=0.74$, respectively. There is no phase coexistence found, indicating a continuous phase transition, in contrast to the histograms shown in panels (a) and (b).

Figure 15

Finite-size analysis of the 1D histogram at $\alpha =0.97$. The inset shows the 1D $m^2$ histogram at the temperature where the probabilities of the ordered and disordered phases are equal for various system sizes. In the main panel, the blue solid dots show $m_{\max }^2\left(L\right)$ having the maximum probability in the ordered phase in the $m^2$ histogram. The blue line is a polynomial fit to the data, giving a value $m_{\max }^2(L=\infty )=0.308$. While we are not concerned the precise value of $m_{\max }^2(L=\infty )$, our analysis unambiguously indicates it is nonzero, allowing us to infer a first-order transition for this $\alpha$.