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On a relation of the angular frequency to the Aharonov–Casher geometric phase in a quantum dot
Annals of Physics 372 (2016) 457–467
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On a relation of the angular frequency to the Aharonov–Casher geometric phase in a quantum dot P.M.T. Barboza, K. Bakke ∗ Departamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008, 58051-900, João Pessoa-PB, Brazil
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Article history: Received 23 February 2016 Accepted 2 June 2016 Available online 16 June 2016 Keywords: Aharonov–Casher effect Geometric quantum phases Quantum dot Persistent spin current Linear plus Coulomb-type potential Biconfluent Heun function
By analysing the behaviour of a neutral particle with permanent magnetic dipole moment confined to a quantum dot in the presence of a radial electric field, Coulomb-type and linear confining potentials, then, an Aharonov–Bohm-type effect for bound states and a dependence of the angular frequency of the system on the Aharonov–Casher geometric phase and the quantum numbers associated with the radial modes, the angular momentum and the spin are obtained. In particular, the possible values of the angular frequency and the persistent spin currents associated with the ground state are investigated in two different cases. © 2016 Elsevier Inc. All rights reserved.
1. Introduction In mesoscopic systems, the dependence of the energy levels on geometric quantum phases [1–4] is a well-known quantum effect in the literature [5–18]. A particular case is the dependence of the spectrum of energy on the Aharonov–Bohm geometric quantum [19] when the movement of an electron is restricted to two concentric cylinders [20,21], or an one-dimensional ring [22], or two-dimensional rings [15–17,23] without interacting with the magnetic field. This particular case is called as the Aharonov–Bohm effect for bound states [24]. Further, analogues of the Aharonov–Bohm effects for bound states have been investigated in mesoscopic systems [5–8] associated with the Berry phase [1] and the Aharonov–Anandan quantum phase [2]. With respect to neutral particles,
∗
Corresponding author. E-mail address: [email protected] (K. Bakke).
http://dx.doi.org/10.1016/j.aop.2016.06.005 0003-4916/© 2016 Elsevier Inc. All rights reserved.
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Aharonov–Bohm-type effects for bound states have been explored in quantum rings [9–14,25,26] due to the interaction of the permanent magnetic dipole moment of the neutral particle with an electric field that gives rise to the Aharonov–Casher effect [27]. The objective of this work is to investigate Aharonov–Bohm-type effects for bound states [24] that arise in a two-dimensional quantum dot [15–17] due to the interaction of the permanent magnetic dipole moment of a neutral particle with a radial electric field under the influence of scalar potentials. By searching for bound state solutions, we show that the energy levels depend on the Aharonov–Casher geometric quantum phase [27] and there exists a restriction of the values of the angular frequency of the system, where the possible values of this angular frequency depend on the quantum numbers associated with the radial modes, the angular momentum and the spin, and the Aharonov–Casher geometric phase [27]. Due to this dependence of the energy levels on the geometric quantum phase, we show that persistent spin currents [9–14,18] can arise in the two-dimensional quantum dot. A particular contribution to the persistent spin currents in the quantum dot comes from the dependence of the angular frequency on the geometric quantum phase. The structure of this paper is as follows: in Section 2, we investigate an Aharonov–Bohm-type effect for bound states [24] or an Aharonov–Casher effect for bound states [26] that arises in a twodimensional quantum dot [15–17] under the influence of a linear scalar potential. In particular, we obtain the persistent spin currents associated with the ground state of the system; in Section 3, we investigate an Aharonov–Casher effect for bound states [26] in a quantum dot under the influence of a Coulomb-type and a linear scalar potentials. As a particular case, we show that the possible values of the angular frequency associated with the ground state of the system are determined by a third degree algebraic equation in order that bound state solutions can be obtained; in Section 4, we present our conclusions. 2. Influence of a linear scalar potential In this section, we investigate quantum effects on a neutral particle with a permanent magnetic dipole moment confined to a quantum dot under the influence of a linear scalar potential when the magnetic dipole moment of the neutral particle interacts with an electric field. As shown in Refs. [25,27–31], the quantum dynamics of a neutral particle with a permanent magnetic dipole moment that interacts with electric and magnetic fields is described by (with h¯ = c = 1):
πˆ 2 µ2 E 2 µ ⃗ ⃗ ∂ψ = ψ− ψ+ ∇ · E ψ + µ σ⃗ · B⃗ ψ + V ψ, (1) ∂t 2m 2m 2m where σ i are the Pauli matrices that satisfy the relation σ i σ j + σ j σ i = 2 ηij , ηij = diag(+ + +) and V is a scalar potential. Besides, the operator πˆ is defined as 1 3 πˆ k = −i∂k − σ δϕ k + µ σ⃗ × E⃗ . (2) k 2ρ i
The second term of the right-hand-side of Eq. (2) stems from the curvilinear coordinates system (cylindrical coordinates) adopted [25,31,32]. In quantum field theory in curved spacetime, this term stems from the spinorial connection [25,33]. Now, let us introduce the Aharonov–Casher effect [27]. In 1984, Aharonov and Casher [27] showed a particular case of the interaction between the permanent magnetic dipole moment and external fields, where the magnetic moment of the neutral particle interacts with an electric field produced by a linear distribution of electric charges, that is, E⃗ = ρλ ρˆ , where ρ = x2 + y2 , ρˆ is a unit vector in the radial direction and λ is a constant associated with the linear charge distribution along the z-axis; thus, Aharonov and Casher [27] proposed an experiment of interferometry for neutral particles in the presence of the electric field E⃗ = ρλ ρˆ and showed that the wave function of the neutral particle acquires a geometric quantum phase given by
φAC = µ
σ⃗ × E⃗ · d⃗r = ±2π µλ,
(3)
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where the ± signs correspond to the projections of the magnetic dipole moment on the z-axis and µ is the magnetic dipole moment of the neutral particle. Therefore, the thirdterm of the right-hand-side φ
⃗ AC = µ σ⃗ × E⃗ = ± AC ϕˆ , with ϕˆ as being of Eq. (2) plays the role of an effective vector potential µ A 2π ρ a unit vector in the azimuthal direction. Based on the experimental verification reported in Ref. [34], the Aharonov–Casher geometric phase [27] can be observed in any system with neutral particle with a magnetic moment, for instance, in an atomic system. By following Refs. [35–39], relativistic quantum particles can have a position-dependent mass, which can be described by modifying the mass term as m → m + S (ρ), where S (ρ) plays the role of a scalar potential and m is a constant that corresponds to the mass of the free particle. For instance, in ¯ Ref. [37], the position-dependent mass of the relativistic particle is given by m (ρ) = m + ρλ , where λ¯ is a constant. In Ref. [36], the position-dependent mass of the relativistic quantum particle is given ˜ ρ 2 , where λ˜ is a constant. From this perspective, let us follow Refs. [38,39] and by m (ρ) = m + λ
thus describe a relativistic neutral particle with a position-dependent mass by modifying the mass term of the Dirac equation as m (ρ) = m + η ρ , where η is a constant. For a neutral particle with a permanent magnetic dipole moment, then, this position-dependent mass system is described by the Dirac equation [25,26,31,32]: µ
i γ Dµ Ψ + where Dµ =
1 hµ
3 i
2
k=1
γ
k
Dk ln
h1 h2 h3 hk
Ψ +
µ 2
Fµν (x) Σ µν Ψ = (m + η ρ) Ψ ,
(4)
∂µ is the derivative of the corresponding coordinate system and the parameter
hk corresponds to the scale factors of this coordinate system. In our case (cylindrical coordinates), the scale factors are h0 = 1, h1 = 1, h2 = ρ and h3 = 1. Moreover, the tensor Fµν (x) is the electromagnetic tensor and Σ ab = 2i γ a , γ b , where γ a corresponds to the standard Dirac matrices [35]. Note that the second term of the left-hand side of the Dirac equation stems from the curvilinear coordinate system and gives rise to the spinorial connection that we have mentioned in Eq. (2). However, our focus is on a nonrelativistic neutral particle confined to a mesoscopic system, therefore, by taking the nonrelativistic limit of the Dirac equation (4), for instance, through the Foldy–Wouthuysen approximation up to terms of order m−1 [35], then, the Schrödinger–Pauli equation (1) becomes:
πˆ 2 µ2 E 2 µ ⃗ ⃗ ∂ψ ∇ · E ψ + µ σ⃗ · B⃗ ψ + η ρ ψ. (5) = ψ− ψ+ ∂t 2m 2m 2m Hence, the modification of the mass term S (ρ) = η ρ in the Dirac equation (4) plays the role of a i
linear scalar potential in the nonrelativistic limit (up to terms of order m−1 ). As shown in Ref. [34], the Aharonov–Casher geometric quantum phase [27] can be observed in an atomic system. Then, the interaction that stems from relativistic corrections up to terms of order m−1 described by Eq. (5), therefore, could be achieved in an atomic system. It is worth mentioning that the interest in the linear scalar potential comes from the studies of confinement of quarks [38,39], where experimental data show a behaviour of the confinement to be proportional to the distance between the quarks [40–43]. It has also been explored in studies of the quark–antiquark interaction, where the mass term acquires a contribution given by an interaction potential that consists of a linear and a harmonic confining potential plus a Coulomb potential term [36]. Besides, molecular and atomic physics [44–50] and relativistic quantum systems [51–62] have shown a great interest in the linear scalar potential. In what follows, our focus is on the quantum effects associated with the Aharonov–Casher geometric phase (3) when a neutral particle is confined to a quantum dot under the influence of a linear scalar potential. Since the Aharonov–Casher effect [27] has been investigated in mesoscopic systems [9–14, 25,26], let us describe the confinement of a neutral particle with a permanent magnetic dipole moment to a quantum dot by introducing the following scalar potential into the Schrödinger Eq. (1): V (ρ) = a2 ρ 2 ,
(6)
where this scalar potential represents the Tan–Inkson model for a two-dimensional quantum dot [15–17]. The parameter a2 is a constant that characterizes the Tan–Inkson model for a quantum
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dot [15–17]. Note that the Tan–Inkson model [15–17] describes a mesoscopic system and it has a particular interest in studies of the Aharonov–Bohm effect for bound states [5–8,24]. Since the neu⃗ · E⃗ is null, where we assume tral particle is placed into the region where ρ ̸= 0, thus, the term ∇ that the wave function of the neutral particle is well-behaved at the origin. We also assume that the permanent magnetic dipole moment of the neutral particle is aligned in the z-direction, therefore, by introducing the Tan–Inkson potential (6) into the Schrödinger–Pauli equation (5), we have
2 1 ∂ ψ 1 ∂ψ 1 ∂ 2ψ ∂ 2ψ i σ 3 ∂ψ 1 φAC ∂ψ =− + + + + − ψ 2 2 2 2 2 ∂t 2m ∂ρ ρ ∂ρ ρ ∂ϕ ∂z 2m ρ ∂ϕ 2mρ 2 2π i φAC σ 3 ∂ψ 1 φAC 2 1 − + ψ+ ψ + η ρ ψ + a2 ρ 2 ψ. (7) m 2π ρ 2 ∂ϕ 2mρ 2 2π 8mρ 2 ⃗ AC = µ σ⃗ × E⃗ = ± φAC ϕˆ Note that the presence of the term φAC in Eq. (7) stems from the term µ A 2π ρ i
that plays the role of an effective vector potential as shown in Eqs. (2) and (3). In the Schrödinger–Pauli equation (7), we have that ψ is an eigenfunction of σ 3 , where eigenvalues ˆ are s = ±1.Moreover,with the Hamiltonian operator H given on right-hand side of Eq. (7), we have
= 0 and Hˆ , Jˆz = 0, where pˆ z = −i∂z and Jˆz = −i∂ϕ [32]. Therefore, we can write a
ˆ , pˆ z that H
solution to the Schrödinger–Pauli equation (7) as
ψ (t , ρ, ϕ, z ) = e
1 −iE t i l+ 2 ϕ ikz
e
e
G (ρ) ,
(8)
where l = 0, ±1, ±2, . . . and k is a constant. Let us substitute Eq. (8) into the Schrödinger–Pauli equation (7), and thus consider k = 0 from now on in order to get a planar system. Thereby, we obtain a radial equation given by
τ2 2 + − 2 − 2mη ρ − 2ma2 ρ + 2mE G = 0, dρ 2 ρ dρ ρ d2
1 d
(9)
where τ is defined in equation above as
τ =l+
1 2
( 1 − s) + s
φAC . 2π
(10)
Let us perform a change of variables given by: ξ = (2ma2 )1/4 ρ ; thus, Eq. (9) becomes d2 G dξ 2
+
1 dG
ξ dξ
τ2 Gs − µ ξ G − ξ 2 G + β G = 0, ξ2
−
(11)
where we have defined the parameters
µ= β=
2m η
(2ma2 )3/4 2m E (2ma2 )
; (12)
. 1/2
Let us analyse the asymptotic behaviour of the possible solutions to Eq. (11). This asymptotic behaviour is determined for ξ → 0 and ξ → ∞, hence, based on Refs. [60–63], the possible solutions to Eq. (11) at ξ → 0 and ξ → ∞ permit us to write the function G (ξ ) in terms of an unknown function H (ξ ) as it follows: 2
µξ ξ G (ξ ) = e− 2 e− 2 ξ |τ | H (ξ ) .
(13)
Substituting (13) into (11), we obtain the second order differential equation: d2 H dξ 2
+
2 |τ | + 1
ξ
− µ − 2ξ
dH dξ
+ g−
h
ξ
H = 0,
(14)
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where the parameters g and h are defined as g =β+ h=
µ
µ2 4
− 2 − 2 |τ | ; (15)
(2 |τ | + 1) .
2 Hence, Eq. (14) is called in the literature as the biconfluent Heun equation [63], and the function H (ξ ) is known as the biconfluent Heun function [62–64]: H (ξ ) = HB
2 |τ | , µ, β +
µ2 4
, 0, ξ .
(16)
Henceforth, we write the biconfluent Heun function H (ξ ) as a power series expansion around the origin [62,65,66], i.e., ∞
H (ξ ) =
cj ξ j ,
(17)
j=0
and then, by substituting Eq. (17) into Eq. (14), we obtain the recurrence relation:
µ (j + 1) + h g − 2j cj+1 − cj , (j + 2) (j + 2 + 2 |τ |) (j + 2) (j + 2 + 2 |τ |)
cj+2 =
(18)
µ
and c1 = 2 c0 . By starting with c0 = 1 and using the relation (18), we can calculate other coefficients of the power series expansion (17). As examples, the coefficients c1 and c2 are given by c1 =
µ 2
;
c2 =
µ (µ + h) g − . 4 (2 + 2 |τ |) 2 (2 + 2 |τ |)
(19)
In the search for bound state solutions, we must impose that the biconfluent Heun series (17) becomes a polynomial of degree n [31,62,66]. Through the expression (18), we can see that the biconfluent Heun series becomes a polynomial of degree n if we impose the conditions: g = 2n and
cn+1 = 0,
(20)
where n = 1, 2, 3, . . . . From the condition g = 2n, we can obtain the following expression for the energy levels for bound states:
En, l, s = ω [n + |τ | + 1] −
η2 , 2mω2
(21)
where n = 1, 2, 3, . . . is the quantum number associated with the radial modes, l = 0, ±1, ±2, . . . is the angular momentum quantum number, s = ±1 is the spin quantum number and the parameter ω corresponds to the angular frequency of the system, which is given by
ω=
2a2 m
.
(22)
Note that the angular frequency of the Tan–Inkson model for a quantum dot [15–17] is given by
ω0 =
8a2 , m
then, we have that the influence of the linear scalar potential on the confinement of a
neutral particle to a quantum dot modifies the angular frequency of the Tan–Inkson model, whose result is given in Eq. (22). Moreover, the spectrum of energy of a neutral particle confined to a quantum dot changes in contrast to Ref. [25], where we have in the present case that the ground state is determined by the quantum number n = 1 instead of n = 0 as we can see in Eq. (21). We can also observe that the energy levels (21) depend on the Aharonov–Casher geometric quantum phase φAC due to the presence of the parameter τ . This dependence of the geometric quantum phase corresponds to an Aharonov–Bohm-type effect for bound states or the Aharonov–Casher effect for bound states [26,67].
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Next, let us analyse the condition cn+1 = 0 given in Eq. (20) by using, as an example, the ground state of the system. For this purpose, let us consider the angular frequency ω can be adjusted in such a way that the condition cn+1 = 0 can be satisfied. This is possible because we can adjust the parameter a2 of the Tan–Inkson model [15–17,31] in order to satisfy the condition cn+1 = 0. In this way, from the condition cn+1 = 0, we have c2 = 0 and then
ω1, l, s =
η2 2m
(2 |τ | + 3)
1/3
.
(23)
This example shows us that both conditions imposed in Eq. (20) are satisfied and a polynomial solution to the function H (ξ ) is obtained. Further, it shows us that only specific values of the angular frequency ω are allowed and depend on the quantum numbers {n, l, s}. For this reason, we label [31]
n, l, s
2a2
ωn, l, s =
m
.
(24)
In the quantum mechanics context, what we have behind Eqs. (23) and (24) is a quantum effect characterized by a dependence of the angular frequency on the quantum numbers {n, l, s} of the system and the Aharonov–Casher geometric phase φAC . This dependence of the angular frequency on the Aharonov–Casher geometric phase can be seen through the parameter τ defined in Eq. (10). The origin of this dependence of the angular frequency on the Aharonov–Casher geometric phase and the quantum numbers of the systems stems from the influence of the linear scalar potential on the confinement of a neutral particle to a quantum dot. Recent studies have obtained this quantum effect in different quantum mechanical contexts [62,64,68]. In Ref. [31], a dependence of the angular frequency on the quantum numbers and the Aharonov–Casher geometric phase [27] has been obtained in a quantum ring. Besides, from Eqs. (21) and (23), the energy level associated with the ground state is given by
E1, l, s =
η2 2m
(2 |τ | + 3)
1/3 × [|τ | + 2] −
η2 2m
2m
η2 [2 |τ | + 3]
2/3
.
(25)
Observe that, due to the presence of the parameter τ in Eq. (25), we have that the energy level associated with the ground state of the system depends on the Aharonov–Casher geometric quantum phase, where it has a periodicity φ0 = ±2π ; thus, we have that E1, l, s (φAC + φ0 ) = E1, l+1, s (φAC ). This dependence of the energy levels on the geometric quantum phase yields the arising of persistent spin currents in the quantum dot [9–14,18,25,69]. By following Refs. [6–18,23,69], the expression for the total persistent spin currents that arises in the two-dimensional quantum dot is given by ∂ En, l ∂φ
(φ is the geometric quantum phase) is called as the Byers–Yang relation [18]. Therefore, since the energy level (25) depends on the Aharonov–Casher geometric phase φAC , the persistent spin current associated with the ground state of the system is
I=
n, l
In, l , where In, l = −
I1, l = −
∂ E1, l, s ∂φAC
=−
τ 2π |τ | s
η
2
2m
1/3 [2 |τ | + 3]
η (|τ | + 2) η 2 + 2 2 + 1 . η 3m η 2 |τ | [2 + 3] 3m 2m [2 |τ | + 3] 2m 2
4
(26) Hence, we can observe the arising of persistent spin currents in a quantum dot. In particular, the dependence of the angular frequency on the Aharonov–Casher quantum phase yields new contributions to the persistent spin current associated with the ground state in contrast to that obtained in Ref. [25]. This kind of contribution has been investigated in a two-dimensional quantum ring under the influence of a Coulomb-like potential in Ref. [31]. We must also observe that the
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expressions for both the angular frequency and the persistent spin current change for other values of the quantum number n, that is, for other energy levels. Note that, for n = 1 we have the simplest case of the function H (ξ ) which corresponds to a polynomial of first degree. Thereby, the radial wave function associated with the ground state is
µξ ξ2 µ ξ . (27) G1, l, s (r ) = e− 2 e− 2 ξ |τ | 1 + 2 Let us finalize this section by rewriting the general expression for the energy levels of the bound states (21) En, l, s = ωn, l, s [n + |τ | + 1] −
η2 , 2 m ωn2, l, s
(28)
The spectrum of energy (28) corresponds to the energy levels of a neutral particle confined to a quantum dot [15–17] under the influence of a linear confining potential in the presence of an electric field E⃗ = ρλ ρˆ . This general expression shows us that the energy levels depend on the Aharonov–Casher geometric phase [27] and the angular frequency of the system depends on the quantum numbers {n, l, s} and also the Aharonov–Casher geometric quantum phase. 3. Influence of Coulomb-type and linear scalar potentials In this section, we focus on the quantum effects associated with the Aharonov–Casher geometric phase (3) when a neutral particle is confined to a quantum dot under the influence of a linear scalar potential and a Coulomb-like potential. From Eqs. (4) and (5) and based on Refs. [38,39], let us consider a position-dependent mass given by m (ρ) = m + η ρ + αρ , where α is a constant. Thereby, through
the Foldy–Wouthuysen approximation up to terms of order m−1 [35], then, the Schrödinger–Pauli equation (1) becomes:
∂ψ πˆ 2 µ2 E 2 µ ⃗ ⃗ α = ψ− ψ+ (29) ∇ · E ψ + µ σ⃗ · B⃗ ψ + η ρ ψ + ψ. ∂t 2m 2m 2m ρ Thereby, the modification of the mass term S (ρ) = η ρ + αρ in the Dirac equation (4) plays the role i
of a linear and a Coulomb-type scalar potentials [50,51] in the nonrelativistic limit (up to terms of order m−1 ). In particular, the Coulomb-type potential has attracted interests in studies of quark models [39,62], chemical physics [70,71], the Kratzer potential [72–75], the Mie-type potential [76,77], position-dependent mass systems [37,78,79] and topological defects in solids [80–83]. Recently, the two-dimensional Coulomb potential has been investigated under the presence of the Aharonov–Bohm effect in Refs. [84–86]. Therefore, let us confine the neutral particle with a permanent magnetic dipole moment to a quantum dot described by the Tan–Inkson model [15–17], as given in Eq. (6), under the influence of a linear confining potential and a Coulomb-like potential. Again, we assume that the wave function of the neutral particle is well-behaved at the origin and the permanent magnetic dipole moment of the neutral particle to be aligned in the z-direction as in the previous section, therefore, by introducing the Tan–Inkson potential (6) into the Schrödinger–Pauli equation (29), we obtain i
2 i σ 3 ∂ψ 1 ∂ ψ 1 ∂ψ 1 ∂ 2ψ ∂ 2ψ 1 φAC ∂ψ =− + + + + − ψ 2 2 2 2 ∂t 2m ∂ρ ρ ∂ρ ρ ∂ϕ ∂z 2m ρ 2 ∂ϕ 2mρ 2 2π i φAC σ 3 ∂ψ 1 φAC 2 1 α − + ψ+ ψ + ψ + η ρ ψ + a2 ρ 2 ψ. 2 2 2 m 2π ρ ∂ϕ 2mρ 2π 8mρ ρ
(30)
By following the steps from Eq. (7) to Eq. (10), we have that the solution to the Schrödinger–Pauli equation (30) has the same form of Eq. (8), and thus we obtain the following radial equation:
τ2 2mα 2 + − 2− − 2mη ρ − 2ma2 ρ + 2mE G¯ = 0, dρ 2 ρ dρ ρ ρ d2
1 d
(31)
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¯ (ρ). Now, let us where τ has been defined in Eq. (10) and we have replaced the function G (ρ) with G perform the same change of variables made in the previous section, that is, ξ = (2ma2 )1/4 ρ ; thus, we obtain τ2 ¯ θ ¯ ¯ − ξ 2 G¯ + β G¯ = 0, G − G − µξ G (32) dξ ξ dξ ξ2 ξ where the parameters µ and β have been defined in Eq. (12) and the new parameter θ is given by ¯ d2 G 2
θ=
+
¯ 1 dG
−
2m α . (2ma2 )1/4
(33)
From the discussion about the asymptotic behaviour of the wave function made from Eq. (13) to ¯ (ξ ) in terms of an unknown function H¯ (ξ ) Eq. (16), hence, we have that we can write the function G as in Eq. (13), that is, 2
ξ µξ ¯ (ξ ) = e− 2 e− 2 ξ |τ | H¯ (ξ ) . G
(34)
By substituting (34) into (32), we have
¯ d2 H dξ 2
+
2 |τ | + 1
ξ
− µ − 2ξ
¯ dH dξ
+ g−
f
ξ
¯ = 0, H
(35)
where the parameter g has been defined in Eq. (3) and the parameter f is f =
µ
(2 |τ | + 1) + θ .
2
(36)
¯ (ξ ) is the biconfluent Eq. (35) is called as the biconfluent Heun equation [63], and the function H Heun function [62–64]: ¯ (ξ ) = HB H
2 |τ | , µ, β +
µ2 4
, 2θ , ξ .
(37)
Again, we follow the steps from Eq. (17) to Eq. (20) in order to obtain the bound state solutions. In this case, the recurrence relation becomes
µ (j + 1) + f (g − 2j) c¯j+1 − c¯j , (j + 2) (j + 2 + 2 |τ |) (j + 2) (j + 2 + 2 |τ |)
c¯j+2 =
(38)
f
and c¯1 = (1+2|τ |) c¯0 . We also start with c¯0 = 1 and with the relation (38), we can calculate other coefficients of the biconfluent Heun series (17). For instance, c¯1 =
f
(1 + 2 |τ |)
;
c¯2 =
f (f + µ) 2 (2 + 2 |τ |) (1 + 2 |τ |)
−
g 2 (2 + 2 |τ |)
.
(39)
With the conditions established in Eq. (20) we have that the biconfluent Heun series becomes a polynomial of degree n. By analysing the condition g = 2n, we obtain the same expression for the energy levels for bound states obtained in Eq. (21) and the same angular frequency (22), that is,
η2 , (40) 2mω2 where ω is given in Eq. (22) and n = 1, 2, 3, . . . . Again, we can see that the angular frequency of the En, l, s = ω [n + |τ | + 1] −
system differs from that one of the Tan–Inkson model [15–17] and the ground state is determined by the quantum number n = 1 instead of n = 0 as in the previous case. We have also that the energy levels (40) depend on the Aharonov–Casher geometric phase [27], which is an Aharonov–Bohmtype effect for bound states [26]. However, the energy levels (40) seem to have no influence of the parameter α that characterizes the Coulomb-like potential. For this reason, let us analyse the condition
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c¯n+1 = 0 [87]. We also assume that the parameter a2 of the Tan–Inkson model [15–17] can be adjusted in such a way that the condition c¯n+1 = 0 can be satisfied. By considering the ground state of the system (n = 1), we have that c¯2 = 0 from the condition c¯n+1 = 0. The condition c¯2 = 0 imposes that the possible values of the angular frequency ω1, l, s are determined by the third-degree algebraic equation [62,64]:
ω13, l, s −
2 m α2
(1 + 2 |τ |)
ω12, l, s − 2 η α
η2 (2 + 2 |τ |) ω1, l, s − (3 + 2 |τ |) = 0. 2m (1 + 2 |τ |)
(41)
Despite Eq. (41) has at least one real solution, we do not write it because it’s expression is very long. Observe that the solution to Eq. (41) depends on the parameter τ , which means that the angular frequency associated with the ground state of the system depends on the Aharonov–Casher geometric phase φAC . This kind of behaviour of the angular frequency can be expected for other energy levels. Therefore, there exists a quantum effect characterized by the dependence of the angular frequency on the quantum numbers of the system {n, l, s} and the Aharonov–Casher geometric phase φAC . ¯ (ξ ), which corresponds to a Moreover, for n = 1 we have the simplest case of the function H polynomial of first degree: 2
ξ µξ ¯ 1, l, s (ξ ) = e− 2 e− 2 ξ |τ | G
1+
µ (2 |τ | + 1) + 2θ ξ . 2 (1 + 2 |τ |)
(42)
Finally, we can rewrite expression for the energy levels of the bound states (40) as
En, l, s = ωn, l, s [n + |τ | + 1] −
η2 , 2 m ωn2, l, s
(43)
Hence, the general expression for energy levels (43) is obtained for a neutral particle with a permanent magnetic dipole moment that interacts with a radial electric field confined to a quantum dot under the influence of a Coulomb-like and linear scalar potentials. Observe that there exists the dependence of the energy levels on the Aharonov–Casher geometric phase [27] which gives rise to the arising of the persistent spin current in the quantum dot. Due to the difficulty of obtaining the analytical expression for the angular frequency associated with each energy level, we decided not to calculate the persistent spin current even in the case of the ground state, since the expression for the angular frequency is too long as we have seen in Eq. (41). 4. Conclusions We have analysed quantum effects associated with the Aharonov–Casher geometric phase [27] when a neutral particle is confined to a quantum dot under the influence of a linear scalar potential and a Coulomb-like potential. In both cases analysed here, we have seen that the energy levels of the bound states depend on the Aharonov–Casher geometric [27], which is called as an analogue of the Aharonov–Bohm effect for bound states [24] or the Aharonov–Casher effect for bound states [26]. We have also shown that the angular frequency of the Tan–Inkson model for a quantum dot [15–17] is modified by the influence of the linear and Coulomb-like scalar potentials. Besides, we have seen that there is a restriction on the values of the angular frequency of the system in order that bound state solutions can be achieved, where these possible values of the angular frequency of the system depend on the quantum numbers of the system and the Aharonov–Casher geometric quantum phase. As examples, we have calculated the possible values of the angular frequency of the system associated with the ground state of the system in the two cases discussed in this work. In the first case, where the neutral particle is confined to a quantum dot in the presence of the linear scalar potential, we have obtained the analytical expression for the possible values of the angular frequency and the persistent spin currents. On the other hand, when the neutral particle is confined to a quantum dot in the presence of both linear and Coulomb-type scalar potentials, we have seen that the possible values of the angular frequency of the system are determined by third-degree algebraic equation.
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On a relation of the angular frequency to the Aharonov–Casher geometric phase in a quantum dot P.M.T. Barboza, K. Bakke ∗ Departamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008, 58051-900, João Pessoa-PB, Brazil
article
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Article history: Received 23 February 2016 Accepted 2 June 2016 Available online 16 June 2016 Keywords: Aharonov–Casher effect Geometric quantum phases Quantum dot Persistent spin current Linear plus Coulomb-type potential Biconfluent Heun function
By analysing the behaviour of a neutral particle with permanent magnetic dipole moment confined to a quantum dot in the presence of a radial electric field, Coulomb-type and linear confining potentials, then, an Aharonov–Bohm-type effect for bound states and a dependence of the angular frequency of the system on the Aharonov–Casher geometric phase and the quantum numbers associated with the radial modes, the angular momentum and the spin are obtained. In particular, the possible values of the angular frequency and the persistent spin currents associated with the ground state are investigated in two different cases. © 2016 Elsevier Inc. All rights reserved.
1. Introduction In mesoscopic systems, the dependence of the energy levels on geometric quantum phases [1–4] is a well-known quantum effect in the literature [5–18]. A particular case is the dependence of the spectrum of energy on the Aharonov–Bohm geometric quantum [19] when the movement of an electron is restricted to two concentric cylinders [20,21], or an one-dimensional ring [22], or two-dimensional rings [15–17,23] without interacting with the magnetic field. This particular case is called as the Aharonov–Bohm effect for bound states [24]. Further, analogues of the Aharonov–Bohm effects for bound states have been investigated in mesoscopic systems [5–8] associated with the Berry phase [1] and the Aharonov–Anandan quantum phase [2]. With respect to neutral particles,
∗
Corresponding author. E-mail address: [email protected] (K. Bakke).
http://dx.doi.org/10.1016/j.aop.2016.06.005 0003-4916/© 2016 Elsevier Inc. All rights reserved.
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Aharonov–Bohm-type effects for bound states have been explored in quantum rings [9–14,25,26] due to the interaction of the permanent magnetic dipole moment of the neutral particle with an electric field that gives rise to the Aharonov–Casher effect [27]. The objective of this work is to investigate Aharonov–Bohm-type effects for bound states [24] that arise in a two-dimensional quantum dot [15–17] due to the interaction of the permanent magnetic dipole moment of a neutral particle with a radial electric field under the influence of scalar potentials. By searching for bound state solutions, we show that the energy levels depend on the Aharonov–Casher geometric quantum phase [27] and there exists a restriction of the values of the angular frequency of the system, where the possible values of this angular frequency depend on the quantum numbers associated with the radial modes, the angular momentum and the spin, and the Aharonov–Casher geometric phase [27]. Due to this dependence of the energy levels on the geometric quantum phase, we show that persistent spin currents [9–14,18] can arise in the two-dimensional quantum dot. A particular contribution to the persistent spin currents in the quantum dot comes from the dependence of the angular frequency on the geometric quantum phase. The structure of this paper is as follows: in Section 2, we investigate an Aharonov–Bohm-type effect for bound states [24] or an Aharonov–Casher effect for bound states [26] that arises in a twodimensional quantum dot [15–17] under the influence of a linear scalar potential. In particular, we obtain the persistent spin currents associated with the ground state of the system; in Section 3, we investigate an Aharonov–Casher effect for bound states [26] in a quantum dot under the influence of a Coulomb-type and a linear scalar potentials. As a particular case, we show that the possible values of the angular frequency associated with the ground state of the system are determined by a third degree algebraic equation in order that bound state solutions can be obtained; in Section 4, we present our conclusions. 2. Influence of a linear scalar potential In this section, we investigate quantum effects on a neutral particle with a permanent magnetic dipole moment confined to a quantum dot under the influence of a linear scalar potential when the magnetic dipole moment of the neutral particle interacts with an electric field. As shown in Refs. [25,27–31], the quantum dynamics of a neutral particle with a permanent magnetic dipole moment that interacts with electric and magnetic fields is described by (with h¯ = c = 1):
πˆ 2 µ2 E 2 µ ⃗ ⃗ ∂ψ = ψ− ψ+ ∇ · E ψ + µ σ⃗ · B⃗ ψ + V ψ, (1) ∂t 2m 2m 2m where σ i are the Pauli matrices that satisfy the relation σ i σ j + σ j σ i = 2 ηij , ηij = diag(+ + +) and V is a scalar potential. Besides, the operator πˆ is defined as 1 3 πˆ k = −i∂k − σ δϕ k + µ σ⃗ × E⃗ . (2) k 2ρ i
The second term of the right-hand-side of Eq. (2) stems from the curvilinear coordinates system (cylindrical coordinates) adopted [25,31,32]. In quantum field theory in curved spacetime, this term stems from the spinorial connection [25,33]. Now, let us introduce the Aharonov–Casher effect [27]. In 1984, Aharonov and Casher [27] showed a particular case of the interaction between the permanent magnetic dipole moment and external fields, where the magnetic moment of the neutral particle interacts with an electric field produced by a linear distribution of electric charges, that is, E⃗ = ρλ ρˆ , where ρ = x2 + y2 , ρˆ is a unit vector in the radial direction and λ is a constant associated with the linear charge distribution along the z-axis; thus, Aharonov and Casher [27] proposed an experiment of interferometry for neutral particles in the presence of the electric field E⃗ = ρλ ρˆ and showed that the wave function of the neutral particle acquires a geometric quantum phase given by
φAC = µ
σ⃗ × E⃗ · d⃗r = ±2π µλ,
(3)
P.M.T. Barboza, K. Bakke / Annals of Physics 372 (2016) 457–467
459
where the ± signs correspond to the projections of the magnetic dipole moment on the z-axis and µ is the magnetic dipole moment of the neutral particle. Therefore, the thirdterm of the right-hand-side φ
⃗ AC = µ σ⃗ × E⃗ = ± AC ϕˆ , with ϕˆ as being of Eq. (2) plays the role of an effective vector potential µ A 2π ρ a unit vector in the azimuthal direction. Based on the experimental verification reported in Ref. [34], the Aharonov–Casher geometric phase [27] can be observed in any system with neutral particle with a magnetic moment, for instance, in an atomic system. By following Refs. [35–39], relativistic quantum particles can have a position-dependent mass, which can be described by modifying the mass term as m → m + S (ρ), where S (ρ) plays the role of a scalar potential and m is a constant that corresponds to the mass of the free particle. For instance, in ¯ Ref. [37], the position-dependent mass of the relativistic particle is given by m (ρ) = m + ρλ , where λ¯ is a constant. In Ref. [36], the position-dependent mass of the relativistic quantum particle is given ˜ ρ 2 , where λ˜ is a constant. From this perspective, let us follow Refs. [38,39] and by m (ρ) = m + λ
thus describe a relativistic neutral particle with a position-dependent mass by modifying the mass term of the Dirac equation as m (ρ) = m + η ρ , where η is a constant. For a neutral particle with a permanent magnetic dipole moment, then, this position-dependent mass system is described by the Dirac equation [25,26,31,32]: µ
i γ Dµ Ψ + where Dµ =
1 hµ
3 i
2
k=1
γ
k
Dk ln
h1 h2 h3 hk
Ψ +
µ 2
Fµν (x) Σ µν Ψ = (m + η ρ) Ψ ,
(4)
∂µ is the derivative of the corresponding coordinate system and the parameter
hk corresponds to the scale factors of this coordinate system. In our case (cylindrical coordinates), the scale factors are h0 = 1, h1 = 1, h2 = ρ and h3 = 1. Moreover, the tensor Fµν (x) is the electromagnetic tensor and Σ ab = 2i γ a , γ b , where γ a corresponds to the standard Dirac matrices [35]. Note that the second term of the left-hand side of the Dirac equation stems from the curvilinear coordinate system and gives rise to the spinorial connection that we have mentioned in Eq. (2). However, our focus is on a nonrelativistic neutral particle confined to a mesoscopic system, therefore, by taking the nonrelativistic limit of the Dirac equation (4), for instance, through the Foldy–Wouthuysen approximation up to terms of order m−1 [35], then, the Schrödinger–Pauli equation (1) becomes:
πˆ 2 µ2 E 2 µ ⃗ ⃗ ∂ψ ∇ · E ψ + µ σ⃗ · B⃗ ψ + η ρ ψ. (5) = ψ− ψ+ ∂t 2m 2m 2m Hence, the modification of the mass term S (ρ) = η ρ in the Dirac equation (4) plays the role of a i
linear scalar potential in the nonrelativistic limit (up to terms of order m−1 ). As shown in Ref. [34], the Aharonov–Casher geometric quantum phase [27] can be observed in an atomic system. Then, the interaction that stems from relativistic corrections up to terms of order m−1 described by Eq. (5), therefore, could be achieved in an atomic system. It is worth mentioning that the interest in the linear scalar potential comes from the studies of confinement of quarks [38,39], where experimental data show a behaviour of the confinement to be proportional to the distance between the quarks [40–43]. It has also been explored in studies of the quark–antiquark interaction, where the mass term acquires a contribution given by an interaction potential that consists of a linear and a harmonic confining potential plus a Coulomb potential term [36]. Besides, molecular and atomic physics [44–50] and relativistic quantum systems [51–62] have shown a great interest in the linear scalar potential. In what follows, our focus is on the quantum effects associated with the Aharonov–Casher geometric phase (3) when a neutral particle is confined to a quantum dot under the influence of a linear scalar potential. Since the Aharonov–Casher effect [27] has been investigated in mesoscopic systems [9–14, 25,26], let us describe the confinement of a neutral particle with a permanent magnetic dipole moment to a quantum dot by introducing the following scalar potential into the Schrödinger Eq. (1): V (ρ) = a2 ρ 2 ,
(6)
where this scalar potential represents the Tan–Inkson model for a two-dimensional quantum dot [15–17]. The parameter a2 is a constant that characterizes the Tan–Inkson model for a quantum
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dot [15–17]. Note that the Tan–Inkson model [15–17] describes a mesoscopic system and it has a particular interest in studies of the Aharonov–Bohm effect for bound states [5–8,24]. Since the neu⃗ · E⃗ is null, where we assume tral particle is placed into the region where ρ ̸= 0, thus, the term ∇ that the wave function of the neutral particle is well-behaved at the origin. We also assume that the permanent magnetic dipole moment of the neutral particle is aligned in the z-direction, therefore, by introducing the Tan–Inkson potential (6) into the Schrödinger–Pauli equation (5), we have
2 1 ∂ ψ 1 ∂ψ 1 ∂ 2ψ ∂ 2ψ i σ 3 ∂ψ 1 φAC ∂ψ =− + + + + − ψ 2 2 2 2 2 ∂t 2m ∂ρ ρ ∂ρ ρ ∂ϕ ∂z 2m ρ ∂ϕ 2mρ 2 2π i φAC σ 3 ∂ψ 1 φAC 2 1 − + ψ+ ψ + η ρ ψ + a2 ρ 2 ψ. (7) m 2π ρ 2 ∂ϕ 2mρ 2 2π 8mρ 2 ⃗ AC = µ σ⃗ × E⃗ = ± φAC ϕˆ Note that the presence of the term φAC in Eq. (7) stems from the term µ A 2π ρ i
that plays the role of an effective vector potential as shown in Eqs. (2) and (3). In the Schrödinger–Pauli equation (7), we have that ψ is an eigenfunction of σ 3 , where eigenvalues ˆ are s = ±1.Moreover,with the Hamiltonian operator H given on right-hand side of Eq. (7), we have
= 0 and Hˆ , Jˆz = 0, where pˆ z = −i∂z and Jˆz = −i∂ϕ [32]. Therefore, we can write a
ˆ , pˆ z that H
solution to the Schrödinger–Pauli equation (7) as
ψ (t , ρ, ϕ, z ) = e
1 −iE t i l+ 2 ϕ ikz
e
e
G (ρ) ,
(8)
where l = 0, ±1, ±2, . . . and k is a constant. Let us substitute Eq. (8) into the Schrödinger–Pauli equation (7), and thus consider k = 0 from now on in order to get a planar system. Thereby, we obtain a radial equation given by
τ2 2 + − 2 − 2mη ρ − 2ma2 ρ + 2mE G = 0, dρ 2 ρ dρ ρ d2
1 d
(9)
where τ is defined in equation above as
τ =l+
1 2
( 1 − s) + s
φAC . 2π
(10)
Let us perform a change of variables given by: ξ = (2ma2 )1/4 ρ ; thus, Eq. (9) becomes d2 G dξ 2
+
1 dG
ξ dξ
τ2 Gs − µ ξ G − ξ 2 G + β G = 0, ξ2
−
(11)
where we have defined the parameters
µ= β=
2m η
(2ma2 )3/4 2m E (2ma2 )
; (12)
. 1/2
Let us analyse the asymptotic behaviour of the possible solutions to Eq. (11). This asymptotic behaviour is determined for ξ → 0 and ξ → ∞, hence, based on Refs. [60–63], the possible solutions to Eq. (11) at ξ → 0 and ξ → ∞ permit us to write the function G (ξ ) in terms of an unknown function H (ξ ) as it follows: 2
µξ ξ G (ξ ) = e− 2 e− 2 ξ |τ | H (ξ ) .
(13)
Substituting (13) into (11), we obtain the second order differential equation: d2 H dξ 2
+
2 |τ | + 1
ξ
− µ − 2ξ
dH dξ
+ g−
h
ξ
H = 0,
(14)
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461
where the parameters g and h are defined as g =β+ h=
µ
µ2 4
− 2 − 2 |τ | ; (15)
(2 |τ | + 1) .
2 Hence, Eq. (14) is called in the literature as the biconfluent Heun equation [63], and the function H (ξ ) is known as the biconfluent Heun function [62–64]: H (ξ ) = HB
2 |τ | , µ, β +
µ2 4
, 0, ξ .
(16)
Henceforth, we write the biconfluent Heun function H (ξ ) as a power series expansion around the origin [62,65,66], i.e., ∞
H (ξ ) =
cj ξ j ,
(17)
j=0
and then, by substituting Eq. (17) into Eq. (14), we obtain the recurrence relation:
µ (j + 1) + h g − 2j cj+1 − cj , (j + 2) (j + 2 + 2 |τ |) (j + 2) (j + 2 + 2 |τ |)
cj+2 =
(18)
µ
and c1 = 2 c0 . By starting with c0 = 1 and using the relation (18), we can calculate other coefficients of the power series expansion (17). As examples, the coefficients c1 and c2 are given by c1 =
µ 2
;
c2 =
µ (µ + h) g − . 4 (2 + 2 |τ |) 2 (2 + 2 |τ |)
(19)
In the search for bound state solutions, we must impose that the biconfluent Heun series (17) becomes a polynomial of degree n [31,62,66]. Through the expression (18), we can see that the biconfluent Heun series becomes a polynomial of degree n if we impose the conditions: g = 2n and
cn+1 = 0,
(20)
where n = 1, 2, 3, . . . . From the condition g = 2n, we can obtain the following expression for the energy levels for bound states:
En, l, s = ω [n + |τ | + 1] −
η2 , 2mω2
(21)
where n = 1, 2, 3, . . . is the quantum number associated with the radial modes, l = 0, ±1, ±2, . . . is the angular momentum quantum number, s = ±1 is the spin quantum number and the parameter ω corresponds to the angular frequency of the system, which is given by
ω=
2a2 m
.
(22)
Note that the angular frequency of the Tan–Inkson model for a quantum dot [15–17] is given by
ω0 =
8a2 , m
then, we have that the influence of the linear scalar potential on the confinement of a
neutral particle to a quantum dot modifies the angular frequency of the Tan–Inkson model, whose result is given in Eq. (22). Moreover, the spectrum of energy of a neutral particle confined to a quantum dot changes in contrast to Ref. [25], where we have in the present case that the ground state is determined by the quantum number n = 1 instead of n = 0 as we can see in Eq. (21). We can also observe that the energy levels (21) depend on the Aharonov–Casher geometric quantum phase φAC due to the presence of the parameter τ . This dependence of the geometric quantum phase corresponds to an Aharonov–Bohm-type effect for bound states or the Aharonov–Casher effect for bound states [26,67].
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Next, let us analyse the condition cn+1 = 0 given in Eq. (20) by using, as an example, the ground state of the system. For this purpose, let us consider the angular frequency ω can be adjusted in such a way that the condition cn+1 = 0 can be satisfied. This is possible because we can adjust the parameter a2 of the Tan–Inkson model [15–17,31] in order to satisfy the condition cn+1 = 0. In this way, from the condition cn+1 = 0, we have c2 = 0 and then
ω1, l, s =
η2 2m
(2 |τ | + 3)
1/3
.
(23)
This example shows us that both conditions imposed in Eq. (20) are satisfied and a polynomial solution to the function H (ξ ) is obtained. Further, it shows us that only specific values of the angular frequency ω are allowed and depend on the quantum numbers {n, l, s}. For this reason, we label [31]
n, l, s
2a2
ωn, l, s =
m
.
(24)
In the quantum mechanics context, what we have behind Eqs. (23) and (24) is a quantum effect characterized by a dependence of the angular frequency on the quantum numbers {n, l, s} of the system and the Aharonov–Casher geometric phase φAC . This dependence of the angular frequency on the Aharonov–Casher geometric phase can be seen through the parameter τ defined in Eq. (10). The origin of this dependence of the angular frequency on the Aharonov–Casher geometric phase and the quantum numbers of the systems stems from the influence of the linear scalar potential on the confinement of a neutral particle to a quantum dot. Recent studies have obtained this quantum effect in different quantum mechanical contexts [62,64,68]. In Ref. [31], a dependence of the angular frequency on the quantum numbers and the Aharonov–Casher geometric phase [27] has been obtained in a quantum ring. Besides, from Eqs. (21) and (23), the energy level associated with the ground state is given by
E1, l, s =
η2 2m
(2 |τ | + 3)
1/3 × [|τ | + 2] −
η2 2m
2m
η2 [2 |τ | + 3]
2/3
.
(25)
Observe that, due to the presence of the parameter τ in Eq. (25), we have that the energy level associated with the ground state of the system depends on the Aharonov–Casher geometric quantum phase, where it has a periodicity φ0 = ±2π ; thus, we have that E1, l, s (φAC + φ0 ) = E1, l+1, s (φAC ). This dependence of the energy levels on the geometric quantum phase yields the arising of persistent spin currents in the quantum dot [9–14,18,25,69]. By following Refs. [6–18,23,69], the expression for the total persistent spin currents that arises in the two-dimensional quantum dot is given by ∂ En, l ∂φ
(φ is the geometric quantum phase) is called as the Byers–Yang relation [18]. Therefore, since the energy level (25) depends on the Aharonov–Casher geometric phase φAC , the persistent spin current associated with the ground state of the system is
I=
n, l
In, l , where In, l = −
I1, l = −
∂ E1, l, s ∂φAC
=−
τ 2π |τ | s
η
2
2m
1/3 [2 |τ | + 3]
η (|τ | + 2) η 2 + 2 2 + 1 . η 3m η 2 |τ | [2 + 3] 3m 2m [2 |τ | + 3] 2m 2
4
(26) Hence, we can observe the arising of persistent spin currents in a quantum dot. In particular, the dependence of the angular frequency on the Aharonov–Casher quantum phase yields new contributions to the persistent spin current associated with the ground state in contrast to that obtained in Ref. [25]. This kind of contribution has been investigated in a two-dimensional quantum ring under the influence of a Coulomb-like potential in Ref. [31]. We must also observe that the
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expressions for both the angular frequency and the persistent spin current change for other values of the quantum number n, that is, for other energy levels. Note that, for n = 1 we have the simplest case of the function H (ξ ) which corresponds to a polynomial of first degree. Thereby, the radial wave function associated with the ground state is
µξ ξ2 µ ξ . (27) G1, l, s (r ) = e− 2 e− 2 ξ |τ | 1 + 2 Let us finalize this section by rewriting the general expression for the energy levels of the bound states (21) En, l, s = ωn, l, s [n + |τ | + 1] −
η2 , 2 m ωn2, l, s
(28)
The spectrum of energy (28) corresponds to the energy levels of a neutral particle confined to a quantum dot [15–17] under the influence of a linear confining potential in the presence of an electric field E⃗ = ρλ ρˆ . This general expression shows us that the energy levels depend on the Aharonov–Casher geometric phase [27] and the angular frequency of the system depends on the quantum numbers {n, l, s} and also the Aharonov–Casher geometric quantum phase. 3. Influence of Coulomb-type and linear scalar potentials In this section, we focus on the quantum effects associated with the Aharonov–Casher geometric phase (3) when a neutral particle is confined to a quantum dot under the influence of a linear scalar potential and a Coulomb-like potential. From Eqs. (4) and (5) and based on Refs. [38,39], let us consider a position-dependent mass given by m (ρ) = m + η ρ + αρ , where α is a constant. Thereby, through
the Foldy–Wouthuysen approximation up to terms of order m−1 [35], then, the Schrödinger–Pauli equation (1) becomes:
∂ψ πˆ 2 µ2 E 2 µ ⃗ ⃗ α = ψ− ψ+ (29) ∇ · E ψ + µ σ⃗ · B⃗ ψ + η ρ ψ + ψ. ∂t 2m 2m 2m ρ Thereby, the modification of the mass term S (ρ) = η ρ + αρ in the Dirac equation (4) plays the role i
of a linear and a Coulomb-type scalar potentials [50,51] in the nonrelativistic limit (up to terms of order m−1 ). In particular, the Coulomb-type potential has attracted interests in studies of quark models [39,62], chemical physics [70,71], the Kratzer potential [72–75], the Mie-type potential [76,77], position-dependent mass systems [37,78,79] and topological defects in solids [80–83]. Recently, the two-dimensional Coulomb potential has been investigated under the presence of the Aharonov–Bohm effect in Refs. [84–86]. Therefore, let us confine the neutral particle with a permanent magnetic dipole moment to a quantum dot described by the Tan–Inkson model [15–17], as given in Eq. (6), under the influence of a linear confining potential and a Coulomb-like potential. Again, we assume that the wave function of the neutral particle is well-behaved at the origin and the permanent magnetic dipole moment of the neutral particle to be aligned in the z-direction as in the previous section, therefore, by introducing the Tan–Inkson potential (6) into the Schrödinger–Pauli equation (29), we obtain i
2 i σ 3 ∂ψ 1 ∂ ψ 1 ∂ψ 1 ∂ 2ψ ∂ 2ψ 1 φAC ∂ψ =− + + + + − ψ 2 2 2 2 ∂t 2m ∂ρ ρ ∂ρ ρ ∂ϕ ∂z 2m ρ 2 ∂ϕ 2mρ 2 2π i φAC σ 3 ∂ψ 1 φAC 2 1 α − + ψ+ ψ + ψ + η ρ ψ + a2 ρ 2 ψ. 2 2 2 m 2π ρ ∂ϕ 2mρ 2π 8mρ ρ
(30)
By following the steps from Eq. (7) to Eq. (10), we have that the solution to the Schrödinger–Pauli equation (30) has the same form of Eq. (8), and thus we obtain the following radial equation:
τ2 2mα 2 + − 2− − 2mη ρ − 2ma2 ρ + 2mE G¯ = 0, dρ 2 ρ dρ ρ ρ d2
1 d
(31)
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¯ (ρ). Now, let us where τ has been defined in Eq. (10) and we have replaced the function G (ρ) with G perform the same change of variables made in the previous section, that is, ξ = (2ma2 )1/4 ρ ; thus, we obtain τ2 ¯ θ ¯ ¯ − ξ 2 G¯ + β G¯ = 0, G − G − µξ G (32) dξ ξ dξ ξ2 ξ where the parameters µ and β have been defined in Eq. (12) and the new parameter θ is given by ¯ d2 G 2
θ=
+
¯ 1 dG
−
2m α . (2ma2 )1/4
(33)
From the discussion about the asymptotic behaviour of the wave function made from Eq. (13) to ¯ (ξ ) in terms of an unknown function H¯ (ξ ) Eq. (16), hence, we have that we can write the function G as in Eq. (13), that is, 2
ξ µξ ¯ (ξ ) = e− 2 e− 2 ξ |τ | H¯ (ξ ) . G
(34)
By substituting (34) into (32), we have
¯ d2 H dξ 2
+
2 |τ | + 1
ξ
− µ − 2ξ
¯ dH dξ
+ g−
f
ξ
¯ = 0, H
(35)
where the parameter g has been defined in Eq. (3) and the parameter f is f =
µ
(2 |τ | + 1) + θ .
2
(36)
¯ (ξ ) is the biconfluent Eq. (35) is called as the biconfluent Heun equation [63], and the function H Heun function [62–64]: ¯ (ξ ) = HB H
2 |τ | , µ, β +
µ2 4
, 2θ , ξ .
(37)
Again, we follow the steps from Eq. (17) to Eq. (20) in order to obtain the bound state solutions. In this case, the recurrence relation becomes
µ (j + 1) + f (g − 2j) c¯j+1 − c¯j , (j + 2) (j + 2 + 2 |τ |) (j + 2) (j + 2 + 2 |τ |)
c¯j+2 =
(38)
f
and c¯1 = (1+2|τ |) c¯0 . We also start with c¯0 = 1 and with the relation (38), we can calculate other coefficients of the biconfluent Heun series (17). For instance, c¯1 =
f
(1 + 2 |τ |)
;
c¯2 =
f (f + µ) 2 (2 + 2 |τ |) (1 + 2 |τ |)
−
g 2 (2 + 2 |τ |)
.
(39)
With the conditions established in Eq. (20) we have that the biconfluent Heun series becomes a polynomial of degree n. By analysing the condition g = 2n, we obtain the same expression for the energy levels for bound states obtained in Eq. (21) and the same angular frequency (22), that is,
η2 , (40) 2mω2 where ω is given in Eq. (22) and n = 1, 2, 3, . . . . Again, we can see that the angular frequency of the En, l, s = ω [n + |τ | + 1] −
system differs from that one of the Tan–Inkson model [15–17] and the ground state is determined by the quantum number n = 1 instead of n = 0 as in the previous case. We have also that the energy levels (40) depend on the Aharonov–Casher geometric phase [27], which is an Aharonov–Bohmtype effect for bound states [26]. However, the energy levels (40) seem to have no influence of the parameter α that characterizes the Coulomb-like potential. For this reason, let us analyse the condition
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c¯n+1 = 0 [87]. We also assume that the parameter a2 of the Tan–Inkson model [15–17] can be adjusted in such a way that the condition c¯n+1 = 0 can be satisfied. By considering the ground state of the system (n = 1), we have that c¯2 = 0 from the condition c¯n+1 = 0. The condition c¯2 = 0 imposes that the possible values of the angular frequency ω1, l, s are determined by the third-degree algebraic equation [62,64]:
ω13, l, s −
2 m α2
(1 + 2 |τ |)
ω12, l, s − 2 η α
η2 (2 + 2 |τ |) ω1, l, s − (3 + 2 |τ |) = 0. 2m (1 + 2 |τ |)
(41)
Despite Eq. (41) has at least one real solution, we do not write it because it’s expression is very long. Observe that the solution to Eq. (41) depends on the parameter τ , which means that the angular frequency associated with the ground state of the system depends on the Aharonov–Casher geometric phase φAC . This kind of behaviour of the angular frequency can be expected for other energy levels. Therefore, there exists a quantum effect characterized by the dependence of the angular frequency on the quantum numbers of the system {n, l, s} and the Aharonov–Casher geometric phase φAC . ¯ (ξ ), which corresponds to a Moreover, for n = 1 we have the simplest case of the function H polynomial of first degree: 2
ξ µξ ¯ 1, l, s (ξ ) = e− 2 e− 2 ξ |τ | G
1+
µ (2 |τ | + 1) + 2θ ξ . 2 (1 + 2 |τ |)
(42)
Finally, we can rewrite expression for the energy levels of the bound states (40) as
En, l, s = ωn, l, s [n + |τ | + 1] −
η2 , 2 m ωn2, l, s
(43)
Hence, the general expression for energy levels (43) is obtained for a neutral particle with a permanent magnetic dipole moment that interacts with a radial electric field confined to a quantum dot under the influence of a Coulomb-like and linear scalar potentials. Observe that there exists the dependence of the energy levels on the Aharonov–Casher geometric phase [27] which gives rise to the arising of the persistent spin current in the quantum dot. Due to the difficulty of obtaining the analytical expression for the angular frequency associated with each energy level, we decided not to calculate the persistent spin current even in the case of the ground state, since the expression for the angular frequency is too long as we have seen in Eq. (41). 4. Conclusions We have analysed quantum effects associated with the Aharonov–Casher geometric phase [27] when a neutral particle is confined to a quantum dot under the influence of a linear scalar potential and a Coulomb-like potential. In both cases analysed here, we have seen that the energy levels of the bound states depend on the Aharonov–Casher geometric [27], which is called as an analogue of the Aharonov–Bohm effect for bound states [24] or the Aharonov–Casher effect for bound states [26]. We have also shown that the angular frequency of the Tan–Inkson model for a quantum dot [15–17] is modified by the influence of the linear and Coulomb-like scalar potentials. Besides, we have seen that there is a restriction on the values of the angular frequency of the system in order that bound state solutions can be achieved, where these possible values of the angular frequency of the system depend on the quantum numbers of the system and the Aharonov–Casher geometric quantum phase. As examples, we have calculated the possible values of the angular frequency of the system associated with the ground state of the system in the two cases discussed in this work. In the first case, where the neutral particle is confined to a quantum dot in the presence of the linear scalar potential, we have obtained the analytical expression for the possible values of the angular frequency and the persistent spin currents. On the other hand, when the neutral particle is confined to a quantum dot in the presence of both linear and Coulomb-type scalar potentials, we have seen that the possible values of the angular frequency of the system are determined by third-degree algebraic equation.
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