We formulate a collective field approach to a planar nonrelativistic system of bosons interacting via the Chern-Simons interaction. Adding the attractive $\ensuremath{\delta}$ potential this system can be solved and the vortex soliton appears. The ground state of this system is scale and inversion invariant. In the harmonic well this system is not stable against the creation of vortices. Leading and next to leading terms in the
N1 expansion of the ground-state energy are calculated.
Based on the observation that a particle motion in one dimension maps to a two-dimensional motion of a charged particle in a uniform magnetic field, constrained in the lowest Landau level, we formulate a system of one-dimen- sional nonrelativistic fermions by using a Chern-Simons field theory in 2+1 dimensions. Using a hydrodynamical formulation we obtain a two-dimensional droplet picture of one-dimensional fermions. The dynamics involved is that of the boundary between a uniform density of particles and vortices. We use the sharp boundary approximation. In the case of well separated boundaries we derive the one-dimensional collective field Hamiltonian. Symmetries of the theory are also discussed as properties of curves in two dimensions.
Abelian Chern-Simons-Maxwell theory can emerge from the bosonization of the (2+1)-dimensional Thirring model that describes interacting Dirac fermions. Here we show how the Thirring model manifests itself in the low energy limit of a two-dimensional tight-binding model of spinless fermions. To establish that, we employ a modification of Haldane's model, where the "doubling" of fermions is rectified by adiabatic elimination. Subsequently, fermionic interactions are introduced that lead to the analytically tractable Thirring model. By local density measurements of the lattice fermions we can establish that for specific values of the couplings the model exhibits the confining (2+1)-dimensional QED phase or a topological ordered phase that corresponds to the Chern-Simons theory. The implementation of the model as well as the measurement protocol are accessible with the current technology of cold atoms in optical lattices.
Here, we provide a simple Hubbard-like model of spin-
1/2 fermions that gives rise to the SU(2)-symmetric Thirring model that is equivalent, in the low-energy limit, to the Yang-Mills-Chern-Simons model. First, we identify the regime that simulates the SU(2) Yang-Mills theory. Then, we suitably extend this model so that it gives rise to the SU(2) Chern-Simons theory with level $k\ensuremath{\ge}2$ that can support non-Abelian anyons. This is achieved by introducing multiple fermionic species and modifying the Thirring interactions, while preserving the SU(2) symmetry. Our proposal provides the means to theoretically and experimentally probe non-Abelian SU(2) level
k topological phases.
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.
Topological gauge theories provide powerful effective descriptions of certain strongly correlated systems, a prime example being the Chern-Simons gauge theory of fractional quantum Hall states.Engineering topological gauge theories in controlled quantum systems is of both conceptual and practical importance, as it would provide access to systems with exotic excitations such as anyons without the need for strong correlations.Here, we discuss a scheme to engineer the chiral BF theory, a minimal model of a topological gauge theory corresponding to a one-dimensional reduction of the Chern-Simons theory, with ultracold atoms.Using the local conservation laws of the theory, we encode its quantum Hamiltonian into an ultracold quantum gas with chiral interactions.Building on a seminal proposal by Edmonds et al. [Phys. Rev. Lett. 110, 085301 (2013)], we show how to implement it in a Raman-coupled Bose-Einstein condensate with imbalanced scattering lengths, as we have recently realized experimentally [Frölian et al., Nature (London) 608, 293 (2022)]. We discuss the properties of the chiral condensate from a gauge theory perspective and assess the validity of the effective quantum description for accessible experimental parameters via numerical simulations.Our approach lays the foundation for realizing topological gauge theories in higher dimensions with Bose-Einstein condensates.