The author considers the quantum mechanics coupled to the gauge field which has the Chern-Simons term as the action. The generalized Hamiltonian formalism of the system is constructed and canonical and path-integral quantizations are performed. In solving the Gauss law constraint, the gauge field is represented by a multivalued function and the charge density operator of the particle. The gauge potential is singular and also induces a singular magnetic field through the magnetic flux is finite. By a phase transformation, the singular gauge field is eliminated. Then the transformed wavefunction absorbing the gauge field induces a novel phase factor under 2 pi -rotation in space. Thus the author shows that the transformed wavefunction describes exotic spin state. The extension of these analyses to the relativistic case is also presented.
Frustrated spin systems on kagome lattices have long been considered to be a promising candidate for realizing exotic spin-liquid phases. Recently, there has been a lot of renewed interest in these systems with the discovery of materials such as volborthite and herbertsmithite that have kagomelike structures. In the presence of an external magnetic field, these frustrated systems can give rise to magnetization plateaus of which the plateau at
m=31 is considered to be the most prominent. Here, we study the problem of the antiferromagnetic spin-
21 quantum XXZ Heisenberg model on a kagome lattice by using a Jordan-Wigner transformation that maps the spins onto a problem of fermions coupled to a Chern-Simons gauge field. This mapping relies on being able to define a consistent Chern-Simons term on the lattice. Such a lattice Chern-Simons term had previously only been written for the square lattice and was used to successfully study the unfrustrated Heisenberg antiferromagnet on the square lattice. At a mean-field level, these ideas have also been applied to frustrated systems by ignoring the details of the Chern-Simons term. However, fluctuations are generally strong in these models and are expected to affect the mean-field physics. Using a recently developed method to rigorously extend the Chern-Simons term to the frustrated kagome lattice, we can now formalize the Jordan-Wigner transformation on the kagome lattice. We then discuss the possible phases that can arise at the mean-field level from this mapping and focus specifically on the case of
31 filling (
m=31 plateau) and analyze the effects of fluctuations in our theory. We show that in the regime of
XY anisotropy, the ground state at the
31 plateau is equivalent to a bosonic fractional quantum Hall Laughlin state with filling fraction
21 and that at the
95 plateau it is equivalent to the first bosonic Jain daughter state at filling fraction
32.
We propose a novel mechanism for the protection against spin gapped states in doped antiferromagnets. It requires the presence of a Chern-Simons term that can be generated by a fermion-gauge field coupling. We first demonstrate that in the presence of this term the vortex loop excitations of the spin sector behave as anyons with fractional statistics. To generate such term, the fermions should have massive Dirac spectrum coupled to the emergent gauge field of the spin sector. The Dirac spectrum can be realized by a planar spin configuration arising as the lowest-energy configuration of a square lattice antiferromagnet Hamiltonian involving a Dzyaloshinskii-Moriya interaction. The mass is provided by a combination of dimerization and staggered chemical potential. We finally show that for realistic parameters, anyonic vortex loop condensation will never occur and thus the spin gapped state is prevented. We also propose real magnetic materials for an experimental verification of our theory.