Phase structure of self-dual lattice gauge theories in 4d

TL;DR

We investigate phase structure of self-dual U(1) lattice gauge theories in 4d using a modified Villain action that treats electric and magnetic matter on equal footing. A lattice self-duality with coupling relation $\tilde{\beta}=1/(4\pi^2\beta)$ and a central-extension symmetry group $G$ is derived, and a dual worldline Monte Carlo is employed to study the bosonic one-flavor theory on self-dual lines. Numerically, two transitions are found along the self-dual line: a first-order transition at $J_1$ separating photon and Higgs/confined phases, and a second-order 4d Ising-type transition at $J_2$ signaling restoration of the self-dual symmetry through a Gaussian fixed point; a triple point connects these regimes on the self-dual line. For larger flavor content, mixed ’t Hooft anomalies constrain the phase structure and can allow or forbid trivially gapped phases, with RG analysis indicating a possible second-order fixed point for $N_f\ge 183$, suggesting potential interacting CFTs. Overall, the work highlights how self-duality and 1-form anomalies shape the nonperturbative phase diagram of abelian gauge theories on the lattice and points toward intriguing continuum limits and fixed points.

Abstract

We discuss U(1) lattice gauge theory models based on a modified Villain formulation of the gauge action, which allows coupling to bosonic electric and magnetic matter. The formulation enjoys a duality which maps electric and magnetic sectors into each other. We propose several generalizations of the model and discuss their 't~Hooft anomalies. A particularly interesting class of theories is the one where electric and magnetic matter fields are coupled with identical actions, such that for a particular value of the gauge coupling the theory has a self-dual symmetry. The self-dual symmetry turns out to be a generator of a group which is a central extension of $\mathbb Z_4$ by the lattice translation symmetry group. The simplest case amenable to numerical simulations is the case when there is exactly one electrically and one magnetically charged boson. We discuss the phase structure of this theory and the nature of the self-dual symmetry in detail. Using a suitable worldline representation of the system we present the results of numerical simulations that support the conjectured phase diagram.

Figures (13)

• Figure 1: Examples for the action of the $\star$-operator in 2d (left) and 3d (right).
• Figure 2: A cartoon of the phase diagram in the $J$, $\log(2\pi\beta)$ plane.
• Figure 3: The conjectured phase diagram for sufficiently large $N_f^e=N_f^m=N_f$. The limits $\beta\sim 0$ or $\beta\sim +\infty$ can have a 2nd order phase transition if $N_f\ge 183$, as discussed in Sec. \ref{['sec:limits']}.
• Figure 4: Vacuum expectation value (top row), susceptibility (middle row) and Binder cumulant (bottom row) for the gauge field order parameter $M_g$ (lhs. column) and the matter order parameter $M_m$ (rhs. column). The results are for the self-dual value $\beta = \beta^* \equiv 1/2\pi$ and are plotted as a function of $J$. We compare different volumes as indicated in the legends.
• Figure 5: The gauge field order parameter (lhs.) and the matter order parameter (rhs.) as a function of the rescaled gauge coupling $\ln (2\pi \beta)$, for values across the self-dual value $\beta^* \equiv 1/2\pi$. The matter field coupling is set to $J = 0.6$ here.