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Coherent States of Graphene Layer with and without a PT-symmetric Chemical Potential

Fabio Bagarello, Francesco Gargano, Lidia Saluto

Abstract

In this paper we construct different classes of coherent and bicoherent states for the graphene tight-binding model in presence of a magnetic field, and for a deformed version where we include a $\mathcal{P}\mathcal{T}$-symmetric chemical potential $V$. In particular, the problems caused by the absence of a suitable ground state for the system is taken into account in the construction of these states, for $V=0$ and for $V\neq0$. We introduce ladder operators which work well in our context, and we show, in particular, that there exists a choice of these operators which produce a factorization of the Hamiltonian. The role of broken and unbroken $\mathcal{P}\mathcal{T}$-symmetry is discussed, in connection with the strength of $V$.

Coherent States of Graphene Layer with and without a PT-symmetric Chemical Potential

Abstract

In this paper we construct different classes of coherent and bicoherent states for the graphene tight-binding model in presence of a magnetic field, and for a deformed version where we include a -symmetric chemical potential . In particular, the problems caused by the absence of a suitable ground state for the system is taken into account in the construction of these states, for and for . We introduce ladder operators which work well in our context, and we show, in particular, that there exists a choice of these operators which produce a factorization of the Hamiltonian. The role of broken and unbroken -symmetry is discussed, in connection with the strength of .

Paper Structure

This paper contains 10 sections, 118 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Coherent states for graphene with $V=0$
  3. Coherent states
  4. A different decomposition of $\mathcal{H}_2$
  5. Working with tensor products
  6. Chemical potential: $0<V<1$
  7. A different class of bicoherent states
  8. Chemical potential: $V>1$
  9. Gain and loss phenomena for large $V$
  10. Conclusions

Figures (7)

  • Figure 1: (a) Schematic representation of the action of operators $\mathcal{A}_{2},\mathcal{A}^\dagger_{2}$ on the vectors $c_{n,p}$ (b) Schematic representation of the action of operators $\mathcal{B}_{2},\mathcal{B}^\dagger_{2}$ on the vectors $c_{n,p}$.
  • Figure 2: (a) Schematic representation of the action of the operators $\mathfrak{A}_{\mathcal{K}}(V),\mathfrak{B}_{\mathcal{K}}(V)$ on the vectors $x_{n,p}$ (b) Schematic representation of the action of the operators $\mathfrak{A}^\dagger_{\mathcal{K}}(V),\mathfrak{B}^\dagger_{\mathcal{K}}(V)$ on the vectors $y_{n,p}$.
  • Figure 3: Probability density for $|\Phi^+_\mathcal{A}(z_1,z_2)|^2$ (a) for its first component $|\Phi^+_\mathcal{A}(z_1,z_2)\{1\}|^2$ (b) and its second component $|\Phi^+_\mathcal{A}(z_1,z_2)\{2\}|^2$ (c) . Parameters are $V=0$ and $z_1=0,z_2=1-i$.
  • Figure 4: Probability density for $|\varphi^+(z_1,z_2)|^2$ (a) for its first component $|\varphi^+(z_1,z_2)\{1\}|^2$ (b) and its second component $|\varphi^+(z_1,z_2)\{2\}|^2$ (c) . Parameters are $V=9.5$ and $z_1=0,z_2=1-i$.
  • Figure 5: Probability density for $|\psi^-(z_1,z_2)|^2$ (a) for its first component $|\psi^-(z_1,z_2)\{1\}|^2$ (b) and its second component $|\psi^- (z_1,z_2)\{2\}|^2$ (c) . Parameters are $V=9.5$ and $z_1=0,z_2=1-i$.
  • ...and 2 more figures