Charged fermion in two-dimensional wormhole with axial magnetic field

We investigate the effects of magnetic field on a charged fermion in the two-dimensional wormhole. Applying external magnetic field along the axis direction of the wormhole, the Dirac equation is set up and analytically solved in two scenarios, constant magnetic flux and constant magnetic field through the throat of the wormhole. For the constant magnetic flux scenario, the system can be solved analytically and exact solutions are found. For the constant magnetic field scenario, with the short wormhole approximation, the quantized energies and eigenstates are obtained. The system exhibits both the spin-orbit coupling and the Landau quantization for the stationary states in both scenarios. The intrinsic curvature of the surface induces the spin-orbit and spin-magnetic Landau couplings that generate imaginary energy. Imaginary energy can be interpreted as the energy dissipation and instability of the states. Generically, the states of charged fermion in wormhole are quasinormal modes~(QNMs) that could be unstable for positive imaginary frequencies and decaying for negative imaginary ones. For the constant flux scenario, the fermions in the wormhole can behave like bosons and have arbitrary statistics depending on the flux.

to carbon nanotubes properties [11,22], and applications in curved graphene [23][24][25][26][27][28]. Graphene is an ideal place to study behaviour of confined charged fermions such as electrons in a two-dimensional surface since its thickness is only roughly one-carbon-atom diameter. A sheet of graphene can be curved, rolled, stretched, twisted and deformed or even punctured holes into. The holes can be connected to a nanotube and become a wormhole bridging two graphene sheets. Multiple graphene sheets can be connected with one another by multiple wormholes forming a network of entangled electronic structure. Wormholes can even be built into a cage structure of schwarzite with many promising properties [29].
There have been many studies concerning the behavior of electron on curved graphene surfaces. Gonzalez et al. [30] consider a wormhole attached to two graphene sheets via 12 heptagonal defects, the defects act like effective non-Abelian gauge flux that swaps two Dirac points on the graphene lattice. Garcia et al. [31] investigate the charged fermion in two-dimensional spherical space in a rotating frame, study the change in the spectrum of the C 60 molecule when it is crossed by a magnetic flux tube in the z-direction, and the appearance of an analogue of the Aharonov-Carmi phase in the system [32]. Cariglia et al. [24] consider Dirac fermions on an essentially smooth simplified spacetime, namely a Bronnikov-Ellis wormhole. In Ref. [33], the surface of the graphene wormhole is realized by a two-dimensional axially-symmetric curved space of a constant Gaussian curvature. The graphene wormhole dynamics of a Dirac fermion is then discussed in the (1+2)-dimensional spacetime. In Ref. [34], a charged fermion in curved surface subject to external electric field is analyzed in the stationary optical metric conformal to the BTZ black hole. Rojjanason and Boonchui [35] showed that the exact solution of the Dirac fermion on the graphene wormhole can be expressed in terms of Jacobi polynomials and the spin-orbit coupling is generated by the curvature of the wormhole.
In this work, we study physical properties of a charged fermion confined on the surface of wormhole in the presence of the external magnetic field along the axis direction of the wormhole. In Section II, basic geometric and gauge setup are established. In Section III, the Dirac equation in curved spacetime is used to analyze the (1+2)-dimensional stationary state of the charged fermion in the wormhole. To solve for the energy and wave function, two scenarios of constant flux and constant field are considered. Analysis in special cylindrical wormhole case is compared to the general case to identify the crucial role of surface curvature. A simple interpretation of the results is given in terms of the angle between the spin and orbital angular momentum of the surface-confined fermion. The special cases of Beltrami and elliptic wormhole are considered in Section IV. We summarize and discuss our results in Section V.

II. GEOMETRIC AND GAUGE SETUP OF THE WORMHOLE
The wormhole is defined geometrically as in Figure 1 with points on the surface parameterized by The shape of the wormhole is generically described by the radius function R(u). The constraint on z follows from the relation ds 2 = dx 2 + dy 2 + dz 2 = du 2 + R(u) 2 dv 2 . It gives the Hilbert horizon at 1 = R (u). Embedding (1+2)-D wormhole into higher dimensional (1+3)-D space generates effective gravity or effective curvature to the reduced "spacetime". Any particle or quasiparticle living on the reduced spacetime will experience the spacetime curvature. We define dx µ = {cdt, dx, dy, dz} as the (1+3)-D Minkowski spacetime coordinates, and dx µ = {cdt, du, dv} as the (1+2)-D wormhole coordinates. The transformation matrix FIG. 1: Geometric structure of a wormhole surface where a is a radius of the wormhole at the mid-point, u = 0, between the two ends. And r is the radius of curvature of the wormhole surface along u direction.
between the two coordinates is then Since the line element of the wormhole space is in the following form where the (1 + 2)-D metric is g µυ . The dreibein e A µ is then defined as where g µν ≡ e A µ e B ν η AB , and η AB = diag(−1, 1, 1) in (1+2)-dimensions and A, B ∈ {0, 1, 2}. We consider an electron (or charged fermion) in Minkowski space subject to the wormhole embedding constraints. The fermion will experience the effective curvature that can be addressed by considering the Dirac equation in the locally flat (1 + 2)-D spacetime The canonical momentum of fermion in the presence of an electromagnetic potential is therefore the coordinate-space Dirac equation can be written as Ψ = Ψ(u, v, t) represents the Dirac spinor field and M represents the rest mass of the particle, c is the speed of light in the curved spacetimes, e is electric charge, and A µ is the electromagnetic four-potential. The γ A are the Dirac matrices given by where σ a are the Pauli matrices defined by They obey the Clifford algebra The Pauli matrices has a useful identity that we will use later where abc is Levi-Civita symbol. The covariant derivative of the spinor interaction with gauge field in the curved space is given by where the spin connection Γ µ [36] is where β, µ, ν ∈ {t, u, v} and the Christoffel symbols Γ kji are defined by Then and zero otherwise. The spin connections are then In this work, we will apply an external magnetic field such that the z-component B z = B(z) is uniform with respect to the plane (x, y) in two different ways: a.) the magnetic flux through the circular area enclosed by the wormhole at a fixed z is constant, namely B z ∼ 1/R 2 and b.) the magnetic field is uniform and constant. Due to the axial symmetry, the electromagnetic four-potential can be expressed in the axial gauge as and in the wormhole coordinates as x, y, z).
Specifically by components, they are The magnetic field is then given by having all x, y, z components for the constant-magnetic-flux case and having only z component for the constant-magnetic-field case.

III. THE DIRAC EQUATION IN MAGNETIZED WORMHOLE
Utilizing the results from above equations, the Dirac equation Eq.(4) can be written in the form By using relationships from Eq.(5) then where D is a differential operator We can define the pseudo vector potential as The effective gauge potential in the u direction, Aũ, is generated by the curvature along the v direction, Γ v . In this sense, wormhole gravity connection manifests itself in the form of (imaginary) gauge connection in the perpendicular direction on the surface. The second term in Eq.(16) is similar to a spin-orbit-curvature coupling potential. A similar setup has been used to study nanotubes under a sinusoidal potential [30]. Here we consider the dispersion relation for the two-dimensional fermions described by the Dirac equation in the presence of the effective potential arising from the wormhole geometrical structure and the gauge field. Consider a stationary state of the Dirac spinor Ψ(t, u, v) in the form where ψ I(II) (u, v) are two-component spinors. Eq.(15) can be rewritten in the form of coupled equations for the 2-spinors In the presence of external magnetic field B = × A along the z direction, the charged fermion moving in v direction is expected to form a stationary state with quantized angular momentum and energy, i.e. the Landau levels in the curved space with hole. To show this, we need to solve for the stationary states of the system.
Now we will solve the equation of motion in two cases.

A. Constant magnetic flux solution
In type II superconductors, the magnetic flux can be trapped in a magnetic vortex (i.e. Abrikosov vortex) in quantized units of the magnetic flux quantum φ 0 ≡ hc/e. The flux will be kept constant along the vortex which can be thought of as a wormhole if the charged carriers are confined to the boundary of the vortex by other constraint. For example, the confining surface can be created as the wormhole connecting two graphene sheets. In such situation, our analysis in this section would be applicable. The generic results demonstrate that the statistics of charged fermion in the wormhole is determined solely by the magnetic flux and not the shape or curvature of the wormhole via the effective orbital angular momentum. The fermion can behave like boson when the flux is half-integer and have arbitrary statistics for arbitrary fluxes.
The presence of axial magnetic field along the wormhole gives rise to a number of interesting effects notably the Landau quantization of energy due to the interaction between charge of the fermion and the external magnetic field. On the other hand, the wormhole intrinsic curvature ("gravity") is responsible for the spinorbit coupling of the fermion as we will see subsequently in this Section and later on in Section III C (where the spin-orbit disappears with the intrinsic curvature). It costs energy to tilt the spin along the wormhole space even in the absence of the magnetic field.
We will start with the situation where the magnetic flux is constant along the throat of the wormhole.
The operator D 2 in the equation of motion now takes the form where φ = B · d a = πR 2 B and φ 0 the magnetic flux quantum. We have used the identity (8), For zero magnetic field, the operator D 2 satisfies the Lichnerowicz-Weitzenböck formula where D µ = ∂ µ − Γ µ is the covariant derivative including the spin connection and R is the Ricci scalar. In our case, R = 2R /R. For stationary states, the wave function needs to be single-valued at every point in spacetime, ψ II must be a periodic function in v, where the orbital angular momentum quantum number m = 0, ±1, ±2, ..
where m = m − φ φ 0 , the new orbital angular momentum in the presence of magnetic flux [37]. Notably by adjusting the flux φ/φ 0 to be half an odd integer, the charged fermions such as electrons in the wormhole can behave like bosons, e.g. they can condensate and flow like superfluid along the wormhole. We have used the momentum parameter k 2 ≡ (E 2 − M 2 c 4 )/ 2 c 2 and σ is a spin-state index corresponding to spin up (σ=+1) or down (σ = −1) of the fermion for each eigenvalue of σ 3 . The σ 3 -spin component is pointing along the direction of the normal vector of the wormhole surface since the dreibein e A µ is defined on the tangent space of the wormhole. Also because R (u) = cos θ where θ is the angle between the σ 3 -spin component and the z axis, the spin-orbit coupling term can thus be rewritten as ∼ σmR = σm cos θ = − → σ 3 ·ẑm. The spin-orbit coupling vanishes when R = 0 = cos θ or θ = π/2, i.e. when the normal vector of the surface is perpendicular toẑ.
To be specific, we will choose a deformed hyperbolic wormhole described by R(u) = a cosh q (u/r) where a is the radius of wormhole at the mid-point between the two sheets and r is the radius of curvature of the wormhole connecting the sheets [33]. They are based on a q-deformation of the usual hyperbolic functions which are defined by [38,39] Definitions same to else hyperbolic functions but note that almost all relations known from the usual hyperbolic functions have been modified, for example They reduce to hyperbolic functions when q = 1. With this choice of R(u), we perform the transformation X = sinh q (u/r) to obtain Define weighting function solution ϕ(X) = ( , the equation of motion can be rewritten as where we assume leading to The coefficients are defined by Depending on the sign choices of α, β, the resulting equation of motion and the corresponding energy levels will be dependent or independent of the spin-orbit coupling term ∼ σmr/a √ q.
Define X = −i √ qY , the equation then takes the form Eq.(33) is the Jacobi Differential Equation for integer n and For the choice the energy levels become independent of the magnetic flux and spin-orbit term. On the other hand, for another sign choice the energy levels depend on the spin-orbit term a complex quantity which can be interpreted as the quasi-normal modes (QNMs). For other combinations with (α, β) ↔ −(α, β), the energy becomes and respectively. All of these solutions are part of the Hilbert space of the system with their own corresponding wave functions. The energy from Eq. (39) and (41) contains the interaction between the spin-orbit coupling ∼ σm (independent of the magnetic field), and the Landau coupling between the magnetic field and the spin (orbital) angular momentum ∼ σB (mB). Notably, it also contains the term proportional to σmφ, m 2 , φ 2 which could be interpreted as the spin-orbit-flux, orbit-orbit and the flux-flux couplings respectively. The spinorbit and m 2 terms are purely gravitational and kinematical since they are independent of the magnetic field. The orbit-orbit m 2 term is actually the kinetic energy from the angular momentum as we will see in Section III C where the curvature vanishes. Another interesting aspect of the energies given by Eq. (39) and (41) is the complexity, i.e., they are quasinormal modes (QNMs). Regardless of the magnetic field, the imaginary parts in the energy expression have the gravitational origin. They are originated from the curvature of the wormhole and they will vanish when the curvature is zero as we can see again in Section III C. For the QNMs with negative imaginary parts, the curvature effects leak the energy of the fermion away from the wormhole as long as the angle θ = arccos R between the σ 3 -spin component and orbital angular momentum is not π/2. For the choice (39) (and (41)), the positive-energy solution is unstable with positive imaginary part for σm < 0 (and σm > 0) respectively. For these states, the fermion will either slowly decay away or be spun off the wormhole due to the curvature effect. A special case occurs when m = φ/φ 0 where the imaginary spin-orbit coupling term vanishes.
It is challenging to give physical interpretation to the states from the choice in (37) and (40). They have negative momentum square p 2 along u direction, they do not feel the magnetic field and do not have the angular momenta. It is most natural to identify them with diffusive modes (due to imaginary momentum along the wormhole direction u) with m = 0. However, the energy of these modes can be either real or purely imaginary depending on the quantum number n.
Finally, the solutions of Eq. (25) is where X(u) = sinh q (u/r). Note that the solutions have the following properties, For α = β * , the Jacobi polynomial is real (imaginary) for even (odd) n. Notably for this case, the spatial for even n is also real and the energy given by (37) depends only on the quantum number n. This energy is independent of the spin-orbit and magnetic field.

B. Constant magnetic field solution
Another physical situation that can give insight to the role of magnetic field on the fermion is the uniform magnetic field environment. We will show that again the spin-orbit coupling is induced by the curvature of space or "gravity" of the wormhole. Unstable modes and QNMs will be generated again from such interaction. However, there is additional momentum-diffusive modes (negative p 2 u ) that also have dependency on the coupling between orbital angular momentum and the magnetic field. This leads to a new p u -diffusive modes that depend on the spin of the fermion in the wormhole. The p u -diffusive modes can have either real or imaginary energy depending on the quantum number n and the magnetic field in comparison to the rest-mass energy M c 2 .
For constant uniform magnetic field, the equation of motion is modified since now the operator D 2 is given by With the same separation of variables (25), we obtain Again, a shape of deformed hyperbolic wormhole R(u) = a cosh q (u/r) is chosen for further calculation. After performing the similar transformations using X(u) = sinh q (u/r) and ϕ(X) = ( the equation of motion becomes where we again assume leading to The coefficient parameters are defined as the following In order to find the solution to Eq. (47), we first obtain the asymptotic solution for large X, Eq. (47) now becomes where the parameters are defined as and for α = −β * = 1 4 respectively.

Small X approximation
The wave function can be solved exactly for small X in terms of the Jacobi polynomials as we will show in the following. For small X, Eq. (52) takes the form 0 ≈ (q + X 2 )Φ (X) + F + GX Φ (X) + I + k 2 r 2 Φ (X).
And the solution isΦ The quantization of energy for α = β * case is For the other possibility α = −β * , the energy is The energy given by Eq. (63) has an energy splitting between the spin up (σ = 1) and down (σ = −1) proportional to 2mr 2 eB/ c. This is the spin-orbit-magnetic coupling. Notably, the spin-down state does not feel the magnetic field. For sufficiently large n, m, B, the energy becomes purely imaginary since the negative interaction energy is larger than the rest-mass energy M c 2 . On the other hand, the energy given by Eq. (64) is complex with the imaginary part depending on both the spin-orbit and the external magnetic field. QNMs always exist for nonzero m and magnetic field in this case. For (α , β ) → (−α , −β ) cases, the energies become

C. Cylindrical Wormhole
To understand essential physics of the magnetized charged fermion in the wormhole, consider a simple case when R(u) is constant, i.e. a wormhole tube. In this case, the intrinsic (or Gaussian) curvature of the wormhole is zero so we can identify which effects are induced by the wormhole "gravity". The two separate magnetic scenarios of constant flux and field reduce to the same physical system as both (26) and (47) become (suppressing subscript and superscript) Assuming the solution in the form ϕ m (u) = ϕ m (0) exp(ik u u) to obtain Setting 2r as the length of the cylinder with the boundary conditions ϕ(u = 0) = ϕ(u = 2r) = 0, we get k u = nπ/2r where n = 0, 1, 2, ... The quantization of energy via Eq. (68) is then The energy is purely real and only normal modes exist. The spin-orbit coupling disappears together with the Landau coupling between spin and the magnetic field. The only remaining interaction is the orbital-magnetic Landau coupling.
As discussed in Ref. [37], the fermions can have arbitrary orbital angular momentum which results in various possible statistics. For φ/φ 0 = integer + 1/2, the fermions become bosons in the magnetized wormhole. They can condensate and flow like superfluid along the hole.

IV. BELTRAMI AND ELLIPTIC PSEUDOSPHERE WORMHOLE
There are special cases when the deformation parameter q = 0, −1, i.e. the Beltrami and elliptic pseudosphere wormhole that are not captured in the general analysis, we address them in this section.
The general solution can be expressed in the form where Z = 4m re −u/r a that takes the value Z = 4m r/a, 0 for u = 0, ∞ respectively. U (a, b, Z) is the confluent hypergeometric function of the second kind.
Stationary-states solutions require finite wave functions at Z = 4m r/a, 0 even though the wormhole actually ends at the Hilbert horizon u H = r log(2r/a), R H = r, Z H = 2m (where R = 1). Regularity at Z = 4m r/a > 1 demands that the series of the hypergeometric function truncates at finite power of Z giving for non-negative integer n. This leads to the energy quantization Remarkably, the energies do not depend on B and m at all, only the wave functions have m dependence. All m states degenerate in each energy level E n .
A special solution for m = 0 where the magnetic flux is quantized to integer values φ/φ 0 = m = 0, 1, 2, ... can be obtained from (26), giving The solutions are decaying plane wave travelling in the u direction, in and out of the wormhole. The wave has zero effective angular momentum. For the constant field scenario, the equation of motion (46) takes the form where L ≡ c/eB is the magnetic length. To solve (75), we approximate by considering the situation when the terms containing a/L 2 is negligible. The resulting equation of motion takes the form which is exactly the same as (46) with replacement m → m, k 2 → k 2 + m/L 2 . The solutions are thus the same with the replacement above. Note that the energy formulae becomes for non-negative integer n, leading to the energy quantization There is an orbit-magnetic coupling contributing to the energy. The modes are normal for small n and coupling but become QNMs with purely imaginary values for large coupling and/or highly excited states.

B. Ellliptic wormhole
For elliptic wormhole with q = −1, all formulae of the hyperbolic cases can be used. Notably since √ q = i, all of the parameters α ( , ) , β ( , ) become real and we can simply make replacement √ q → i in the results of the hyperbolic cases, i.e. (39), (41) and (64), (66). The spin-orbit and all magnetic induced coupling terms become real. The QNMs only occur for highly excited states where the coupling terms are larger than the rest energy term. When QNMs emerge, they are purely imaginary (or diffusive) with zero energy and contribute only in the form of dissipation. Modes given by (37), (40) and (63), (65) are not affected by the wormhole geometry, they are leaking solutions in the u direction for highly excited states. Topologically, the elliptic wormhole is distinctively different from the hyperbolic and Beltrami ones. The space starts at u = 0 with R = 0 so the modes cannot leak out through the hole, resulting in the absence of QNMs for low n states in contrast to the hyperbolic and Beltrami cases.

V. CONCLUSIONS AND DISCUSSIONS
We consider charged fermion in a two-dimensional wormhole in the presence of the external magnetic field with axial symmetry. Assuming uniform field in the plane perpendicular to the direction of the field, we consider energy levels of fermion in two scenarios, constant flux through the wormhole throat and constant field. The curvature connection of wormhole generates effective gauge connection resulting in the induced spin-orbit coupling of the fermion on the wormhole. The coupling is genuinely "gravitational" since it exists even in the absence of the magnetic field and it is vanishing when the wormhole is flat, e.g. cylindrical wormhole. When the magnetic field is turned on along the wormhole axis, the spin-orbit-magnetic coupling is also generated in addition to the conventional Landau coupling between the angular momentum of the fermion and the magnetic field. This new interaction is the combined effect of gravity and gauge field on the charged fermion.
A simple picture to help understanding these results is the following. When a fermion is confined to the curved space like a two-dimensional wormhole considered here, its σ 3 -spin component is perpendicular to the surface (since the dreibein is locally defined in the tangent space of wormhole) while the orbital angular momentum is pointing along the z-direction. The spin-orbit coupling ∼ − → σ 3 · mẑ is thus generated for generic wormhole with curvature. For cylindrical wormhole tube, the surface is always parallel toẑ and the σ 3 -spin component is always perpendicular to the surface so the spin-orbit coupling naturally vanishes.
For both constant flux and constant field scenarios in every choice of solution parameters, sufficiently highly excited states with large n will always give QNMs. The energy naturally leaks out of the wormhole when the fermion is sufficiently excited. This is consistent with the existence of Hilbert horizons [33] at finite u H ≡ r log r a ± r a 1 + qa 2 r 2 where the wormhole geometry ends. Highly excited fermion lives at larger u and it will leak out of the wormhole through the Hilbert horizons.
On the other hand, the spin-orbit coupling always generate QNMs since the coupling (on the wormhole) itself is imaginary ∼ iσm. The origin of this term can be traced back to the pseudo gauge connection Aũ from Eq. (17) which is purely imaginary. Remarkably, the curvature connection Γ v (i.e. "gravity") generates an effective (pseudo) gauge connection that is purely imaginary resulting in the complexity of the energy and the existence of the QNMs. Physically, imaginary energy should be interpreted as the energy dissipation and instability. Energy dissipation corresponds to the case with Im(E) < 0. Instability stems from the enhancement in time of the wave function when Im(E) > 0. A state with high orbital angular momentum m tends to leak energy faster due to larger imaginary part of the QNMs. Note that the spin-orbit coupling term in the equation of motion is zero when R = 0 (at midpoint of the wormhole throat or in the case of cylindrical wormhole) and maximum when R = cos θ = 1 at the Hilbert horizon. At Hilbert horizon, the surface is merging to the plane and perpendicular toẑ.
The interplay between the curvature connection of the wormhole and the induced (pseudo) gauge connection demonstrates an interesting kind of gauge-gravity duality. The real gravity connection can be interpreted as the imaginary (effective) gauge connection (in the locally perpendicular direction on the surface) that leads to the complexity of the energy and the emergence of the QNMs and unstable modes. Adding external magnetic field induces a new imaginary coupling term proportional to the field that only exists when there is curvature, i.e. last term in (22) and (45). The new curvature-spin-magnetic field coupling similarly leads to the emergence of QNMs and unstable modes.
The gauge field in the wormhole can change the total angular momentum of the charged fermions, altering their statistics accordingly. It is possible to store condensated boson-like (when the flux is half-integer) fermions in the wormhole connecting e.g. two graphene sheets and control their behaviour by changing either the magnetic field or the shape of the wormhole. This could potentially lead to a number of profound electronic properties and future applications.