Mass and Charge of the Quantum Vortex in the $(2+1)$-d $O(2)$ Scalar Field Theory

Vortices are topological excitations that arise in superfluids, superconductors, and Bose-Einstein condensates [1]. In three spatial dimensions vortices are linedefects, including cosmic strings [2], that sweep out a world-sheet during their time-evolution, while in two dimensions vortices are point-defects. In the 2-d classical XY model, they drive the Berezinskii-Kosterlitz-Thouless phase transition [3, 4]. Popov was first to note that vortices and phonons in a (2 + 1)-d superfluid are dual to charged particles and photons in scalar QED [5]. He concluded that the mass of a vortex corresponds to its rest energy divided by the square of the speed of sound. Duan found the mass to diverge logarithmically with the volume, but attributed a finite mass to the vortex core [6]. According to Baym and Chandler, the core-mass corresponds to the mass of the superfluid within the core [7]. An equivalent concept, the Kopnin-mass exists for superconductors and fermionic superfluids [8–11]. Thouless and Anglin studied the reaction of a vortex to an external force by a pinning potential by means of the Gross-Pitaevskii equation [12]. They confirmed that the vortex mass receives an infinite contribution. A recent Monte Carlo study in the (2 + 1)-d O(2) model, using boundary conditions that break translation invariance, concluded that the vortex mass is finite in the infinite volume limit [13], thus contraditing the previous results. A finite vortex mass in (2 + 1)-d is also inconsistent with the fact that vortex loops and global cosmic strings in the (3 + 1)-d O(2) model, which are again surrounded by massless Goldstone bosons, have a tension that increases logarithmically with the string length [14, 15].