Vortex Mass in the Three-Dimensional O(2) Scalar Theory

We study the spontaneously broken phase of the $XY$ model in three dimensions, with boundary conditions enforcing the presence of a vortex line. Comparing Monte Carlo and field-theoretic determinations of the magnetization and energy density profiles, we numerically determine the mass of the vortex particle in the underlying O(2)-invariant quantum field theory. The result shows, in particular, that the obstruction posed by Derrick’s theorem to the existence of stable topological particles in scalar theories in more than two dimensions does not in general persist beyond the classical level.

Figure 1

Geometry considered for the $XY$ model below $T_c$. Boundary spins point outwards orthogonally to the vertical external surfaces. On the top and bottom surfaces they are fixed as indicated so that a vortex line (one configuration is shown) runs between the central points of these surfaces.

Figure 2

Analytic values of the magnetization (10) (continuous curves) and corresponding Monte Carlo results (data points). In order of decreasing slope at $x_1=0$, the curves refer to ($T=2.0$, $R=31$), ($T=2.1$, $R=61$), and ($T=2.18$, $R=61$). The Monte Carlo curves were obtained for $L=61$, 101, 161, respectively.

Figure 3

Analytic values of the energy density profile (14) (continuous curves) and corresponding Monte Carlo results (data points). The top and bottom profiles refer to ($T=2.0$, $R=31$) and ($T=2.16$, $R=91$), respectively. The profiles in between are obtained for $T=2.1$ and refer to $R=31$ (deeper minimum) and $R=81$. Simulations were performed with $L=91$, 101, 151 for $T=2.0$, 2.1, 2.16, respectively.