Narrow Results

Electron hopping models on the honeycomb lattice are studied. The lattice consists of two triangular sublattices, and it is non-Bravais. The dual space has non-trivial topology. The gauge fields of Bloch electrons have the U(1) symmetry and thus represent superconducting states in the dual space. Two quantized Abrikosov fluxes exist at the Dirac points and have fluxes 2π and -2π, respectively. We define the non-Abelian SO(3) gauge theory in the extended 3d dual space and it is shown that a monopole and anti-monoplole solution is stable. The SO(3) gauge group is broken down to U(1) at the 2d boundary. The Abrikosov fluxes are related to quantized Hall conductance by the topological expression. Based on this, monopole confinement and deconfinement are discussed in relation to time reversal symmetry and QHE. The Jahn–Teller effect is briefly discussed.

We develop a basic formulation of the spin (SU(2)) coherent state path integrals based not on the conventional highest or lowest weight vectors but on arbitrary fiducial vectors. The coherent states, being defined on a 3-sphere, are specified by a full set of Euler angles. They are generally considered as states without classical analogues. The overcompleteness relation holds for the states, by which we obtain the time evolution of general systems in terms of the path integral representation; the resultant Lagrangian in the action has a monopole-type term à la Balachandran et al. as well as some additional terms, both of which depend on fiducial vectors in a simple way. The process of the discrete path integrals to the continuous ones is clarified. Complex variable forms of the states and path integrals are also obtained. During the course of all steps, we emphasize the analogies and correspondences to the general canonical coherent states and path integrals that we proposed some time ago. In this paper we concentrate on the basic formulation. The physical applications as well as criteria in choosing fiducial vectors for real Lagrangians, in relation to fictitious monopoles and geometric phases, will be treated in subsequent papers separately.

From the viewpoint of the SU(2) coherent states (CS) and their path integrals (PI) labeled by a full set of Euler angles (ϕ, θ, ψ) which we developed in the previous paper, we study the relations between gauge symmetries of Lagrangians and allowed quantum states; We investigate permissible types of fiducial vectors (FV) in the full quantum dynamics in terms of SU(2) CS for typical Lagrangians. We propose a general framework for a Lagrangian having a certain gauge symmetry with respect to one of the Euler angles ψ. We find that for the case FV are so restricted that they belong to the eigenstates of Ŝ₃ or to the orbits of them under the action of the SU(2); And the strength of a fictitious monopole, which appears in the Lagrangian, is a multiple of ½. In this case Dirac strings are permitted. Our formulations and results deepen those of the preceding work by Stone that has piloted us; We illustrate the relation between the two methods. The reasoning here does not work for a Lagrangian without the gauge symmetry. This suggests a new possibility about monopole charge quantization. Besides analogies to field theory and entanglements in quantum information (QI) are briefly mentioned.