Self-dual criticality in three-dimensional $\mathbb{Z}_2$ gauge theory with matter

Abstract

The simplest topologically ordered phase in 2+1D is the deconfined phase of Z2 gauge theory, realized for example in the toric code. This phase permits a duality that exchanges electric and magnetic excitations ("e" and "m" particles). Condensing either particle while the other remains gapped yields a phase transition with 3D Ising exponents. More mysterious, however, is the transition out of the deconfined phase when self-duality symmetry is preserved. If this transition is continuous, which has so far been unclear, then it may be the simplest critical point for which we still lack any useful continuum Lagrangian description. This transition also has a soft matter interpretation, as a multicritical point for classical membranes in 3D.We study the self-dual transition with Monte Carlo simulations of the Z2 gauge-Higgs model on cubic lattices of linear size L ≤ 96. Our results indicate a continuous transition: for example, cumulants show a striking parameter-free scaling collapse. We estimate scaling dimensions by using duality symmetry to distinguish the leading duality-odd/duality-even scaling operators A and S. All local operators have large scaling dimensions, making standard techniques for locating the critical point ineffective. We develop an alternative using "renormalization group trajectories" of cumulants. We check that two-and three-point functions, and temporal correlators in the Monte-Carlo dynamics, are scale-invariant, with scaling dimensions xA and xS and dynamical exponent z.We also give a picture for emergence of 1-form symmetries, in some parts of the phase diagram, in terms of "patching" of membranes/worldsurfaces. We relate this to the percolation of anyon worldlines in spacetime. Analyzing percolation yields a fourth exponent for the self-dual transition. We propose variations of the model for further investigation.

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

“…This is sketched in figure 3. A similar phenomenon is known to occur for Z 2 gauge theory in 3d, where it is believed that two continuous Ising lines meet at a novel self-dual critical point [34].…”

Mean string field theory: Landau-Ginzburg theory for 1-form symmetries

“…Within this formalism it should be possible to explicitly derive the functional form of this crossover. Note that as r → 0 the scale of this crossover is pushed out to arbitrarily large areas, again showing that the deformation is generically irrelevant at the critical point (see a recent discussion of this point on the lattice at [31]).…”

Persistent Homology of $\mathbb{Z}_2$ Gauge Theories

“…Such a model possesses two distinct deconfinement transitions characterized by the condensation of e particles (for h X 1) and m particles (for h Z 1). In the 2+1D membrane picture and the σ x basis, h X can be viewed as a parameter for the surface tension of membranes (since σ x is a string tension term) and h Z can be viewed as a parameter for the frequency of holes in the membranes (since σ z is a generator for pairs of e particles) [25]. In this case, we expect that the frequency and persistence values for H 1 and H 2 homologies may serve as useful tools for detecting and distinguishing the deconfinement transitions.…”