Quantum phase transitions of the Dirac oscillator in a minimal length scenario

Abstract

We obtain exact solutions of the (2+1) dimensional Dirac oscillator in a homogeneous magnetic field within a minimal length (∆x0 = √ β), or generalized uncertainty principle (GUP) scenario. This system in ordinary quantum mechanics has a single left-right chiral quantum phase transition (QPT). We show that a non zero minimal length turns on a infinite number of quantum phase transitions which accumulate towards the known QPT when β → 0. It is also shown that the presence of the minimal length modifies the degeneracy of the states and that in this case there exist a new class of states which do not survive in the ordinary quantum mechanics limit β → 0.

Exact Solutions of the (2+1)-Dimensional Dirac Oscillator under a Magnetic Field in the Presence of a Minimal Length in the Non-commutative Phase Space

“…Firstly, by adopting the same procedure that used by Menculini et al [29], we have solved the problem only in the case of NC space. The results found are in well agreement with those obtained in the literature.…”

“…In this section, we have studied the solutions of the two-dimensional DO with or without an external magnetic field by using the same way described in [29]: the authors work within a momentum space representation of the Heisenberg algebra, and by an appropriate transformation, the problem is identified as a Kummer differential equation where the solutions are well known. The solutions that we have found are in wellagreement with those obtained in the literature.…”

“…The electromagnetic potential associated with the DO has been found by Benitez et al [5]. The DO has attracted a lot of interest because not only it provides one of the examples of the Dirac equation exact solvability but also for its numerous physical applications [6][7][8][9][10]. Recently, Franco-Villafane et al [11] exposed the proposal of the first experimental microwave realisation of the onedimensional DO.…”

“…In this context, many papers were published where a different quantum system in space with the Heisenberg algebra was studied. They are the Abelian Higgs model [19], the thermostatics with minimal length [20], the one-dimensional hydrogen atom [21], the casimir effect in minimal length theories [22], the effect of minimal lengths on electron magnetism [23], the DO in one and three dimensions [24][25][26][27][28], the noncommutative (NC) (2+1)-dimensional DO and quantum phase transition [10], the solutions of a two-dimensional Dirac equation in the presence of an external magnetic field [29], the NC phase space Schrödinger equation [30], and the Schrödinger equation with harmonic potential in the presence of a magnetic field [31].…”

Symmetry operators and separation of variables in the (2 + 1)-dimensional Dirac equation with external electromagnetic field

“…Exact solutions of the (2 + 1)-dimensional Dirac equation for an external field in the presence of a minimal length were studied by Menculini et al [11,12].…”