Fermionic currents in topologically nontrivial braneworlds
We investigate the influence of a brane on the vacuum expectation value (VEV) of the current density for a charged fermionic field in background of locally AdS spacetime with an arbitrary number of toroidally compact dimensions and in the presence of a constant gauge field. Along compact dimensions the field operator obeys quasiperiodicity conditions with arbitrary phases and on the brane it is constrained by the bag boundary condition. The brane is parallel to the AdS boundary and it divides the space into two regions with different properties for the fermionic vacuum. In both these regions, the VEVs for the charge density and the components of the current density along uncompact dimensions vanish. The components along compact dimensions are decomposed into the brane-free and brane-induced contributions. The behavior of the latter in various asymptotic regions of the parameters is investigated. It particular, it is shown that the brane-induced contribution is mainly located near the brane and vanishes on the AdS boundary and on the horizon. An important feature is the finiteness of the current density on the brane. Applications are given to -symmetric braneworlds of the Randall-Sundrum type with compact dimensions for two classes of boundary conditions on the fermionic field. For the second one we show that the contribution of the brane to the current does not vanish when the location of the brane tends to the AdS boundary. In odd spacetime dimensions, the fermionic fields realizing two inequivalent irreducible representations of the Clifford algebra and having equal phases in the periodicity conditions give the same contribution to the vacuum current density. Combining the contributions from these fields, the current density in odd-dimensional -,- and -symmetric models is obtained. In the special case of three-dimensional spacetime, the corresponding results are applied for the investigation of the edge effects on the ground state current density induced in curved graphene tubes by an enclosed magnetic flux.
PACS numbers: 04.62.+v, 03.70.+k, 98.80.-k, 61.46.Fg
In a variety of quantum field-theoretical problems the fields are defined on a manifold with a boundary and one must take care imposing suitable boundary conditions on the corresponding hypersurfaces. The boundaries may have different physical origins. Examples are interfaces between two media with different electromagnetic properties in condensed matter physics (e.g., media with different dielectric permittivities), various sorts of horizons in gravitational physics and in non-inertial reference frames, boundaries separating the spatial regions with different gravitational backgrounds (for example, de Sitter bubbles in Minkowski spacetime), domain walls in the theory of phase transitions, and branes in higher-dimensional cosmologies and in string theories. In a number of physical problems the model is formulated in non-globally hyperbolic manifolds possessing a timelike boundary at spatial infinity. In order to preserve the information to be lost to, or gained from, spatial infinity, appropriate boundary conditions should be imposed. A well known example of this kind is anti-de Sitter (AdS) spacetime . Another class of conditions imposed on fields appear in models with compact spatial dimensions. The latter are an inherent feature of high-energy theories unifying physical interactions, like Kaluza-Klein and string theories. Depending on the periodicity conditions along compact dimensions different topologically inequivalent field configurations may arise . The quantum effects arising from the nontrivial topological structure of the background spacetime include symmetry breaking, topological quantum phase transitions, instabilities in interacting field theories, and topological mass generation. The topological issues also play an important role in effective theories describing a number of condensed matter systems .
In the present paper we consider the combined effects of background gravitational field and of two sorts of boundary conditions on the local properties of the vacuum state for a charged fermionic field. As the bulk geometry we take a locally AdS spacetime with an arbitrary number of toroidally compactified spatial dimensions (in Poincaré coordinates). The first kind of boundary condition is related to the presence of a brane parallel to the AdS boundary and the second one is related to the compactification of a part of spatial dimensions to a torus. We impose bag boundary condition on the brane and quasiperiodicity conditions with general phases along compact dimensions. The results are generalized for a boundary condition arising in -symmetric braneworld models of the Randall-Sundrum type with extra compact dimensions.
Our choice of AdS spacetime as a local bulk geometry has several motivations. First of all, AdS spacetime is maximally symmetric and a large number of problems in quantum field theory on curved backgrounds is exactly solvable. That is the case in the problem at hand. The corresponding investigations may help developing the research tools and insights to deal with less symmetric geometries. The AdS spacetime naturally appears as a ground state in extended supergravity and Kaluza-Klein theories and also as the near horizon geometry of the extremal black holes and domain walls. Moreover, the AdS spacetime has a constant negative curvature and the related length scale can serve as a regularization parameter for infrared divergences in interacting quantum field theories without reducing the number of symmetries . The AdS geometry plays a crucial role in two exciting developments of the last two decades: the gauge/gravity duality and the braneworld scenario with large extra dimensions (for reviews see [5, 6]). Braneworlds naturally appear in the string/M-theory context and provide an interesting alternative to address various problems in cosmological and particle physics. A number of particularly important implications of AdS geometry recently appeared in condensed matter physics (see, e.g., ).
Both types of constraints, induced by the presence of boundaries and by the compactification of spatial dimensions, give rise to the modification of the spectrum for vacuum fluctuations of quantum fields. As a result, the vacuum expectation values (VEVs) of physical quantities are shifted by an amount depending on the bulk and boundary geometries, and also on the boundary conditions imposed. This is the familiar Casimir effect (for reviews see ). The vacuum energy and the forces acting on the boundaries were among the main physical quantities of interest in the studies of this effect. In particular, motivated by the radion stabilization in braneworld models of the Randall-Sundrum type, the investigations of these quantities in the geometry of two parallel branes in AdS spacetime have attracted a great deal of attention (see, for instance, the references in [9, 10]). In particular, the fermionic Casimir effect has been considered in [10, 11, 12, 13] (for a recent discussion of the renormalised fermion expectation values on AdS spacetime in the absence of branes see, for example, ). The vacuum energy, the Casimir forces and the VEV of the energy-momentum tensor in higher-dimensional generalizations of the AdS spacetime with compact internal spaces have been investigated in .
As another important local characteristic of the vacuum state for charged fields, bilinear in the field operator, here we consider the VEV of the current density for a fermionic field in background of locally AdS spacetime with compact dimensions in the presence of a brane. For a flat background geometry with an arbitrary number of toroidally compact dimensions, the zero and finite temperature expectation values of the charge and current densities for scalar and fermionic fields were investigated in Refs. [16, 17, 18]. The corresponding results for a special case of a three-dimensional spacetime with one and two compact spatial dimensions have been applied to the electronic subsystem of cylindrical and toroidal carbon nanotubes described within the framework of the effective Dirac model. The influence of additional boundaries on the vacuum charges and currents with applications to finite length carbon nanotubes is studied in [19, 20]. This is the analog of the Casimir effect for the charge and current densities. The VEVs of the current densities for scalar and Dirac spinor fields in de Sitter and AdS spacetimes with toroidally compact subspace have been discussed in  and [22, 23], respectively, for scalar and fermionic fields. The effects of the branes in background of locally AdS bulk on the VEV of the current density for a scalar field with general curvature coupling parameter are investigated in [24, 25]. The general case of the Robin boundary conditions on the branes was discussed and applications were given to Randall-Sundrum type braneworlds.
The organization of the present paper is as follows. In the next section we describe the bulk geometry and the fields under consideration. The boundary and periodicity conditions are specified for a fermionic field. A complete set of positive and negative energy fermionic modes are described in Section 3. By using the corresponding mode functions, in Section 4 we investigate the brane-induced effects on the current density along compact dimensions in the region between the brane and AdS boundary (L-region). The behavior of the current density in different asymptotic regions of the parameters is discussed in detail. Similar investigations for the region between the brane and AdS horizon (R-region) are presented in Section 5. In Section 6, the vacuum currents are considered in -symmetric braneworlds with a single brane and with an arbitrary number of toroidally compact spatial dimensions for two types of boundary conditions on the brane. The numerical results are given for the simplest generalization of the Randall-Sundrum model with a single extra compact dimension. The fermionic current density in parity and time-reversal symmetric models in odd-dimensional spacetimes is considered in Section 7. Applications are given to deformed carbon nanotubes described within the framework of the effective Dirac model in three-dimensional spacetime. The main results of the paper are summarized in Section 8.
2 Background geometry and the fields
The background geometry we consider is described by the -dimensional line element
where and is the Minkowskian metric tensor in -dimensional subspace with the coordinates . The local geometrical characteristics corresponding to (2.1) coincide with those for AdS spacetime with the curvature radius . In particular, for the curvature scalar and the Ricci tensor one has and . However, the global geometry we shall be concerned about is different. Namely, it will be assumed that the spatial dimension , , is compactified to a circle with the length , . For the remaining coordinates one has , , and . Hence, in the problem at hand the subspace has the topology , , where stands for a -dimensional torus (for a discussion of physical effects in models with toroidal dimensions, see ). For the further consideration, it is convenient, in addition to the coordinate , to use the conformal coordinate , defined as with the range . In terms of the latter, the metric tensor is written in a conformally flat form
with the spacetime coordinates . The hypersurfaces and correspond to the AdS boundary and horizon, respectively. Note that for the proper length of the th compact dimension, measured by an observer with a fixed coordinate , one has .
We are interested in combined effects of the nontrivial topology and boundaries on the local characteristics of the vacuum state for a massive fermionic field . Assuming the presence of an external abelian gauge field , the corresponding field equation reads
where is the coupling between the fermionic and gauge fields and is the spin connection. For the curved spacetime Dirac matrices one has , with being the corresponding flat spacetime matrices and are the tetrad fields. For a fermionic field realizing the irreducible representation of the Clifford algebra the matrices are matrices with , where the square brackets stand for the integer part of the enclosed expression. For odd the irreducible representation is unique up to a similarity transformation, whereas for even there are two inequivalent irreducible representations (see Section 7 below). In the conformal coordinates , with the metric tensor (2.2), we can take the tetrad fields in the form . The corresponding spin connection has the components for , and .
In the discussion below we assume the presence of a boundary, parallel to the AdS boundary and located at , on which the field operator is constrained by the bag boundary condition
where is the corresponding normal. The respective value of the -coordinate we shall denote by , . Note that the physical distance from the boundary is given by . Though the boundary under consideration my have different physical origins (for example, in carbon nanotubes it corresponds to the edge of the tube), for the convenience of the discussion below we shall use the term ’brane’. It divides the background space into two regions: and . We shall refer to them as L- and R-regions (left and right regions), respectively. For the normal one has in the L-region and in the R-region. From (2.4) it follows that the normal component of the fermionic current vanishes on the brane. This feature is used in bag models of hadrons for confinement of quarks. Note that, though the geometrical characteristics of the background geometry do not depend on , the boundary under consideration has a nonzero extrinsic curvature tensor with nonzero components , where the upper and lower signs correspond to the L- and R-regions, respectively. Related to this, the physical properties of the vacuum will be different in these regions.
The topology of the background space is nontrivial and, in addition to the boundary condition at , we need to specify the periodicity conditions imposed on the field operator along compact dimensions. For the spatial dimension , , we take the quasiperiodicity condition
with a constant phase . The special cases of the most frequently used conditions with and correspond to untwisted and twisted fields. As for the gauge field, we assume the simplest configuration with . The corresponding effects on quantum properties of the vacuum are of the Aharonov-Bohm type and they are related to the nontrivial topology of the background space. The components of the vector potential along noncompact dimensions are simply removed by a gauge transformation and only the components along compact dimensions are physically relevant. Hence, our model is specified by the set of parameters with .
Under the gauge transformation of the field variables , , with the function , we obtain a new set of parameters . In particular, in the gauge with the vector potential vanishes and for the new phases in the quasiperiodicity conditions for the field one gets
Hence, the effects of and are not physically independent: the physical effects depend on these parameters in the form of the combination (2.6) which is invariant under the gauge transformation. In particular, a constant gauge field induces nontrivial effective phases for twisted and untwisted fields and vice versa: the nontrivial phases can be interpreted in terms of a constant gauge filed (or in terms of the magnetic flux). In what follows we will work in the gauge omitting the primes. Along the th compact dimension the field obeys the condition (2.5) with replaced by from (2.6). The part in the definition of the latter coming from the vector potential can be interpreted in terms of the magnetic flux enclosed by the th dimension: (the minus sign comes from the fact that is the covariant component of the -vector and it is related to the th component of the spatial vector by ), with being the flux quantum. Of course, this flux is fictive, it lives in the embedding space. However, it can be real flux if the model under consideration is realized as a brane in a higher dimensional spacetime. Another problem where the magnetic flux has the real physical sense will be considered in Section 7.
3 Fermionic modes
The VEVs of physical observables bilinear in the field operator are expressed in terms of the sums over a complete set of positive and negative energy fermionic modes , where the set of quantum numbers specifies the solution. These modes obey the field equation (2.3) (with in the gauge under consideration), the boundary condition (2.4) and the quasiperiodicity conditions (2.5) with replaced by . In order to find the solutions to the field equation one needs to specify the representation of the flat spacetime Dirac matrices (for the construction of the Dirac matrices in an arbitrary number of spacetime dimensions see, for example, ). We find it convenient to use the representation (see also )
with matrices , . In even dimensional spacetimes the irreducible representation is unique (up to a similarity transformation) and one can take . In odd-dimensional spacetimes, the values and correspond to two inequivalent irreducible representations of the Clifford algebra. From the anticommutation relations for the Dirac matrices we obtain the following relations
with . In the special case , taking , we get , , , where are the Pauli matrices.
With the flat spacetime matrices (3.1), substituting in the field equation (2.3) the Dirac matrices , the complete set of the positive and negative energy solutions of the field equation can be found in a way similar to that we have described in Appendix of Ref. . In accordance with the symmetry of the problem, the dependence of the mode functions on the coordinates can be taken in the form of plane waves , , with the momentum and the energy . The mode functions are presented as
where , , are one-column matrices having rows and the elements . In (3.3), , , , and
is a linear combination of the Bessel and Neumann functions and . The coefficients and depend on the region under consideration and will be determined below separately in the L- and R-regions.
For the components of the momentum along uncompact spatial dimensions, as usual, one has , . The eigenvalues of the components along compact dimensions are quantized by the periodicity conditions:
where . For , with being an integer, the parameter is removed from the problem by the redefinition of the quantum number . Therefore, only the fractional part of is physically relevant. The set of quantum numbers specifying the modes is given by , where is the momentum in the non-compact subspace and determines the momentum in the compact subspace. The orthonormalization condition for the mode functions reads
where is understood as the Dirac delta function for the continuous components of and the Kronecker delta for discrete ones.
We are interested in the VEV of the current density , where for the Dirac conjugate one has . Expanding the field operator in terms of the complete set of modes and using the anticommutation relations for the annihilation and creation operators, the VEV is presented in the form of the mode sum
Here, stands for the integration over the continuous components of the collective index and for the summation over the discrete components. The functions in (3.3) and the eigenvalues for are different in the L- and R-regions and we investigate the corresponding current densities separately.
4 Current density in the L-region
First we consider the region between the brane and the AdS boundary, corresponding to . In the range of the mass and for in (3.4) the modes (3.3) are not normalizable. Hence, for this range, from the normalizability condition it follows that and the mode functions are given by (3.3) with . From the boundary condition (2.4) it follows that the eigenvalues for the quantum number are roots of the equation
for both the cases . We shall denote the corresponding positive roots with respect to by , , assuming that they are numerated in the ascending order, . Note that the roots do not depend on the location of the brane.
Now the mode functions are written as
where . For a massless field one has .
The normalization coefficients are determined from the condition (3.6), where the integration over is done in the region and on the right-hand side the Kronecker delta appears. By using the standard integral for the square of the Bessel function one finds
where is the volume of the compact subspace. As seen, the normalization constants are the same for both the representations and .
For the range of masses the modes with in (3.4) are normalizable. In this case, in order to determine the additional coefficient in the mode function one needs to specify a boundary condition for the field on the AdS boundary. This kind of boundary conditions for fermions have been discussed, for example, in Refs. [28, 29, 30]. Here we shall consider a special type of boundary condition when the bag boundary condition is imposed on the hypersurface and then the limiting transition is taken. As it will be shown in the next section, this procedure leads to the mode functions which are given by (4.1) for all .
with being the momentum in the uncompact subspace. Now we note that for a matrix one has . By taking into account the relations (3.2) it can be seen that and, hence, . From here we conclude that the VEV of the charge density vanishes.
For , the integrand is an odd function with respect to the momentum and the corresponding component of the current density is zero, . Hence, a nonzero current density may appear along the compact dimensions only. This is a purely topological effect of the Aharonov-Bohm type and is induced by the nontrivial phases in the quasiperiodicity conditions (or, alternatively, by the enclosed magnetic fluxes). For , with being an integer, after passing to the summation over , we see that the contributions in (4.5) coming from the modes with positive and negative values of cancel each other and the resulting current density vanishes. Another important conclusion following from (4.5) is that the current densities for the representations with and coincide. We will continue the investigation of the current density in the L-region for the case .
The representation (4.5) contains the eigenvalues which are given implicitly, as the zeros of the Bessel function. In order to obtain a representation more convenient for the asymptotic and numerical analysis, and for explicit extraction of the brane-induced contribution, we apply to the series over a variant of the generalized Abel-Plana formula 
valid for a function analytic in the right half-plane of the complex variable (function may have branch points on the imaginary axis, for the conditions imposed on this function see ). In (4.6), and are the modified Bessel functions. In the problem under consideration the function is specified as
and has branch points .
where the term
is the contribution induced by the brane. For a fixed , the latter goes to zero in the limit .
The current density in the brane-free geometry has been investigated in . An alternative representation is given by
where, , . The function is expressed in terms of the hypergeometric function as
Note that for (this corresponds to odd values of in (4.11)) one has with being the Legendre function of the second kind. For (even values of in (4.11)) the function is expressed in terms of the elementary functions. In what follows we will be mainly concerned about the brane-induced effects in the current density.
By taking into account that , with
the contribution (4.10) is further simplified by using the relation
for a given function . This leads to the following expression for the brane-induced contribution to the current density:
with the notation
Note that the integrand in (4.15) is always negative. Both the brane-free and brane-induced contributions in the th component of the vacuum current density are odd periodic functions of the phase and even periodic functions of with , with the period . In particular, they are periodic functions of the magnetic flux with the period equal to the flux quantum . The charge flux through the hypersurface is given by , where is the normal to that hypersurface. The product depends on the variables having the dimension of length in the form of the dimensionless combinations , , . This feature is a consequence of the maximal symmetry of the AdS spacetime. Note that the ratio is the proper length of the th compact dimension in units of the curvature radius .
In order to further clarify the behavior of the current density we pass to the investigation of the VEV (4.15) in special cases and in various asymptotic regions of the parameters. First we consider the current density of a massless fermionic field. In this case the modified Bessel functions in (4.15) are expressed in terms of the elementary functions and one gets
with the function
The second representation in (4.17) is obtained from the first one by using the expansion . The massless fermionic field is conformally invariant in an arbitrary number of spatial dimensions and the result (4.17) is obtained from the expression for the current density in the region between two boundaries at and on a locally Minkowskian bulk with compact dimensions by using the conformal relation . Note that the boundary in the Minkowski bulk is the conformal image of the AdS boundary. We can see that obtained from (4.17) coincides with the result from  (the sign difference is related to the fact that in  corresponds to in the present paper).
The Minkowskian limit corresponds to for fixed and . In this limit the conformal coordinates and are large, , , and, consequently, both the order and the argument of the modified Bessel functions in (4.15) are large. By using the corresponding uniform asymptotic expansions , to the leading order we get
with the notation . This expression coincides with the result from  for a boundary in a flat bulk with topology (again, with the sign difference related to definition of the parameters ). For a massless field the current density induced by a single boundary in flat spacetime vanishes.
Now let us consider the behavior of the current density near the AdS boundary and near the brane for the fixed location of the brane. For points close to the AdS boundary one has and the main contribution to the integral in (4.15) comes from the region of the integration where the argument of the functions is small. By using the corresponding asymptotic expression, the leading order term reads
and on the AdS boundary the brane-induced contribution vanishes as . Note that the brane-free contribution behaves in a similar manner, .
The representation (4.15) for the current density is not well suited for the investigation of the near-brane asymptotic. In order to obtain an alternative representation, we apply to the series over in the initial expression (4.5) the Abel–Plana-type formula 
for given functions , and with defined in (3.5) (formula (4.21) is reduced to the standard Abel-Plana formula in the special case , ). For the series in (4.5) one has and the first integral in (4.21) is zero. By making use of the relation
in the last term in (4.21), the integral over is expressed in terms of the modified Bessel function . Evaluating the remaining integral over by using the formula from , the VEV of the current density is presented as (as it has been shown above, the current densities for the representations and are the same and we consider the case )
where the function is defined by (4.18) and
Note that in the representation (4.23) the terms of the series over decay exponentially for large . In the case of a massless field we have and the ratio of the Bessel functions in (4.23) is equal to . In this case we get the standard conformal relation with the corresponding representation of the current density between two boundaries in locally Minkowskian spacetime with compact dimensions.
The total current, per unit surface along the uncompact dimensions, is obtained by integration of (4.23):
Note that the dependence on the curvature radius of the background spacetime, on the mass of the field and on the location of the brane appears through the ratio . We recall that the roots are completely determined by the parameter and do not depend on the location of the brane.
In the model with a single compact dimension with the length (, ) the formula (4.23) is specified to