Symmetries and Supersymmetries of the Dirac Hamiltonian with Axially-Deformed Scalar and Vector Potentials
aa r X i v : . [ nu c l - t h ] J u l Symmetries and Supersymmetries of the Dirac Hamiltonian withAxially-Deformed Scalar and Vector Potentials
A. Leviatan
Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel (Dated: October 29, 2018)We consider several classes of symmetries of the Dirac Hamiltonian in 3+1 dimensions, withaxially-deformed scalar and vector potentials. The symmetries include the known pseudospin andspin limits and additional symmetries which occur when the potentials depend on different variables.Supersymmetries are observed within each class and the corresponding charges are identified.
PACS numbers: 24.10.Jv, 11.30.Pb, 21.60.Cs, 24.80.+y
The Dirac equation plays a key role in microscopicdescriptions of many-fermion systems, employing covari-ant density functional theory and the relativistic meanfield approach. The relevant mean-field potentials are ofCoulomb vector type in atoms, and a mixture of Lorentzvector and scalar potentials in nuclei and hadrons [1].Recently, symmetries of Dirac Hamiltonians with suchmixed Lorentz structure have been shown to be rele-vant for explaining the observed degeneracies of certainshell-model orbitals in nuclei (“pseudospin doublets”) [2],and the absence of quark spin-orbit splitting (“spin dou-blets”) [3], as observed in heavy-light quark mesons.Supersymmetric patterns have been identified in spe-cific limits of such spherical potentials [4, 5]. In thepresent Letter we further explore classes of symmetriesand supersymmetries when these potentials are axially-deformed. Such a study is significant in view of the factthat mean-field Hamiltonians often break the rotationalsymmetry. Cylindrical geometries are relevant to a num-ber of problems, including electron channeling in crystals,structure of axially-deformed nuclei and quark confine-ment in spheroidal flux-tubes.The Dirac Hamiltonian, H , for a fermion of mass M moving in external scalar, V S , and vector, V V , potentialsis given by H = ˆ α · ˆ p + ˆ β ( M + V S ) + V V [6]. When thepotentials are axially-symmetric, i.e. , independent of theazimuthal angle φ , V S,V = V S,V ( ρ, z ) , ρ = p x + y ,then the z -component of the angular momentum opera-tor, ˆ J z , commutes with H and its half-integer eigenval-ues Ω are used to label the Dirac wave functions Ψ = (cid:0) g + e − iφ/ , g − e iφ/ , if + e − iφ/ , if − e iφ/ (cid:1) e i Ω φ . Here g ± ≡ g ± ( ρ, z ) and f ± ≡ f ± ( ρ, z ) are the radial wavefunctions of the upper and lower components, respec-tively. Henceforth, such a wave function will be denotedby Ψ Ω : { g + , g − , f + , f − } . The potentials enter the Diracequation through the combinations A ( ρ, z ) = E + M + V S ( ρ, z ) − V V ( ρ, z ) (1a) B ( ρ, z ) = E − M − V S ( ρ, z ) − V V ( ρ, z ) . (1b)For each solution with Ω >
0, there is a degenerate time-reversed solution with − Ω <
0, hence, we confine the dis-cussion to solutions with Ω >
0. Of particular interest are bound Dirac states with | E | < M and normalizable wavefunctions in potentials satisfying ρV S ( ρ, z ) , ρV V ( ρ, z ) → ρ → V S ( ρ, z ) , V V ( ρ, z ) → ρ → ∞ or z → ±∞ . The boundary conditions imply that the ra-dial wave functions fall off exponentially for large dis-tances and behave as a power law for ρ →
0. Further-more, for z = 0 and ρ → ∞ , f − /g + ∝ ( M − E ) > g − /f + ∝ ( M + E ) >
0, while for z = 0 and ρ → f − /g + ∝ B (0) ρ and g − /f + ∝ − A (0) ρ . These propertieshave important implications for the structure of radialnodes. In particular, it follows that for potentials withthe indicated asymptotic behaviour and A (0) , B (0) > g − = 0 or f + = 0 . (2)The Dirac equation, H Ψ = E Ψ, leads to a set of fourcoupled partial differential equations involving the radialwave functions. Their solutions are greatly simplified inthe presence of symmetries. We now discuss four classesof relativistic symmetries and possible supersymmetrieswithin each class.The symmetry of class I, referred to as pseudospinsymmetry, occurs when the sum of the scalar and vec-tor potentials is a constant, V S ( ρ, z ) + V V ( ρ, z ) = ∆ .The symmetry generators, ˆ˜ S i , commute with the DiracHamiltonian and span an SU(2) algebra [7, 8]ˆ˜ S i = (cid:18) U p ˆ s i U p
00 ˆ s i (cid:19) i = x, y, z U p = σ · p p . (3)Here ˆ s i = σ i / S z Ψ (˜ µ )Ω = ˜ µ Ψ (˜ µ )Ω ˜ µ = ± / SU (2) doublets. Their wave func-tions have been shown to be of the form [9]Ψ ( − / =˜Λ − / : { g + , − g, , f } , (5a)Ψ (1 / =˜Λ+1 / : { g, g − , f, } , (5b)where ˜Λ = Ω − ˜ µ ≥ J z − ˆ˜ S z . Therelativistic pseudospin symmetry has been tested in nu-merous realistic mean field calculations of nuclei and werefound to be obeyed to a good approximation, especiallyfor doublets near the Fermi surface [9, 10]. The domi-nant upper components of the states in Eq. (5), involv-ing g + and g − , correspond to non-relativistic pseudospindoublets with asymptotic (Nilsson) quantum numbers[ N, n , Λ]Ω = Λ + 1 / N, n , Λ + 2]Ω = Λ + 3 / µ = ± / ± /
2. Such dou-blets play a crucial role in explaining features of de-formed nuclei, including superdeformation and identicalbands [9, 11].The symmetry of class II, referred to as spin symme-try, occurs when the difference of the scalar and vectorpotentials is a constant, V S ( ρ, z ) − V V ( ρ, z ) = Ξ . Thesymmetry group is again SU (2) and its generators [7]ˆ S i = (cid:18) ˆ s i U p ˆ s i U p (cid:19) i = x, y, z (6)commute with the Dirac Hamiltonian. The Dirac eigen-functions in the spin limit satisfyˆ S z Ψ ( µ )Ω = µ Ψ ( µ )Ω µ = ± / SU (2) doublets. Their wave func-tions are of the form [9]Ψ (1 / =Λ+1 / : (cid:8) g, , f, f − (cid:9) , (8a)Ψ ( − / =Λ − / : (cid:8) , g, f + , − f (cid:9) , (8b)where Λ = Ω − µ ≥ J z − ˆ S z .The upper components of the two states in Eq. (8) formthe usual non-relativistic spin doublet with a commonradial wave function, g , an orbital angular momentumprojection, Λ, and two spin orientations Ω = Λ ± /
2. Therelativistic spin symmetry has been shown to be relevantto the structure of heavy-light quark mesons [3].The Dirac Hamiltonian has additional symmetrieswhen the scalar and vector potentials depend on differ-ent variables. The symmetry of class III occurs when thepotentials are of the form V S = V S ( z ) and V V = V V ( ρ ).In this case, the Dirac Hamiltonian commutes with thefollowing Hermitian operatorˆ R z = [ M + V S ( z ) ] ˆ β ˆΣ + γ ˆ p z , (9)where ˆΣ i = (cid:16) σ i σ i (cid:17) . The Dirac eigenfunctions satisfyˆ R z Ψ ( ǫ )Ω = ǫ Ψ ( ǫ )Ω . (10)A separation of variables is possible by choosing the Diracwave function in the formΨ ( ǫ )Ω : { u h + , u h − , u h − , − u h + } / √ ρ , (11) where u i ≡ u i ( ρ ), h ± ≡ h ± ( z ) and, for simplicity, wehave omitted the label ǫ from these wave functions. TheDirac equation then reduces to a set of two coupled first-order ordinary differential equations in the variable ρ ,[ d/dρ − Ω /ρ ] u ( ρ ) − [ E − V V ( ρ ) + ǫ ] u ( ρ ) = 0 (12a)[ d/dρ + Ω /ρ ] u ( ρ ) + [ E − V V ( ρ ) − ǫ ] u ( ρ ) = 0 (12b)and a separate set in the variable z [ M + V S ( z ) + d/dz ] h ( z ) = ǫ h ( z ) (13a)[ M + V S ( z ) − d/dz ] h ( z ) = ǫ h ( z ) (13b)where h ± ( z ) = h ( z ) ± h ( z ). The separation constant, ǫ ,plays the role of a mass for the transverse motion and isdetermined from imposed boundary conditions. A specialcase within the symmetry class III, with V S ( z ) = 0 and ǫ = ± p M + p z , was considered for electron channelingin crystals [12]. For V S ( z ) = 0, ˆ R z of Eq. (9), reduces tothe transverse polarization operator relevant to studiesof synchrotron radiation in storage rings and QED pro-cesses in magnetic flux tubes ( e.g. , e + e − production andBremsstrahlung) [13].The symmetry of class IV occurs when the potentialsare of the form V S = V S ( ρ ) and V V = V V ( z ). In thiscase, the following Hermitian operatorˆ R ρ = [ M + V S ( ρ ) ] ˆΣ − i ˆ β γ ( ˆΣ × ˆ p ) (14)commutes with the Dirac Hamiltonian and the Diraceigenfunctions satisfyˆ R ρ Ψ (˜ ǫ )Ω = ˜ ǫ Ψ (˜ ǫ )Ω . (15)Again, a separation of variables is possible with the choiceof wave function,Ψ (˜ ǫ )Ω : { ξ w + , − iξ w − , iξ w − , − ξ w + } / √ ρ , (16)where ξ i ≡ ξ i ( ρ ) and w ± ≡ w ± ( z ). The Dirac equationthen reduces to a set of ordinary differential equations inthe variable ρ ,[ d/dρ − Ω /ρ ] ξ ( ρ ) − [˜ ǫ + M + V S ( ρ ) ] ξ ( ρ ) = 0 (17a)[ d/dρ + Ω /ρ ] ξ ( ρ ) + [˜ ǫ − M − V S ( ρ ) ] ξ ( ρ ) = 0 (17b)and a separate set in the variable z [ E − V V ( z ) − id/dz ] w ( z ) = ˜ ǫ w ( z ) (18a)[ E − V V ( z ) + id/dz ] w ( z ) = ˜ ǫ w ( z ) (18b)where w ± ( z ) = w ( z ) ± w ( z ). The quantum number, ˜ ǫ ,plays the role of an energy for the transverse motion.A particular selection of potentials within the symmetryclass IV was encountered in the study of the Schwingermechanism for particle-production in a strong confinedfield ( V V ( z ) = α V z ) [14, 15], q ¯ q pair-creation in a flux TABLE I:
Conserved, anticommuting operators for Dirac Hamiltonians ( H ) exhibiting a supersymmetric structure. SUSY ˆ R ˆ B ˆ B = f ( H ) V S ( ρ, z ) + V V ( ρ, z ) = ∆ ˆ˜ S z (3) 2( M + ∆ − H ) ˆ˜ S x ( M + ∆ − H ) V S ( ρ, z ) − V V ( ρ, z ) = Ξ ˆ S z (6) 2( M + Ξ + H ) ˆ S x ( M + Ξ + H ) V S = V S ( z ), V V = α V ρ ˆ R z (9) ˆ β ˆΣ { i ˆ J z γ [ H − ˆΣ ˆ R z ] − α V ρ ( ˆΣ · ρ ) ˆ R z } ˆ J z ( H − ˆ R z ) + α V ˆ R z V S = α S ρ , V V = V V ( z ) ˆ R ρ (14) ˆΣ { i ˆ J z γ [ M − ˆΣ ˆ R ρ ] − α S ρ ( ˆΣ · ρ ) ˆ β ˆ R ρ } ˆ J z ( ˆ R ρ − M ) + α S ˆ R ρ tube ( V S ( ρ ) = 0 , ˜ ǫ = ±√ M + k ) [16], and the canon-ical quantization in cylindrical geometry of a free Diracfield ( V S ( ρ ) = V V ( z ) = 0 , E = M + k + p z ) [17].Dirac Hamiltonians with selected external fields areknown to be supersymmetric [4, 5, 6, 18]. It is, therefore,natural to inquire whether a supersymmetric structurecan develop within each of the above symmetry classes.The essential ingredients of supersymmetric quantummechanics [18] are the supersymmetric Hamiltonian, H ,and charges Q + , Q − = Q † + , which generate the super-symmetry (SUSY) algebra [ H , Q ± ] = { Q ± , Q ± } = 0, { Q − , Q + } = H . Accompanying this set is an Hermitian Z -grading operator satisfying [ H , P ] = { Q ± , P} = 0and P = 1l. The +1 and − P define the“positive-parity”, H + , and “negative-parity”, H − , sec-tors of the spectrum, with eigenvectors Ψ (+) and Ψ ( − ) ,respectively. The SUSY algebra imply that if Ψ (+) is aneigenstate of H , then also Ψ ( − ) = Q − Ψ (+) is an eigen-state of H with the same energy, unless Q − Ψ (+) vanishesor produces an unphysical state, ( e.g. , non-normalizable).The resulting spectrum consists of pairwise degeneratelevels with a non-degenerate single state (the groundstate) in one sector when the supersymmetry is exact.If all states are pairwise degenerate, the supersymmetryis said to be broken. Typical spectra for good and bro-ken SUSY are shown in Fig. 1. Degenerate doublets,signaling a supersymmetric structure, can emerge in aquantum system with a Hamiltonian H , from the exis-tence of two Hermitian, conserved and anticommutingoperators, ˆ R and ˆ B [ H, ˆ R ] = [ H, ˆ B ] = { ˆ R, ˆ B } = 0 . (19) Q − Q + H + H − H + H − good SUSY broken SUSY FIG. 1: Typical spectra of good and broken SUSY. The op-erators Q − and Q + connect degenerate states in the H + and H − sectors. The operator ˆ R has non-zero eigenvalues, r , which comein pairs of opposite signs. ˆ B = ˆ B † ˆ B = f ( H ), isa function of the Hamiltonian. A Z -grading opera-tor, P r = ˆ R/ | r | , and Hermitian supercharges Q = ˆ B , Q = iQ P r can now be constructed. The triad of oper-ators Q ± = ( Q ± iQ ) / H = Q = f ( H ) form thestandard SUSY algebra. In the present analysis, f ( H ) isa quadratic function of the Dirac Hamiltonian, H , andthe relevant ˆ R and ˆ B operators are listed in Table I.In the pseudospin symmetry limit, the relevant opera-tor ˆ B , connects the doublet states of Eq. (5). The spec-trum, for each ˜Λ = 0, consists of twin towers of pairwisedegenerate pseudospin doublet states, with Ω = ˜Λ − / = ˜Λ + 1 /
2, and an additional non-degeneratenodeless state at the bottom of the Ω = ˜Λ − / N, n , Λ = N − n ]Ω = Λ + 1 /
2, which, empirically,are found not to be part of a doublet [11]. The latterproperty follows from the fact that a nodeless boundDirac state satisfies the criteria of Eq. (2), hence hasa wave function as in Eq. (5a) with g + , g, f = 0 and f /g + >
0. Its pseudospin partner state has a wavefunction as in Eq. (5b). The radial components sat-isfy Bg − = [ B − /ρ ) f /g + ] g + , where B is defined inEq. (1b). This relation is satisfied, to a good approxima-tion, for mean-field potentials relevant to nuclei, and ther.h.s. is non-zero and, consequently, g − = 0. If so, thenthe partner state (5b) is also nodeless, but it cannot be abound eigenstate since its radial components do not ful-fill the condition of Eq. (2). Altogether, the ensemble ofDirac states with Ω − Ω = 1 exhibits a supersymmetricpattern of good SUSY, as illustrated in Fig. (2a).In the spin symmetry limit, the relevant operator ˆ B connects the doublet states of Eq. (8). The spectrum, foreach Λ = 0, consists of twin towers of pairwise degeneratespin-doublet states with Ω = Λ − / = Λ + 1 / g, f, f − = 0 and g/f − >
0. Its spin partner hasa wave function as in Eq. (8b). The radial componentssatisfy Af + = [ A − /ρ ) g/f − ] f − , where A is definedin Eq. (1a). For relevant potentials the r.h.s. of thisrelation can vanish, hence f + has a node. Therefore, the [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / L = L = L = L = L = L = [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / [ ] / L = L = L = L = L = FIG. 2: Grouping of deformed shell-model states [ N = 4 , n , Λ]Ω, exhibiting a pattern of (a) good SUSY, relevant to thepseudospin symmetry limit, and (b) broken SUSY, relevant to the spin symmetry limit. N and n are harmonic oscillatorquantum numbers. Λ (˜Λ) is the orbital (pseudo-orbital) angular momentum projection along the symmetry z -axis. spin-partner of a nodeless state is not nodeless and canbe a bound eigenstate, since the restrictions of Eq. (2)do not apply. Altogether, the ensemble of Dirac stateswith Ω − Ω = − V V ( ρ ) = α V /ρ and V S ( z ) arbitrary. Theenergy eigenvalues are E ( ǫ ) n ρ , Ω = | ǫ | / p α V / ( n ρ + γ ) ( n ρ = 0 , , , . . . ), with γ = p Ω − α V . From Eqs. (13)we see that if [ h ( z ) , h ( z )] are solutions with ǫ >
0, then[ h ( z ) , − h ( z )] are solutions with − ǫ <
0. Accordingly,the doublet wave functions are as in Eq. (11), with thereplacements, u i u ( ǫ ) i ( ρ ) for Ψ ( ǫ ) n ρ , Ω , and u i u ( − ǫ ) i ( ρ ), h ±
7→ − h ∓ ( z ) for Ψ ( − ǫ ) n ρ , Ω . For n ρ ≥
1, the states Ψ ( ± ǫ ) n ρ , Ω are degenerate. For n ρ = 0 only one state is an accept-able solution, which has ǫ > α V <
0) andis annihilated by the relevant operator ˆ B . For each Ωand ǫ the spectrum resembles a supersymmetric patternof good SUSY, with the towers H + ( H − ) of Fig. 1 corre-sponding to states with ǫ > ǫ < V S ( ρ ) = α S /ρ ( α S <
0) and V V ( z ) arbitrary.The allowed values are ˜ ǫ = ± M p − α S / ( n ρ + ˜ γ ) ( n ρ = 0 , , , . . . ), where ˜ γ = p Ω + α S . From Eqs. (18)we see that if [ w ( z ) , w ( z )] are solutions with ˜ ǫ >
0, then[ w ( z ) , − w ( z )] are solutions with − ˜ ǫ < E . Accordingly, the doublet wave-functions areas in Eq. (16), with the replacements, ξ i ξ (˜ ǫ ) i ( ρ ) forΨ (˜ ǫ ) n ρ , Ω , and ξ i ξ ( − ˜ ǫ ) i ( ρ ), w ±
7→ − w ∓ ( z ) for Ψ ( − ˜ ǫ ) n ρ , Ω . For n ρ ≥ ( ± ˜ ǫ ) n ρ , Ω are degenerate. For n ρ = 0 onlyone state, with ˜ ǫ >