Quantum phase transition in the chirality of the (2+1)-dimensional Dirac oscillator

Abstract

We study the (2+1)-dimensional Dirac oscillator in the presence of an external uniform magnetic field (B). We show how the change of the strength of B leads to the existence of a quantum phase transition in the chirality of the system. A critical value of the strength of the external magnetic field (B c ) can be naturally defined in terms of physical parameters of the system.While for B = B c the fermion can be considered as a free particle without defined chirality, for B < B c (B > B c ) the chirality is left (right) and there exist a net potential acting on the fermion.For the three regimes defined in the quantum phase transition of chirality, we observe that the energy spectra for each regime is drastically different. Then, we consider the z-component of the orbital angular momentum as an order parameter that characterizes the quantum phase transition.

Quantum phase transitions in the noncommutative Dirac oscillator

“…We point out that in previous discussions of the left-right chiral phase transition of the Dirac oscillator in a constant homogeneous magnetic field the presence of the classes of solutions with opposite angular momentum average (with finite degeneracy in the left phases and infinite degeneracy in the right phase) was apparently overlooked [28,[35][36][37][38][39][40].…”

“…Our explicit solutions in the three chiral phases (left-right-left) show that for every energy level (with the exception of the singlet-like zero modes) there is a class of states with finite degeneracy and opposite average angular momentum L z relative to the class of states with infinite degeneracy. Apparently, these classes of states with finite degeneracy in the left phases and infinite degeneracy in the right phase, were previously overlooked and the sign of the angular momentum was proposed as an order parameter of the quantum phase transition [28,[35][36][37][38][39][40].…”

“…It is interesting to note that a further interaction in the form of a homogeneous magnetic field can still be incorporated in the Dirac oscillator keeping the system still exactly solvable [28,[35][36][37][38]] and this combined system has quite interesting properties. In some recent papers it has been shown that for this combined system there is a chirality phase transition if the magnitude of the magnetic field either exceeds or is less than a critical value B cr (which also depends on the oscillator strength) [39,40]. A consequence of this chirality phase transition is that the spectrum is different for B > B cr and B < B cr , B being the magnetic field strength.…”

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

“…In particular the Dirac oscillator model, one of the few relativistic systems exactly solvable [8][9][10][11][12][13][14], has recently attracted attention [15,16] also in view of the possible applications to the physics of graphene [17]. In [17] the authors conjecture that in graphene a Dirac oscillator coupling may arise as a consequence of the effective internal magnetic field generated by the motion of the charge carriers in the planar hexagonal lattice of the carbon atoms.…”

“…This model also turns out to be exactly solvable and we find explicit analytic solutions discussing in detail the degeneracy with respect to the angular momentum quantum number. In this context it may be recalled that this system in ordinary quantum mechanics (no minimal length or β → 0 limit) is characterized by a left-right chiral quantum phase transition at a given critical value (B cr ) of the external magnetic field [16,19]. Such a phase transition is characterized, apart from other things, by the spectrum which is different for B ≷ B cr .…”