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A Bound Quantum Particle in a RiemannCartan space with Topological Defects andPlanar Potential
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Physics Letters A 366 (2007) 315–323www.elsevier.com/locate/pla
A bound quantum particle in a Riemann–Cartan space with topologicaldefects and planar potential
S.A. Alix
a
, C. Cafaro
a
, S. Capozziello
b
,
∗
, Ch. Corda
c
a
Department of Physics, University at AlbanySUNY, 1400 Washington Avenue, Albany, NY 12222, USA
b
Dipartimento di Scienze Fisiche, Università di Napoli “Federico II” and INFN Sez. di Napoli, Complesso Universitario Monte S. Angelo, Ed.N, Via Cinthia, I80126 Napoli, Italy
c
INFN Sez. di Pisa and Università di Pisa, Via F. Buonarroti 2, I56127 Pisa, Italy
Received 9 February 2007; accepted 18 February 2007Available online 21 February 2007Communicated by V.M. Agranovich
Abstract
Starting from a continuum theory of defects, that is the analogous to threedimensional Einstein–Cartan–Sciama–Kibble gravity, we consider acharged particle with spin
12
propagating in a uniform magnetic ﬁeld coincident with a wedge dispiration of ﬁnite extent. We assume the particleis bound in the vicinity of the dispiration by long range attractive (harmonic) and short range (inverse square) repulsive potentials. Moreover,we consider the effects of spintorsion and spinmagnetic ﬁeld interactions. Exact expressions for the energy eigenfunctions and eigenvalues aredetermined. The limit, in which the defect region becomes singular, is considered and comparison with the electromagnetic Aharonov–Bohmeffect is made.
©
2007 Elsevier B.V. All rights reserved.
PACS:
02.40.Ky; 61.72.Lk; 03.65.Ge
Keywords:
Topological defects; Riemann–Cartan geometry; Screw dislocation; Bounded quantum mechanical particle
1. Introduction
The investigation of quantum mechanical systems with nontrivial boundary conditions is an active ﬁeld of research. A fertileground for such systems is provided by particles moving in a background space with nonvanishing (positive or negative) curvatureand/or nontrivial topology. By nontrivial topologywe meanmultiply connectedspaces. Indeed, the prototyperepresenting the caseof nontrivial topology is the wellknown electromagnetic Aharonov–Bohm (AB) effect [1]. In the context of the electromagnetic
ABeffect,theeffectoftopologymanifestsasthephasefactorinthewavefunctionofanelectronmovingaroundamagneticﬂuxline.The gravitational analogue of this effect has been investigated in [2–4]. We are going to investigate a space with topological defects
characterized by vanishing Riemann–Christoffel curvature and torsion [5] everywhere except on the defects. Such defects arise ingauge theories with spontaneous symmetry breaking and may have played some signiﬁcant role in the formation of large scalecosmological structure. Some examples in cosmology are domain walls [6], cosmic strings [6,7] and monopoles [8]. Analogues
of such defects in condensed matter physics include vortices in superﬂuids and superconductors [9], domain walls in magnetic
materials, dislocations in solids and disclinations in liquid crystals or twodimensional graphite [10].Quantum effects on particles moving in a crystalline media with topological defects have been a subject of investigation sincethe early 1950s [11]. In the geometric approach of Katanaev and Volovich [12], the theory of defects in solids is translated into
*
Corresponding author.
Email address:
capozziello@na.infn.it (S. Capozziello).03759601/$ – see front matter
©
2007 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2007.02.057
316
S.A. Alix et al / Physics Letters A 366 (2007) 315–323
the language of threedimensional Einstein–Cartan–Sciama–Kibble gravity [13,14] but it could be extended considering moregeneral forms of torsion [15]. This formalism corresponds to viewing the continuous limit of a crystalline solid in which the defect
conﬁguration is characterized by a nontrivial metric describing a static threespace. In this way, elastic deformations introduced inthe medium by defects are incorporated into the metric manifold. The boundary conditions required by the presence of such defectsare accounted for by the associated nonEuclidean metric. In the continuum limit, that is, for distances much larger than the Bravaislattice spacing, the theory describes a nonRiemannian manifold where curvature and torsion are associated with disclinations anddislocations in the medium, respectively.In the case of solid crystals, a topological defect can be thought as consisting of a core region characterized by the absence of regular lattice order and an ordered (undistorted) farﬁeld region [16]. Although in the continuum approximation in which we work
the core region is usually shrunk to a singularity, we choose to relax this condition by smearing the singularity over a ﬁnite region.By virtue of having a ﬁnite defect region, we consider the effects of spintorsion and spinmagnetic ﬁeld interactions on the system.In this work, we study a charged spin
12
particle moving in the space of a defect comprised of a combined screw dislocation anda wedge disclination. Such a combined defect is called a wedge dispiration, a defect possessing nonvanishing local curvature andtorsion. A homogeneous, ﬁnite magnetic ﬁeld, concentric with the wedge dispiration line, is included. As a working hypothesis, weassume the particle is bound in the vicinity of the wedge dispiration by a planar potential given by a long range attractive (harmonic)and short range (inverse square) repulsive terms.The Letter is organized as follows: In Section 2, we introduce the topological defect. In Section 3, we consider a simple ﬁnite
defect distribution describing a nonsingular dispiration. In Section 4, the corresponding timeindependent Schrödinger equation is
taken into account. In Section 5, exact expressions for the energy eigenfunctions of the particle moving both inside and outside thedefect core of the medium are obtained. Boundary conditions on the surface of the defect region are implemented in order to getthe wavefunction deﬁned over the whole space. The energy spectrum is determined and we consider the limit in which the defectregion is shrunk to a null size. Conclusions are drawn in Section 6.
2. The magnetic wedge dispiration
In the framework of traditional elasticity theory, a Cartesian reference frame
x
i
is attached to the undistorted medium of anelastic solid with Euclidean metric
δ
ab
. The deformation of the medium is described locally in terms of a continuous displacementfunction
u(x)
. In this way, after a deformation has occurred, the point
x
i
will have coordinates
x
i
→
y
i
(x)
=
x
i
+
u
i
(x)
. The initialmetric
δ
ab
is transformed into [17,18](1)
g
ij
:=
∂x
a
∂y
i
∂x
b
∂y
j
δ
ab
, i,j
=
1
,
2
,
3
.
We consider a topological defect in three dimensions whose geometry is characterized by the spatial line element [12](2)
dl
2
=
g
ij
dx
i
dx
j
=
dρ
2
+
κ
2
ρ
2
dϕ
2
+
(dz
+
τ dϕ)
2
where
(ρ,ϕ,z)
arecylindricalcoordinates,with
ρ
0, 0
ϕ
2
π
,
−∞
z
∞
,
κ,τ
∈
R
.Theparameter
κ
isrelatedtotheFrank vector
Ω
[19] of the disclination (describing curvature, i.e. the angular deﬁcit in the manifold) while
τ
is related to the Burgers’vector
b
[17,19] of the dislocation (describing torsion). The threedimensional geometry of the medium is therefore characterizedby nonvanishing torsion and curvature. When
κ
=
0 and
τ
=
0 we have a wedge disclination; when
κ
=
0 and
τ
=
0 we have ascrew dislocation. The metric tensor
g
ij
and its inverse
g
ij
=
(g
ij
)
−
1
are given by [20](3)
g
ij
=
1 0 00
κ
2
ρ
2
+
τ
2
τ
0
τ
1
, g
ij
=
1 0 00
1
κ
2
ρ
2
−
τ κ
2
ρ
2
0
−
τ κ
2
ρ
2
1
+
τ
2
κ
2
ρ
2
.
The line element (2) describes an inﬁnitely long linear wedge dispiration oriented along the
z
axis. We introduce the dual (1form)basis vectors
ϑ
i
=
e
ij
dx
j
which describe the background (2), with coframe components [18]
(4)
ϑ
1
≡
ϑ
ρ
=
dρ, ϑ
2
≡
ϑ
ϕ
=
κρdϕ
and
ϑ
3
≡
ϑ
z
=
dz
+
τ dϕ.
The metric
g
ij
and triad components
e
ia
are related via
e
ia
e
j b
δ
ab
=
g
ij
, where the triads satisfy
e
aj
e
j b
=
δ
ab
. In a Riemanniangeometry without torsion, the Ricci
R
ij
and Riemann curvature tensor
R
lijk
are given by [5](5)
R
ij
=
g
ab
R
aibj
=
∂
k
Γ
kij
−
∂
j
Γ
kik
+
Γ
kij
Γ
nkn
−
Γ
mik
Γ
kjm
and(6)
R
lijk
=
∂
i
Γ
ljk
−
∂
j
Γ
lik
+
Γ
lim
Γ
mjk
−
Γ
ljm
Γ
mik
,
S.A. Alix et al / Physics Letters A 366 (2007) 315–323
317
respectively. The Christoffel symbols
Γ
kij
appearing in (5) or (6) are deﬁned by [5]
(7)
Γ
kij
:=
12
(∂
k
g
ij
+
∂
i
g
jk
−
∂
j
g
ki
)
where
∂
k
:=
∂/∂x
k
. The nonvanishing components of the Christoffel symbols in the space with metric (3) are [18]
(8)
Γ
zϕρ
=
Γ
zρϕ
=−
τ ρ, Γ
ρϕϕ
=−
κ
2
ρ, Γ
ϕρϕ
=
Γ
ϕϕρ
=
1
ρ.
By contrast, a nonRiemannian geometry is one that is characterized by nonvanishing curvature and torsion. The torsion tensor
T
iab
is deﬁned by(9)
T
kij
:=
∂
i
e
kj
−
∂
j
e
ki
+
Γ
kiν
e
νj
−
Γ
kjν
e
νi
.
The Ricci scalar is given by [16,21](10)
R
1212
=
R
11
=
R
22
=
2
π
1
−
κκ
δ
2
(ρ),
where
κ
=
1
+
φ/
2
π
and
δ
2
(ρ)
is a twodimensional Dirac
δ
function. This
δ
function revels the conic singularity in the curvaturegiven in (10). The dispiration characterized by (2) can be thought as arising from a “cut and paste” process, known as the Volterra
process [20]. From the perspective of the Volterra process, the disclination is generated by removing
(κ <
1
)
or inserting
(κ >
1
)
a wedge of material with deﬁcit angle
φ
=
2
π(κ
−
1
)
. For 0
<κ <
1 the disclination carries positive curvature while for 1
<κ <
∞
it carries negative curvature. The only nonvanishing component of torsion 2form
T
k
=
T
kij
dx
i
∧
dx
j
is given by [16,21](11)
T
z
=
2
πτδ
2
(ρ)dρ
∧
dϕ.
It is clear that the space described by metric (3) features two conical singularities at the srcin as seen in (10) and (11). The Burger
vector can be viewed as a ﬂux of torsion and the Frank vector as a ﬂux of curvature. The Burger vector is calculated by integratingaround a closed path
C
encircling the dislocation [17,18](12)
b
3
=
C
ϑ
3
=
S
dρ
∧
dϕT
3
ρϕ
=
2
πτ,
implying that 2
πτ
is the ﬂux intensity of the torsion source passing through a closed loop
C
in the
ˆ
e
z
direction. Thus, the parameter
τ
is the modulus of the Burger vector. In a similar manner, the Frank vector is determined by [10,18]
Ω
:=
ijk
Ω
jk
ˆ
e
i
, with(13)
Ω
12
=
S
dρ
∧
dϕR
12
ρϕ
=
2
π
1
−
κκ
,
where
ijk
is the LeviCivita symbol
(
123
=
1
)
,
Ω
12
=−
Ω
21
and
S
is a surface perpendicular to the defect line, implying the ﬂuxof curvature is the surface density of the Frank vector ﬁeld.
3. Finite defect distribution
To avoid the singular nature of (10) and (11), we choose a simple exactly solvable model of ﬁnite torsion and curvature. In the
former case, we choose a torsion ﬁeld with a homogeneous ﬂux distribution within the dislocation region. In particular, we choosea torsion ﬁeld speciﬁed by(14)
T
z
=
T
z
(ρ,ϕ)
=
τ
Θ(R
c
−
ρ)dρ
∧
dϕ,
where
R
c
denotes the radius of the defect core,
τ
=
s
T
b
z
πR
2
c
and
s
T
=±
1 denotes the handedness of the screw, where
(s
T
=−
1
)
indicates a lefthanded screw in which a clockwise rotation of
ϕ
=
2
πκ
(
κ
is deﬁned in Eq. (17)) relative to the
(
+
) x
axisinduces a shift in the
(
−
) z
direction; similarly,
(s
T
=+
1
)
describes a righthanded screw in which a counterclockwise rotation of
ϕ
=
2
πκ
relative to the
(
+
) x
axis induces a shift in the
(
+
) z
direction. In (14),
Θ
denotes the Heaviside theta function(15)
Θ(R
c
−
ρ)
=
1 for
ρ <R
c
,
0 for
ρ >R
c
.
In the case of nonsingular curvature, we consider a disclination characterized by the deﬁcit angle(16)
¯
φ
=
12
φR
2
c
Θ(R
c
−
ρ).
318
S.A. Alix et al / Physics Letters A 366 (2007) 315–323
Under
φ
→ ¯
φ
transformation, the angular deﬁcit
κ
=
1
+
φ/
2
π
transforms into(17)
κ
=
1
+
12
φR
2
c
Θ(R
c
−
ρ)
=
κ
in
=
1
+
12
φR
2
c
for
ρ <R
c
,κ
out
=
1 for
ρ >R
c
.
By the change of variables
ρ
→ ¯
ρ
, where(18)
ρ
→ ¯
ρ
=
ρ
κ
κ
=
ρ
<
=
ρ
κ
in
κ
in
for
ρ <R
c
,ρ
>
=
ρ
for
ρ >R
c
,
the metric describing the nonsingular wedge dispiration is given by [16](19)
dl
2
=
g
ij
dx
i
dx
j
=
d
¯
ρ
2
+
κ
2
¯
ρ
2
dϕ
2
+
(dz
+
τ
dϕ)
2
.
Observe that distributions (14) and (17) are chosen such that the total torsion and curvature ﬂux within the dispiration region areequivalent to initial values 2
πτ
and 2
π(
1
−
κκ
)
, respectively. Furthermore, we assume the particle is moving in the electromagneticvector potential
A(
¯
ρ)
[22](20)
A(
¯
ρ)
=
B
0
2
πκ
¯
ρ
¯
ρ
2
R
2
c
Θ(R
c
−
ρ)
+
Θ(ρ
−
R
c
)
ˆ
e
ϕ
,
where
B
0
is the total magnetic ﬂux. Vector potential (20) gives rise to a uniform magnetic ﬁeld
B
within the dispirated region.
4. The Schrödinger equation
We study the nonrelativistic quantum dynamics of a charged spin
12
particle propagating in a space with ﬁnite dispiration defect(19) in presence of a uniform magnetic ﬁeld
B
= ∇×
A
and planar potential(21)
V(
¯
ρ)
=
12
Mω
2
¯
ρ
2
Θ(R
c
−
ρ)
+
f
¯
h
2
2
M
¯
ρ
2
,
where
f
is a real, dimensionless constant characterizing the medium being probed. Observe that the harmonic potential does notextend beyond the medium
(R
m
)
being considered, where
R
m
R
c
and
R
m
is the linear dimension of the medium. Furthermore,following [23], the spintorsion interaction term
−
18
σ
·
T
is considered, where
σ
=
g
e
µ
¯
h
ˆ
e
S
,
ˆ
e
S
denotes the spin direction,
µ
=
q
¯
h
2
Mc
is the Bohr magneton and
g
e
is the gyromagnetic ratio of the electron. Finally, the timeindependent Schrödinger equation becomes,(22)
12
M
+
σ
·
B
−
18
T
+
V(
¯
ρ)
ψ(
¯
ρ,ϕ,z)
=
Eψ(
¯
ρ,ϕ,z).
The wavefunction satisfying (22) has periodicity
ψ(
¯
ρ,ϕ,z)
=
ψ(
¯
ρ,ϕ
+
2
πκ
,z)
rather than the usual situation
ψ(ρ,ϕ,z)
=
ψ(ρ,ϕ
+
2
π,z)
in ﬂat space. The Laplace–Beltrami operator
is deﬁned by(23)
:=
1
√
g
¯
hi
C
∂
i
−
qcA
i
g
ij
√
g
¯
hi
C
∂
j
−
qcA
j
=−¯
h
2
∇
2
+
q
2
c
2
A
2
−
q
¯
hci
C
(
∇·
A
+
A
· ∇
),
where
i
C
is the imaginary unit,
c
is the speed of light,
q
is the charge of the electron and
g
=
det

g
ij

is the determinant of themetric. The Laplacian in the space (2) is given by(24)
∇
2
=
1
¯
ρ∂∂
¯
ρ
¯
ρ∂∂
¯
ρ
+
1
κ
2
¯
ρ
2
∂∂ϕ
−
T
z
∂∂z
2
+
∂
2
∂z
2
.
The divergence appearing in (23) is computed according to(25)
∇·
A
=
1
¯
ρ∂∂
¯
ρ(
¯
ρA
¯
ρ
)
+
1
κ
¯
ρ
∂∂ϕ
−
T
z
∂∂z
A
ϕ
+
∂A
z
∂z
and is vanishing since
∇·
A
=
1
κ
¯
ρ
(
∂∂ϕ
−
T
z ∂∂z
)
B
0
2
πκ
¯
ρ
=
0. With the aid of (24) and (25), operator (23) can be written as
=−¯
h
2
∇
2
+
q
2
c
2
B
0
2
πκ
¯
ρ
2
¯
ρ
2
R
2
Θ(R
c
−
ρ)
+
Θ(ρ
−
R
c
)
2
(26)
−
qB
0
2
πcκ
¯
ρ
¯
hi
C
¯
ρ
2
R
2
Θ(R
c
−
ρ)
+
Θ(ρ
−
R
c
)
∂∂ϕ
−
s
T
b
z
πR
2
Θ(R
c
−
ρ)∂∂z
.