On quantum Hall effect, Kosterlitz-Thouless phase transition, Dirac magnetic monopole, and Bohr–Sommerfeld quantization

Abstract

We addressed quantization phenomena in open systems and confined motion in low-dimensional systems, as well as quantized sources in 3-dimensions. The thesis of the paper is that if we simply cast the Bohr–Sommerfeld (B-S) quantization condition as a U(1) gauge theory, like the gauge field of Chern-Simons gauge theory or as in topological band theory (TBT) of condensed matter physics in terms of Berry connection and Berry curvature to make it self-consistent, then the quantization method in all the physical phenomena treated in this paper are unified in the sense of being traceable to the self-consistent B-S quantization. These are the stationary quantization of due to oscillatory dynamics in compactified space and time for steady-state systems (e.g., particle in a box or torus, Brillouin zone, and Matsubara time zone or Matsubara quantized frequencies), and the quantization of sources through the gauge field. Thus, the self-consistent B-S quantization condition permeates the quantization of integer quantum Hall effect (IQHE), fractional quantum Hall effect (FQHE), the Berezenskii-Kosterlitz-Thouless vortex quantization, Aharonov–Bohm effect, the Dirac magnetic monopole, the Haldane phase, contact resistance in closed mesoscopic circuits of quantum physics, and in the monodromy (holonomy) of completely integrable Hamiltonian systems of quantum geometry. In transport of open systems, we introduced a novel phase-space derivation of the quantized conductance of the IQHE based on nonequilibrium quantum transport and lattice Weyl transform approach.

On Fractional Quantum Hall Effect (FQHE): A Chern-Simons and nonequilibrium quantum transport Weyl transform approach

“…= Φ φo . This can be inferred simply by a Bohr-Sommerfeld quantization condition [8] which amounts to counting of Planck states ('pixels' of action) in phase space using Berry's curvature and connection, i.e., magnetic field and vector potential, respectively.…”

“…Moreover, this is also realization of the B-S quantization condition given in Eq. ( 6) [8]. The resulting quantization of orbital motion leads to edge states and integer quantum Hall effect under uniform magnetic fields.…”

“…Another interesting approach to FQHE, which makes use of Bohr-Sommerfeld quantization, was given by Jacak [29,30]. This approach may be related to our use here of Maxwell Chern-Simons U (1) gauge theory, which may also related to the Bohr-Sommerfeld quantization [8].…”

“…The last term indicates the operation of the curl of the Berry connection which is related to the quantization of Hall conductivity. This quantization is due to the uniqueness of the parallel-transported wavefunction, which may also have bearing on the self-consistent Bohr-Sommerfeld quantization [8].…”

“…We find that the quantization of Hall effect occurs strictly not to first order in the electric field per se but rather to first-order gradient expansion in the nonequilibrium quantum transport equation. The Berry connection and Berry curvature is the fundamental physics [8] behind the exact quantization of Hall conductance in units of e 2 h , which also happens to coincide with the source and drain contact conductance per spin in a closed circuit of mesoscopic quantum transport [5].…”