Lensing of Dirac monopole in Berry's phase
aa r X i v : . [ h e p - t h ] F e b Lensing of Dirac monopole in Berry’s phase
Kazuo Fujikawa and Koichiro Umetsu Interdisciplinary Theoretical and Mathematical Sciences Program,RIKEN, Wako 351-0198, Japan Laboratory of Physics, College of Science and Technology, and Junior College,Funabashi Campus, Nihon University, Funabashi, Chiba 274-8501, Japan
Abstract
Berry’s phase, which is associated with the slow cyclic motion with a finiteperiod, looks like a Dirac monopole when seen from far away but smoothlychanges to a dipole near the level crossing point in the parameter space in anexactly solvable model. This topology change of Berry’s phase is visualizedas a result of lensing effect; the monopole supposed to be located at the levelcrossing point appears at the displaced point when the variables of the modeldeviate from the precisely adiabatic movement. The effective magnetic fieldgenerated by Berry’s phase is determined by a simple geometrical considera-tion of the magnetic flux coming from the displaced Dirac monopole.
The notion of topology and topological phenomena have become common in variousfields in physics. Among them, topological Berry’s phase arises when one analyzeslevel crossing phenomena in quantum mechanics by a careful use of the adiabatictheorem [1, 2, 3]. The basic mechanism of the phenomenon is very simple and itis ubiquitous in quantum physics. It is thus surprising that one encounters Dirac’smagnetic monopole-like topological phase [4] essentially at each level crossing pointfor the sufficiently slow cyclic motion in quantum mechanics [2, 5]. The generalaspects of the monopole-like topological Berry’s phase in the adiabatic limit andthe smooth change of Berry’s phase to a dipole in the nonadiabatic limit have beenanalyzed in [6] using an exactly solvable version of Berry’s model [5]. We here reporton a more quantitative description of the magnetic field generated by Berry’s phase,which is essential to understand the motion of a particle placed in the monopole-like field, together with a surprising connection of the topology change of Berry’s1hase with the formal geometrical movement of Dirac’s monopole in the parameterspace caused by the nonadiabatic variation of parameters. This movement is alsocharacterized as an analogue of the lensing effect of Dirac’s monopole in Berry’sphase.We first briefly summarize the essential setup of the problem for the sake ofcompleteness. Berry originally analyzed the Schr¨odinger equation [2] i ~ ∂ t ψ ( t ) = ˆ Hψ ( t ) (1)for the Hamiltonian ˆ H = − µ ~ ~σ · ~B ( t ) describing the motion of a magnetic moment µ ~ ~σ placed in a rotating magnetic field ~B ( t ) = B (sin θ cos ϕ ( t ) , sin θ sin ϕ ( t ) , cos θ ) (2)with ~σ standing for Pauli matrices. The level crossing takes place at the vanishingexternal field B = 0. It is explained later that this parameterization (2) describesthe essence of Berry’s phase. It has been noted that the equation (1) is exactlysolved if one restricts the movement of the magnetic field to the form ϕ ( t ) = ωt with constant ω , and constant B and θ [5]. The exact solution is then written as ψ ± ( t ) = w ± ( t ) exp (cid:20) − i ~ Z t dtw †± ( t ) (cid:0) ˆ H − i ~ ∂ t (cid:1) w ± ( t ) (cid:21) = w ± ( t ) exp (cid:20) − i ~ Z t dtw †± ( t ) ˆ Hw ± ( t ) (cid:21) exp " − i ~ Z t A ± ( ~B ) · d ~Bdt dt (3)where w + ( t ) = (cid:18) cos ( θ − α ) e − iϕ ( t ) sin ( θ − α ) (cid:19) , w − ( t ) = (cid:18) sin ( θ − α ) e − iϕ ( t ) − cos ( θ − α ) (cid:19) . (4)It is important that these solutions differ from the so-called instantaneous solutionsused in the adiabatic approximation, which are given by setting α = 0; the followinganalysis of topology change is not feasible using the instantaneous solutions. Theparameter α ( θ, η ) is defined by µ ~ B sin α = ( ~ ω/
2) sin( θ − α ) or equivalently [5]cot α ( θ, η ) = η + cos θ sin θ (5)with η = 2 µ ~ B/ ~ ω for 0 ≤ θ ≤ π , which specifies the branch of the cotangentfunction. The second term in the exponential of the exact solution (3) is customarilycalled Berry’s phase which is defined by a potential-like object (or connection) A ± ( ~B ) ≡ w †± ( t )( − i ~ ∂∂ ~B ) w ± ( t ) . (6)2his potential describes an azimuthally symmetric static magnetic monopole-likeobject in the present case.The solution (3) is confirmed by evaluating i ~ ∂ t ψ ± ( t ) = { i ~ ∂ t w ± ( t ) + w ± ( t )[ w †± ( t ) (cid:0) ˆ H − i ~ ∂ t (cid:1) w ± ( t )] }× exp (cid:20) − i ~ Z t dt ′ w †± ( t ′ ) (cid:0) ˆ H − i ~ ∂ t ′ (cid:1) w ± ( t ′ ) (cid:21) = { i ~ ∂ t w ± ( t ) + w ± ( t )[ w †± ( t ) (cid:0) ˆ H − i ~ ∂ t (cid:1) w ± ( t )]+ w ∓ ( t )[ w †∓ ( t ) (cid:0) ˆ H − i ~ ∂ t (cid:1) w ± ( t )] }× exp (cid:20) − i ~ Z t dt ′ w †± ( t ′ ) (cid:0) ˆ H − i ~ ∂ t ′ (cid:1) w ± ( t ′ ) (cid:21) = ˆ Hψ ± ( t ) (7)where we used w †∓ (cid:0) ˆ H − i ~ ∂ t (cid:1) w ± = 0 by noting (5), and the completeness relation w + w † + + w − w †− = 1.The parameter η ≥ η = 2 µ ~ B ~ ω = µBTπ (8)when one defines the period T = 2 π/ω . The parameter η is a ratio of the two differentenergy scales appearing in the model, namely, the static energy 2 µ ~ B of the dipolemoment in an external magnetic field and the kinetic energy (rotation energy) ~ ω : η ≫ T → ∞ for any finite B ) corresponds to the adiabatic limit,and η ≪ T → B ) corresponds to the nonadiabatic limit.In a mathematical treatment of the adiabatic theorem, the precise adiabaticity isdefined by T → ∞ with fixed B [3].The parameter α ( θ, η ) in (5) is normalized as α (0 , η ) = 0 by definition. Thenthe topology of the monopole-like object is specified by the valuelim θ → π α ( θ, η ) = 0 , π, π, (9)for η > η = 1 and η <
1, respectively, as is explained later.The extra phase factor for one period of motion is written as,exp " − i ~ I A ± ( ~B ) · d ~Bdt dt = exp {− i I − ∓ cos( θ − α ( θ, η ))2 dϕ } = exp {− i I ∓ cos( θ − α ( θ, η ))2 dϕ + 2 iπ } = exp {− i ~ Ω ± } , (10)3ith the monopole-like integrated fluxΩ ± = ~ I [1 ∓ cos( θ − α ( θ, η ))]2 B sin θ B sin θdϕ. (11)In (10), we adjusted the trivial phase 2 πi for the convenience of the later analysis;this is related to a gauge transformation of Wu and Yang [7, 6]. The correspondingenergy eigenvalues are E ± = w †± ( t ) ˆ Hw ± ( t ) = ∓ ( µ ~ B cos α ) . (12)From now on, we concentrate on Ω + associated with the energy eigenvalue E + ; themonopole Ω − associated with the nergy eigenvalue E − is described by − Ω + up toa gauge transformation of Wu and Yang. We then have an azimuthally symmetric monopole-like potential [6] A ϕ = ~ B sin θ [1 − cos Θ( θ, η )] (13)and A θ = A B = 0, where we definedΘ( θ, η ) = θ − α ( θ, η ) . (14)The standard Dirac monopole [4] is recovered when one sets α ( θ, η ) = 0 (or in theideal adiabatic limit η = ∞ in (5)), namely, Θ = θ in (13) and when B is identifiedwith the radial coordinate r in the real space. The crucial parameter Θ( θ, η ) isshown in Fig.1 [6].Figure 1: The relation between θ and Θ( θ, η ) = θ − α ( θ, η ) parameterized by η . Wehave the exact relations Θ( θ, ∞ ) = θ , Θ( θ,
1) = θ/ θ,
0) = 0, respectively,for η = ∞ , η = 1 and η = 0. Topologically, η > η = 1corresponds to a half-monopole, and η < θ = − η with η <
1, for which ∂ Θ( θ, η ) /∂θ = 0.4he Dirac string appears at the singularity of the potential (13). There exists nosingularity at θ = 0 since Θ( θ, η ) → θ →
0. The singularity does not appear atthe origin B = 0 with any fixed T since α ( θ, η ) → θ for B →
0, namely, if one usesΘ( θ, η ) → η = µBT /π → B = 0 forany finite T ; we have a useful relation in the non-adiabatic domain η = µBT /π ≪ A ϕ ≃ ~ B ( µT B/π ) sin θ (15)that has no singularity associated with the Dirac string at θ = π near B = 0 andvanishes at B = 0. Thus the Dirac string can appear only at θ = π and only whenΘ( π, η ) = 0, namely, η ≥ B ≥ πµT (16)for any fixed finite T [6]; the end of the Dirac string is located at πµT and θ = π .The total magnetic flux passing through a small circle C around the Dirac string atthe point B and θ = π is given by the potential (13) I C A ϕ B sin θdϕ = e M − cos Θ( π, η )) (17)with e M = 2 π ~ . This flux agrees with the integrated flux outgoing from a spherewith a radius B covering the monopole due to Stokes’ theorem, since no singularityappears except for the Dirac string. For η >
1, one sees from Fig.1 that the aboveflux is given by e M = 2 π ~ and thus Dirac’s quantization condition is satisfied in thesense exp[ − ie M / ~ ] = 1. On the other hand, the flux vanishes for η < i.e., B < πµT )and thus the object changes to a dipole [6]. T configurations We analyze the behavior of the magnetic monopole-like object (13) for fixed T andvarying B; this is close to the description of a monopole in the real space if oneidentifies B with the radial variable r of the real space. The topology and topologychange of Berry’s phase when regarded as a magnetic monopole defined in the spaceof ~B is specified by the parameter η , as is suggested by a discrete jump of the endpoint lim θ → π Θ( θ, η ) in Fig.1 [6].Using the exact potential (13) we have an analogue of the magnetic flux in the parameter space ~B = B (sin θ cos ϕ, sin θ sin ϕ, cos θ ), B ≡ ∇ × A = ~ ∂ Θ( θ,η ) ∂θ sin Θ( θ, η )sin θ B e B − ~ ∂ Θ( θ,η ) ∂B sin Θ( θ, η ) B sin θ e θ (18)5or θ = π and B = 0 with e B = ~BB , and e θ is a unit vector in the direction θ in thespherical coordinates. We have ∂ Θ( θ, η ) ∂θ = η ( η + cos θ )1 + η + 2 η cos θ , (19)by noting ∂α ( θ,η ) ∂θ = η cos θ ( η +cos θ ) +sin θ in (5), and thus ∂ Θ( θ,η ) ∂θ = 0 at cos θ = − η for η <
1. The factor in the second term in (18) is given by recalling η = µT B/π , ∂ Θ( θ, η ) ∂B = µTπ ∂ Θ( θ, η ) ∂η = ηB sin θ η + 2 η cos θ (20)using (5) and (14). Thus we have (by setting e M = 2 π ~ ) B ≡ ∇ × A = e M π sin Θ( θ, η )sin θ ηB
11 + η + 2 η cos θ [( η + cos θ ) e B − sin θ e θ ] (21)We also have from (5),cos α = η + cos θ p η + 2 η cos θ , sin α = sin θ p η + 2 η cos θ , (22)and thus sin Θ( θ, η ) = sin( θ − α )= sin θ cos α − cos θ sin α = η sin θ p η + 2 η cos θ , (23)and similarly cos Θ( θ, η ) = [1 + η cos θ ] / p η + 2 η cos θ .We finally have the azimuthally symmetric magnetic field from (21) B = e M π η B η + 2 η cos θ ) / [( η + cos θ ) e B − sin θ e θ ] . (24)We note that B/η = π/µT and θ = π define the end point of the Dirac string inthe fixed T picture. The magnetic field B is not singular at θ = π for η > η → ∞ ( π/µT → B ) in (24), theoutgoing magnetic flux agrees with that of the Dirac monopole B = e M π B e B (25)located at the origin (level crossing point) in the parameter space. This is the com-mon magnetic monopole field associated with Berry’s phase in the precise adiabaticapproximation. At the origin B = 0 with fixed finite T , which corresponds to thenonadiabatic limit η = µBT /π →
0, the magnetic field (24) approaches a constantfield parallel to the z-axis B = e M π ( µTπ ) [cos θ e B − sin θ e θ ] . (26)A view of the magnetic flux generated by the monopole-like object (24) is shown inFig.2.In passing, we comment on the notational conventions: ~B ( t ) stands for theexternally applied magnetic field to define the original Hamiltonian in (1) and ~B isused to specify the parameter space to define Berry’s phase, and B stands for the“magnetic field” generated by Berry’s phase in the parameter space. The calligraphicsymbols A , B , ∇ and the bold e stand for vectors without arrows.Figure 2: Arrows indicating the direction and magnitude of the magnetic flux fromthe azimuthally symmetric monopole-like object associated with Berry’s phase (24)in the fixed T picture. Two spheres with radii B > π/µT (i.e., η >
1) and
B < π/µT (i.e., η <
1) are shown. The wavy line stands for the Dirac string with the endlocated at B = π/µT and θ = π from which the magnetic flux is imported. Onlyin the ideal adiabatic limit T → ∞ , the end of the Dirac string and the geometricalcenter of Berry’s phase which is located at the origin agree.7 .2 Lensing of Dirac monopole in Berry’s phase We show that the monopole associated with Berry’s phase is mathematically re-garded as a Dirac monopole moving away from the level crossing point of the pa-rameter space driven by the force generated by the nonadiabatic rotating externalfield with finite period T = 2 π/ω < ∞ in Berry’s model. We consider the configu-ration in Fig.3.Figure 3: A geometric picture in the 3-dimensional parameter space ~B with a spherecentered at O and the radius B by assuming azimuthal symmetry. We suppose thata genuine azimuthally symmetric Dirac monopole is located at the point O ′ in theparameter space. The distance between O and O ′ is chosen at OO ′ = B/η . Thethree angles θ , α and Θ = θ − α are shown. The observer is located at the point P .The wavy line indicates the Dirac string.We then have O ′ P = B + ( Bη ) − B ( Bη ) cos( π − θ )= B η [1 + η + 2 η cos θ ] , (27)and the unit vector e in the direction of ~O ′ P is e = cos α e B − sin α e θ (28)with e B = ~B/B and e θ is a unit vector in the direction of θ in the spherical coordi-nates. Then the magnetic flux of Dirac’s monopole located at O ′ when observed at8he point P is given by B ′ = e M π O ′ P e = e M π η B
11 + η + 2 η cos θ (cos α e B − sin α e θ ) . (29)Next we fix the parameter α . We have ( B/η ) = B + O ′ P − BO ′ P cos α whichgivescos α = 12 B ( B/η ) p η + 2 η cos θ [ B + ( Bη ) (1 + η + 2 η cos θ ) − ( Bη ) ]= η + cos θ p η + 2 η cos θ (30)and from the geometrical relation B sin αB sin θ = B/η ( B/η ) √ η +2 η cos θ ,sin α = sin θ p η + 2 η cos θ . (31)The parameter α agrees with the parameter in (22). The azimuthally symmetricflux (29) is thus given by B ′ = e M π η B η + 2 η cos θ ) / [( η + cos θ ) e B − (sin θ ) e θ )] (32)which agrees with the flux given by Berry’s phase (24).This agreement of two expressions (24) and (32) shows that the Dirac monopoleoriginally at the level crossing point in the parameter space formally appears todrift away by the distance B/η = π/µT in the parameter space when the preciseadiabaticity condition T = ∞ [3] is spoiled by the finite T . It is interesting that twodynamical parameters, the strength of the external magnetic field and the periodin Berry’s model, are converted to very different geometrical parameters in Berry’sphase, namely, the shape of the monopole and the distance of the deviation of themonopole from the level crossing point. The observed magnetic field on the spherewith a radius B , which is controlled by the observer, thus changes when one changesthe parameter T that determines the end of the Dirac string located at π/µT inthe parameter space. This geometrical picture is useful when one draws the precisemagnetic flux from the monopole-like object for finite T as in Fig.4 and it is essentialwhen one attempts to understand the motion of a particle in the magnetic field.9igure 4: Arrows indicating the direction and magnitude of the magnetic flux ob-served at the point P with fixed B and θ when the end of Dirac string at π/µT isvaried from the point π/µT < B (Fig.4a) to the boundary π/µT = B (Fig.4b) andthen to the point π/µT > B (Fig.4c), which correspond to the change of the basicparameter η = µBT /π from the adiabatic domain η = 2 . > η = 1 and then to the nonadiabatic domain η = 0 . <
1, respectively. The wavyline stands for the Dirac string with the end at B = π/µT and θ = π . These fig-ures after a suitable rescaling may also be interpreted as the results with the end ofthe Dirac string kept fixed at π/µT and θ = π and varying the distance B , startingwith a large B > π/µT (Fig.4a) toward a small
B < π/µT (Fig.4c) in the parameterspace, such as two spheres in Fig.2.In terms of the original physical setting of a magnetic dipole placed in a givenrotating magnetic field described by the Hamiltonian (1), the cone drawn by thedipole becomes sharper compared to the cone of the given magnetic field, whichsubtends the solid angle Ω = 2 π (1 − cos θ ), when the rotating speed of the externalmagnetic field becomes larger and the dipole moment is “left behind”, namely, [5] ψ † + ( t ) ~σψ + ( t ) = w † + ( t ) ~σw + ( t )= (sin Θ cos ϕ ( t ) , sin Θ sin ϕ ( t ) , cos Θ)= − ψ †− ( t ) ~σψ − ( t ) (33)that subtends the solid angle Ω = 2 π (1 − cos Θ) with Θ = θ − α ; this sharper coneis effectively recognized as the drifting monopole in Berry’s phase by an observerlocated at the point P in Fig.3.One may thus prefer to understand that Fig.3 implies an analogue of the effect oflensing of Dirac’s monopole, since the movement of the monopole in the parameterspace is a mathematical one. In the precise adiabatic limit with T = ∞ [3], themonopole is located at the level crossing point O , but when the effect of nonadiabatic10otation with finite T < ∞ is turned on, the image of the monopole is displaced tothe point O ′ located at π/µT by keeping the topology and strength of the point-likemonopole intact. In this picture, it is important that the topological monopole itselfis not resolved in the nonadiabatic domain but it disappears from observer’s viewlocated at the point P for fixed B when π/µT = B/η → large with fixed B (i.e., η → small). In the middle, the formal topology change takes place when π/µT touches the sphere with the fixed radius B (i.e., η = 1). Even in the picture oflensing, the “magnetic flux” generated by Berry’s phase measured at the point inthe parameter space specified by ( B, θ ) is the real flux. It will be interesting toexamine the possible experimental implications of these aspects of Berry’s phase,which is expressed by the magnetic field (24), in a wider area of physics.As for the smooth transition from a monopole to a dipole, it appears in theprocess of shrinking of the sphere with a radius B covering the end of the Dirac stringlocated at π/µT to a smaller sphere for which B < π/µT as in Fig.2. When thesphere touches the end of the Dirac string (at η = 1) in the middle, one encounters ahalf monopole with the outgoing flux which is half of the full monopole e M / π ~ .See Stokes’ theorem (17) with Θ( π, η = 1) = π/ η = 1 is interesting, but it is natural to incorporate it as a part of a dipole. Themonopole-like object (13) is always a dipole if one counts the Dirac string as inFig.2 and Stokes’ theorem (17) always holds. In this sense, no real topology changetakes place for the movement of B , from large B to small B , except for the factthat the unobservable Dirac string becomes observable at B = π/µT and triggersthe topology change from a monopole to a dipole. The topology or singularity in Berry’s phase arises from the well-known adiabatictheorem [14, 15], namely, no level crossing takes place in the precise adiabatic limit T → ∞ . This theorem implies the appearance of some kind of obstruction orbarrier to the level crossing in the precise adiabatic limit; the appearance of Dirac’smonopole singularity in the adiabatic limit may be regarded as a manifestation ofthis obstruction or barrier in the parameter space. Off the precise adiabatic limitwith finite T , which is physically relevant for the applications of Berry’s phase as11as noted by Berry [2], no more obstruction to the level crossing appears. This is abasis of our expectation of the topology change in Berry’s phase in the nonadiabaticdomain.The topology change in Berry’s phase in the exactly solvable model has beenanalyzed in detail including the appearance of a half-monopole in [6]. The analysisis essentially based on Fig.1 that is a result of solving the relation (5), which is inturn a result of the Schr¨odinger equation (1). Because of this complicated logicalprocedure, the exact “magnetic field” generated by Berry’s phase was not verytransparent. In the present paper, we remedied this short coming in [6] by givinga more explicit representation of the magnetic field. In this attempt, we recognizedthat the magnetic field is in fact given by a very simple geometrical picture in Fig.3.We thus encountered an interesting mathematical description of the topology changein Berry’s phase in terms of a geometrical movement of Dirac’s monopole caused bythe nonadiabatic variations of parameters in Berry’s model. It is remarkable that themonopole by itself remains in tact without being resolved even in the nonadiabaticdomain, except for the constraint coming from the uncertainty principle. In theexactly solvable model (1), there is a physical lower bound on T given by an energy-time uncertainty principle for the difference of energy eigenvalues in (12) | ~ µB cos α | T ≥ ~ α in (22), gives( η + cos θ ) ( η − ( 14 π ) ) ≥ ( sin θ π ) . (35)Namely, we have a lower bound η ≥ π . (36)For the values of η smaller than (36), one cannot physically recognize the differenceof two energy eigenvalues in (12); in such a case, the distinction of two monopolesassoiated with E ± in (12) will not be possible, besides Ω ± → π ~ or 0 in (11). Ouranalysis of the exactly solvable model indicates that the dipole-like bahavior forboth Ω ± will appear for the values of η between the boundary η = 1 and the bound(36), namely, 1 > η > π .Traditionally, we are accustomed to understanding the topology change in termsof the winding and unwinding of some topological obstruction. The present geomet-rical description of topology change in terms of the moving monopole is a hithertounknown mechanism. This new mechanism partly arises from the fact that Berry’s12hase is not a simple monopole but rather a complex of the monopole and thelevel crossing point located at the origin of coordinates. If one instead understandsBerry’s phase as a simple monopole, one will find a novel class of monopoles [10, 11].A notable application of Berry’s phase in momentum space, which is defined bythe effective Hamiltonian by replacing ~B ( t ) → ~p ( t ) in the original model of Berryˆ H = − µ~σ · ~p ( t ) , (37)is known in the analyses of the anomalous Hall effect [12] and the spin Hall effect[13]. This effective Hamiltonian of the two-level crossing for the generic ~p ( t ) (Blochmomentum) has been analyzed in detail in [6], and it has been shown that Berry’sphase for (37) is determined by the time derivative of the azimuthal angle ˙ ϕ ( t ) inboth adiabatic (monopole) and nonadiabatic (dipole) limits, and thus our parame-terization (2) describes an essential aspect of the topology of Berry’s phase. To bemore precise, Berry’s phase becomes trivial, namely, either 0 or 2 π , in the model(37) for the nonadiabatic limit [6]( µ | ~p | ) T / ~ ≪ η ≪ η < π in (36). Ourpresent analysis implies that one may be able to observe experimentally the effectivemovement of the monopole in momentum space, as is represented by the magneticfield in (24) (by replacing B → | ~p | ), at away from the precise adiabaticity in themodel (37). Also, it will be interesting to examine the implications of the presentanalysis on the very basic issue if Berry’s phase associated with (37) deforms theprinciple of quantum mechanics by giving rise to anomalous canonical commutators[16].In conclusion, the analysis of an exactly solvable model has revealed that thetopology change in Berry’s phase is mathematically visualized as the geometricalmovement or the lensing of Dirac’s monopole in the parameter space. This will helpbetter understand both Berry’s phase and Dirac’s monopole.The present work is supported in part by JSPS KAKENHI (Grant No.18K03633). References [1] H. Longuet-Higgins, Proc. Roy. Soc. A , 147 (1975).[2] M. V. Berry, Proc. R. Soc. Ser. A , 45 (1984).133] B. Simon, Phys. Rev. Lett. (1983) 2167.[4] P.A.M. Dirac, Proc. Roy. Soc. London , 60 (1931).[5] K. Fujikawa, Ann. of Phys. , 1500 (2007). Earlier works on the basic aspectsof Berry’s phase are quoted in this reference.[6] S. Deguchi and K. Fujikawa, Phys. Rev. D , 025002 (2019).[7] T. T. Wu and C. N. Yang, Phys. Rev. D , 3845 (1975).[8] Y. Aharonov and D. Bohm, Phys. Rev. , 485 (1959).[9] A. Tonomura, N. Osakabe, T. Matsuda, T. Kawasaki, J. Endo, S. Yano, andH. Yamada, Phys. Rev. Lett. , 792 (1986).[10] S. Deguchi and K. Fujikawa, Phys. Lett. B , 135210 (2020).[11] A recent review of the magnetic monopole is found in N. E. Mavromatos andV. A. Mitsou, “Magnetic monopoles revisited: Models and searches at collidersand in the Cosmos”, Int. J. Mod. Phys. A (2020) 2030012.[12] T. Jungwirth, Q. Niu, A.H. MacDonald, Phys. Rev. Lett. (2002) 207208.Z. Fang, et al., Science (2003) 92, and references therein.[13] J.E. Hirsch, Phys. Rev. Lett. (1999) 1834; S.-F. Zhang, Phys. Rev. Lett. 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