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
C4 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.
We consider the thermal phase transition from a paramagnetic to a stripe-antiferromagnetic phase in the frustrated two-dimensional square-lattice Ising model with competing interactions
J1<0 (nearest neighbor, ferromagnetic) and
J2>0 (second neighbor, antiferromagnetic). The striped phase breaks a
Z4 symmetry and is stabilized at low temperatures for
g=J2/∣J1∣>1/2. Despite the simplicity of the model, it has proved difficult to precisely determine the order and the universality class of the phase transitions. This was done convincingly only recently by Jin et al. [Phys. Rev. Lett. 108, 045702 (2012)]. Here, we further elucidate the nature of these transitions and their anomalies by employing a combination of cluster mean-field theory, Monte Carlo simulations, and transfer-matrix calculations. The
J1-
J2 model has a line of very weak first-order phase transitions in the whole region
1/2<g<g∗, where ${g}^{*}=0.67\ifmmode\pm\else\textpm\fi{}0.01$. Thereafter, the transitions from
g=g∗ to $g\ensuremath{\rightarrow}\ensuremath{\infty}$ are continuous and can be fully mapped, using universality arguments, to the critical line of the well-known Ashkin-Teller model from its four-state Potts point to the decoupled Ising limit. We also comment on the pseudo-first-order behavior at the Potts point and its neighborhood in the Ashkin-Teller model on finite lattices, which in turn leads to the appearance of similar effects in the vicinity of the multicritical point
g∗ in the
J1-
J2 model. The continuous transitions near
g∗ can therefore be mistaken for first-order transitions, and this realization was the key to understanding the paramagnetic-striped transition for the full range of
g>1/2. Most of our conclusions are based on Monte Carlo calculations, while the cluster mean-field and transfer-matrix results provide useful methodological benchmarks for weakly first-order behaviors and Ashkin-Teller criticality.
Spiral spin liquids possess a subextensively degenerate ground-state manifold, represented by a continuum of energy minima in reciprocal space. Since a small change of the spiral state wave vector requires a global change of the spin configuration in real space, it is a priori unclear how such systems can fluctuate within the degenerate ground-state manifold. Only recently it was proposed that momentum vortices are responsible for the liquidity of the spiral phase and that these systems are closely related to an emergent rank-2 U(1) gauge theory H. Yan et al. [Phys. Rev. Res. 4, 023175 (2022)]. As a consequence of this gauge structure, fourfold pinch-point singularities were found in a generalized spin correlator. In this paper, we use classical Monte Carlo and molecular dynamics calculations to embed the previously studied spiral spin liquid into a broader phase diagram of the square lattice XY model. We find a multitude of unusual phases and phase transitions surrounding the spiral spin liquid such as an effective four-state Potts transition into a collinear double-striped phase resulting from the spontaneous breaking of two coupled
Z2 symmetries. Since this phase is stabilized by entropic effects selecting the momenta away from the spiral manifold, it undergoes a second phase transition at low temperatures into a nematic spiral phase which only breaks one
Z2 symmetry. We also observe a region of parameters where the phase transition into the spiral spin liquid does not break any symmetries and where the critical exponents do not match those of standard universality classes. We study the importance of momentum vortices in driving this phase transition and discuss the possibility of a Kosterlitz-Thouless transition of momentum vortices. Finally, we explore the regime where the rank-2 U(1) gauge theory is valid by investigating the fourfold pinch-point singularities across the phase diagram.
We study the challenging thermal phase transition to stripe order in the frustrated square-lattice Ising model with couplings J(1) < 0 (nearest-neighbor, ferromagnetic) and J(2) > 0 (second-neighbor, antiferromagnetic) for g = J(2)/|J(1| > 1/2. Using Monte Carlo simulations and known analytical results, we demonstrate Ashkin-Teller criticality for g ≥ g*; i.e., the critical exponents vary continuously between those of the 4-state Potts model at g = g* and the Ising model for g → ∞. Thus, stripe transitions offer a route to realizing a related class of conformal field theories with conformal charge c = 1 and varying exponents. The transition is first order for g < g* = 0.67 ± 0.01, much lower than previously believed, and exhibits pseudo-first-order behavior for |g* ≤ g </~1.
Spiral spin liquids possess a subextensively degenerate ground-state manifold, represented by a continuum of energy minima in reciprocal space. Since a small change of the spiral state wavevector requires a global change of the spin configuration in real space, it is a priori unclear how such systems can fluctuate within the degenerate ground state manifold. Only recently it was proposed that momentum vortices are responsible for the liquidity of the spiral phase and that these systems are closely related to an emergent rank-2 U(1) gauge theory [H. Yan and J. Reuther, Phys. Rev. Research 4, 023175 (2022)]. As a consequence of this gauge structure, four-fold pinch-point singularities were found in a generalized spin correlator. In this article, we use classical Monte Carlo and molecular dynamics calculations to embed the previously studied spiral spin liquid into a broader phase diagram of the square lattice XY model. We find a multitude of unusual phases and phase transitions surrounding the spiral spin liquid such as an effective four-state Potts transition into a colinear double-striped phase resulting from the spontaneous breaking of two coupled
Z2 symmetries. Since this phase is stabilized by entropic effects selecting the momenta away from the spiral manifold, it undergoes a re-entrance phenomenon at low temperatures into a nematic spiral phase. We also observe a region of parameters where the phase transition into the spiral spin liquid does not break any symmetries and where the critical exponents do not match those of standard universality classes. We study the importance of momentum vortices in driving this phase transition and discuss the possibility of a Kosterlitz-Thouless transition of momentum vortices. Finally, we explore the regime where the rank-2 U(1) gauge theory is valid by investigating the four-fold pinch point singularities across the phase diagram
The measurements of the magnetic and nematic correlation lengths in a generalization of the two-dimensional XY model on the square lattice are presented using classical Monte Carlo simulation. The full phase diagram is reexamined based on these correlation lengths, demonstrating their power in studying generalized XY models. The ratio between the correlation length and the lattice size has distinctive behaviors which can be used to distinguish different types of phase transition. More importantly, the magnetic correlation length give more insights into the tricritical region where the paramagnetic, nematic, and quasi-long-range phases meet. It shows signatures for the intermediate region starting from the tricritical point, where the transition line is neither of the same physics as the Ising transition below nor the Berezinskii-Kosterlitz-Thouless transition far above the tricritical point.