Contact attraction drives the self-generated vortex lattice of a trapped anyon gas towards self-duality, but integrated energies do not reveal where the remaining departure from it is located. We address this question in the almost-bosonic Chern--Simons--Schrödinger average-field model using a positive, gauge-invariant local defect that measures the covariant kinetic energy in excess of the magnetic bound. Numerically computed low-energy states at three statistical fluxes show that, as the stability line is approached, the integrated defect falls by about two orders of magnitude while the density modulation grows, so that magnetic saturation and density uniformity separate. The residual defect concentrates in a vortex-free outer shell that carries unscreened statistical flux. There it reduces to a smooth radial mismatch between the statistical flow and the density decay, which the interior lattice cannot cancel, whereas the covariant kinetic and amplitude-gradient energies show no comparable enrichment. The effect is robust to grid and box refinement, while its magnitude depends on flux and on the low-energy branch. Exact dilation and contact-response identities relate the squared energy to the density shape and provide consistency checks. Repulsion produces the opposite trend: a smoother density with a larger defect.
In this work we consider the
N-body Hamiltonian describing the microscopic structure of a quantum gas of almost-bosonic anyons. This description includes both extended magnetic flux and spin-orbit/soft-disk interaction between the particles which are confined in a scalar trapping potential. We study a physically well-motivated ansatz for a sequence of trial states, consisting of Jastrow repulsive short-range correlations and a condensate, with sufficient variational freedom to approximate the ground state (and possibly also low-energy excited states) of the gas. In the limit
N→∞, while taking the relative size of the anyons to zero and the total magnetic flux
2πβ to remain finite, we rigorously derive the stationary Chern-Simons-Schrödinger/average-field-Pauli effective energy density functional for the condensate wave function. This includes a scalar self-interaction parameter
γ which depends both on
β, the diluteness of the gas, and the spin-orbit coupling strength
g, but becomes independent of these microscopic details for a particular value of the coupling
g=2 in which supersymmetry is exhibited (on all scales, both microscopic and mesoscopic) with
γ=2π∣β∣. Our findings confirm and clarify the predictions we have found in the physics literature.
I consider a gas of ``free'' anyons with statistical parameter \ensuremath{\delta}, with hard cores, on a two-dimensional square lattice. Using a recently derived Jordan-Wigner transformation, I map this problem onto a gas of fermions on the same lattice coupled to a Chern-Simons gauge theory with coupling \ensuremath{\theta}=1/2\ensuremath{\delta}. At the semiclassical level, the system is found to be equivalent to a gas of fermions, with the same density, in an average effective magnetic field \ensuremath{\rho}/\ensuremath{\theta}. I consider the case in which an integer number of the Landau bands of the saddle-point problem are completely filled. If \ensuremath{\delta}=\ensuremath{\pi}/m and the density \ensuremath{\rho}=r/q, with m, r, and q integers, the system is a superfluid, provided that q is larger than twice the largest common factor of m and r. If q is even and the system is half filled, the state may be either a superfluid or a quantum Hall phase. For all other values of \ensuremath{\rho} and \ensuremath{\delta}, compatible with integer filling of the Landau bands, the system is in a quantum Hall phase. The dynamical stability of the superfluid state is ensured by the topological invariance of the quantized Hall conductance of the fermion problem. I find a close analogy between anyon superconductivity and the Schwinger mechanism. The effective Lagrangian for the low-energy modes coupled to the electromagnetic field is derived. The energies of fermion and flux states are logarithmically divergent, but finite for the anyon state. The system has flux quantization, a zero-temperature Hall effect with a quantized Hall conductance, Meissner effect, charged vortices, screening with induced magnetic fields for static charges, and different masses for the longitudinal and transverse components of the electromagnetic field.
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.
Electric-magnetic self-duality of the
Z2 gauge theory, realized microscopically as a half-lattice-translation exchanging electric charge and magnetic flux, has been an influential example of a duality symmetry with an exact lattice realization. We construct the first non-Abelian generalization of this construction: a lattice model of the
S3 quantum double
D(S3) on a tensor product Hilbert space in which the
Z2em anyon-permutation symmetry, exchanging the non-Abelian chargeon
C and fluxon
F, is realized via lattice translation. Consequently we find that the zigzag boundary termination of the model realizes, without fine-tuning, a gapless critical edge state described by the tetracritical Ising CFT. The bulk admits three independent
Z2em-preserving bosonic perturbations, driving
D(S3) into distinct gapped phases. We analyze these transitions by three independent methods: category-theoretic anyon condensation, microscopic lattice Hamiltonians, and Chern-Simons-Higgs theory, which all agree, yielding a unified picture. These examples motivate a minimal-condensation principle: proliferating a bosonic anyon generically drives condensation of a minimal condensable algebra containing it, with symmetry-related condensates appearing as degenerate vacua that spontaneously break the anyon-permutation symmetry. Our model construction extends to an infinite family of self-dual dihedral quantum doubles
D(D2n). Notably, each model is sign-problem-free, opening the door to large-scale numerical exploration of the phases of non-Abelian Chern-Simons-Higgs theories.