Topological lattice actions for the 2d XY model

Abstract

We consider the 2d XY Model with topological lattice actions, which are invariant against small deformations of the field configuration. These actions constrain the angle between neighbouring spins by an upper bound, or they explicitly suppress vortices (and anti-vortices). Although topological actions do not have a classical limit, they still lead to the universal behaviour of the Berezinskii-Kosterlitz-Thouless (BKT) phase transition -at least up to moderate vortex suppression. Thus our study underscores the robustness of universality, which persists even when basic principles of classical physics are violated. In the massive phase, the analytically known Step Scaling Function (SSF) is reproduced in numerical simulations. In the massless phase, the BKT value of the critical exponent η c is confirmed. Hence, even though for some topological actions vortices cost zero energy, they still drive the standard BKT transition. In addition we identify a vortex-free transition point, which deviates from the BKT behaviour.

U(1) lattice gauge theory with a topological action

“…We have, in this specific case, not found any concrete advantages which would motivate the choice of this action over the Wilson action although at a given value of the effective coupling in the Coulomb phase there are significantly fewer monopoles (lattice artifacts). This is in line with other known cases where a topological action reduces discretization errors [10][11][12]. Perhaps the most interesting approach is to search for optimized combinations of a standard action and constrained fields.…”

“…Several studies have investigated different spin models [10][11][12], and it has been shown in analytically solvable O(N ) models that the continuum limit is that associated with the usual, sigma-model action. In higher dimensions numerical investigations also support this claim very strongly.…”

Berezinskii–Kosterlitz–Thouless transition with a constraint lattice action

“…Instead there is an enormous degeneracy, since all allowed configurations have the same action. For increasing δ a transition from a massless to a massive phase was observed [16]. In particular, at δ > ∼ δ c the correlation length could be fitted well to the behaviour analogous to relation (1.3),…”

“…On the L = 128 lattice, and δ close to -or above -δ c , we cannot check all possibilities, since N is of O(100). 16 Hence we resort to the technique of simulated annealing [43]. We start from one arbitrary pairing, measure D 2 and suggest a minimal modification by exchanging the partners among two pairs (which are randomly selected).…”

Aspects of topological actions on the lattice

“…While it is intuitively clear that the correlation length will keep increasing as the constraint selects weaker and weaker fields, it is not at all obvious which continuum theory will be approached: a gradient expansion of the lattice action is not possible, since the action is either zero or infinite. Nevertheless, it has been shown numerically [7] that in a 2d XY spin model, the same Coulomb phase is obtained as with the standard action. Here, we study the case of a 4d U(1) gauge theory, and compare the Wilson action S W = −β ∑ P cos θ P with the topological action…”