Quantum phase transitions in the noncommutative Dirac oscillator

We study the (2 + 1)-dimensional Dirac oscillator in a homogeneous magnetic field in the noncommutative plane. It is shown that the effect of noncommutativity is twofold: (i) momentum noncommuting coordinates simply shift the critical value $\left(B_{\text{cr}}\right)$ of the magnetic field at which the well known left-right chiral quantum phase transition takes place (in the commuting phase); (ii) noncommutativity in the space coordinates induces a new critical value of the magnetic field, $B_{\text{cr}}^{*}$, where there is a second quantum phase transition (right-left): this critical point disappears in the commutative limit. The change in chirality associated with the magnitude of the magnetic field is examined in detail for both critical points. The phase transitions are described in terms of the magnetization of the system. Possible applications to the physics of silicene and graphene are briefly discussed.

Figure 1

Energy eigenvalues of the positive branch $E_N^{+}$, in units of $m{c}^2$, of the ground state, solid line (blue) and of the first few excited states, dashed (red), dotted (light green) and dot-dashed (green), as a function of the dimensionless ratio $\xi =\hslash (\tilde{\omega }-\omega )/\left(m{c}^2\right)$. This is achieved combining the expressions derived in the text in Eqs. (31), (50), (62). There are two values of the magnetic field (or $\tilde{\omega })$ for which there is a quantum phase transition (a point of nonanalyticity in the energy eigenvalues). In the commutative limit $\left(\eta ,\theta \to 0\right)$ ${\omega }_{\eta }\to 0$ and ${\omega }_{\theta }\to \infty$ and the spectrum goes smoothly into the one discussed in [39] with only one phase transition at $\xi =0$.

Figure 2

The magnetization of the energy levels of the positive branch of the spectrum $E_N^{+}$ is shown for the ground state, solid line (blue) and first few excited energy levels, dashed (red), dotted (light green), and dot-dashed (green), in units of the Bohr magneton ${\mu }_B=\frac{e\hslash }{2m_{e}c}$ (and assuming $m_e/m=1)$, as a function of the dimensionless ratio $\xi =\hslash (\tilde{\omega }-\omega )/\left(m{c}^2\right)$. This is achieved combining the expressions derived in the text in Eqs. (31), (50), (62). We note that at the two critical points $(\xi =\frac{\hslash {\omega }_{\eta }}{m{c}^2}$ and $\xi =\frac{\hslash {\omega }_{\eta }}{m{c}^2})$ the magnetization presents a discontinuity for each energy level. We remark also that there is a particular value of the magnetic field, which corresponds to the midpoint of the two critical points at which the magnetization vanishes for each energy level.