Eur. Phys. J. C (2020) 80:1111 https://doi.org/10.1140/epjc/s10052-020-08681-6

Regular Article - Theoretical Physics

Scattering of charged fermion to two-dimensional wormhole with constant axial magnetic flux

Kulapant Pimsamarn1,a, Piyabut Burikham2,b, Trithos Rojjanason2,c 1 Department of Physics, Faculty of Science, Kasetsart University, Bangkok 10900, Thailand 2 High Energy Physics Theory Group, Department of Physics, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand

Received: 14 April 2020 / Accepted: 18 November 2020 / Published online: 3 December 2020 © The Author(s) 2020

Abstract Scattering of charged fermion with (1 + 2)- boundary phenomena that have no bulk analogs emerge, e.g. dimensional wormhole in the presence of constant axial mag- quantum (spin) hall effects, topological matters, and more netic flux is explored. By extending the class of fermionic recently graphene physics [1]. Two dimensional surface can solutions of the Dirac equation in the curved space of worm- be manipulated in the laboratory so that it can be curved, hole surface to include normal modes with real energy and strained, and twisted. A cage structure of graphene worm- momentum, we found a quantum selection rule for the scat- holes, the schwarzite, can even be produced with promising tering of fermion waves to the wormhole. The newly found properties [2,3]. The gauge fields such as electric [4] and momentumÐangular momentum relation implies√ that only magnetic fields [5] can be applied to the system, leading to fermion with the quantized momentum k = m/a q can be unique intriguing 2D phenomena, notably the well-known transmitted through the hole. The allowed momentum is pro- Landau quantization of fermionic states on a plane. It has portional to an effective angular momentum quantum number been shown that effects of strain and gauge fields could be m and inversely proportional to the radius of the throat of similar in 2D [6,7]. Behaviour of quantum particle on the the wormhole a. Flux dependence of the effective angular curved surface in the presence of gauge fields can be con- momentum quantum number permits us to select fermions siderably different from the flat situation [8,9]. Similar to that can pass through according to their momenta. A conser- the strain, curvature effects [10Ð12] can mimic gauge fields vation law is also naturally enforced in terms of the unitar- [13,14], specifically curvature and gauge connection appears ity condition among the incident, reflected, and transmitted in the equation of motion with equal role. For example, waves. The scattering involving quasinormal modes (QNMs) fermions on two-dimensional sphere and wormhole experi- of fermionic states in the wormhole is subsequently explored. ence spin-orbit coupling induced from the surface curvature It is found that the transmitted waves through the wormhole [15Ð18] even in the absence of the gauge fields. Addition of for all scenarios involving QNMs are mostly suppressed and axial magnetic field generates Landau quantization distinc- decaying in time. In the case of QNMs scattering, the uni- tive from the planar case. For the wormhole, the fermionic tarity condition is violated but a more generic relation of states can be in quasinormal modes with complex energies, the scattering coefficients is established. When the magnetic its quantum statistics can be altered by the magnetic flux flux φ = mhc/e, i.e., quantized in units of the magnetic flux through the hole [18,19]. quantum hc/e, the fermion will tunnel through the wormhole Dirac fermion in graphene wormhole without gauge field with zero reflection. has been discussed in Refs. [20,21]. In Ref. [18], we investi- gate the effects of axial magnetic field on a charged fermion in a (1 + 2)-dimensional wormhole. This two-dimensional 1 Introduction wormhole is fundamentally different from the (1 + 3)- dimensional wormhole in general relativity (see e.g. [22]), The physics of fermion on two-dimensional surface has been there is no time dilation in the two-dimensional wormhole one of the major research topics in recent years. Interesting under consideration. For the constant magnetic flux scenario, the system can be solved analytically and exact solutions are a e-mail: [email protected] found to contain “normal” (real energy but complex momen- b e-mail: [email protected] (corresponding author) tum) and quasinormal modes (QNMs). c e-mail: [email protected] 123 1111 Page 2 of 13 Eur. Phys. J. C (2020) 80 :1111

where  = (t, u,v) represents the Dirac spinor field on the wormhole and M represents the rest mass of the particle, c is the speed of light, e is electric charge, and Aμ is the electromagnetic four-potential. The γ a are the Dirac matrices given by     σ k γ 0 = i 0 ,γk = 0 i , 0 −i −iσ k 0

where σ k are the Pauli matrices defined by       − σ 1 = 01 ,σ2 = 0 i ,σ3 = 10 . 10 i 0 0 −1 They obey the Clifford algebra Fig. 1 Geometric structure of a wormhole surface where a is a radius {γ a,γb}=2ηab. (2.3) at the radius function R(u = u0 = ln q/2).Andr is the radius of curvature of the wormhole surface along u direction The Pauli matrices have a useful identity that we will use later In this work the scattering of fermions to the worm- σ i σ j = δij + i ijkσ k, (2.4) hole is explored. The more generic solutions to the equation of motion of the fermion in the magnetized wormhole are where ijk is Levi-Civita symbol. constructed in terms of hypergeometric functions. The nor- The covariant derivative of the spinor interaction with mal modes are actually found when the (effective) angular gauge field in the curved space is given by quantum number m (see definition below) is related to the  ∇ ≡ ∂ − , m μ μ μ (2.5) momentum of the fermion by k = √ ≡ km . Such quan- a q where the spin connection μ [23]is tized momentumÐangular momentum relation is unique to     the 2D wormhole under consideration. =−1γ aγ b ν ∂ β − β , μ ea μ gνβ eb eb βμν (2.6) In Sect. 2, the mathematical formulation of the wormhole 4 and fermion in curved space is established. In Sect. 3 as a where β,μ,ν ∈{t, u,v} and the Christoffel symbols βμν review of the main results of Ref. [18], the Dirac equation in are defined by the magnetized wormhole is written and solved analytically, 1 βμν = (∂μgβν + ∂ν gβμ − ∂β gμν). then the general solutions in the upper and lower plane con- 2 nected to the wormhole is discussed. Matching conditions of Then for the metric (2.1), the scattering of fermionic waves to the wormhole is con- sidered in Sect. 4. Section 5 discusses the use of Wronskian 1 2  − uvv = vuv = vvu = ∂u R = RR , (2.7) to derive a general relation between scattering coefficients. 2 The scattering scenarios are categorized into normal-modes and zero otherwise. It is then straightforward to show that and QNMs scattering and some of the results are presented the spin connections are in Sect. 6. Section 7 concludes our work. 1 1 2  t = 0, u = 0, v = γ γ R . (2.8) 2 2 Geometric and gauge setup of the wormhole We will apply an external magnetic field such that the z- component Bz = B(z) is uniform with respect to the ( , ) In the (1 + 2)-dimensional curved spacetime, the line ele- plane x y and the magnetic flux through the circular area ment on the surface of wormhole (Fig. 1) can be cast in the enclosed by the wormhole at a fixed z is constant, namely ∼ / 2 following form Bz 1 R . Due to the axial symmetry, the electromagnetic four-potential can be expressed in the axial gauge as 2 μ υ 2 2 2 2 2 ds = gμυ dx dx =−c dt + du + R (u)dv . (2.1)   1 1 Aμ (t, x, y, z) = 0, − By, Bx, 0 , The fermion will experience the effective curvature that can 2 2 be addressed by considering the Dirac equation in curved and in the wormhole coordinates as spacetime        ∂xν 1 a μ e ( , ,v)=  ( , , , ) = , , 2 . γ e −h¯ ∇μ + i Aμ − Mc  = 0, (2.2) Aμ t u μ Aν t x y z 0 0 BR (2.9) a c ∂x 2 123 Eur. Phys. J. C (2020) 80 :1111 Page 3 of 13 1111

The magnetic field is then given by of the effective potential arising from the wormhole geo-   x y metrical structure. In the presence of external magnetic field B = − ∂z B, − ∂z B, B . (2.10) =× 2 2 B A along the z direction, the charged fermion mov- ing in v direction is expected to form a stationary state with We will now consider the Dirac equation of fermion in the quantized angular momentum and energy, i.e. the Landau wormhole in the presence of constant magnetic flux. levels in the curved space with hole. To show this, we need to solve for the stationary states of the system. We can start by considering −iD×Eq. (3.5)- E + Mc2 /hc¯ ×Eq. (3.6)to 3 The Dirac equation in magnetized wormhole obtain   Utilizing the results from above equations, the Dirac equation E2 − M2c4 D2 + ϕ(u) = 0. (3.7) Eq. (2.2) can be written in the form then h¯ 2c2 ⎛   ⎞ ∂ + Mc For constant magnetic flux, the operator D2 in the equation i ct h¯ iD ⎝   ⎠  = 0, (3.1) of motion now takes the form −iD −i∂ + Mc     ct h¯   2  φ 2 2 2 R R R 1 D = ∂ + ∂ − + + ∂v − i u u 2 where D is a differential operator R 2R 2R R φ0         φ 3 R 1 R 2 1 ie −iσ ∂v − i . (3.8) D ≡ σ ∂u + + σ ∂v − BR . (3.2) 2 φ 2R R 2hc¯ R 0 We can define the pseudo-vector potential as Substitute Eq. (3.8) into Eq. (3.7) to obtain       R  1 e 2 1 e 0 = ϕ (u) + ϕ (u) D = σ ∂u − i Au˜ + σ ∂v − i Av ; hc¯ R hc¯ ⎡ R   ⎤   2 hc¯ R 1  σ  − 2 − R ≡ , = , = 2. ⎢ R m R m 2 ⎥ Au˜ i Au 0 Av BR (3.3) + ⎣ + + k2⎦ ϕ(u), (3.9) e 2R 2 2R R2 Note that the effective gauge potential in the u direction, Au˜ , is generated by the curvature along the v direction, v.Inthis  φ sense, wormhole “gravity” or curvature connection manifests where m = m − , and the magnetic flux quantum φ0 ≡ φ0 itself in the form of gauge connection in the perpendicular hc/e. We have used the momentum parameter k2 ≡ (E2 − direction. M2c4)/h¯ 2c2 and σ is a spin-state index corresponding to spin Consider a stationary state of the Dirac spinor needs to be up (σ =+1) or down (σ =−1). ( , ,v) single-valued at every point in spacetime, t u must be To be specific, we parametrise the shape of wormhole a periodic function in v with period v ∈[0, 2π],intheform ( ) = ( / )   by setting R u a coshq u r . They are based on a q- − i Et v χ(u) deformation of the usual hyperbolic functions defined by [24] (t, u,v)= e h¯ eim , (3.4) ϕ( ) − u ex + qe x coshq (x) ≡ , where the orbital angular momentum quantum number m = 2 x −x 0, ±1, ±2,.... χ(u), ϕ(u) are two-component spinors. Eq. e − qe sinhq (x) sinhq (x) ≡ , tanhq (x) = . (3.10) (3.1) can be rewritten in the form of coupled equations for 2 coshq (x) the 2-spinors   Note the relevant properties E + Mc2 χ(u) + iDϕ(u) = 0 (3.5) 2 ( ) − 2 ( ) = , d ( ) = ( ), hc¯ coshq x sinhq x q sinhq x coshq x   dx E − Mc2 d q ϕ( ) + χ( ) = tanh (x) = . (3.11) u iD u 0 (3.6) q 2 ( ) hc¯ dx coshq x From Eq. (3.2), the first term is equivalent to the Dirac The deformed functions reduce to hyperbolic functions ( ) operator with the pseudo gauge potential Au˜ u in the u direc- when q = 1. For this choice, the wormhole will possess   tion. This term is generated from the change in the radius of r 2 r the hole. The second term contains gauge potential that gen- two Hilbert horizons at u , = r ln q + ± where p m a2 a erates a spin-orbit coupling. A similar setup has been used   R (u , ) =±1. Then define a variable X(u) ≡ rR(u)/a = to study nanotubes under a sinusoidal potential [20]. Here p m sinh (u/r) to obtain we consider the dispersion relation for the two-dimensional q   fermions described by the Dirac equation in the presence 0 = (q + X 2)ϕ (X) + 2Xϕ (X) + k2r 2ϕ(X) 123 1111 Page 4 of 13 Eur. Phys. J. C (2020) 80 :1111   2  q + r σ  − r   for integer n and α0 = 2α, β0 = 2β. Finally, the solutions 4 a m X a m 1 + + ϕ(X). (3.12) of Eq. (3.9)is (q + X 2) 4 √ ϕ  (κ ,κ , ) ϕ( ) = ( + n,m ,σ 1 2 u Define√ weighting function solution X q √  √  √  )α( − )β ( ) = + α − β (2β,2α) / , iX q iX X , the equation of motion can be rewrit- q iX q iX Pn iX q (3.19) ten as where X(u) = sinh (u/r) and κ ,κ =±1. These are the  q 1 2 0 = (q + X 2) (X) + 2 [(α + β + 1)X √  solutions with QNMs found in Ref. [18]. The momentum k +i(α − β) q (X) for these solutions is generally a complex quantity, so they     are not exactly normal modes even when the energy E is real. 1 2 + α + β + + k2r 2 (X), As will be shown subsequently in Sects. 6.1 and 6.2.2, we can 2 analytically continue the solutions to the general n < 0 cases   0 = (q + X 2) (X) +[A + BX] (X) where the normal modes with real energy and momentum can =− / +[C + k2r 2](X), (3.13) be found as a special case with n 1 2. Other quasinormal n < 0 solutions are also relevant in the scattering processes. whereweassume √ r  2i(α2 − β2) q + σm = 0, 3.1 Solutions in the upper and lower plane a   2   q r  In the flat upper (R (u) = 1) and lower (R (u) =−1) plane −2(α2 + β2)q + − m = 0, 4 a region outside the wormhole, the equation of motion, Eq. (3.9), takes the form leading to        2 1 i σ m r  1  m ∓ σ/2 α = κ + √ , 0 = ϕ (u) ± ϕ (u) + k2 − ϕ(u), 1 4 q 2a R(u) (R(u))2    1 i σ m r (3.20) β = κ − √ , where κ ,κ =±1. (3.14) 2 4 q 2a 1 2 with The coefficients are defined as the following  √ u − u p + Rp, for u > u p A = 2i(α − β) q, B = 2(α + β + 1), R(u) = (3.21) −(u − um) + Rm, for u < um, 1 C = (α + β)(α + β + 1) + . (3.15) 4 respectively. Rp,m ≡ R(u p,m) is the corresponding radial distance on the upper and lower plane. Wave solutions of Eq. Depending on the sign choices of κ1,κ2, the resulting equa- tion of motion and the corresponding energy levels will be (3.20) are the Hankel function of the first and second kind. Generically the solutions can be expressed as dependent√ or independent of the spin-orbit coupling term ∼ σ mr/a q. (+) (2) (+) (1) √ ϕ (u) = Im,σ H  (kR(u)) + R ,σ H  (kR(u)), Define X =−i qY, the equation then takes the form m −σ/2 m m −σ/2 (3.22)   0 = (1 − Y 2) (Y ) + 2[(α − β) − (α + β + 1)Y ] (Y )   ≥ 1 for u u p in the upper plane, and − (α + β)(α + β + 1) + k2r 2 + (Y ) (3.16) 4 ϕ(−)( ) = (1) ( ( )) + (−) (2) ( ( )), u Tm,σ Hm+σ/2 kR u Rm,σ Hm+σ/2 kR u Equation (3.16) is the Jacobi differential equation, the energy (3.23) levels become ≤ 2 2 4 2 2 2 for u um in the lower plane. The Hankel function of the E ,  = M c + h¯ c k ,  n m n m  first (second) kind corresponds to the waves propagating in h¯ 2c2 1 2 +(−)uˆ direction respectively. = M2c4 − n + + α + β . (3.17) r 2 2 The solutions to Eq. (3.16) are the Jacobi polynomials [25] (α ,β ) 4 Matching conditions in the scattering  ( ) = 0 0 ( ) n Y Pn Y n n (−1) −α −β d = (1 − Y ) 0 (1 + Y ) 0 The matching conditions of the wave functions are the equal- 2nn! dYn ity of the complex energy (E) and angular momentum (m), α β ×[(1 − Y ) 0 (1 + Y ) 0 (1 − Y 2)n] (3.18) and the smooth continuity (i.e., C1 continuity) of the wave 123 Eur. Phys. J. C (2020) 80 :1111 Page 5 of 13 1111

Table 1 The wave vector along the u direction kn,m , the energy levels En,m , and the solutions ϕ(u) of the Dirac equation in two dimensional wormhole with constant magnetic flux through the throat of the wormhole

κ κ  2 ϕ( ) 1 2 kn,m rEn,m u

    √  σm√r     / i 1 + σm√ r , 1 − σm√ r 2 2 1 4 q + iX 2a q i i √  ++i (n + 1) M2c4 − h¯ c (n + 1)2 q + X 2 √ P 2 a q 2 a q iX/ q r2 q − iX n     √  σm√r    − / i − 1 − σm√ r ,− 1 + σm√ r 2 2 1 4 q − iX 2a q i i √  −−in M2c4 − h¯ c n2 q + X 2 √ P 2 a q 2 a q iX/ q r2 q + iX n     √        σm√r 1/4 1 + σm√ r ,− 1 + σm√ r   2 2  2 i q + iX i i √  +−i n + 1 − σ m√ r M2c4 − h¯ c n + 1 + i σ m√ r q + X 2 2a q √ P 2 a q 2 a q iX/ q 2 a q r2 2 a q q − iX n     √       − σm√r 1/4 − 1 − σm√ r , 1 − σm√ r   2 2  2 i q − iX i i √  −+i n + 1 + σ m√ r M2c4 − h¯ c n + 1 − i σ m√ r q + X 2 2a q √ P 2 a q 2 a q iX/ q 2 a q r2 2 a q q + iX n functions between the inner and outer region of the worm- hole. The momentum k will also be equal due to the relation (3.17). Consider the incoming waves propagating in the upper plane region scatter with the wormhole. At the upper Hilbert horizon u p, the waves will be partially reflected back and partially transmitted into the inner region of the wormhole. The transmitted waves will be again partially reflected and partially transmitted into the lower plane region at the lower Hilbert horizon um. In this scenario, there is no incoming (−) waves at the lower Hilbert horizon, i.e., Rm,σ = 0. We can (+) normalize Im,σ ≡ 1 and Rm,σ ≡ Rm,σ where the Hankel function of the first and second kind correspond to outgoing and incoming waves, respectively. In the wormhole region um < u < u p, the general solution from Eq. (3.19) can be expressed as (in) ϕ (u) = ϕm,n,σ (+, −, u) + ϕm,n,σ  (−, +, u) √ α √ −α∗ 0 0 = Am,n,σ q + iX q − iX ∗ Fig. 2 A wormhole connected smoothly to two flat planes at Hilbert (−2α ,2α ) √  × 0 0 / = 1 Pn iX q horizons u p and um , the midpoint of wormhole is at u0 2 ln q where ( ) √ −α √ α∗ R u is minimum. The wormhole is symmetric with respect to u0 0 0 +Bm,n,σ  q + iX q − iX ( α∗,− α ) √  2 0 2 0 ×P  iX/ q , (4.1)   n (2) + (1) Hm−σ/ kRp Rm,σ Hm−σ/ kRp ∗ 2 2 = ( / ), α = κ α ,β= κ α α ≡ √  √  ∗ where X sinhq u r 1 0 2 0 , and 0 α0 −α  = , ,σ + − 0 1 √i σm r Am n q iX q iX + and the condition En,σ = En,σ  is required. 4 q 2a (−2α∗,2α ) √  0 0 / Note that the parameter α0 also depends on m,σ. Pn iX q √ −α √ α∗ ( α∗,− α ) 0 0 2 0 2 0 +Bm,n,σ q + iX q − iX Pn 4.1 The matching condition: at upper surface √  iX/ q , (4.2) The boundary between the inner and outer region of the   / = / = 2 + 2 wormhole is at R(u p) ≡ Rp, see Fig. 2. The wave function where sinhq u p r r a, and Rp qa r .The and the first derivative of the wave function must be con- derivative of the wave function at the boundary u = u p tinuous at the boundary. From conservation law of energy- ( ) ( ) ( ) ( ) momentum, E in = E out and k in = k out . The first (+) dϕ (u) (2)  (1)  boundary condition at the upper plane is | = k(H  (kR ) + R , H  (kR )). du u p m −σ/2 p m n m −σ/2 p (+) (in) (4.3) ϕ (u p) = ϕ (u p) 123 1111 Page 6 of 13 Eur. Phys. J. C (2020) 80 :1111

4.2 The matching condition: at lower surface the equation of motion of one solution with the other solution then subtracting the two equations we obtain ≥ Even for general q 0, the wormhole is symmetric with     ϕ ϕ − ϕ ϕ + f (u)(ϕ ϕ − ϕ ϕ ) = 0, (5.4) respect to the midpoint u0 = ln q/2. The Hilbert horizon at 1 2 2 1 1 2 2 1 the lower surface is at u where R(u ) ≡ R = R .The   m m m p or in terms of the Wronskian W ≡ ϕ ϕ − ϕ ϕ , second boundary condition takes the form 12 1 2 2 1  (−) (in) W ϕ (um) = ϕ (um) 12 + f (u) = 0. (5.5) (1) ( ) W12 Tm,σ Hm+σ/2 kRm √ α √ −α∗ By integrating this equation in the region covering the worm- = − 0 + 0 Am,n,σ q iX q iX hole, (− α∗, α ) √  2 0 2 0   " ×P −iX/ q u p n W(u p) √ −α √ α∗ ln =− f (u) du, (5.6) 0 0 ( ) +Bm,n,σ q − iX q + iX W um um ( α∗,− α ) √  2 0 2 0 ×Pn −iX/ q . (4.4) is the resulting general relation and the subscript of the Wron- skian has been omitted since the relation is valid for any pairs The derivative of the wave function at the second boundary of solutions. The RHS of (5.6) can be calculated explicitly, u = u , R = R is m m " # $ u um ϕ(−)( ) p coshq d u (1)  − f (u) du = r  . | =− ,σ  ( ) . ln u p (5.7) um kTm Hm +σ/2 kRm (4.5) du um coshq r From the matching conditions (4.2)Ð(4.5), the scattering ( ) = ( ) = Since the wormhole is symmetric with R um R u p coefficients A , ,σ , B , ,σ , T ,σ , R , can be solved for m n m n m m n r 2 + qa2, the Wronskian at the two Hilbert horizons are which here and henceforth the subscripts will be suppressed. always equal, W(um) = W(u p). The general relation (5.6) can be applied to the two solu- ∗ tions ϕ and ϕ , specifically at um, u p where the wave func- 5 Analytic relation between reflection and transmission tions and their first derivatives in the two connecting regions coefficients are equal. In terms of the solutions in the outer regions, we have the relation (with the subscripts suppressed) In this section a very general analytic relation between the 2 (2) (2)∗ reflection and transmission coefficients are derived from the −|T | (H (kRm)H (kRm) − cc.) equation of motions. The relation can be expressed in terms (2) (2)∗ = (H (kRp)H (kRp) − cc.) of Wronskian of the solutions in each region. The resulting  +| |2( (1)( ) (1)∗ ( ) − .) relation is then verified numerically for each scattering sce- R H kRp H kRp cc (1) (2)∗ nario. +R(H (kRp)H (kRp) The generic form of the equation of motion can be (1) (2)∗ −H (kRp)H (kRp)) − cc. (5.8) expressed as   0 = ϕ (u) + f (u)ϕ (u) + g(u)ϕ(u), (5.1) 6 Scattering for normal and quasi-normal modes where   ∓ 1 , for u < u , u > u ( ) = R = R(u) m p In this section we consider scattering of fermion in two- f u tanh ( u ) (5.2) R q r , < < , dimensional planar surface with (1 + 2)-dimensional worm- r for um u u p hole. Even this “wormhole” is not the actual spacetime worm- and ⎡   ⎤ hole as in GR and other gravity theories where time dilata-  2  σ  − 2 − R tion exists, its scattering still reveals a number of interesting ⎢ R m R m 2 ⎥ g(u) = ⎣ + + k2⎦ properties. As discussed in details below, scattering of phys- 2R R2 ⎧   ical fermion with real momentum and energy has quantized ⎪ ∓mσ −m2− 1 behaviour that relates momentum and angular momentum of ⎨⎪ 4 + 2 , < , > ( )2 k for u um u u p =  R u  the scattered fermion. For scattering involving QNMs of the σ (5.3) ⎪ (m− a sinh ( u ))2 ⎩⎪ 1 − 2r q r + 2 , < < , 2 2 2 u k for um u u p fermion, enhancement (and attenuation) of scattered fermion 2r a coshq ( ) r is a possibility. The wormhole and spin parameters are set to respectively. Note that R(u) in the planes is given by (3.21). r = a = q = 1 = σ for all of the numerical results in this For any two solutions ϕ1,ϕ2 satisfying (5.1), by multiplying section. 123 Eur. Phys. J. C (2020) 80 :1111 Page 7 of 13 1111

6.1 Scattering for real momentum k and outer regions of the wormhole. Generically since the momentum given by (3.17) is always a complex quantity with Naturally, incoming waves from outside region are gener- an exception of km discussed in Sect. 6.1, the momentum ated with real momentum and energy. When scattered with of wave functions in the outer regions will also need to be the wormhole, only real momentum and energy states (i.e., complex as well. Consequently, there are spatial attenuation normal modes) will be allowed to propagate through the hole and enhancement of waves in the outer regions as the waves and transmitted to the other outside region. The scattering are concurrently decaying in time. The physical energy to waves will be partially reflected back and partially transmit- be measured in experiments is the real part of the (complex) ted through the hole to the other side. From Eq. (3.17), the energy used in the matching conditions. only possibility for real positive momentum k (in units of    1 iσm r  hc¯ )iswhenα =−β∗ =± + √ and n =−1/2. 6.2.1 Scattering with m = 0 4 2a q This is the analytic continuation of the equation of motion The scattering in this scenario corresponds to the states in the (3.16) (with (3.17) substituted) to n < 0 cases. The momen- wormhole with quantized flux φ = mφ0, m = 1, 2, 3,... tum then takes the value and the orbital angular momentum of the states is mh¯ .The  m quantized magnetic flux is required specifically for the scat- k = √ ≡ k  , (6.1) a q m tering to occur. In this case the energy could be real for mas- sive fermion and small n. On the other hand, the momentum notably a momentumÐangular momentum relation. Only the 1 waves with quantized momentum k  and energy given by becomes k = i n + /r, purely imaginary. The waves % m 2 = 2 4 + ¯ 2 2 2 E M c h c km are allowed to pass through the in the outer regions to be matched with the ones inside the (2)∗ wormhole thus need to be attenuating or enhancing along wormhole to the other side. For real k since Hν (kR) = (1) (1) (2) the u direction. Since the source of the fermion can locate at Hν (kR), and using the identity Hν (z)Hν (z) (2) (1) finite distance from the hole in experimental situation, this − Hν (z)Hν (z) = i/πz 4 , the relation between transmis- scattering scenario is still physically relavant. sion and reflection coefficients from Eq. (5.8)issimplified The fermionic states inside the wormhole consist of two to waves with n = n, ϕ(+, −, n) and ϕ(−, +, n) where 2 2 (κ ,κ ) = (+, −), (−, +) +ˆ, −ˆ |T | +|R| = 1, (6.2) 1 2 represents u u going modes respectively. The scattering coefficients for n = 0, 1, 2, 3 the unitarity condition. Unitarity is the consequence of real- are presented in Table 3. The transmission and reflection ity of momentum in the scattering. Table 2 shows scattering coefficients obey general relation (5.8) but not the unitar-  coefficients for m = 1, 2, 3, 4 cases, all of which the uni- ity condition (6.2) due to the complexity of momentum k. tarity relation (6.2) is numerically verified. Wave function Remarkably, the waves simply tunnel through the wormhole profile of the matching is shown in Fig. 7a. with no reflection, i.e., R = 0. The scattering coefficients for  The m -dependence of the transmission coefficient T and n = 0, 1, 2, 3 are given in Table 3. Wave function profile of   R are shown in Fig. 3. Interestingly, T (m ) and R(m ) con- the matching is shown in Fig. 7b. verge to oscillating functions for large m. The unitarity con- dition (6.2) is always obeyed. 6.2.2 Scattering with nonzero m

6.2 Scattering involving QNMs  For m = 0 the waves inside the wormhole consist of ϕ(+, −, n) and ϕ(−, +, n). According to the energy for- Scattering involving QNMs usually violates unitarity since  mula (3.17), the −ˆu-travelling wave ϕ(−, +, n ) must have the waves are decaying in time. It will satisfy a more general  the wave function associated with n =−n − 1 in order to relation given by (5.8). There are three possible scenarios for have the same energy, En = En . Since the Jacobi polyno- such scattering. mials Pn are zero for non-positive integer n, we need a more general form of the solutions. From the equation of motion  = 1. scattering with m 0 (3.16) with (3.17) substituted, 2. scattering with nonzero m  3. scattering with spin-flip σ ↔−σ in the inner region 0 = (1 − Y 2) (Y ) + (−2α + 2β + Y (−(2α  +2β + 2))) (Y ) − n(2α + 2β + n + 1)(Y ), (6.3) There are three scenarios for scattering involving QNMs of the wormhole. The matching condition requires the energy we can analytically continue the solution to the case where and inevitably the momentum k to be the same value for inner n < 0 by solving this equation as the hypergeometric equa- 123 1111 Page 8 of 13 Eur. Phys. J. C (2020) 80 :1111

Table 2 Scattering coefficients for real momentum scenario m AB T R

10.486599 + 0.867254i 0.0131312 + 0.0626096i − 0.292892 + 0.699874i 0.600951 + 0.251493i 20.434897 + 0.900917i − 0.0245842 + 0.0137852i − 0.408684 + 0.576578i 0.577195 + 0.409121i 30.334517 + 0.942154i − 0.0147694 − 0.00833657i − 0.516515 + 0.462697i 0.480745 + 0.536663i 40.219563 + 0.975395i − 0.00110507 − 0.0115601i − 0.612787 + 0.328918i 0.339825 + 0.633107i

Fig. 3 The m-dependence of the transmission coefficient T and reflection coefficient R in the scattering with real momentum scenario

Table 3 Scattering coefficients for m = 0 scenario with r = a = q = out through both horizons into the outer planes. The leak- 1 = σ ing waves are attenuated or enhanced depending on the sign (2) (1) nA B T Rof Im(k).Forthe−ˆu (+ˆu) moving Hν (Hν ),thewave function is spatially enhanced (attenuated) with respect to 0 − 1.04147i 1.04147 1.70376i 0 the increase of u respectively. Wave function profile of the 1 − 2.04891 − 2.04891i 4.94571i 0 matching is shown in Fig. 7c. The scattering coefficients for 23.60535i − 3.60535 14.3565i 0 certain parameters are given in Table 4.Them-dependence . . . 362298 6 2298i 41 6741i 0 of the transmission coefficient T and R are shown in Fig. 4. Remarkably, both T and R show a peak at m =±6. Scattering of waves to the wormhole with QNMs exci- tion. There are two solutions, tation is analogous to interaction of oscillating free springs  with damped spring. The springs in all regions will eventu- 2α −2α 1 1 1 2 (Y − 1) 2 F1 − α + β − γ, − α + β ally decay to zero due to the imaginary part of QNMs in the 2 2 2  factor e−iEt for Im(E)<0. For QNMs with Im(E)>0, the 1 1 − Y + γ ; 1 − 2α; , (6.4) system will be unstable, the backreaction to the curve surface 2 2 from the fermion will be large. and   1 1 1 1 1 − Y 6.2.3 Scattering with spin flip σ ↔−σ in the inner region F + α + β − γ, + α + β + γ ; 1 + 2α; , (6.5) 2 1 2 2 2 2 2 An interesting scenario is the spin-flip scattering while main- where % taining the same energy. Instead of the reflected waves with γ = −4n(n + 2α + 2β + 1) + (2α + 2β + 1)2. (6.6) different n’s as in Sect. 6.2.2, the reflected waves in the inner region of the wormhole could get spin-flipped σ →−σ The first solution has kinks or sharp turns in the inner region with (α, β) →−(α, β) and still have the same energy and of wormhole while the second solution is smooth through- momentum. From Eq. (4.1), this corresponds to n = n,σ = out. Therefore we use only the second solution (6.5)inthe −σ . The scattering coefficients for m = 0, 1aregivenin scattering analysis. Tables 5 and 6 respectively. Note the exact values of the scat- The incoming waves are scattered by the wormhole, thus tering coefficients in m = 0 case to the values in Table 3. excite the fermionic modes in the process. The reflected out- Wave function profile of the matching is shown in Fig. 7d.  comes occurred at both Hilbert horizons where u p and um.At The m -dependence of the transmission and reflection coef- the same time, some of the waves in the inner region leaks ficients are shown in Fig. 5. 123 Eur. Phys. J. C (2020) 80 :1111 Page 9 of 13 1111

  Table 4 Scattering coefficients for the wormhole states with En = En where n =−n − 1, m = 1 nA B T R

0 − 0.955236 + 0.304246i 0.357174 + 0.524487i − 0.946488 + 0.27418i 2.178 + 4.25366i 1 − 0.517527 − 0.00380051i − 0.0447471 + 0.569521i − 0.695048 − 4.20998i 20.8443 + 62.0839i 2 − 0.436724 − 0.524891i − 0.304297 + 0.350316i 67.6094 − 33.8419i 408.353 + 1083.51i 30.124303 − 0.795473i − 0.382871 + 0.0366021i 409.066 + 930.761i 7201.99 + 18,766.6i

(a)(b)

(c) (d)

Fig. 4 The m-dependence of the transmission coefficient T and reflection coefficient R in the scattering with nonzero m scenario

Table 5 Scattering coefficients for the spin-flip σ ↔−σ scenario where m = 0 nA B T R

0 − 1.04147i 1.04147 1.70376i 0 1 − 2.04891 − 2.04891i 4.94571i 0 23.60535i − 3.60535 14.3565i 0 36.2298 6.2298i 41.6741i 0

Table 6 Scattering coefficients for σ ↔−σ scenario where m = 1 nA B T R

00.893023 + 0.432957i − 0.618941 + 0.390062i 0.421126 + 0.610085i 0.331071 + 0.331983i 10.115499 − 1.60794i 1.11401 + 0.323024i 3.55076 − 1.43318i − 0.385585 + 2.55466i 2 − 2.98693 + 0.177235i 0.0839698 − 2.34635i − 0.757591 + 7.39531i − 9.83135 + 17.1882i 30.572121 + 5.37043i − 4.47751 + 0.239986i 129.784 − 108.923i − 120.673 + 140.163i

123 1111 Page 10 of 13 Eur. Phys. J. C (2020) 80 :1111

Fig. 5 The m-dependence of the transmission coefficient T and reflection coefficient R in the scattering with spin-flip scenario

Another possibility for scattering that results in the outgo- tion (the ones with QNMs) actually diverge in the wormhole ing waves in the lower plane is to replace H (1)(kR(u)) with space if it is continued to infinity. The back reaction should H (2)(−kR(u)) in the lower plane region of Eqs. (4.2) and be large but results of Ref. [31] suggests that this divergence (4.3). All of the scattering coefficients numerically solved might actually be consistent with different wormhole geom- turn out to be the same except the change in the transmission etry. This interesting aspect will be explored in the future coefficients T →−iT for each corresponding case. The work. wave function profile in the three regions are also identical in every case. 7 Discussions and conclusions 6.3 Comments on fermionic QNMs of wormhole The scattering of fermion to the two-dimensional wormhole Table 1 shows QNMs/energies of the fermion in the magne- is calculated by considering both normal and quasinormal modes of the fermionic states in the wormhole. For incom- tized wormhole for all combinations (κ1,κ2) = (+1, +1), (−1, −1), (+1, −1), (−1, +1). We can see from the expres- ing waves with real momentum and energy, we found the sion that the imaginary and real parts of energy depend on quantum selection of the states allowed to propagate through  the wormhole, i.e., the momentum-spin relation given by n, m , M as well as the spin σ.For(κ1,κ2) = (+1, +1), Mcr Eq. (6.1). Only the states with the quantized momentum (−1, −1) regardless of the spin, E is real for n+1, n ≤ h¯ related to its angular momentum can be transmitted through and pure imaginary otherwise. Note the reality of E depends the hole via the normal modes and thus be maintained station- on the ratio of the radius of curvature in u direction, r, and ary for a long period of time. The waves will also be partially the Compton wavelength h¯ /Mc of the fermion. reflected with the same momentum and energy. The unitarity Figure 6 shows dependency of E on each parame- condition (6.2) is valid in this case. Notably, the main result, ter when the spin is chosen to σ = 1for(κ1,κ2) = Eq. (6.1), is also valid for uncharged or zero-flux fermion (+1, −1), (−1, +1). It has certain similar characteristics to scattering with a replacement m → m. the QNMs investigated in Refs. [26Ð30]. Namely for some For scattering involving QNMs of the wormhole, the pro- range of parameters, Im(E) could be very small. These long- cess will be decaying or growing in time due to the imaginary lived fermionic QNMs are quite common for both black holes part of the frequency. For QNMs with zero imaginary part, and wormholes. For our wormhole, the long-lived modes are e.g. when m = 0 (this is not exactly a normal mode since the ones close to the Re(E) axis in each plot of Fig. 6. While the momentum is not real), the energy can be real for mas- Re(E) decreases with n and increases with m&M, |Im(E)| sive fermion and sufficiently small n. Even in such cases, the increases with n&m and decreases with M. momentum will be pure imaginary resulting in spatial atten- In Ref. [31], massless and massive wormhole supported uation and enhancement of the wave functions in the upper by fermionic fields are constructed with diverging field pro- and lower plane regions. The scattering with QNMs of the file (divergent total fermion density integral). Interestingly, wormhole violates unitarity but satisfies more generic rela- the massless wormhole has no time dilation similar to the tion (5.8). Only the incoming waves with the right complex wormhole we consider in this work. However, in our case energy and momentum E = En,m , k = kn,m will undergo a the wormhole is assumed as background presumably con- resonant scattering with the hole resulting in partially trans- structed from other materials where the back reaction to the mitted and reflected waves. The physical energy of the parti- surface is neglected. Certain classes of our fermionic solu- cle, however, is given by the real part of E. Remarkably, there 123 Eur. Phys. J. C (2020) 80 :1111 Page 11 of 13 1111

Fig. 6 The QNMs E = h¯ ω of fermion on the wormhole from Table 1 in natural units for (κ1,κ2) = (+1, −1), (−1, +1)

123 1111 Page 12 of 13 Eur. Phys. J. C (2020) 80 :1111 will be only transmitted (tunneling) waves with no reflec- Acknowledgements P.B. is supported in part by the Thailand Research tion for m = 0 fermion scattering, i.e., when the magnetic Fund (TRF), Office of Higher Education Commission (OHEC) and flux is quantized in integer multiples of the magnetic flux Chulalongkorn University under Grant RSA6180002.  quantum φ = mhc/e. For general nonzero m scenario, the Data Availability Statement This manuscript has no associated data or transmission and reflection coefficients have curious peaks the data will not be deposited. [Authors’ comment: This is a theoretical at m =±6 research, therefore there is no data.]. Lastly following the argument in Ref. [18], a few com- Open Access This article is licensed under a Creative Commons Attri- ments on fermion scattering in the graphene wormhole bution 4.0 International License, which permits use, sharing, adaptation, 6 can be made. With c → vF  10 m/s and M = 0 distribution and reproduction in any medium or format, as long as you replacement, the normal-modes energy of the fermion in the give appropriate credit to the original author(s) and the source, pro- h¯ v m 0.658 nm m vide a link to the Creative Commons licence, and indicate if changes graphene wormhole is E = √F = √ eV. were made. The images or other third party material in this article a q a q are included in the article’s Creative Commons licence, unless indi- Only quasi-fermion with this quantized energy can be trans- cated otherwise in a credit line to the material. If material is not mitted through the graphene wormhole as a stationary state, included in the article’s Creative Commons licence and your intended and it will be accompanied by the reflected waves obeying use is not permitted by statutory regulation or exceeds the permit- ted use, you will need to obtain permission directly from the copy- the unitarity condition. The effects of stitching the graphene right holder. To view a copy of this licence, visit http://creativecomm wormhole to the plane include the exchange of inequivalent ons.org/licenses/by/4.0/. Dirac points that can be taken into account by the effective Funded by SCOAP3. flux of gauge field [20,32]. This effective flux can simply be added to the magnetic flux in our work resulting in the change  φ of the effective angular momentum number m = m − total φ0 and otherwise the same results. Appendix A: Wave function profiles

See Fig. 7.

(a) (b)

(c) (d)

Fig. 7 Wave function profiles of scattering fermion for a real momentum, b m = 0, c general m,andd spin-flip scenario. The real and imaginary parts of the wave functions in the wormhole region um < u < u p are depicted in red

123 Eur. Phys. J. C (2020) 80 :1111 Page 13 of 13 1111

References 17. T. Rojjanason, J. Phys. Conf. Ser. 1380(1), 012165 (2019). https:// doi.org/10.1088/1742-6596/1380/1/012165 1. K.S. Novoselov, A.K. Geim, S.V.Morozov, D. Jiang, M.I. Katsnel- 18. T. Rojjanason, P. Burikham, K. Pimsamarn, Eur. Phys. J. C 79(8), son, I.V. Grigorieva, S.V. Dubonos, A.A. Firsov, Nature 438, 197 660 (2019). https://doi.org/10.1140/epjc/s10052-019-7156-y. (2005). https://doi.org/10.1038/nature04233 arXiv:1810.12668 [hep-th] 2. E. Braun et al., Generating carbon schwarzites via zeolite- 19. F. Wilczek, Magnetic flux, angular momentum, and statistics. Phys. templating. Proc. Natl. Acad. Sci. USA 115, E8116ÐE8124 (2018) Rev. Lett. 48, 1144 (1982) 3. A. Lherbier, H. Terrones, J.-C. Charlier, Three-dimensional mass- 20. J. Gonzalez, J. Herrero, Graphene wormholes: a condensed matter less Dirac fermions in carbon schwarzites. Phys. Rev. B 90, 125434 illustration of Dirac fermions in curved space. Nucl. Phys. B 825, (2014) 426Ð443 (2010) 4. K.S. Novoselov et al., Electric field effect in atomically thin carbon 21. A. Iorio, G. Lambiase, Quantum field theory in curved graphene films. Science 306, 666Ð669 (2004) spacetimes, Lobachevsky geometry, Weyl symmetry, Hawking 5. L.J. Yin, K.K. Bai, W.X. Wang, S.Y. Li, Y. Zhang, L. He, Lan- effect, and all that. Phys. Rev. D 90, 025006 (2014) dau quantization of Dirac fermions in graphene and its multilay- 22. R.A. Konoplya, C. Molina, Ringing wormholes. Phys. Rev. D 71, ers. Front. Phys. (Beijing) 12(4), 127208 (2017). https://doi.org/ 124009 (2005) 10.1007/s11467-017-0655-0 23. J. Yepez, Einstein’s vierbein field theory of curved space. 6. F. Guinea, M.I. Katsnelson, A.K. Geim, Energy gaps, topological arXiv:1106.2037 [gr-qc] (2011) insulator state and zero-field quantum Hall effect in graphene by 24. A. Arai, Exactly solvable supersymmetric quantum mechanics. J. strain engineering. Nat. Phys. 6, 30Ð33 (2010) Math. Anal. Appl. 158, 63 (1991) 7. F. De Juan, J.L. Manes, M.A.H. Vozmediano, Gauge fields from 25. H. Egrifes, D. Demirhan, F. Buyukkilic, Exact solutions of the strain in graphene. Phys. Rev. B 87, 165131 (2013) Schrodinger equation for the deformed hyperbolic potential well 8. Y.-L. Wang, L. Du, C.-T. Xu, X.-J. Liu, H.-S. Zong, Pauli equation and the deformed four-parameter exponential type potential. Phys. for a charged spin particle on a curved surface in an electric and A 275, 229Ð237 (2000) magnetic field. Phys. Rev. A 90, 042117 (2014) 26. A. Lasenby, C. Doran, J. Pritchard, A. Caceres, S. Dolan, Phys. 9. Y.-L. Wang, H. Jiang, H.-S. Zong, Geometric influences of a par- Rev. D 72, 105014 (2005). https://doi.org/10.1103/PhysRevD.72. ticle confined to a curved surface embedded in three-dimensional 105014. arXiv:gr-qc/0209090 Euclidean space. Phys. Rev. A 96, 022116 (2017) 27. H.T. Cho, Phys. Rev. D 68, 024003 (2003). https://doi.org/10.1103/ 10. R.C.T. da Costa, Quantum mechanics of a constrained particle. PhysRevD.68.024003. arXiv:gr-qc/0303078 Phys. Rev. A 23, 1892 (1981) 28. J.L. Jing, Phys. Rev. D 71, 124006 (2005). https://doi.org/10.1103/ 11. R.C.T. da Costa, Constraints in quantum mechanics. Phys. Rev. A PhysRevD.71.124006. arXiv:gr-qc/0502023 25, 2893 (1982) 29. J.L. Blázquez-Salcedo, C. Knoll, Phys. Rev. D 97(4), 12. M. Burgess, B. Jensen, Fermions near two-dimensional surfaces. 044020 (2018). https://doi.org/10.1103/PhysRevD.97.044020. Phys. Rev. A 48, 1861 (1993) arXiv:1709.07864 [gr-qc] 13. P. Castro-Villarreal, R. Ruiz-Sánchez, Pseudo-magnetic field in 30. J.L. Blázquez-Salcedo, C. Knoll, Class. Quantum Grav- curved graphene. Phys. Rev. B 95, 125432 (2017) ity 36(10), 105012 (2019). https://doi.org/10.1088/1361-6382/ 14. G.H. Liang, Y.L. Wang, M.Y. Lai, H. Liu, H.S. Zong, S.N. ab1882. arXiv:1811.02014 [gr-qc] Zhu, Pseudo-magnetic field and effective spin-orbit interaction 31. J.L. Blázquez-Salcedo, C. Knoll, Eur. Phys. J. C 80(2), 174 for a spin-1/2 particle confined to a curved surface. Phys. Rev. (2020). https://doi.org/10.1140/epjc/s10052-020-7706-3. A 98(6), 062112 (2018). https://doi.org/10.1103/PhysRevA.98. arXiv:1910.03565 [gr-qc] 062112. arXiv:1808.09959 [quant-ph] 32. G. Garcia, P. Porfírio, D. Moreira, C. Furtado, Nucl. Phys. B 950, 15. M.V. Entin, L.I. Magarill, Spin-orbit interaction of electrons on a 114853 (2020). https://doi.org/10.1016/j.nuclphysb.2019.114853. curved surface. Phys. Rev. B 64, 085330 (2001) [arXiv:1906.09195 [hep-th] 16. T. Rojjanason, S. Boonchui, The geometrical parameters condi- tion for bound and unbound state in graphene wormhole, in Siam Physics Congress, Rayong, Thailand, 24–26 May 2017 (2017)

123