We revisit the issue of worldline formulations for the q-state Potts model and discuss a worldline representation in arbitrary dimensions which also allows for magnetic terms. For vanishing magnetic field we implement a Hodge decomposition for resolv
ing the constraints with dual variables, which in two dimensions implies self-duality as a simple corollary. We present exploratory 2-d Monte Carlo simulations in terms of the worldlines, based on worm algorithms. We study both, vanishing and non-zero magnetic field, and explore q between q = 2 and q = 30, i.e., Potts models with continuous, as well as strong first order transitions.
We study 2d U(1) gauge Higgs systems with a $theta$-term. For properly discretizing the topological charge as an integer we introduce a mixed group- and algebra-valued discretization (MGA scheme) for the gauge fields, such that the charge conjugation
symmetry at $theta = pi$ is implemented exactly. The complex action problem from the $theta$-term is overcome by exactly mapping the partition sum to a worldline/worldsheet representation. Using Monte Carlo simulation of the worldline/worldsheet representation we study the system at $theta = pi$ and show that as a function of the mass parameter the system undergoes a phase transition. Determining the critical exponents from a finite size scaling analysis we show that the transition is in the 2d Ising universality class. We furthermore study the U(1) gauge Higgs systems at $theta = pi$ also with charge 2 matter fields, where an additional $Z_2$ symmetry is expected to alter the phase structure. Our results indicate that for charge 2 a true phase transition is absent and only a rapid crossover separates the large and small mass regions.
We simulate the 2d U(1) gauge Higgs model on the lattice with a topological angle $theta$. The corresponding complex action problem is overcome by using a dual representation based on the Villain action appropriately endowed with a $theta$-term. The
Villain action is interpreted as a non-compact gauge theory whose center symmetry is gauged and has the advantage that the topological term is correctly quantized so that $2pi$ periodicity in $theta$ is intact. Because of this the $theta = pi$ theory has an exact $Z_2$ charge-conjugation symmetry $C$, which is spontaneously broken when the mass-squared of the scalars is large and positive. Lowering the mass squared the symmetry becomes restored in a second order phase transition. Simulating the system at $theta = pi$ in its dual form we determine the corresponding critical endpoint as a function of the mass parameter. Using a finite size scaling analysis we determine the critical exponents and show that the transition is in the 2d Ising universality class, as expected.
