Phase diagram and Ashkin-Teller universality in the classical square-lattice Heisenberg-compass model

Abstract

We determine the finite-temperature phase diagram and critical behavior of the classical square-lattice Heisenberg-compass model using large-scale Monte Carlo simulations and finite-size scaling. Six symmetry distinct ordered phases are identified. The four phases that simultaneously break the spin-lattice $C_4$ and in-plane spin-inversion symmetries undergo continuous transitions in the Ashkin-Teller universality class, with the associated critical lines terminating at four-state Potts points, beyond which the transitions become first order. In contrast, the two $z$-polarized phases display conventional two-dimensional Ising criticality. Our results reveal how the interplay between Heisenberg exchange and compass anisotropy organizes these distinct critical regimes, thereby completing the characterization of the model's thermal phase transitions.

Figures (9)

• Figure 1: (a) Finite-temperature phase diagram of the classical Heisenberg--compass model in the $(t,\phi)$ plane, showing six ordered phases: x-y Néel, Stripe $\parallel$, z FM, x-y FM, Stripe $\perp$, and z AFM. Stripe$\parallel$ (Stripe$\perp$) denotes stripe states with spins aligned parallel (perpendicular) to the stripe direction. The low-temperature order parameters as functions of $\phi$ are summarized in Appendix \ref{['app:order']}. Cyan dots mark points where only thermal crossovers occur, and the corresponding crossover temperatures are extracted from the broad specific-heat maximum Deng2025. In particular, the cyan dots at $\phi=0$ and $\phi=\pi$ correspond to explicitly $O(3)$-symmetric limits, while those at $\phi=\pi-\tan^{-1}(2)$ and $\phi=2\pi-\tan^{-1}(2)$ possess a hidden $O(3)$ symmetry under Klein duality PhysRevB.110.104426. As the $O(3)$-symmetric limits are approached, the anisotropy-driven transition temperature is expected to vanish asymptotically ($t_c \to 0$) in the thermodynamic limit. Dashed cyan lines only serve as guides to the crossover trajectories. At $\phi=\pi/2$ and $3\pi/2$ (orange), the model reduces to the pure compass limit with an Ising-type transition PhysRevLett.93.207201. (b) Enlarged view near $\phi\!\approx\!\pi/2$: solid dark magenta lines denote continuous Ashkin--Teller transitions (with a $\phi$-dependent exponent $\nu$), the bright magenta stars mark the four-state Potts points, and dashed dark magenta lines indicate first-order transitions.
• Figure 2: Finite-size scaling of the Binder cumulants at $\phi = 0.35\pi$ across the transition into the x-y Néel phase. (a,c) Nematic ($U_N$) and magnetic ($U_M$) cumulants exhibit crossings at a common critical temperature $t_c$ (zoomed-in insets highlight the crossing regions). (b,d) Corresponding scaling collapses according to $U = F[(t - t_c)L^{1/\nu}]$ give a consistent exponent $\nu = 1.70(5)$. A shallow negative dip in $U_N$ is attributed to strong fluctuations of the bond-based nematic order parameter, which can produce nonmonotonic Binder behavior even for a continuous transition, unlike the pronounced negative minimum that deepens systematically with $L$ in the first-order regime discussed later.
• Figure 3: Finite-size scaling of the longest-distance correlations at $\phi = 0.35\pi$. (a) Magnetic anomalous dimension $\eta_M$ extracted from $C_M$ for system sizes $L = 48$–$160$, yielding $\eta_M = 0.214(2),\,0.240(3),\,0.273(3),\,0.321(6)$ at $t = 0.57,\,0.575,\,0.58,\,0.585$, respectively. (b) Nematic anomalous dimension $\eta_N$ extracted from $C_N$ over the same system sizes, giving $\eta_N = 0.64(1),\,0.74(2),\,0.88(2),\,1.06(4)$.
• Figure 4: Correlation-length exponent $\nu$ along the continuous phase boundaries of the x-y Néel and Stripe $\parallel$ phases, extracted from finite-size scaling of Binder cumulants. For $\phi>\pi$, the values coincide with those at $\phi-\pi$. The continuous variation of $\nu$ with $\phi$ is consistent with the Ashkin--Teller universality class.
• Figure 5: Identification of the four-state Potts critical point. (a) Specific heat $C(t)$ for several system sizes at $\phi = 0.485\pi$. (b) Scaling of the specific-heat peak $C_{\mathrm{max}}(L)$, plotted as $C_{\mathrm{max}}(\ln L)^{3/2}/L$, together with the same quantity for the four-state Potts model (black symbols). (c) Binder cumulant $U_N$ for several lattice sizes $L$ at $\phi = 0.485\pi$, with the crossings yielding the critical temperature $t_c = 0.2508(3)$. (d) Scaling collapse of $U_N$ yields a correlation-length exponent $\nu = 0.67(2)$, consistent with the four-state Potts value.