Applications of the potential algebras of the two-dimensional Dirac-like operators

0.44 MB·English

by  Vit Jakubsky

Potential algebras of low-dimensional Dirac operators: applications
V´ıt Jakubsk´y
Nuclear Physics Institute, Rˇeˇz near Prague, 25068, Czech Republic
E-mail: [email protected]
Abstract
2
Potential algebras can be used effectively in the analysis of quantum systems described by (2+
1
0 1) dimensional Dirac equation. We find integrals of motion of Dirac operator which close centrally
2 extended so(3), so(2,1) or oscillator algebra. The algebraic framework is used in construction of
physically interesting solvable models. It is applied in description of open-cage fullerenes where the
t
c
energiesandwavefunctionsoflow-energychargecarriersarecomputed. Besides,thepotentialalgebras
O
are employed in the context of one-dimensional, both relativistic and nonrelativistic, shape-invariant,

5 quantum systems. We show that shape-invarianceof the consideredDirac operatorsis associatedwith
the nonlinear N = 2 supersymmetry. The potential algebra is used in construction of nonrelativistic

] shape-invariantsystemswithpositiondependentmass. Wedemonstratethatsolvabilityofthesemodels
h
p is associated with their N =8 supersymmetry.
-
h
t 1 Introduction
a
m
[ Exactly solvable models play exceptional role in physics. They are simple enough to be solvable, yet

sufficiently complicated to grab the essence of physical reality. In quantum mechanics, exact solvability
2
v usually means that the eigenfunctions and eigenvalues of the Hamiltonian can be found explicitly. They
0
can be computed analytically by solving directly corresponding differential equations.
8
Alternatively, they can be obtained in algebraic manner with the use of the integrals of motion;
9
4 when the symmetries of the quantum system form Lie algebra, the spectrum and eigenstates of the
.
9 energy operator can be deduced from the analysis of the admissible unitary representations [1], [2]. The
0 symmetry operators preserve domain of the Hamiltonian by transforming one physical state into another.
2
When they are time-dependent, they do not commute with the energy operator and, hence, they do
1
: not preserve energy. The associated algebra of the integrals of motion is called dynamical or spectrum
v
i generating, see [3], [4]. When the integrals of motion commute with the Hamiltonian, they rather reflect
X
spectral degeneracy of the system. The algebraic structure, where the Hamiltonian plays the role of
r
a central element, is denoted as potential algebra. In the current article, we will focus on the analysis and
applications of the potential algebras of therelativistic quantum systems describedby (2+1) dimensional
Dirac equation.
The low-dimensional Dirac equation appears in surprisingvariety of physical settings that are studied
bothinquantumfieldtheoryandinthecondensedmatter physics. Letusmention the(1+1) dimensional
variant of the famous Nambu-Jona-Lasinio (chiral Gross-Neveu) model [5], [6], [7], analysis of fractionally
charged solitons [8], [9] as well as description of inhomogeneous superconductors [10], carbon nanotubes
[11], [12], or linear molecules like polyacetylene [13]. The (2+1) dimensional massless Dirac Hamiltonian
appearsin thelow-energy approximation of dynamicsof charge carriers in grapheneandin related carbon
nanostructures [14], [15], [16], [17].
The work is organized as follows: in the next section, the potential algebras are constructed for a
wide class of Dirac operators with the use of a specific ansatz for both the ladder operators and for
1
the structure of the algebra. Relevant aspects of the representations based on the lowest/highest weight
vectors are discussed in more detail, as well as the application of the algebraic structure in description
of Dirac fermions in the curved space. The results are used directly in the third section, where solvable
models describing low-energy charge carriers on the surface of open-cage fullerenes are analyzed. Two
configurations are considered, with one and two holes in the surface of the crystal. In the fourth section,
we apply the potential algebras in construction of one-dimensional shape-invariant systems. First, we
focus on the shape-invariance of one-dimensional Dirac Hamiltonians and show that it is associated
with N = 2 nonlinear supersymmetry. Next, nonrelativistic systems with position dependent mass are
studied. The algebraic background of the shape-invariant superextended Hamiltonian is identified as
N = 8 superalgebra. Illustrative example of solvable system is presented. In the last section, we discuss
brieflytwospecifictwo-dimensional systems; wemakefewcomments onDiracoscillator andits dynamical
symmetries. Additionally, we show that the so(2,1) potential algebra can be used for analysis of (1+1)-
dimensional quantum system whose metric can be identified with the (restricted) BTZ black hole metric.
We conclude with short discussion of the results and outlook to possible future research.
2 Potential algebras of two-dimensional Dirac operator
Let us consider the operator h which is given in terms of the Pauli matrices σ and real functions g ,
D a 1
g and g ,
2 3
g
h = iσ g ∂ + 1′ +σ ( ig ∂ +c g ), g = g (x ), a = 1,2,3. (1)
D 1 1 x1 2 2 − 2 x2 3 3 a a 1
(cid:18) (cid:19)
Weleavethedomainofh aswellastherangeoftherealvariablesx andx unspecifiedatthemoment. If
D 1 2
not stated otherwise, the coordinates x and x are considered to bespace-like. Then (1) can beregarded
1 2
as Dirac Hamiltonian of (2+1)-dimensional Dirac equation. Thefunction g = g (x ) represents external
3 3 1
magnetic field with real coupling constant c . The operator is manifestly hermitian with respect to the
3
standard scalar product. It anticommutes with σ , which implies that its spectrum will be symmetric
3
with respect to zero.
The system represented by (1) has integral of motion J ,
3
J = i∂ , [h ,J ] = 0. (2)
3 − x2 D 3
We are interested in the settings where two additional integrals of motion J exist, such that
±
[J ,J ] J , [J ,J ] J , [J ,h ] = 0. (3)
3 + 3 D
± ∼ ± ± − ∼ ±
Let us make the ansatz in the form J = eix2 2 (A + σ A )∂ +C +σ C ) where the coeffi-
+ k=1 k,1 3 k,2 xk 1 3 2
cients Ak,a and Ca (a,k = 1,2) depend on x1 o(cid:16)nlPy. The operator J is defined as J = J+†. Inserting the
− −
ansatz into (3) and comparing the coefficients of corresponding derivatives, we find that the coefficients
of J can be expressed in terms of the functions g , g and g as
1 2 3
±
g g g ( J + 1) g g g
J = ie ix2 g ∂ + 1′ + 1 2′ ± 3 2 1 3′ 2σ . (4)
± ± 1 x1 2 g2 ± g2 ∓ 2 3!
It suggests that the function g should be node-less to avoid singularities in definition of these operators.
2
Additionally, the functions g , g and g have to solve the following set of differential equations,
1 2 3
g3g +g2g g g g (g g +g g ) = 0,
2 3 1 2′ 3′ − 1 2 1′ 3′ 1 3′′
g4+g2g2 g g (g g +g g ) = 0,
2 1 2′ − 1 2 1′ 2′ 1 2′′
g g (g g ) 2g g2 ′
1 2 1 2′ ′− 1 2′ = 0,
g2
(cid:0) 2 (cid:1)!
g1 (g2(g1g3′)′−2g1g2′g3′) ′ = 0.
g2
(cid:18) 2 (cid:19)
2
Quick inspection suggests that the system can be reduced to two equations as long as g = g . However,
3 2
wewillomitthissolution; looking at(1), itwouldjustshifttheoperator J by aconstant c . Thesolution
3 3
of the unreduced system can be found such that g as well as g are given in terms of g in the following
1 3 2
manner,
c2(g2 +c ), for c = 0,
g2 = g24+c1g22, g2 = 3 2 1 1 6 (5)
1 (g )2 3 
2′ c2g 2, for c =0,
 3 2− 1
where the c and c have to be real valued to maintain hermicity of the Hamiltonian.
1 3 
The operators J and J form Lie algebra. Its explicit form depends on the value of c . For c = 0,
3 1 1
± 6
it can be identified either as so(3) (c < 0) or so(2,1) (c > 0), while the oscillator algebra is restored for
1 1
c = 0. h commutes with both J and J , and, hence, it is the central element of the following algebra
1 D 3
±
2c J for c = 0,
[J3,J±]= ±J±, [J−,J+] = 14c33 for c11 =6 0, (6)
(cid:26) −
[h ,J ]= [h ,J ]= 0. (7)
D D 3
±
Let us notice that nonrelativistic systems with similar algebraic background were classified as superin-
tegrable in [18]. The representation space of (6)-(7) is spanned by the eigenvectors of h . It can be
D
decomposed into subspaces where h acquires constant value and wherethe algebra (6)-(7) is irreducible.
D
These subspaces can have either finite or infinite dimension, dependently on the type of the algebra.
The square of (1) can be written as a second-order polynomial in the generators of (6). When c = 0,
1
6
it coincides up to an additive constant with the standard Casimir operator = J J c J (J 1) =
+ 1 3 3
C − − −
J J c J (J +1) of the algebra so(3) or so(2,1). When c = 0, the algebra is not semi-simple (Killing
+ 1 3 3 1
− −
form is degenerate) and the standard (quadratic) Casimir operator does not exist. There holds
J J c J 1 2 c2 = J J c J + 1 2 c2 for c = 0,
h2D = ( J++J−− −+41c3(cid:16)(cid:0)J33−−122(cid:1)=−J−3J(cid:17)+ +4−c3+J−3+1(cid:16)12(cid:0) 3 2(cid:1) − 3(cid:17) for c11 =6 0. (8)
The operator h commutes w(cid:0)ith1 =(cid:1) R R σ . Th(cid:0)e reflec(cid:1)tion operators R and R are defined as
D R x2 c3 1 x2 c3
R f(x ,x ,c ) = f(x , x ,c ) and R f(x ,x ,c ) = f(x ,x , c ). For generators of the potential
x2 1 2 3 1 − 2 3 c3 1 2 3 1 2 − 3
algebra, there holds
J = J , J = J .
+ 3 3
R −R R R −
In our analysis, we will focus on the representations that can be constructed from the lowest (highest)
weight vectors, the zero-modes of ladder operator J (or J ). These vectors, denoted as ψ (or ψ+), can
+ m− m
−
be fixed as the mutual eigenstates of h and J ,
D 3
J ψ = 0, J ψ = mψ , h ψ = E ψ . (9)
m± 3 m± m± D m± m± m±
±
The energies E can be found with the use of the relations (8),
m±
c m 1 2 c2 for c = 0,
(Em±)2 = ( − 1(cid:16)(cid:0) ±4c32(cid:1)m−±321(cid:17) for c11 =6 0. (10)
To keep the energies real, there must hold m 1 (cid:0) c for(cid:1)c < 0. For c > 0, we get m 1 c .
± 2 ≥ 3 1 1 ± 2 ≤ 3
The other eigenstates of h corresponding to the same energy can be found by repeated action
D (cid:12) (cid:12) (cid:12) (cid:12)
of the ladder operator J (J ) on the vect(cid:12)ors ψ (cid:12)(or ψ+). These eigenstates then est(cid:12)ablish(cid:12)irreducible
+ m− m
−
representations of (6)-(7)that is specifiedby theenergies E of thecorrespondinglowest (highest) weight
m±
1Similar operator that switches the sing of coupling constant was employed in [20] in the analysis of quantum systems
with nonlinear supersymmetry.
3
vector. These representations are usually associated with the bound states of the quantum system, see
e.g. [19].
The unitary representations of so(3) are finite dimensional. Consequently, the lowest weight vector,
let us fix it as ψ , has be to annihilated by specific power of the ladder operator J . There holds
−m +
(J )dψ =0 wh−ere d denotes dimension of the corresponding representation. The quantum number m
+ −m
has to b−e either integer or semi- integer; we have ψ+ (J )d 1ψ so that J Jd 1ψ = mJd 1ψ .
Additionally, we can use (6) and write J Jd 1ψ =m(∼ m++d− 1−−)mJd 1ψ . C3om+−bin−i−nmg these+re−lati−−omns
together, there must hold 2m + d 13=+−0. −−Smince d−is a po−sitive+−inte−−gmer, we get that m has to be
− −
non-negative integer or semi-integer. The admissible values of the coupling constant c depend on m;
3
when m acquires integer (half-integer) values, c has to be half-integer (integer). We refer to Appendix
3
A for details of the proof. Let us notice in this context that the representations of odd dimension for two
different half-integer values of c were considered in [21]. Representations of so(3) and so(2,1) were also
3
discussed in [22] where they were obtained analytically with the use of the master equation.
In the end of the section, we shall discuss shortly the situation when the spin 1/2 particle is living in
−
the two-dimensional curved space. Let us consider the space where the metric tensor acquires diagonal
form and the non-vanishing components depend on one coordinate only,
g (x ) 0
g = 11 1 .
µν 0 g (x )
22 1
(cid:18) (cid:19)
In the most of the text, we will deal with the systems where both x and x are considered to be
1 2
space-like (g > 0, g > 0), i.e. the metric tensor corresponds to a curved surface embedded into
11 22
three-dimensional euclidean space. Quantum settings of this kind appear frequently in the analysis of
carbon nanostructures where deformations of graphene crystal are considered [24]. The massless Dirac
Hamiltonian can be written in the following form (see Appendix B for details),
1 1 J
3
h = iσ ∂ +σ +c g σ . (11)
D 1g1/4 x1g1/4 2√g22 3 3 2
11 11
It is manifestly hermitian with respect to the standard scalar product. The operator (11) is a special case
of (1); when g and g in (1) are both positive, we can fix g = 1 , g = 1 . The operators (1) and
1 2 1 √g11 2 √g22
(11) then coincide. Employing the same substitution in (4) and (5), we find that the Hamiltonian (11)
has the integrals of motion
J = ie ix2 1 ∂ 1 g2′2 J + 1 c √g22g3′ 1 σ , J = i∂ , (12)
± ± g111/4 x1g111/4 − 2g22√g11 (cid:18)± 3 2(cid:19)± 3 g22 ∓ 2√g11 3! 3 − x2
provided that g and the external potential g are fixed in the following way
11 3
c 1 +c , for c = 0,
g11 = 1 (g2′2)2 , g3 =  3qg22 1 1 6 (13)
4g (1+c g )
22 1 22  c g1/2, for c = 0.
3 22 1
Concludingthesection, letusnotice thatthegenerally non-constant coefficients of∂ and∂ in(11)can
x1 x2
beinterpreted as thespace-dependentmodulation of Fermi velocity of massless Dirac particle ingraphene
induced by the strains and ripples of the crystal [25].
3 Solvable models of open-cage fullerenes
Fullerenes are spherical molecules made of carbon atoms. The spherical shape is due to twelve pentagons
that are inserted into hexagonal carbon lattice. The most famous example is the C , “Buckminster-
60
fullerene” [26], which can be classified as a truncated icosahedron with 60 vortices and 32 faces. Fixing
4
the number of pentagons in the lattice while increasing the number of hexagons, we can get not only
bigger fullerenes like C or C , but also objects of more complicated shapes, e.g. capped nanotubes,
240 540
elliptic fullerenes etc.. However, we will be interested just in the systems where the surface of the crystal
can be approximated by the spherical geometry.
In quantum chemistry, the situation is considered where small enough atoms or molecules (e.g. of
hydrogen) are inserted inside the fullerenes. During the process called “chemical surgery”, some bonds
between the carbon atoms of the fullerene are broken, making the hole into the surface. The alien atoms
are then inserted into the opened cage and the hole is closed again [27]. In this section, we will focus
on the electronic properties of the open-cage fullerenes with one or two holes in the surface. We will
construct exactly solvable models that can serve as the low-energy approximation of these systems.
3.1 Spectrum of fullerenes
First, let us go briefly through the analysis of the electronic properties of the (unopened) fullerenes. The
low-energy excitations ofchargecarriersinthecrystalcanbeinvestigated efficiently withintheframework
of the Dirac equation on the spherical surface [21]. The surface embedded into three-dimensional space
can be parametrized in spherical coordinates as x = sinx cosx , y = sinx sinx , z = cosx where
1 2 1 2 1
x 0,π and x = 0,2π . The associated metric tensor is explicitly
1 2
∈h i h i
1 0
g = .
µν 0 sin2x
1
(cid:18) (cid:19)
The entries of the metric tensor satisfy (13) for c = 1. Employing the formula (11), we can write down
1
−
immediately the Dirac Hamiltonian corresponding to the spin 1/2 particle on the sphere,
−
J
3
h = iσ ∂ +σ +c σ cotx , x (0,π). (14)
D 1 x1 2sinx 3 2 1 1 ∈
1
The potential term corresponds to Dirac monopole that is situated in the center of the sphere. It occurs
due to the topological defects, twelve pentagons, that have to be inserted into the hexagonal lattice of
graphene to close the spherical surface [16], [17], [21]. The actual value of the coupling constant is c = n
3 8
where n corresponds to the number of conical defects in the crystal. As n = 12 for fullerenes, we get
c = 3, see [16], [21].
3 2
We willnotwritedowntheexplicit formofJ , itcanbeextracted easily from(12). Thelowest weight
±
vectors can be found explicitly by solving (9). They are
1
ψm− = eimx2wF−c3(sinx1)−m+12  qmm−+211++c3cwwF−122 , wF = 1−sincoxs1x1, (15)
− − 2 3 F
 q 
2
1
h ψ = m c2ψ . (16)
D m− s − 2 − 3 m−
(cid:18) (cid:19)
We require the eigenfunctions of (14) to be squareintegrable and vanishing at x = 0,π. The vectors (15)
1
are physically acceptable as long as m c + 1. In that case, they are regular and correspond to real
≤ −| 3| 2
energies, see the restriction below (10). Degeneracy of the zero-energy level is equal to the dimension of
the representation of so(3) spanned by zero modes of the Hamiltonian. For c = 3, the zero-energy has
3 2
triple degeneracy.
For purposes of the forthcoming analysis, it is convenient to define transformation G
F
1/2
w 0
G = wc3 sinx F σ ,
F F 1 1/2 1
0 w− !
F
p
5
and rewrite the lowest weight vector in the factorized form
ψ = eimx2y mG − −m+ 12 +c3wFku , y = sinx , w (y) = 1−√y1−y2, x1<π/2, (17)
m− − F  q m− 12 +c3wFkd  1 F (1+√y1−y2, x1≥π/2,
 q 
where
k = 1 2c , k = 2c 1. (18)
u 3 d 3
− − −
With the use of this notation, the ladder operator can be rewritten as
J
G 1J G = ieix2(∂ cotx J ) = ieix2(1 yw (y)) ∂ 3 . (19)
− + x1 − 1 3 − F y − y
(cid:18) (cid:19)
It would be necessary to check that a multiple action of J on ψ does not violate regularity of the wave
+ m−
functions. However, we can skip this task as the considered wave functions are known to be regular and
based on the Jacobi polynomials, see [22], [28]. It means that the action of (19) on (17) preserves the
required boundary conditions. We will use this fact in the analysis of the open-cage fullerenes.
3.2 Open-cage fullerene with one hole
Let us consider the fullerene molecule with one hole carved into its surface (see Fig.1). We suppose that
the twelve pentagons allowing the crystal to close the spherical shell are still present in the lattice. To
meet this requirement, we can suppose that the model describes larger molecules like C where one or
240
few hexagons can beextracted from thecrystalwithoutaffecting thepentagons. Theconsidered spherical
surface can be parametrized as
x = sin(πdn(x ,k))cosx , y = sin(πdn(x ,k))sinx z = cos(πdn(x ,k)),
1 2 1 2 1
where x 0,K , x = π,π . The components of the associated metric tensor
1 2
∈h i h− i
πk2cn(x ,k)sn(x ,k) 2 0
g = 1 1
µν 0 sin2(πdn(x ,k))
(cid:18) (cid:0) (cid:1) 1 (cid:19)
satisfy (13) for c = 1.
1
− π
The functions dn(x ,k), cn(x ,k) and sn(x ,k) are Jacobi elliptic functions and K = 2(1
1 1 1 0 −
k2sin2θ) 1dθ is the complete elliptic integral of the first kind with 0 k 1. Let us briefly refresh rele-
−
≤ ≤ R
vant properties of these functions. dn(x ,k) is periodic with periodicity 2K while cn(x ,k) and sn(x ,k)
1 1 1
have periodicity 4K. Thereholds0 < dn(x ,k) 1forallrealx andallvalues ofthemodularparameter
1 1
≤
k 0,1 . The functions cn(x ,k) and sn(x ,k) can be understood as a generalization of trigonomet-
1 1
∈ h i
ric functions; they satisfy the relation cn(x ,k)2 +sn(x ,k)2 = 1. There holds cn(x ,0) = cosx and
1 1 1 1
sn(x ,0) = sinx , dn(x ,0) = 1. When k 1, the periodof theJacobi elliptic functions tends to infinity.
1 1 1
→
There holds sn(x ,1) = tanhx , whereas sn(x ,1) = dn(x ,1) = sechx . In the cases where the actual
1 1 1 1 1
valueof themodularparameter isnotinformative oritis clear fromthecontext, wewillusetheshortened
notation sn(x ) sn(x ,k), cn(x ) cn(x ,k) and dn(x ) dn(x ,k).
1 1 1 1 1 1
≡ ≡ ≡
Dirac Hamiltonian on the considered surface acquires the following explicit form
1 dnx (sn2x cn2x ) J
1 1 1 3
h = iσ ∂ + − +σ +c σ cot(πdnx ), (20)
D 1k2πcnx snx x1 2cnx snx 2sin(πdnx ) 3 2 1
1 1 (cid:18) 1 1 (cid:19) 1
where x (0,K). Likewise in (14), the potential term corresponds to the field generated by Dirac
1
∈
monopole. We will require the eigenstates of h to be regular and vanishing at the edge of the hole (i.e.
D
at x = K). The lowest weight vectors ψ annihilated by J are
1 m−
−
ψm− = eimx2√cnwxI1c3snx1(sin(πdnx1))−m+12  qcc33+− 1122 −+mmwwII1−/12/2 , (21)
 q 
6
where wI = 1−sicno(sπ(dπndxn1x)1) and hDψm− = (m−1/2)2 −c23ψm−.
The wave(cid:16)functions (2(cid:17)1) comply withpthe required boundary condition (cn(K) = 0). However, it is
not clear whether the multiple action of J on (21) does not violate the boundary conditions. We can
+
analyze this point using the close analogy with the model of fullerenes. Let us define the transformation
G as
I
1/2
G = cnx snx sin(πdn(x ))wc3 wI− 0 .
I 1 1 1 I 1/2
0 w !
I
p
The lowest weight vector can be then written as
ψ = eimx2y mG c3+ 21 −mwIku , y = sin(πdnx ), w (y)= 1−√y1−y2, dnx1≤1/2, (22)
m− − I  qc3− 21 +mwIkd  1 I (1+√y1−y2, dnx1>1/2,
 q 
where the constants k and k are those in (18). Performing the “gauge” transformation of J , we get
u d +
1 J
G−I1J+GI = ieix2 k2πcnx snx ∂x1 +cotπdnx1J3 = −ieix2(1−ywI(y)) ∂y − y3 . (23)
(cid:18) 1 1 (cid:19) (cid:18) (cid:19)
Thelatter operator is identical with (19). Thevector (22) coincides with (17) up to gauge factors G and
F
G and the up to the sign of the upper component. Hence, likewise in the case of fullerenes, the multiple
I
action of J on ψ will keep regularity of the wave functions, i.e. only positive powers of y will emerge.
+ m−
3.3 Open-cage fullerene with two holes
Similar results can be obtained for spin-1/2 particle on the spherical surface with two holes centered in
the poles, see Fig.1. The surface can be parametrized as x = dnx cosx , y = dnx sinx , z = ksnx ,
1 2 1 2 1
where x K,K and x π,π . The metric tensor reads this time
1 2
∈h− i ∈ h− i
(kcnx )2 0
g = 1 .
µν 0 dn2x
1
(cid:18) (cid:19)
The explicit form of the Hamiltonian can be written as follows
iσ dnx snx J ksnx
1 1 1 3 1
h = ∂ + +σ +c σ , x ( K,K). (24)
D kcnx x1 2cnx 2dnx 3 2 dnx 1 ∈ −
1 (cid:18) 1 (cid:19) 1 1
Figure 1: The models of open cage fullerenes. We fixed k = .99 in the left figure while k = .9 in the right
figure.
7
Its eigenvectors should be regular and vanishing at the edges of the holes (i.e. for x K). The lowest
1
→ ±
weight vector ψ which satisfies h ψ = (m 1)2 c2ψ has the explicit form
m− D m− − 2 − 3 m−
q
ψm− = √wcnIc3Ix1(dnx1)−m+12  qcc33+− 1122 −+mmwwII1−II/12/2 , wII = (cid:18)1−dnkxsn1x1(cid:19). (25)
 q 
It complies with the required boundary conditions since cn(K) = cn( K) = 0. In order to show that
−
application of J on these vectors does not violate the required properties, let us define the matrix
+
1/2
G = cn(x )dn(x )wc3 wI−I 0 .
II 1 1 II 1/2
0 w !
II
p
The lowest weight vector can be written in the following form
ψ = eimx2y mG c3+ 21 −mwIkIu , y = dnx , w (y) = 1−√y1−y2, snx1>0, (26)
m− − II  qc3− 21 +mwIkId  1 II (1+√y1−y2, snx1≤0,
 q 
while the ladder operator is transformed into
1 J ksnx J
G−II1J+GII = ieix2 kcnx ∂x1 + 3dnx 1 = −ieix2(1−ywII(y)) ∂y − y3 . (27)
(cid:18) 1 1 (cid:19) (cid:18) (cid:19)
Comparing (26) with (17) and (27) with (19), we can conclude that the repeated action of J on ψ will
+ m−
preserve regularity of the wave functions.
The formulas (22), (23), (26) and (27) allow to find the required eigenstates of the corresponding
Hamiltonian as well as energies which are given by (10) for c = 1. Each of the systems has distinct
1
−
topologicalgenusasthenumberoftheholesinthesphericalsurfacevaries. Despitethisdistincttopological
nature, the models are spectrally identical due to the same algebraic background.
In the next section, we will discuss a different application of the potential algebras. We will con-
sider one-dimensional, both relativistic and non- relativistic supersymmetric systems that possess shape
invariance.
4 Shape invariant Dirac Hamiltonians and PDM Schrodinger opera-
tors
Shape invariance underlies exact solvability of the most of the one- dimensional nonrelativistic quantum
mechanical systems. Itoriginates in supersymmetricquantummechanics based on theWitten model[29],
[30], [31]. Within this framework, two hermitian Hamiltonians,
H = ∂2 +V = ∂2 +W(x ,m)2 W (x ,m), (28)
m± − x1 m± − x1 1 ± ′ 1
are intertwined by the operator qm = −i∂x1 −iW(x1,m) and its conjugate qm† , i.e. Hm+qm = qmHm− and
Hm−qm† = qm† Hm+. The function W(x1,m) is called superpotential. In the context of the analysis of differ-
ential equations, the intertwining operators are known as Darboux transformations [32]. Supersymmetric
system is defined in terms of the extended matrix Hamiltonian H˜ and integrals of motion Q˜(1) and Q˜(2),
m m m
H˜ = ∂2 +W(x ,m)2+σ W (x ,m), Q˜(1) = iσ ∂ +σ W(x ,m), Q˜(2) = iσ Q˜(1). (29)
m − x1 1 3 ′ 1 m − 1 x1 2 1 m 3 m
In close analogy to the supersymmetry in quantum field theory, these operators compose a superalgebra.
The generators (29) are classified as bosonic or fermionic provided that they commute or anticommute
8
with a grading operator. In our case, the grading operator can be identified as σ . Then H˜ is bosonic
3 m
([H˜ ,σ ] = 0) and Q˜(a) are fermionic operators ( Q˜(a),σ = 0, a = 1,2). The Hamiltonian H˜ then
m 3 m m 3 m
{ }
possesses N = 2 supersymmetry associated with the superalgebra
H˜ ,Q˜(a) = 0, Q˜(a),Q˜(b) = 2δ H˜ . (30)
m m m m ab m
h i n o
Let us suppose that the potentials V+ and V differ just in the value of the coupling constant and in
m m−
additional constant term, i.e. V+ = V +r(m) where δ and r(m) are real. Then the Hamiltonian H
m m−+δ m−
is called shape-invariant as V+ and V are qualitatively of the same shape. This property can be used
m m−
to compute spectrum and wave functions of H+ very effectively just from the knowledge of the ground
m
state, see e.g. [30]. We will discuss this point in more detailed in the section 4.2.
The condition of shape-invariance (31) can be written in terms of the function W(x ,m) in the
1
following manner
W(x ,m)2+W(x ,m) = W(x ,m+δ)2 W(x ,m+δ) +r(m). (31)
1 1 ′ 1 1 ′
−
This formula proved to be very useful for the systematic search of the shape-invariant systems which has
been done in the literature extensively, see e.g. [33], [34].
Analysis of shape-invariant Dirac operators in the context of supersymmetry is more complicated.
The framework based on the superpotential similar to (28)-(31) is absent. Usually, shape-invariance of
Dirac Hamiltonian is treated such that the square of the operator is identified with a known, nonrela-
tivistic, shape-invariant operator after a series of transformations [35]. In this way, exact eigenstates and
eigenvalues of the original relativistic system can be found.
We will present an alternative approach inspired by Balantekin and Gangopadhyaya [36], [37]. They
found that the shape-invariance of the nonrelativistic systems can be understood in terms of a higher
dimensional Hamiltonian that possesses Lie algebra of integrals of motion. In this framework, the super-
partner Hamiltonians Hm± as well as the associated intertwining operators qm and qm† correspond to an
appropriaterestrictionofthehigher-dimensionalHamiltonianandoftheladderoperatorstothesubspaces
with fixed value of angular momentum.
4.1 Shape-invariance of Dirac operators via N = 2 nonlinear supersymmetry
Let us consider restriction of h in (1) and of the ladder operators (4) to the subspaces where J acquires
D 3
fixed value. The operator h reduces to the one-dimensional Dirac Hamiltonian
D
g
h = h =e imx2h eimx2 =iσ g ∂ + 1′ +σ g m+g σ . (32)
m D|J3=m − D 1 1 x1 2 2 2 3 2
(cid:18) (cid:19)
Theladder operators J exchange the subspaces whereJ = m and J = m+1. Correspondingrestricted
3 3
±
operators j read explicitly
,m
±
g g g (m+ 1) g g g
j+ = e i(m+1)x2J eimx2 = i g ∂ + 1′ + 1 2′ 2 + 1 3′ 2σ ,
m − + 1 x1 2 g g − 2 3
2 2 !
g g g (m+ 1) g g g
j = e imx2J ei(m+1)x2 = i g ∂ + 1′ 1 2′ 2 1 3′ + 2σ . (33)
m− − − 1 x1 2 − g2 − g2 2 3!
Keeping in mind that [h ,J ] = 0, we can write down the following intertwining relations between
D
±
the operators h and h ,
m m+1
h j = j h , h j+ = j+h . (34)
m m− m− m+1 m+1 m m m
These relations establish shape-invariance of the one- dimensional Dirac operator h mediated by the
m
operators j . They also make the link with the supersymmetric quantum mechanics. The operator j+
m± m
9
(j ) can be identified as the generalized matrix Darboux transformation. General properties of these
m−
operators were discussed in [46] in detail. In particular, it was proved that the product j j+ or j+j is
m− m m m−
proportional to the second-order polynomials in h or h respectively,
m m+1
j j+ = h2 , j+j = h2 , (35)
m− m m−Em m m− m+1−Em
where = (E+)2.
Em m
It is convenient to treat the system in the framework of the extended matrix operators,
h 0 m 0 1 τ
h = m , (3) = = m+ 1 1 3,
m 0 hm+1 Jm 0 m+1 2 ⊗ − 2
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
0 j
(1) = m− , (2) = iτ (1), τ = σ 1 =diag(1, 1), (36)
Jm j+ 0 Jm 3Jm 3 3⊗ −
(cid:18) m (cid:19)
where1denotes thetwo-by-two identity matrix. Theoperators actonthebispinorswiththeupperspinor
from the subspace where J = m and with the lower one from the subspace where J = m+1.
3 3
The matrix τ can be identified as the grading operator of the superalgebra, classifying h and (3)
3 m m
J
(1) (2)
as bosonic while and as fermionic operators. The N = 2 superalgebra then reads
m m
J J
[h , (a)] = 0, [h , (3)] = 0, [ (3), (a)]= iε (b), (37)
m Jm m Jm Jm Jm − abJm
(a), (b) = 2δ h2 , a,b = 1,2, (38)
{Jm Jm } ab m−Em
(cid:0) (cid:1)
where ε is completely antisymmetric in indices and ε = 1. The shape-invariance of the relativistic
ab 12
Hamiltonian (32) is encoded in the first commutator of (37). The superalgebra is nonlinear, see [38], as
the anticommutator of the supercharges is a quadratic polynomial in the (extended) Hamiltonian h .
m
Hence, the shape invariance of h is associated with nonlinear N = 2 supersymmetry.
m
4.2 Shape-invariance of nonrelativistic systems with position-dependent mass and
N = 8 supersymmetry
Another interesting application of the potential algebra (6)-(7) can be done in the realm of the nonrela-
tivistic quantum mechanics. The Schro¨dinger operators with the position-dependent mass M(x ),
1
1 1 1
∂ ∂ +V(x ), (39)
− M(x1)1/4 x1 M(x1) x1M(x1)1/4 1
emerge in condensed matter physics, e.g. inpdescription of the semiconductors or heterostructures [39],
[40]. These systems have attracted lots of attention in the literature, see [41]-[42] and references therein.
Let us pick up Dirac Hamiltonian (11). The square of the operator in the subspace of fixed angular
momentum J = m can be written as
3
2
1 1 1 m 1 m ′
H = ∂ ∂ + +c g σ +c g . (40)
m −g111/4 x1√g11 x1g111/4 (cid:18)√g22 3 3(cid:19) − 3√g11 (cid:18)√g22 3 3(cid:19)
We can see that the diagonal entries of the matrix operator are of the type (39) provided that we fix
(a)
M(x ) g (x ). By construction, the Hamiltonian H together with the supercharges Q ,
1 11 1 m m
≡
1 1 m
Q(1) = h = iσ ∂ +σ +c g σ , Q(2) = iσ Q(1), (41)
m m 1g1/4 x1g1/4 2√g22 3 3 2 m 3 m
11 11
close the superalgebra,
H ,Q(a) = 0, Q(a),Q(b) = 2δ H , (42)
m m m m ab m
h i n o
10