Surface worm algorithm for abelian Gauge-Higgs systems on the lattice

TL;DR

The paper introduces the surface worm algorithm (SWA), a generalization of the Prokof'ev Svistunov worm to dual surface–flux representations of abelian Gauge-Higgs models on a lattice, enabling efficient sampling even at finite chemical potential where the conventional formulation suffers from a sign problem. By applying SWA to $Z_3$ and $U(1)$ Gauge-Higgs models and comparing against a local dual-update and, when possible, the standard approach, the authors demonstrate correctness and a pronounced efficiency gain. The work provides detailed constructions of the dual representations, along with quantitative assessments of update metrics and autocorrelation times, showing SWA decorrelates faster across a wide parameter range. This approach broadens the applicability of worm-like methods to surface-based dual theories and offers a robust reference for testing other techniques such as reweighting or series expansions in gauge theories.

Abstract

The Prokof'ev Svistunov worm algorithm was originally developed for models with nearest neighbor interactions that in a high temperature expansion are mapped to systems of closed loops. In this work we present the surface worm algorithm (SWA) which is a generalization of the worm algorithm concept to abelian Gauge-Higgs models on a lattice which can be mapped to systems of surfaces and loops (dual representation). Using Gauge-Higgs models with gauge groups Z(3) and U(1) we compare the SWA to the conventional approach and to a local update in the dual representation. For the Z(3) case we also consider finite chemical potential where the conventional representation has a sign problem which is overcome in the dual representation. For a wide range of parameters we find that the SWA clearly outperforms the local update.

Figures (10)

• Figure 1: Plaquette update: A plaquette occupation number is changed by $+1$ (lhs. plot) or $-1$ (rhs.) and the fluxes at the links of the plaquette are changed simultaneously. We use a full line for an increase by +1 and a dashed line for a decrease by $-1$. The directions $1 \le \nu < \rho \le 4$ indicate the plane of the plaquette.
• Figure 2: Cube update: The plaquette occupation numbers of a 3-cube are changed according to the two patterns we show. The edges of the 3-cube are parallel to the directions $1 \leq \nu < \rho < \sigma \leq 4$.
• Figure 3: 12 of the 24 possible positive (marked with $+$) and negative segments in the $\nu$-$\rho$-plane ($\nu < \rho$). The remaining 12 segments are exactly the same but with the position of the empty and dotted links exchanged. Segments in other planes are constructed equivalently. The plaquette occupation numbers are changed as indicated by the signs. The links marked with full (dashed) lines are changed by $+1$ ($-1$). The empty link shows where the segment is attached to the worm and the dotted link is the new position of the link $L_V$ where the constraints are violated.
• Figure 4: Example of a surface worm algorithm on an initially empty lattice.
• Figure 5: This figure depicts the constraints of the dual partition function. It can be used to determine whether a positive or negative segment will be inserted by the worm: The link $L_V$ where the constraint is violated at the current step of the worm either points in $\nu$, $\rho$ or $\sigma$ direction (plots (a), (b) or (c)), and is marked by a fat link in the corresponding diagrams. Both the old and the new plaquette are attached to the link and need to be identified in the corresponding plot. If they both are surrounded by the same type of line (full versus dashed) the sign of the change of the plaquette variable remains the same, otherwise an extra factor ($-1$) is taken into account.