318
E. Díaz-Bautista et al.
their physical properties, e.g., their stiffness, strength, and optical conductivity by
manipulation of their mechanical properties emerged, as in straintronics [8].
On the other hand, coherent states (CSs) were conceived by Schrödinger [9]
as the most classical states that describe the motion of a particle in a quadratic
potential. Although it is not always possible to obtain these states, they have been
used in many branches of physics, as optics, atomic, nuclear, condensed matter, and
particle physics (see Ref. [10] and references therein). For the harmonic oscillator,
the CSs are constructed from the ladder operators a and a † that together with the
identity operator are generators of the Heisenberg–Weyl (HW) algebra. This algebra
can be f -deformed by replacing them with [11]
A = af (N ) = f (N + 1)a, A
†
= f (N)a
†
= a
† f (N + 1), N = a
† a,
(1)
where f is a well-behaved real function of the number operator N , verifying
[N, A] = −A, [N, A
†
] = A
† , [A, A
+
] = (N + 1)f
2 (N + 1) − Nf
2 (N).
(2)
Thus, nonlinear coherent states (NLCSs) are introduced as eigenstates of A|α f =
α|α f [11]. f (N) is selected to guarantee that such states belong to the Hilbert
space. NLCS are physically realized as stationary states of the center-of-mass
motion of a trapped ion [12] or the vibrations of polyatomic molecules [13]. By
generalizing the results in [14] to anisotropic 2DDMs, we aim to obtain a semiclassical description of the effects of magnetic fields and anisotropy in physical
properties of these materials [10, 15–18].
For that purpose, this contribution is organized as follows. In Sect. 2 the
anisotropic 2D Dirac equation is solved analytically. In Sect. 3 a generalized
annihilation operator is presented and the NLCSs are constructed as its eigenstates.
We obtain their probability density and the Heisenberg uncertainty relation (HUR).
In Sect. 4 we present our conclusions.
2 Anisotropic 2D Dirac Hamiltonian
The isotropic 2D Dirac Hamiltonian H = v F
σ · ·
p, where
σ = (σ x , σ y ) are the
Pauli matrices and
p is the canonical momentum, may be modified either because
the material is inherently anisotropic or it has been mechanically deformed, yielding
a Fermi velocity v F which is no longer isotropic. The anisotropic 2D Dirac equation
(see [19] and references therein) in a magnetic field is
H Ψ (x, y) = =
σ ·
↔
v ·
Π Ψ (x, y) = (v xx σ x π x + v yy σ y π y )Ψ (x, y) = EΨ (x, y),
(3)
where
↔
v is the 2 × 2 symmetric Fermi velocity tensor with non-vanishing diagonal
components v xx and v yy (see Fig. 1) and π x,y = p x,y + eA x,y , with
A denoting the
E. Díaz-Bautista et al.
their physical properties, e.g., their stiffness, strength, and optical conductivity by
manipulation of their mechanical properties emerged, as in straintronics [8].
On the other hand, coherent states (CSs) were conceived by Schrödinger [9]
as the most classical states that describe the motion of a particle in a quadratic
potential. Although it is not always possible to obtain these states, they have been
used in many branches of physics, as optics, atomic, nuclear, condensed matter, and
particle physics (see Ref. [10] and references therein). For the harmonic oscillator,
the CSs are constructed from the ladder operators a and a † that together with the
identity operator are generators of the Heisenberg–Weyl (HW) algebra. This algebra
can be f -deformed by replacing them with [11]
A = af (N ) = f (N + 1)a, A
†
= f (N)a
†
= a
† f (N + 1), N = a
† a,
(1)
where f is a well-behaved real function of the number operator N , verifying
[N, A] = −A, [N, A
†
] = A
† , [A, A
+
] = (N + 1)f
2 (N + 1) − Nf
2 (N).
(2)
Thus, nonlinear coherent states (NLCSs) are introduced as eigenstates of A|α f =
α|α f [11]. f (N) is selected to guarantee that such states belong to the Hilbert
space. NLCS are physically realized as stationary states of the center-of-mass
motion of a trapped ion [12] or the vibrations of polyatomic molecules [13]. By
generalizing the results in [14] to anisotropic 2DDMs, we aim to obtain a semiclassical description of the effects of magnetic fields and anisotropy in physical
properties of these materials [10, 15–18].
For that purpose, this contribution is organized as follows. In Sect. 2 the
anisotropic 2D Dirac equation is solved analytically. In Sect. 3 a generalized
annihilation operator is presented and the NLCSs are constructed as its eigenstates.
We obtain their probability density and the Heisenberg uncertainty relation (HUR).
In Sect. 4 we present our conclusions.
2 Anisotropic 2D Dirac Hamiltonian
The isotropic 2D Dirac Hamiltonian H = v F
σ · ·
p, where
σ = (σ x , σ y ) are the
Pauli matrices and
p is the canonical momentum, may be modified either because
the material is inherently anisotropic or it has been mechanically deformed, yielding
a Fermi velocity v F which is no longer isotropic. The anisotropic 2D Dirac equation
(see [19] and references therein) in a magnetic field is
H Ψ (x, y) = =
σ ·
↔
v ·
Π Ψ (x, y) = (v xx σ x π x + v yy σ y π y )Ψ (x, y) = EΨ (x, y),
(3)
where
↔
v is the 2 × 2 symmetric Fermi velocity tensor with non-vanishing diagonal
components v xx and v yy (see Fig. 1) and π x,y = p x,y + eA x,y , with
A denoting the