Phase Transition of XY Model in Heptagonal Lattice

We numerically investigate the nature of the phase transition of the XY model in the heptagonal lattice with the negative curvature, in comparison to other interaction structures such as a flat two-dimensional (2D) square lattice and a small-world network. Although the heptagonal lattice has a very short characteristic path length like the small-world network structure, it is revealed via calculation of the Binder’s cumulant that the former exhibits a zero-temperature phase transition while the latter has the finite-temperature transition of the mean-field nature. Through the computation of the vortex density as well as the correlation function in the lowtemperature approximation, we show that the absence of the phase transition originates from the strong spinwave-type fluctuation, which is discussed in relation to the usual 2D XY model. A common belief in the complex network researches is that critical phenomena in networks often exhibit meanfield natures. For the small-world network structure proposed by Watts and Strogatz (WS) [1], it is well-known that statistical mechanical model systems such as the Ising and the XY models have phase transitions completely different from their counterparts in oneand two-dimensional (2D) regular lattices [2, 3]. Furthermore, the universality class of the phase transition has been known to undergo a dramatic change as soon as the density of shortcuts has a finite nonzero value, which is accompanied by the structural phase transition from the large world to the small world. These existing findings imply the importance of the topological interaction structure in the determination of the universality class of the phase transition. Recently, the Ising model in the heptagonal lattice, which has complicated geometry with a negative curvature [4], has been shown to possess the mean-field critical exponents, consistently with the common belief [5, 6]. In the present Letter, we take the one of the mostly well-studied statistical mechanical model systems, the XY model, and study the nature of the phase transition in the heptagonal lattice. The comparison with the usual flat 2D square lattice sheds a light on the role of the geometry with different curvatures, while the comparison with the small-world network structure indicates that, different from a naive expectation, the short path length cannot Fig. 1: Schematic representation of a heptagonal lattice upto the 4th level (L = 4), projected on the Poincaré disk [7]. The surrounding circle corresponds to the points at infinity, and the lattice within it is composed of congruent heptagons with respect to the negative curvature metric [8]. solely determine the universality class. Extensive MonteCarlo computations of the Binder’s cumulant and the vortex density, combined with the numerical calculation of the spin-spin correlation function, bring us to the con-