Title: Magnetic properties of the spiral spin liquid and surrounding phases in the square lattice XY model

Abstract: 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 $\mathbb{Z}_2$ 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