aa r X i v : . [ phy s i c s . g e n - ph ] A ug Anomalous Dirac and Majorana states in condensed matter
Boris I. Ivlev
Instituto de F´ısica, Universidad Aut´onoma de San Luis Potos´ı,San Luis Potos´ı, 78000 Mexico
Unexpected electron states, bound to the Coulomb field of the nucleus, are proposed. Theseanomalous states are mediated by positional quantum fluctuations of this nucleus which is a latticesite in a solid. Without that support the states look as formal singular solutions which are, at firstsight, totally useless. The electron binding energy in the
MeV range is surprising in condensedmatter since it usually relates to nuclear processes. Anomalous states are separated from usualelectron ones in a solid by an energy barrier. The lattice distortions, jointly with the electrondegrees of freedom, are responsible for the barrier formation. This contrasts to polaron in a solidwhere lattice distortions form a well but not a barrier. Electron transitions to anomalous levels arepossible under a high energy external perturbation to overcome the barrier. Anomalous state canbe of the Dirac or Majorana type.
PACS numbers: 03.65.Pm, 03.70.+k, 63.20.kdKeywords: wave equations, singularity, electron-phonon interaction
I. INTRODUCTION
The interaction of an electron with lattice deforma-tions in a solid can create a potential well that trapsthe electron. This self-trapping effect (polaron) was pre-dicted by Landau [1] and studied in the known publica-tions [2–4]. The polaron binding energy, roughly 0 . eV ,is determined by energy scales in condensed matter.The M eV energies, which are formally of the nuclearscale, are not supposed to appear in condensed matterprocesses. However this unexpected issue has an under-lying base as shown in this paper.The electron interaction with the nuclear Coulombfield U ( r ) of the lattice site is described by theSchr¨odinger equation. Nothing prevents to use the Diracformalism for that. In this case one of two Dirac spinorsbecomes small at c → ∞ and the other one satisfies theSchr¨odinger equation [5].In the solution, proposed in this paper, the roles of thetwo Dirac spinors are inverted. Namely: the spinor, thatusually disappears in the non-relativistic limit, becomesthe main one. The spinor, that was a solution of theSchr¨odinger equation, becomes singular at r = 0 makingthe solution non-physical. The formal energy of this stateis mc + U (0) where U (0) ≃ − Ze /r N ∼ − M eV . Here r N is the nucleus radius and Ze is the nucleus charge.That singular solution is remarkable since it is a formalmathematical solution of the Dirac equation. This fact isnon-trivial. For example, the equation − ( ~ / m ) ∇ ψ − Eψ = 0 can have a tendency to form the singularity ψ ∼ /r [6]. But this singular solution does not exist evenformally since, analogously to electrostatics, it requiresthe artificial δ ( ~r ) term in the right side of that equation.The electromagnetic interaction does not lead tosmearing of the singularity. This can be interpreted asfollowing. Due to the singularity, the momentum p ∼ ~ /r is infinite at r = 0 and its shift ~p − ~k , by the finite photonmomentum ~k , remains it infinite.When the atomic nucleus, with whom the electron in- teracts, belongs to a lattice site of a solid, the potentialbecomes U ( | ~r − ~u | ) where ~u is the displacement of thelattice site. ~u is not a mean field coordinate but a quan-tum mechanical degree of freedom distributed accordingto quantum fluctuations of the lattice [7, 8]. The latticemotion is very adiabatic compared to the electron dy-namics [7, 8] and the singularity positions at ~r = ~u areaveraged on ~u . Therefore the singularity is smeared outwithin the radius of quantum fluctuations p h u i ∼ − cm. (1)Thus the formal singular solution of the Dirac equation,is cut off within the finite region (1). The resulting stateis smooth and therefore physical. Such anomalous statehas the large binding energy in the M eV range. Despiteit is the nuclear energy scale, that energy is of the electronorigin. This is unexpected in condensed matter.Anomalous states are generally of the Dirac type thatis with distinguished electron and positron [5]. Howeverthose states can be also of the Majorana type when thespin-1/2 particle is its own antiparticle [5, 9]. The amaz-ing feature of these Majorana states is the large energyassociated with them. It can be more than 20
M eV .Usual electron states are separated from anomalousones (Dirac or Majorana) by an energy barrier. Thelattice distortions, jointly with the electron degrees offreedom, are responsible for the barrier formation. Thiscontrasts to polaron in a solid where lattice distortionsform a well but not the barrier.
II. RELATIVISTIC WAVE EQUATIONA. Usual approach
We start with the Dirac equation for electron in thestandard representation when the total bispinor consistsof two spinors Φ( ~r, t ) and Θ( ~r, t ) [5]. We suppose thecentral potential well U ( r ) to satisfy the condition ofharmonic oscillator U ( r ) ≃ U (0) + U ′′ (0) r / r → (cid:20) ∂∂t + iU ( r ) + ~α ∇ + imβ (cid:21) (cid:18) ΦΘ (cid:19) = 0 , (2)where the matrices ~α = (cid:18) ~σ~σ (cid:19) , β = γ = (cid:18) − (cid:19) (3)and we adopt ~ = c = 1. Here ~σ is the Pauli matrix.Using the Fourier transformationΦ( ~r, t ) = Z dε π Φ ε ( ~r ) exp( − itε ) , (4)where ε is the total relativistic energy, one can rewrite(2) in spinor components( ε − U )Φ ε + i~σ ∇ Θ ε = m Φ ε (5)( ε − U )Θ ε + i~σ ∇ Φ ε = − m Θ ε . (6)We express from (6) the spinor Θ ε through Φ ε Θ ε = − i~σ ∇ Φ ε ε − U + m (7)and substitute it into (5). The result is −∇ Φ ε − ∇ Uε − U + m ( ∇ Φ ε − i~σ × ∇ Φ ε )+ m Φ ε = ( ε − U ) Φ ε . (8)The form (8) is convenient to obtain the non-relativistic limit when the energies E = ε − m and U ( r )are small compared to m . In this case the term with ∇ U is small ( ∼ /c in the physical units) and Eq. (8) turnsinto the conventional Schr¨odinger equation for the spinorfunction Φ ε − m ∇ Φ ε + U ( r )Φ ε = E Φ ε . (9) B. Singular case
There is another way to reduce Eqs. (5) and (6) to anequation for one spinor. One should express Φ from (5)and substitute into Eq. (6). In this way one can revealan unusual feature of the solution. It follows thatΦ ε = − i~σ ∇ Θ ε ε − U − m (10)and the equation for the spinor Θ ε , if to introduce thefunction q ( r ) = ε − U ( r ) − m , is − ∇ Θ ε + ∇ qq ( ∇ Θ ε − i~σ × ∇ Θ ε ) + m Θ ε = ( ε − U ) Θ ε . (11) Below we suppose Θ ε to be isotropic. Since U ( r ) isalso isotropic there is no term ~σ × ∇ Θ ε in (11) and thisequation takes the form − qr ∂∂r (cid:18) r q ∂ Θ ε ∂r (cid:19) + m Θ ε = ( ε − U ) Θ ε . (12)In the non-relativistic limit ( E = ε − m ) , U ( r ) ≪ mc Eq. (12) reads − m (cid:20) r ∂∂r (cid:18) r ∂ Θ ε ∂r (cid:19) + 1 E − U ( r ) ∂U∂r ∂ Θ ε ∂r (cid:21) + U ( r )Θ ε = E Θ ε . (13)Due to the extra term with ∂U/∂r , it is not a conven-tional Schr¨odinger equation. This occurs since Φ ε , butnot Θ ε , should be referred to as the wave function. Onecan obtain an equation for Φ ε by applying to (10) anoperator of the type (11) whose action on Θ ε is zero.But this equation for Φ ε is not adequate since it will beredundant solutions for Φ ε .The remarkable feature corresponds to the energy ε coinciding with the bottom of the potential well ε a = m + U (0). In this case q ( r ) /r is finite at r →
0. For thisreason a solution of the wave equation (12), which decaysexponentially on large distances, cannot be singular at r → − ∂ Θ ε ∂r + mU ′′ (0) r Θ ε = 0 , r → . (14)One can easily show from (12) that ∂ Θ ε ∂r ∼ U ′′ (0) + 13 U ′′′ (0) r + ... (15)Now the total bispinor takes the form (cid:18) Φ( ~r, t )Θ( ~r, t ) (cid:19) = (cid:18) − i~σ ∇ w ( r ) U (0) − U ( r ) , w ( r ) (cid:19) e − itε a , (16)where we denote the non-singular solution of (12), with ε = ε a , as Θ ε a = w ( r ). The bispinor (16) corresponds toelectron (not positron). It is clear since one can adiabat-ically switch off the potential U ( r ) and then the energybecomes of the electron type ε = m .Whereas in (16) the spinor Θ( ~r, t ) is smooth and finiteat all r , the spinor Φ( ~r, t ) is singular at r = 0. So we ar-gue that there exists the formal solution (16) of the Diracequation where Φ ∼ ~σ~rw ′ ( r ) /r ∼ /r at r →
0. Due tothe singularity at r → ~r, t ) is of the algebraicorigin (denominator in (16)) but not a direct solution ofa differential equation.Indeed, there is a crucial difference between those twomechanisms of singularity formation. For example, thesingularity 1 /r follows from the equation ∇ /r = 0.But this solution does not exist even formally since theartificial source 4 πδ ( ~r ) is required to support that singu-larity. Like in electrodynamics a point charge supportsthe singular Coulomb potential. In non-relativistic quan-tum mechanics such singularity also does not exist, as aformal solution of a wave equation, since it also requiresa non-existing δ -source. C. Anomalous electron states
The lack of singularities of Θ is a non-trivial fact. Asfollows from (12), at small r the solution can be of theform ∂ Θ /∂r ∼ q ( r ) /r . At ε = ε a , as above, q (0) = 0and therefore Θ is not singular at small r . But at ε > ε a the parameter q (0) is finite and Θ can be singular, as1 /r at small r . Thus one should choose proper energies ε to eliminate those singularities. The resulting anomalousstates do not coincide with the usual Coulomb ones since(13) is not Schr¨odinger equation. The proper wave func-tions are formal solutions of the Dirac equations despitethe spinor Φ (10) remains singular. Electron anomalouslevels are shown in Fig. 1. The lowest anomalous level,with ε = ε a , is negative and marked by the thick line.In contrast, in the usual formalism, related to Eq. (8),the singularity Φ ∼ ~σ~rw ′ ( r ) /r ∼ /r does not existeven formally since it is of the dipole type on the languageof electrostatics. Therefore it requires an artificial sourcein the right side of (8) corresponding to charges of thedipole. D. Charge conjugate anomalous states
The electron state, described by the bispinor (16), canhave its charge conjugate state with the spinors Φ c andΘ c . This state is described by the Dirac equation withthe opposite sign of the potential energy U ( r ) [5] (cid:20) ∂∂t − iU ( r ) + ~α ∇ + imβ (cid:21) (cid:18) Φ c Θ c (cid:19) = 0 (17)The solution of this equation follows from (5) and (6)where one should change the sign of U . In this formalismthe first spinor is Φ ε and the second is given by (7) withthe substitution U ( r ) → − U ( r ). Now we choose ε = − ε a .In this case Φ − ε a ( ~r ) satisfies the same equation as Θ ε a ( ~r )in Sec. II B. Therefore one can take Φ − ε a = ( − σ y w ) ∗ where w ( r ) is the same as in Eq. (16). We remind that w ( r ) is the non-singular spinor. As a result, the solutionof (17) takes the form (cid:18) Φ c ( ~r, t )Θ c ( ~r, t ) (cid:19) = (cid:18) ( − σ y w ) ∗ , i~σ ∇ ( − σ y w ) ∗ U (0) − U ( r ) (cid:19) e itε a . (18)The bispinor (18) is the charge conjugate of the bispinor(16). This follows from the definition of charge conjugate anomalousstateselectroncharge conjugateanomalous states ε FIG. 1: Positive ( m < ε ) and negative ( ε < − m ) usualbranches. The additional (anomalous) levels are shown byhorizontal lines. The energy of the lowest electron anomalouslevel, ε a < − m , is marked by the thick line. Spontaneouspair creation is impossible. Corresponding charge conjugatestates are at positive energies. [5] (cid:18) Φ c Θ c (cid:19) = U c (cid:18) Φ ∗ Θ ∗ (cid:19) γ , U c = (cid:18) − σ y − σ y (cid:19) . (19)Charge conjugate anomalous states are shown in Fig. 1.The upper level, with ε = − ε a , is positive and markedby the thick line. III. CUTTING OFF BY THEELECTROMAGNETIC INTERACTION?
The proposed solutions relate to the hope that it canbe quantum fluctuations ~u of the well position U ( | ~r − ~u | )in space. This may happen due to the interaction withphotons or other fluctuating fields. In this case the re-sulting state will be a continuous superposition of singu-lar wave functions with various singularity positions at ~r = ~u . Such superposition would be smooth in space andtherefore physical.The significant question is that: does the electron-photon interaction smear the singularity in space?To answer this question let us consider electron propa-gator in quantum electrodynamics. It satisfies the equa-tion [5] (cid:0) γ [ ε − U ( r )] + i~γ ∇ − m (cid:1) G ( ε, ~r, ~r ′ ) − Z dr Σ( ε, ~r, ~r ) G ( ε, ~r , ~r ′ ) = δ ( ~r − ~r ′ ) , (20)where Σ is the mass operator.The exact electron propagator has the form [5] G ( ε, ~r, ~r ′ ) = X n (cid:20) h | ψ ( ~r ) | n ih n | ¯ ψ ( ~r ′ ) | i ε − E + n + i h | ¯ ψ ( ~r ′ ) | n ih n | ψ ( ~r ) | i ε + E − n − i (cid:21) , (21)where E ± n are exact energy levels for electrons andpositrons, ¯ ψ = ψ ∗ γ is the Dirac conjugate and γ , ~γ are Dirac matrices in the spinor representation [5]. Thematrix element h | ψ ( ~r ) | n i of the operator ψ is taken be-tween the vacuum and the state n . The state | n i containsone electron (positron), some pairs, and photons.If to ignore the electromagnetic interaction the state,proposed in Sec. II B, is not physical and it cannot beincorporated into the scheme (21). Suppose that theelectron-photon interaction smears the singularity outand the former singular state becomes physical with theexact electron energy E + a . Then this state is automat-ically accounted for in the series (21) as the term with n = a . Without the electron-photon interaction the en-ergy E + a is ε a (Sec. II B).Now one can apply the method of Schwinger when ε isclose to E + a [5]. Under this condition only one resonantterm remains in the series (21). In this case (cid:0) γ [ ε − U ( r )] + i~γ ∇ − m (cid:1) Ψ a ( ~r )= Z dr Σ( ε, ~r, ~r )Ψ a ( ~r ) , (22)where we denote h | ψ ( ~r ) | a i = Ψ a ( ~r ).One takes for the mass operator the expression [5]Σ( ε, ~r, ~r ) = − ie γ µ Z dω π G ( ε + ω, ~r, ~r ) γ ν × D µν ( ω, ~r − ~r ) , (23)where the photon propagator is D µν ( ω, ~r ) = − r exp ( i | ω | r ) g µν (24)and the metric tensor g µν has the signature (+ − −− ).One has to substitute into (23) the series (21) whereonly the resonance term, with n = a , is kept. The resultis Σ( E + a , ~r, ~r ) = ie γ µ Z dω π Ψ a ( ~r ) ¯Ψ a ( ~r ) ω + i γ ν g µν | ~r − ~r | exp ( i | ω || ~r − ~r | ) . (25)Here ¯Ψ a ( ~r ) = h a | ¯ ψ ( ~r ) | i and we put ε = E + a . The ω integration is easily performed and, putting Ψ = ( ϕ, χ ),we obtain from (25)Σ( E + a , ~r, ~r )Ψ a ( ~r ) = (cid:20) (cid:18) χ ( ~r ) ϕ ( ~r ) (cid:19) ¯Ψ a ( ~r ) γ Ψ a ( ~r )+ (cid:18) ~σχ ( ~r ) − ~σϕ ( ~r ) (cid:19) ¯Ψ a ( ~r ) ~γ Ψ a ( ~r ) (cid:21) e | ~r − ~r | . (26)It is clear from comparison of (26) and (22) that therole of Σ in Eq. (22) is equivalent to a renormalizationof U ( r ) and the appearance of the part similar to vectorpotential. Eq. (22) now reads (cid:26) γ [ ε − U ( r ) − P ( r )] + ~γ h i ∇ + ~Q ( ~r ) i − m (cid:27) Ψ a ( ~r ) = 0 , (27) where P ( r ) and Q ( ~r ) are determined by d r integra-tion in (26). In the spinor representation, used in thissection, the bispinor Ψ a consists of two spinors ϕ and χ . It is reasonable to express the spinors in the way ϕ = (Φ ε + Θ ε ) / √ χ = (Φ ε − Θ ε ) / √
2, with ε = ε a , corresponding to the standard representation asin Sec. II. In this notation P ( r ) = Z d r (cid:2) | Φ ε ( ~r ) | + | Θ ε ( ~r ) | (cid:3) e | ~r − ~r | (28) ~Q ( ~r ) = Z d r [Φ ∗ ε ( ~r ) ~σ Θ ε ( ~r ) + Θ ∗ ε ( ~r ) ~σ Φ ε ( ~r )] × e | ~r − ~r | . (29)Substituting here the expression (10) we obtain P ( r ) = Z d r [ U (0) − U ( r )] (cid:12)(cid:12)(cid:12)(cid:12) ∂ Θ ε ∂r (cid:12)(cid:12)(cid:12)(cid:12) e | ~r − ~r | (30) ~Q ( ~r ) = i Z d r U (0) − U ( r ) (cid:2) ( ∇ Θ ∗ ε )Θ ε − Θ ∗ ε ( ∇ Θ ε ) − i ∇ × (Θ ∗ ε ~σ Θ ε ) (cid:3) ( ~r ) e | ~r − ~r | (31)In our case Θ ε ( r ) is real and isotropic. One canshow that in this case ~Q ( ~r ) = ~σ × ~rf ( r ) where f ( r ) isisotropic. Also [ P ( r ) − P (0)] ∼ r at small r . It isnot difficult to show that the equation for Θ ε has thesame form (11) with the only difference: one shouldchange U ( r ) → U ( r ) + P ( r ). In this case the energyis E + a = U (0) + P (0).The radiative correction P and ~Q in the Dirac equa-tion (27) correspond to the first order of the perturbationtheory. These corrections are divergent at small distancesdue to the initial singularity. The account of further or-ders of the perturbation theory, at first sight, should cutoff these singularities making the terms P and ~Q finite.If this occurs, the equation (27), with the definitions (30)and (31) accounting a cut off, can be considered as theself consistency scheme to determine the cut off parame-ters.But that program does not work since, according to theupdated Dirac equation (27), the wave function remainswith the same type of singularity. This means that in anyfinite order of the perturbation theory the singularity isnot cut off.This can be clarified as following. In the phenomenonof the Lamb shift an electron “vibrates” within the cer-tain region in the space under the action of electromag-netic fluctuations [14–16]. The amplitude of these “vi-brations” is much less than a typical spatial scale of thewave function. This allows to apply the perturbationtheory on the electromagnetic interaction. In our case,due to the singularity, the typical spatial scale is zero andtherefore the perturbation theory does not work.Thus the electron-photon interaction does not result incut off the above singularity. This can be interpreted asfollowing. Due to the singularity, the momentum p ∼ /r is infinite at r = 0 and its shift ~p − ~k , by the finite photonmomentum, remains it infinite. IV. CUTTING OFF BY LATTICE VIBRATIONS
So far we have not specified the potential well U ( r ).An atomic electron is acted by the nucleus Coulomb fieldthat can be written in the approximate form U ( r ) = − Ze p r + r N , (32)where r N ∼ − cm is a size of an atomic nucleus and Ze is the nucleus charge. Generally, − U (0) ∼ M eV .We do not specify here detailed nuclear charge distribu-tion. See for example [17]. In the physical units theradiative correction (due to vacuum polarization) to theCoulomb field (2 e / π ~ c ) ln(0 . ~ /mcr ) [5] is negligibleat r ∼ r N . Because of the singularity short distancesare mainly significant and therefore an influence of otheratomic electrons is minor.When an atom is a lattice site of a solid, its fluctuatingdeviation ~u from an equilibrium position is not zero. Thelattice, as a set of oscillators, vibrates due to quantumand thermal fluctuations [7, 8, 18–21]. According to this,the mean displacement h ~u i = 0 but the mean squareddisplacement h u i is finite. See also Appendix.The characteristic time of the lattice motion is the in-verse Debye frequency 1 /ω ∼ − s . But the typicalelectron time in our problem is 1 /m ∼ − s . There-fore the lattice motion is very adiabatic compared to theelectron dynamics.The lattice displacements are ~v l ( t ) where l denotes alattice site. One of the mechanisms of interaction of usualband electrons in solids with the lattice is the deforma-tion potential [7, 8]. In our case the electron is very local-ized at the lattice site with l = 0 and therefore it interactsmainly by the shift of the potential well, U ( | ~r − ~u | ), where ~u = ~v .The lattice displacements are not mean field values butquantum degrees of freedom of the lattice. Quantumfluctuations smear the position of the well (32) withinthe sphere | ~u | . p h u i (A.10). We treat the fluctuationsof the lattice positions to be of the pure quantum naturewhen temperature is much lower than the Debye energy, T ≪ ω . In reality, T ∼ ω but the thermal populationof levels does not lead in our case to qualitatively newphenomena.The solution of the Dirac equation, we obtained inSec. II B, is singular at small distances. Below we studyhow fluctuations of the spatial position of the well cut offthis singularity.The matrix element of the electron Heisenberg oper-ator ψ ( ~r, t ) is expressed through one in the Schr¨odingerrepresentation, ψ ( ~r ), by the relation [5] h | ψ ( ~r, t ) | n i = h | ψ ( ~r ) | n i exp [ − it ( E n − E vac )] . (33) The matrix element (33) can be calculated by the func-tional integration R D~v l ( τ ) on the set of lattice displace-ments. The matrix element Ψ n ( ~r ) = h | ψ ( ~r ) | n i , of theoperator in the Schr¨odinger representation, correspondsto functional integration on trajectories ~v l ( τ ) which ter-minate at τ = 0. This leads to t = 0 in (33). So Ψ n ( ~r )can be calculated by the integration on those trajecto-ries. In the total integral R D~v l ( τ ) one can integrate outall variables with l = 0. As a result,Ψ n ( ~r ) = Z K [ ~r, ~u ( τ )] D~u ( τ ) (34)is expressed through the certain kernel K .Now one can account for that the motion of ~u ( t ) is eightorders of magnitude slower than the electron dynamics.According to Appendix, h ˙ u i ∼ ω h u i ∼ (1 /M ) h u i .Therefore the main trajectories in the ( ~u, τ ) space arewithin the narrow “cigar”, of the diameter δu ∼ p h u i ∼ /M / , that is parallel to the τ axis. The length of thecigar is large, ∼ /ω ∼ M .In other words, the principal trajectories, contributingto the path integral (34), are almost time independent.For n = a this corresponds to the approximationΨ a ( ~r ) ≃ Ψ (0) a ( ~r ) = Z ψ ( ~r − ~u ) F ( ~u ) d u, (35)where F ( ~u ) is the certain function and ~u is just a vari-able but not a trajectory. Here ψ ( ~r − ~u ) is the bispinor(16) corresponding to the potential well U ( | ~r − ~u | ) stat-ically shifted by ~u . The form (35) is not singular since ψ ( ~r ) ∼ /r at small r and the d u integration van-ishes that singularity. Therefore in the exact solutionΨ a ( ~r ) = Ψ (0) a ( ~r ) + δ Ψ( ~r ) the non-adiabatic correction δ Ψ is small.In the propagator (21) the nominators are smoothfunctions of ~r and ~r ′ . In principle, one could use an anal-ogous consideration for the whole G ( ε, ~r, ~r ′ ). But in thiscase in the equation, analogous to (35), the d u integra-tion of the singular product ψ ( ~r − ~u ) ¯ ψ ( ~r ′ − ~u ) would resultin the singularity 1 / | ~r − ~r ′ | . Therefore the non-adiabaticcorrection (analogous to δ Ψ) would be also singular tocompensate 1 / | ~r − ~r ′ | . This is not convenient. Partic-ular forms of K [ ~r, ~u ( τ )] and F ( ~u ) are not significant forour purpose since the goal is not exact calculations butto show an existence of the cut off.We see that the singular solution of the Dirac equationgives rise to the certain anomalous state due to cuttingoff the singularity by quantum fluctuations of the latticesites. In other words, anomalous state is a superpositionof various ones with shifted arguments ( ~r − ~u ) and withproperly distributed ~u according to the lattice quantumfluctuations. The spatial scale of the singularity smearingis r ∼ p h u i (A.10). The energy of anomalous state, m + U (0), can be renormalized within the Debye energy ω ∼ − eV . It is small compared to − U (0) ∼ M eV .The subsequent inclusion of electron-photon interac-tion slightly violates anomalous state. This occurs sinceits size (1) strongly exceeds the quantum electrodynam-ical spatial scale. Analogously, this scale is short com-pared to the Bohr radius resulting in a week Lamb shiftof energy levels in atoms [5, 14–16].
V. MAJORANA STATES
Before we used the standard representation, with thespinors Φ and Θ, when the Dirac equation had the form(2). One can explore the different representation apply-ing the unitary transformation (cid:18) Φ ′ Θ ′ (cid:19) = U (cid:18) ΦΘ (cid:19) (36)In the new representation the Dirac equation takes theform [5] (cid:20) ∂∂t + iU ( r ) + ~α ′ ∇ + imβ ′ (cid:21) (cid:18) Φ ′ Θ ′ (cid:19) = 0 , (37)where one should use the rule β ′ = U βU − . In the newrepresentation the charge conjugation looks analogouslyto (19) [5] (cid:18) Φ ′ c Θ ′ c (cid:19) = U ′ c (cid:18) Φ ′∗ Θ ′∗ (cid:19) γ ′ , (38)where U ′ c = U U c ˜ U . The conjugate bispinor (38) satisfiesthe equation (cid:20) ∂∂t − iU ( r ) + ~α ′ ∇ + imβ ′ (cid:21) (cid:18) Φ ′ c Θ ′ c (cid:19) = 0 , (39)with the inverted sign of U (compared to (37)) corre-sponding to the opposite charge [5].The states, related to various unitary transformations(36), usually belong to the Dirac type that is with dis-tinguished electron and positron. However there is thefamous Majorana transformation U = 1 √ α y + β ) = 1 √ (cid:18) σ y σ y − (cid:19) (40)resulting in the state when the spin-1/2 particle is itsown antiparticle [5]. Perspectives of Majorana fermionsare discussed in the paper by Wilczek [9]. See also thereferences therein.In the Majorana representation the matrices in (39)are α ′ x = − α x , α ′ y = β , α ′ z = − α z , and β ′ = α y [5].Anomalous Dirac bispinor (16) under the transformation(36) and the definition (40) goes over into the Majoranabispinor (cid:18) Φ ′ ( ~r, t )Θ ′ ( ~r, t ) (cid:19) M = 1 √ (cid:18) σ y w ( r ) − i~σ ∇ w ( r ) U (0) − U ( r ) , − w ( r ) − iσ y ~σ ∇ w ( r ) U (0) − U ( r ) (cid:19) e − itε a . (41) It is easy to show that in the Majorana case the Fouriertransformed solution of (39), with ε = − ε a , is the com-plex conjugate spinor (41) with the opposite sign. Inother words, the charge conjugate Majorana bispinor is (cid:18) Φ ′ c ( ~r, t )Θ ′ c ( ~r, t ) (cid:19) M = − (cid:18) Φ ′∗ ( ~r, t )Θ ′∗ ( ~r, t ) (cid:19) M (42)The equivalence of the charge conjugate and the complexconjugate is the fundamental property of the Majoranafermions [9].Since the spinor Φ ∼ /r is singular, this singular-ity transfers to Φ ′ and Θ ′ according to (36). Under theMajorana transformation the singular part of the Diracspinor (16) goes over into the Majorana singular part (cid:18) ΦΘ (cid:19) ∼ r (cid:18) ~σ~r (cid:19) w ′ ( r ) e − itε a (43) → (cid:18) Φ ′ Θ ′ (cid:19) M ∼ r (cid:18) ~σ~ry + i ( σ x z − σ z x ) (cid:19) w ′ ( r ) e − itε a . The sum of two Majorana singular spinors (cid:18) Φ ′ + Φ ′∗ Θ ′ + Θ ′∗ (cid:19) M = (cid:18) ( σ x x + σ z z ) cos tε a − iσ y y sin tε a y cos tε a + ( σ x z − σ z x ) sin tε a (cid:19) w ′ r (44)is real corresponding to the coincidence of the particleand its own antiparticle. In (44) we suppose for simplicitythe spinor w ′ ( r ) to be real.The particle density is the same for the Dirac and Ma-jorana states. Before averaging on positional fluctuationsthe particle density is ρ ( r ) ∼ w ′ (0) w ′∗ (0) r . (45)We use the general relation ρ = | Φ | + | Θ | .The conclusions of Sec. IV about smearing of the sin-gularities is also referred to the Majorana singularities(43). Accordingly, the particle density for the Dirac orMajorana states is localized at r ∼ p h u i ∼ − cm where ρ ∼ / ( h u i ) / . This form accounts for the nor-malization R ρd r = 1.The usual non-relativistic electron state in an atomis described by the wave function ϕ ( ~r ) exp( − itE C ). Itis the solution of the Schr¨odinger equation localized onthe Bohr radius. The usual energy level in the Coulombfield is E C . The corresponding spinors of Sec. II A areΦ = ϕ ( ~r ) exp[ − it ( m + E C )] and Θ ≃
0. The relatedMajorana state corresponds to the real bispinor (cid:18)
Φ + Φ ∗ σ y (Φ − Φ ∗ ) (cid:19) M . (46)It is localized within the Bohr radius and is characterizedby the frequency ( m + E C ). usualstate coordinatesstateenergyanomalous0 FIG. 2: Schematic plot of the energy barrier separating theusual and anomalous states. This barrier exists in the multi-dimensional space of lattice and electron coordinates.
VI. CREATION OF ANOMALOUS STATES
Since h u i ∼ ω the parameter of the perturba-tion theory, with respect to the lattice displacement, is ω / | U (0) | ∼ − . It is also treated as the adiabaticparameter.Anomalous state cannot be obtained by a perturba-tion theory because the bare electron propagator is sin-gular. On the other hand, if to start with the u -averageof some non-singular electron state we will never arriveto anomalous state within the conventional perturbationtheory. This means that anomalous state cannot be ob-tained in any order of the perturbation theory on thelattice displacement.The situation corresponds to an energy barrier sepa-rating the usual and anomalous states. The probabil-ity of tunneling through the barrier is not a series ofpowers of ω / | U (0) | but is of the exponential type likeexp[ −| U (0) | /ω ]. Analogously to a conventional barrier,the exponent depends on the product of the typical en-ergy | U (0) | and the analogue, 1 /ω , of the underbarriertraversal time. The probability of tunneling is small be-cause of heavy lattice sites involved into the process. Re-lated lattice distortions are responsible for the barrierformation. This contrasts to polaron in solids [1–4, 7, 8]where lattice distortions form a well but not a barrier.The energy of anomalous state, see Fig. 2, is below − m .However the spontaneous pair creation is impossible dueto the vanishing barrier penetration as well as for thecreation of anomalous state.The barrier is schematically plotted in Fig. 2. Thecreation of anomalous state occurs by passing the barrierof the M eV height. Tunneling through the barrier isimpossible since the number exp( − ) does not exist innature. Therefore there is the only possibility of creationof anomalous states: to excite them over the barrier topby some external radiation in the M eV energy range.After falling down from the barrier top, with the emissionof quanta, the resulting anomalous state in Fig. 2 is of ageneral Dirac (not necessary Majorana) type.
VII. DISCUSSIONS
We consider the Dirac equation for two spinors Φand Θ in the static central potential U ( r ) which is U (0)+ U ′′ (0) r / r →
0. One can express the spinor Θthrough Φ (Sec. II A). The resulting equation for Φ goesover into the Schr¨odinger equation in the non-relativisticlimit ( c → ∞ ) when the energy slightly differs from m and U ≪ m . In this limit Θ ∼ /c (in physical units) issmall. These facts are well known.In contrast, one can construct a solution, for the elec-tron like particles, in a different way, namely, to expressΦ through Θ to get the equation for Θ (Sec. II B). Thissolution looks awful. The both Θ and Φ are singular.Under the choice of energy ε = m + U (0) the spinor Θbecomes not singular but the singularity of Φ ∼ /r remains.Even in non-relativistic quantum mechanics the solu-tion of the equation − (1 / m ) ∇ ψ − Eψ = 0 can have atendency to form the singularity ψ ∼ /r . But this sin-gular solution does not exist even formally since, analo-gously to electrostatics, it requires the artificial δ ( ~r ) termin the right side of that equation.A quite different situation takes place in our case. Theform Φ ∼ /r is really a formal solution of the Diracequation. No artificial terms, like δ ( ~r ), are required.This occurs since that singularity is not due to a so-lution of a differential equation but is of the algebraicnature. In other words, despite that singular solutionis not physical, it is a formal mathematical solution ofthe Dirac equation. Such singular solution exist also inthe conventional approach (Sec. II A). As follows from(7), the proper energy should be of the positron type ε = − m + U (0).The further question is that: can that singularity becut off by some mechanism? In this case the singular-ity would be smeared within some region and thereforethe state becomes physical. It happens that the elec-tromagnetic interaction does not lead to smearing of thesingularity. This can be interpreted as following. Dueto the singularity, the momentum p ∼ /r is infiniteat r = 0 and its shift ~p − ~k , by the finite photon mo-mentum, remains it infinite. This contrasts to the usual(non-singular) state when, due to electromagnetic fluc-tuations, the electron “vibrates” increasing its positionaluncertainty and resulting in the Lamb shift of levels.It seems, according to the above arguments, that thesingular state can never be turned into a physical one.However there is a mechanism allowing this conversion.Suppose that U ( r ) is a Coulomb potential of a nucleuswhich is cut off on the nuclear size. When this nucleusbelongs to a lattice site in a solid, the potential becomes U ( | ~r − ~u | ) where ~u is the displacement of that lattice site. ~u is not a mean field coordinate but a quantum mechan-ical degree of freedom distributed according to quantumoscillations of the lattice. The lattice motion is very adi-abatic compared to the electron dynamics and the singu-larity positions at ~r = ~u are averaged on ~u . Therefore thesingularity is smeared out within the radius of quantumfluctuations (1) and the state becomes physical.The fluctuations of the lattice positions are of purequantum nature when temperature is much lower thanthe Debye energy, T ≪ ω . In reality, T ∼ ω and thethermal occupation of next vibrational levels does notsignificantly violate h u i .The anomalous electron level is shifted below m by | U (0) | (see Eq. (32)). For usual carbon isotope C( Z = 6) the nuclear radius is r N ≃ . × − cm and therefore U (0) ≃ − . M eV . For aluminum Al( Z = 13) the nuclear radius is r N ≃ . × − cm , and U (0) ≃ − . M eV . For lead
Pb ( Z = 82) the nuclearradius is r N ≃ × − cm , and U (0) ≃ − M eV .Anomalous states cannot be obtained by the perturba-tion theory. The non-perturbative mechanism of anoma-lous states creation is tunneling through the certain bar-rier which separates the usual and anomalous states. Thelattice distortions, jointly with the electron degrees offreedom, are responsible for the barrier formation. Thiscontrasts to polaron in solids where lattice distortionsform a well but not a barrier.The energy of anomalous state, see Fig. 2, is below − m .However the spontaneous pair creation is impossible dueto the vanishing barrier penetration as well as for thecreation of anomalous state.Since the probability of tunneling across the barrier isnegligible the creation of anomalous states can occur bythe action of some external radiation ( γ , α , etc.) of a few M eV energy. This radiation kicks up the electron on thebarrier top with a subsequent quanta emission and anoccupation of the anomalous state. This state is usuallyof the Dirac type that is with distinguished electron andpositron. However, it can be of the Majorana type whenthe spin-1/2 particle is its own antiparticle. The amazingfeature of these Majorana states is the large energy asso-ciated with them. It can be more than 20
M eV as in thecase of the lead atom. Observation in experiments of theproposed Majorana states can relate to the interactionof the electron in the Majorana state with other atomicelectrons.The unexpected scenario, proposed in this paper, isinitiated by the strange solution of the Dirac equation.This state remains strange even under the electromag-netic interaction. Clear, that this state had low chancesto be regarded. It was hard to imagine that the interac-tion with phonons in a solid could bring that anomaloussolution to the category of physical states. Moreover, thebinding energy of an electron in anomalous state is in the
M eV range. This is a non-typical energy scale in solids.For example, the binding energy of a polaron, roughly0 . eV , is seven orders of magnitude lower.There is another aspect. Anomalous states do not ex-ist without quantum vibrations produced by the lattice.This means that outside the medium there is no M eV binding energy per atom. Therefore it costs a large en-ergy to remove the anomalous atom from the medium.Anomalous states of electrons may occur not in bulk solids only. In a molecule there are also quantum mechan-ical uncertainty of nuclei positions resulting in anomalousstates. After a high energy treatment of the atomic sys-tem (for example, big molecules) the formed anomalousstates can substantially change chemical and biologicalproperties. For example, a molecule, with anomalousstates, can dissociate solely under a high energy influ-ence.
VIII. CONCLUSIONS
1. There are singular solutions of the Dirac equation,in a static potential well, which are formally exist butnot physical due to the singularity.2. The electron-photon interaction does not cut off thesingularity.3. It is cut off by the interaction with acoustic vibra-tion of the well position. As a result, the states becomephysical with the electron binding energy in the
M eV range. The states can be of the Dirac or Majorana type.Despite anomalous levels are below − m the spontaneouspair creation does not occur.4. Usual electron states are separated from thoseanomalous ones by the M eV energy barrier. This con-trasts to polaron in a solid where lattice distortions forma well but not a barrier.
Acknowledgments
I thank E. B. Kolomeisky for discussions and remarks.This work was supported by CONACYT through grant237439.
Appendix: LATTICE FLUCTUATIONS
In this section we use the physical units. Suppose thatin a crystal the displacement from an equilibrium posi-tion of the lattice site ~l is ~v l [7, 8]. Then it can be Fouriertransformed ~v l = 1 V X k ~v k exp( i~k~l ) , (A.1)where the volume of the system is V and the summationis restricted by the Brillouin zone. The classical kineticenergy of the system is E kin = M X l ~ ˙ v l = M n V X k ~ ˙ v k ~ ˙ v − k , (A.2)where M is the mass of the site, n = 1 /a is the den-sity of the lattice, and a is the volume of the unit cell.Analogously the elastic energy is E el = M n V X k ω k ~v k ~v − k , (A.3)where ω k is the phonon spectrum. For each independentdegree of freedom ~v k the Schr¨odinger equation followsfrom the classical energies (A.2) and (A.3) − ~ V M n ∂ ψ k ∂~v k + M ω k n V v k ψ k = E k ψ k . (A.4)Here we do not distinguish between ~v k and ~v − k sinceit does not influence the finite result. According tothe probability distribution for oscillator [22], the meansquared value is nV h ~v k i = 3 ~ M ω k coth ~ ω k T , (A.5)where T is the temperature.The mean squared physical displacement [22] is h ~v l i = 1 V X k h ~v k i = Z BZ d k (2 π ) ~ M nω k coth ~ ω k T (A.6)where the sum P k → V R BZ d k/ (2 π ) is reduced tothe integration on the Brillouin zone. In usual crystalsthe Debye energy ( ∼ ~ ω ) is on the same order as theroom temperature. For this reason, quantum and ther-mal effects give comparable contributions to (A.6). Weconsider the quantum contribution, corresponding to thelimit T ≪ ~ ω , to estimate the mean squared displace-ment. We also approximate the phonon spectrum by isotropic one ω k = ω sin πk k , (A.7)where k is the maximal wave number determined by thecondition 1 a = Z d k (2 π ) = k π . (A.8)The mean squared displacement (A.6) of each individualatom does not depend on l and can be denoted as h u i = h ~v l i . Now from (A.6) and (A.7) we obtain h u i = ~ M ω π Z π/ x dx sin x . (A.9)In usual solids ω ∼ ( ~ /ma ) p m/M [7, 8] and it fol-lows that h u i a ∼ r mM . (A.10)The adiabatic parameter p m/M (the ratio of electronand atomic masses) is usually 10 − or less. So the lat-tice sites, due to their large masses, weakly fluctuatearound the equilibrium positions. Generally in solids a ∼ − cm and therefore p h u i ∼ − cm (1). [1] L. D. Landau, Phys. Zs. Sowjet. , 664 (1933).[2] S. I. Pekar, Research in Electron Theory of Crystals (USAEC Transl. AEC-tr-555, 1951).[3] H. Fr¨olich, Adv. Phys. , 325 (1954) .[4] R. P. Feynman, Phys. Rev. , 660 (1955).[5] V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics (Addison-Wesley, 2000).[6] L. D. Landau and E. M. Lifshitz,
Quantum Mechanics (Pergamon, New York, 1977).[7] J. M. Ziman,
Principles of the theory of solids (Cam-bridge at the University Press, 1964).[8] C. Kittel,
Quantum theory of solids (John Wiley andSons, 1963).[9] F. Wilczek, Nature Phys. , 614 (2009).[10] L. Shiff, H. Snyder, and J. Weinberg, Phys. Rev. , 315(1940).[11] V. Alonso, S. Vincenzo, and L. Mondino, Eur. J. Phys., , 315 (1997). [12] G. Esposito, J. Phys. A: General Physics , 5643 (1999).[13] H. Akcay, Phys. Lett. A , 616 (2009).[14] T. A. Welton, Phys. Rev. , 1157 (1948).[15] A. B. Migdal, Qualitative Methods in Quantum Theory (Addison-Wesley, 2000).[16] E. B. Kolomeisky, arXiv:1203.1260.[17] R. C. Barrett, Reports on Progr. in Phys. , 1 (2001).[18] R. P. Feynman, R. W. Hellwarth, C. K. Iddings, andP. M. Platzman, Phys. Rev. , 1004 (1962) .[19] A. O. Caldeira and A. J. Leggett, Annals of Phys. ,374 (1983).[20] B. I. Ivlev, Can. J. Phys. , 1253 (2016).arXiv:1510.01279.[21] U. Weiss, Quantum Dissipative Systems (World Scien-tific, 1993).[22] L. D. Landau and E. M. Lifshitz,