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.
Duality places an important constraint on the renormalization group flows and the phase diagrams. For self-dual theories, the self-duality can be promoted as a symmetry, this leads to the multicriticalities. This work investigates a description of the deconfined quantum criticality, the
Nf=2QED3, as an example of self-dual theories, and its multicritical behavior under perturbative deformations. The multicriticality is described by the theory with Gross-Neveu couplings and falls in a different universality class than the standard deconfined quantum criticality. We systematically calculate the scaling dimensions of various operators in the 3D quantum electrodynamics with the Chern-Simons term and Gross-Neveu couplings by the large-
N renormalization group analysis. Specifically, we find certain Drelativistic four-fermion interactions which correspond to the dimer-dimer interactions in the lattice model will drive the deconfined quantum criticality to the first-order transition, this result is consistent with previous numerical studies.
The deconfined phase of (
2+1)-dimensional
Zk gauge theory exhibits topological order, with
e and
m anyons that have a $2\ensuremath{\pi}/k$ braiding phase. Proliferating either
e or
m drives Higgs or confinement transitions, respectively. At the multicritical point where these transitions meet, the theory enjoys an additional duality symmetry that exchanges
e and
m anyons. This symmetry forces anyons with nontrivial braiding to close their gaps simultaneously, giving rise to a critical theory that mixes strong interactions with mutual statistics. We propose an effective $\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ gauge theory with a mutual Chern-Simons term at level
k to describe the vicinity of the multicritical point for $k\ensuremath{\ge}4$. The emergence of a global $\mathrm{U}{(1)}^{\mathrm{top}}\ifmmode\times\else\texttimes\fi{}\mathrm{U}{(1)}^{\mathrm{top}}$ symmetry at the critical point imposes powerful constraints on universal properties of the phase transition. In particular, we show that (1) the lattice magnetic flux operator embeds as a conserved U(1) current with protected scaling dimension; (2) the first-order line emanating from the critical point for
k=2 disappears generically for sufficiently large
k; (3) the correlation length exponent approaches that of the three-dimensional XY model with corrections of order
1/k2 in the large-
k limit. These predictions can be tested in near-term numerical simulations and pave the way for a more general exploration of topological quantum criticality enriched with anyon-permuting symmetries.