Ground state of a spin-1/2 charged particle in a two-dimensional magnetic field

Abstract

It is investigated that the structure of the kernel of the Dirac–Weyl operator D of a charged particle in the magnetic-field B=B0+B1, given by the sum of a strongly singular magnetic field B0(⋅)=Σνγνδ(⋅−aν) with some singular points aν and a magnetic-field B1 with a bounded support. Here the magnetic field B1 may have some singular points with the order of the singularity less than 2. At a glance, it seems that, following “Aharonov–Casher Theorem” [Phys. Rev. A 19, 2461 (1979)], the dimension of the kernel of D, dim ker D, is a function of one variable of the total magnetic flux (=Σνγν+∫R2B1dxdy) of B. However, since the influence of the strongly singular points works, dim ker D indeed is a function of several variables of the total magnetic flux and each of γν’s.

Conductance fluctuations and disorder induced ν=0 quantum Hall plateau in topological insulator nanowires

“…1 ii) we plot the band structure as a function of the momentum k along the length of the wire, for a magnetic field of B = 4 T. For small momenta k we obtain flat bands that correspond to Landau levels in the top and bottom surfaces; the corresponding probability density |ψ| 2 shown in panel iii), for the points c and f marked in panel ii), is localized in the top and bottom surfaces. The energies of these bulk Landau levels are consistent with that of the half integer quantum Hall effect of Dirac fermions 30,51,52 : E n = sgn (n) 2eB|n|v F . The Landau level center shifts with increasing momentum towards the side surfaces and eventually the modes become dispersive The density of state ρ as a function of the energy E in a disordered, 660nm long nanowire exposed to a 4 T magnetic field for three different disorder strengths g.…”

Zero Modes in a System of Aharonov–bohm Fluxes

“…papers [79,80,81,82]). In the case when the considered magnetic field has a "regular" component in addition to the magnetic field of Aharonov-Bohm solenoids the appearance of zero modes has been analyzed in [83,84]. The results of [84] are applicable also to the case when an infinite number of Aharonov-Bohm solenoids is present in the system but the total magnetic flux is necessarily finite (moreover, after some gauge transformation the total variation of the flux must be finite).…”

On the Dirac and Pauli Operators with Several Aharonov–Bohm Solenoids

“…The Dirac operator with strongly singular magnetic field has been studied before in [6,7,10,15,20]. In [15] a formula for the dimension of the kernel of the Dirac operator was proved for two different asymmetric self-adjoint extensions (i.e. those with different behavior of spin-up and spin-down components), and it was demonstrated that, in fact, this dimension may differ for self-adjoint realizations, although, each of them seems to be quite natural.…”

“…The Dirac operator with strongly singular magnetic field has been studied before in [6,7,10,15,20]. In [15] a formula for the dimension of the kernel of the Dirac operator was proved for two different asymmetric self-adjoint extensions (i.e.…”