Coherent states in the symmetric gauge for graphene under a constant perpendicular magnetic field

Abstract

In this work we describe semiclassical states in graphene under a constant perpendicular magnetic field by constructing coherent states in the Barut–Girardello sense. Since we want to keep track of the angular momentum, the use of the symmetric gauge and polar coordinates seemed the most logical choice. Different classes of coherent states are obtained by means of the underlying algebra system, which consists of the direct sum of two Heisenberg–Weyl algebras. The most interesting cases are a kind of partial coherent states and the coherent states with a well-defined total angular momentum.

Densities and currents for the coherent states \(\varPi _{m,\alpha }\)

The expressions for the probability densities $\rho _{m,\alpha }$ in the coherent states (96) are

$$\begin{aligned} \rho _{m,\alpha }(\xi ,\theta )= & {} \varPi ^{\dagger }_{m,\alpha } \varPi _{m,\alpha } =\frac{1}{2\exp (\vert {\tilde{\alpha }}\vert ^2)-1}\left[ \vert g_m(\xi )\vert ^2\right. \\&\left. +\left| \sum _{n=m+1}^{\infty }\frac{\left( {\tilde{\alpha }}z\right) ^n}{n!}L_{m}^{n-m}(\xi ^2)f_m(\xi )\right| ^2\right. \\&+\left| \sum _{n=m+1}^{\infty }\frac{\left( {\tilde{\alpha }} z\right) ^n}{n!}\frac{\sqrt{n}}{z}\, L_{m}^{n-m-1}(\xi ^2)f_m(\xi )\right| ^2\\&+2\,\text {Re}\left[ \sum _{n=m+1}^{\infty }\frac{({\tilde{\alpha }}z)^n}{n!}f_m(\xi )g_m^*(\xi )\, L_{m}^{n-m}(\xi ^2)\right] \\&+(1-\delta _{0m})\left( \left| \sum _{n=1}^{m}\left( -\frac{{\tilde{\alpha }} }{z^*}\right) ^nL_{n}^{m-n}(\xi ^2)g_m(\xi )\right| ^2\right. \\&\left. +\left| \sum _{n=1}^{m}\left( -\frac{{\tilde{\alpha }}}{z^*}\right) ^n\frac{z^*}{\sqrt{n}}\, L_{n-1}^{m-n+1}(\xi ^2)g_m(\xi )\right| ^2\right. \\&+2\, \text {Re}\left[ \sum _{n=1}^{m}\left( -\frac{{\tilde{\alpha }}}{z^*}\right) ^nL_n^{m-n}(\xi ^2)\vert g_m(\xi )\vert ^2\right. \\&\left. +\left( \sum _{n'=1}^{m}\left( -\frac{{\tilde{\alpha }}^*}{z}\right) ^{n'}L_{n'}^{m-n'}(\xi ^2)g_m^*(\xi )\right) \right. \\&\times \left( \sum _{n=m+1}^{\infty }\frac{\left( {\tilde{\alpha }}z\right) ^n}{n!}\, L_{m}^{n-m}(\xi ^2)f_m(\xi )\right) \\&-\left( \sum _{n'=1}^{m}\left( -\frac{{\tilde{\alpha }}^*}{z}\right) ^{n'}\frac{1}{\sqrt{n'}}\, L_{n'-1}^{m-n'+1}(\xi ^2)g_m^*(\xi )\right) \\&\times \left. \left. \left. \left( \sum _{n=m+1}^{\infty }\frac{\left( {\tilde{\alpha }}z\right) ^n}{n!}\sqrt{n}\, L_{m}^{n-m-1}(\xi ^2)f_m(\xi )\right) \right] \right) \right] . \end{aligned}$$

For the current densities $j_{m,\alpha ,\vec {u}}$ in the coherent states (96) we get

$$\begin{aligned} j_{m,\alpha ,\vec {u}}(\xi )= & {} ev_{\mathrm{F}}\ \varPi ^{\dagger }_{m,\alpha } \,(\vec {\sigma }\cdot \vec {u})_k\, \varPi _{m,\alpha } \\= & {} \frac{2ev_{\mathrm{F}}}{2\exp (\vert \alpha \vert ^2)-1}\, \text {Re}\left[ i(-i)^{k}e^{-i\theta }\left\{ \sum _{n=m+1}^{\infty }\frac{({\tilde{\alpha }}^*z^*)^{n}}{n!}\frac{\sqrt{n}}{z^*}\, L_{m}^{n-m-1}(\xi ^2)f_m^*(\xi )g_m(\xi )\right. \right. \\&+\left( \sum _{n'=m+1}^{\infty }\frac{({\tilde{\alpha }}z)^{n'}}{n'!}\, L_{m}^{n'-m}(\xi ^2)f_m(\xi )\right) \left( \sum _{n=m+1}^{\infty }\frac{({\tilde{\alpha }}^*z^*)^{n}}{n!}\frac{\sqrt{n}}{z^*}\, L_{m}^{n-m-1}(\xi ^2)f_m(\xi )\right) \\&-(1-\delta _{0m})\left( \sum _{n=1}^{m}\left( -\frac{{\tilde{\alpha }}^*}{z}\right) ^n\frac{z}{\sqrt{n}}\, L_{n-1}^{m-n+1}(\xi ^2)\vert g_m(\xi )\vert ^2\right. \\&+\left( \sum _{n'=1}^{m}\left( -\frac{{\tilde{\alpha }}}{z^*}\right) ^{n'}L_{n'}^{m-n'}(\xi ^2)g_m(\xi )\right) \left( \sum _{n=1}^{m}\left( -\frac{{\tilde{\alpha }}^*}{z}\right) ^n\frac{z}{\sqrt{n}}\, L_{n-1}^{m-n+1}(\xi ^2)g_m^*(\xi )\right) \\&-\left( \sum _{n'=1}^{m}\left( -\frac{{\tilde{\alpha }}}{z^*}\right) ^{n'} L_{n'}^{m-n'}(\xi ^2)g_m(\xi )\right) \left( \sum _{n=m+1}^{\infty }\frac{({\tilde{\alpha }}^*z^*)^{n}}{n!}\frac{\sqrt{n}}{z^*}\, L_{m}^{n-m-1}(\xi ^2)f_m^*(\xi )\right) \\&+\left. \left. \left. \left( \sum _{n'=m+1}^{\infty }\frac{({\tilde{\alpha }}z)^{n'}}{n'!}\, L_{m}^{n'-m}(\xi ^2)f_m(\xi )\right) \left( \sum _{n=1}^{m}\left( -\frac{{\tilde{\alpha }}^*}{z}\right) ^{n}\frac{z}{\sqrt{n}}\, L_{n-1}^{m-n+1}(\xi ^2)g_m^*(\xi )\right) \right) \right\} \right] . \end{aligned}$$

In all the cases z is the complex parameter defined in Eq. (76) and

$$\begin{aligned} f_{m}(\xi )=\sqrt{\frac{m!}{2\pi \ell _{\mathrm{B}}^{2}}}\,(-z)^{-m}\exp \left( -\frac{1}{2}\xi ^2\right) , \qquad g_{m}(\xi )=\sqrt{\frac{1}{2\pi \ell _{\mathrm{B}}^{2}\,m!}}\,z^{*m}\exp \left( -\frac{1}{2}\xi ^2\right) . \end{aligned}$$

Some plots of the functions $\rho _{m,\alpha }$ and $j_{m,\alpha ,\vec {u}}$ can be seen on Fig. 7.