Fermionic condensate in a conical space with a circular boundary and magnetic flux
aa r X i v : . [ h e p - t h ] J a n Fermionic condensate in a conical space witha circular boundary and magnetic flux
S. Bellucci ∗ , E. R. Bezerra de Mello † , A. A. Saharian ‡ INFN, Laboratori Nazionali di Frascati,Via Enrico Fermi 40, 00044 Frascati, Italy Departamento de F´ısica, Universidade Federal da Para´ıba58.059-970, Caixa Postal 5.008, Jo˜ao Pessoa, PB, Brazil Department of Physics, Yerevan State University,Alex Manoogian Street, 0025 Yerevan, Armenia
June 11, 2018
Abstract
The fermionic condensate is investigated in a (2+1)-dimensional conical spacetime in thepresence of a circular boundary and a magnetic flux. It is assumed that on the boundary thefermionic field obeys the MIT bag boundary condition. For irregular modes, we consider aspecial case of boundary conditions at the cone apex, when the MIT bag boundary conditionis imposed at a finite radius, which is then taken to zero. The fermionic condensate is aperiodic function of the magnetic flux with the period equal to the flux quantum. For bothexterior and interior regions, the fermionic condensate is decomposed into boundary-free andboundary-induced parts. Two integral representations are given for the boundary-free partfor arbitrary values of the opening angle of the cone and magnetic flux. At distances fromthe boundary larger than the Compton wavelength of the fermion particle, the condensatedecays exponentially with the decay rate depending on the opening angle of the cone. If theratio of the magnetic flux to the flux quantum is not a half-integer number, for a masslessfield the boundary-free part in the fermionic condensate vanishes, whereas the boundary-induced part is negative. For half-integer values of the ratio of the magnetic flux to the fluxquantum, the irregular mode gives non-zero contribution to the fermionic condensate in theboundary-free conical space.
PACS numbers: 03.70.+k, 04.60.Kz, 11.27.+d
Field theoretical models in 2+1 dimensions exhibit a number of interesting effects, such as parityviolation, flavor symmetry breaking, and fractionalization of quantum numbers (see Refs. [1]-[7]). An important aspect is the possibility of giving a topological mass to the gauge bosons ∗ E-mail: [email protected] † E-mail: emello@fisica.ufpb.br ‡ E-mail: [email protected] N c , by the formula 2 π (1 − N c / N c = 1 , , . . . , Let us consider a two-component spinor field ψ on background of a (2 + 1)-dimensional conicalspacetime. The corresponding line element is given by the expression ds = g µν dx µ dx ν = dt − dr − r dφ , (2.1)where r >
0, 0 φ φ , and the points ( r, φ ) and ( r, φ + φ ) are to be identified. In thediscussion below, in addition to φ , we use the notation q = 2 π/φ . (2.2)In the presence of the external electromagnetic field with the vector potential A µ , the dynamicsof the field is governed by the Dirac equation iγ µ ( ∇ µ + ieA µ ) ψ − mψ = 0 , ∇ µ = ∂ µ + Γ µ , (2.3)where γ µ = e µ ( a ) γ ( a ) are the 2 × e µ ( a ) , a = 0 , ,
2, is thebasis tetrad. The operator of the covariant derivative in Eq. (2.3) is defined by the relation ∇ µ = ∂ µ + 14 γ ( a ) γ ( b ) e ν ( a ) e ( b ) ν ; µ , (2.4)where ”;” means the standard covariant derivative for vector fields. In (2 + 1)-dimensionalspacetime there are two inequivalent irreducible representations of the Clifford algebra. Here wechoose the flat space Dirac matrices in the form γ (0) = σ , γ (1) = iσ , γ (2) = iσ , with σ l beingPauli matrices. In the second representation the gamma matrices can be taken as γ (0) = − σ , γ (1) = − iσ , γ (2) = − iσ . The corresponding results for the second representation are obtainedby changing the sign of the mass, m → − m . Note that there is no other 2 × γ ( a ) and, hence, we have no chiral symmetry that would broken by amass term in two-dimensional representation.Our interest in the present paper is the FC, h | ¯ ψψ | i = h ¯ ψψ i , with | i being the vacuumstate, in the conical space with a circular boundary. Here and in what follows ¯ ψ = ψ † γ is theDirac adjoint and the dagger denotes Hermitian conjugation. We assume the magnetic fieldconfiguration corresponding to a infinitely thin magnetic flux located at the apex of the cone.This will be implemented by considering the vector potential A µ = (0 , , A ) for r >
0. Thequantity A is related to the magnetic flux Φ by the formula A = − Φ /φ .3irst we consider the FC in a boundary-free conical space. It can be evaluated by using themode-sum formula h ¯ ψψ i = X σ ¯ ψ ( − ) σ ψ ( − ) σ , (2.5)where { ψ (+) σ , ψ ( − ) σ } is a complete set of positive and negative energy solutions to the Dirac equa-tion specified by quantum numbers σ . As it is well known, the theory of von Neumann deficiencyindices leads to a one-parameter family of allowed boundary conditions in the background ofan Aharonov-Bohm gauge field [22]. Here we consider a special case of boundary conditions atthe cone apex, when the MIT bag boundary condition is imposed at a finite radius, which isthen taken to zero. The FC for other boundary conditions on the cone apex are evaluated in away similar to that described below. The contribution of the regular modes is the same for allboundary conditions and the results differ by the parts related to the irregular modes.In the boundary-free conical space the eigenspinors are specified by the set σ = ( γ, j ) ofquantum numbers with 0 γ < ∞ and j = ± / , ± / , . . . . For j = − e Φ / π , the correspondingnormalized negative-energy eigenspinors have the form [13] ψ ( − )(0) γj = (cid:18) γ E + m φ E (cid:19) / e − iqjφ + iEt γǫ j e − iqφ/ E + m J β j + ǫ j ( γr ) J β j ( γr ) e iqφ/ ! , (2.6)where E = p γ + m , J ν ( x ) is the Bessel function. The order of the Bessel function in (2.6) isgiven by the expression β j = q | j + α | − ǫ j / , q = 2 π/φ , (2.7)with α = eA/q = − e Φ / π, (2.8)and we have defined ǫ j = (cid:26) , j > − α − , j < − α . (2.9)The expression for the positive energy eigenspinor is found from (2.6) by using the relation ψ (+) γj = σ ψ ( − ) ∗ γj , where the asterisk means complex conjugate. Here we assume that the parameter α isnot a half-integer. The special case of half-integer α will be considered separately in Sect. 5.Substituting the eigenspinors (2.6) into the mode-sum (2.5), for the FC in a boundary-freeconical space one finds h ¯ ψψ i = q π X j Z ∞ dγ γE h ( E − m ) J β j + ǫ j ( γr ) − ( E + m ) J β j ( γr ) i , (2.10)where P j means the summation over j = ± / , ± / , . . . . Of course, the expression on theright-hand side of this formula is divergent and needs to be regularized. We introduce a cutofffunction e − sγ with the cutoff parameter s >
0. At the end of calculations the limit s → h ¯ ψψ i , reg = q π X j Z ∞ dγγe − sγ h J β j + ǫ j ( γr ) − J β j ( γr ) i − qm π X j Z ∞ dγ γe − sγ p γ + m h J β j + ǫ j ( γr ) + J β j ( γr ) i . (2.11)4he γ -integral in the first term on the right-hand side is expressed in terms of the modifiedBessel function I ν ( x ). In the second term we use the relation1 p γ + m = 2 √ π Z ∞ dte − ( γ + m ) t , (2.12)and change the order of integrations. After the evaluation of the γ -integral, the regularized FCis presented in the form: h ¯ ψψ i , reg = qe − r / s πs X j [ I β j + ǫ j ( r / s ) − I β j ( r / s )] − qme m s π ) / X j Z r / s dx x − / e − m r / x − x √ r − xs [ I β j + ǫ j ( x ) + I β j ( x )] . (2.13)Before further considering the FC for the general case of the parameters characterizing theconical structure and the magnetic flux, we study a special case, which allows us to obtain asimple expression. In the special case with q being an integer and α = 1 / q − / , (2.14)the orders of the modified Bessel functions in Eq. (2.13) become integer numbers: β j = q | n | , j = n + 1 /
2. The series over n is summarized explicitly by using the formula [23] ∞ X ′ n =0 I qn ( x ) = 12 q q − X k =0 e x cos(2 πk/q ) , (2.15)where the prime means that the term n = 0 should be halved. For the regularized FC we findthe expression h ¯ ψψ i , reg = − πs q − X k =1 sin ( πk/q ) e − r / s ) sin ( πk/q ) − me m s (2 π ) / q − X k =0 cos ( πk/q ) Z r / s dx x − / e − m r / x √ r − xs e − x sin ( πk/q ) . (2.16)The first term on the right-hand side of this formula vanishes in the limit s →
0. In the secondterm the only divergent contribution in the limit s → k = 0 term. This termcoincides with the regularized FC in the Minkowski spacetime in the absence of the magneticflux. Subtracting this contribution and taking the limit s →
0, for the renormalized FC we find h ¯ ψψ i , ren = − m πr q − X k =1 cos ( πk/q )sin( πk/q ) e − mr sin( πk/q ) . (2.17)Note that the renormalized FC vanishes for a massless field and for a massive field in a conicalspace with q = 2. For other cases the FC is negative. As expected, it decays exponentially atdistances larger that the Compton wavelength of the fermionic particle. In Fig. 1 the FC isplotted versus mr for different values of q . The corresponding values of the parameter α arefound from Eq. (2.14). Under the condition (2.14), the induced fermionic current in a higher-dimensional cosmic string spacetimehas been analyzed in [24]. = - - - - m - < ΨΨ > , r e n Figure 1: Fermionic condensate in a boundary-free conical space, as a function of mr for thespecial case of integer values of q with the magnetic flux defined by Eq. (2.14). For the general case of the parameters q and α , as it is seen from (2.13), the regularized FC isexpressed in terms of the series I ( q, α, z ) = X j I β j ( z ) . (2.18)We present the parameter α , related to the magnetic flux by Eq. (2.8), in the form α = α + n , | α | < / , (2.19)with being n an integer number. Now, Eq. (2.18) is written as I ( q, α, z ) = ∞ X n =0 (cid:2) I q ( n + α +1 / − / ( z ) + I q ( n − α +1 / / ( z ) (cid:3) , (2.20)which explicitly shows the independence of the series on n . Note that for the second seriesappearing in the expression of the FC we have X j I β j + ǫ j ( z ) = I ( q, − α , z ) . (2.21)From these relations we conclude that the FC depends on α alone and, hence, it is a periodicfunction of α with period 1.In terms of the function (2.18), the expression (2.13) for the regularized FC is written as h ¯ ψψ i , reg = − qe − r / s πs X δ = ± δ I ( q, δα , r / s ) − qme m s π ) / Z r / s dx x − / e − m r / x − x √ r − xs X δ = ± I ( q, δα , x ) . (2.22)For 2 p < q < p + 2 with p being an integer, we use the representation [13] I ( q, α , z ) = e z q + J ( q, α , z ) , (2.23)6ith the notation J ( q, α , z ) = − π Z ∞ dy e − z cosh y f ( q, α , y )cosh( qy ) − cos( qπ )+ 2 q p X l =1 ( − l cos[2 πl ( α − / q )] e z cos(2 πl/q ) . (2.24)The function in the integrand is defined by the expression f ( q, α , y ) = cos [ qπ (1 / − α )] cosh [( qα + q/ − / y ] − cos [ qπ (1 / α )] cosh [( qα − q/ − / y ] . (2.25)In the case q = 2 p , the term − ( − q/ e − z q sin( qπα ) , (2.26)should be added to the right-hand side of Eq. (2.24). For 1 q <
2, the last term on theright-hand side of Eq. (2.24) is absent.In the limit s →
0, the only divergent contributions to the functions e − r / s I ( q, ± α , r / s ) /s come from the first term in the right-hand side of Eq. (2.23). The contribution of this term tothe FC does not depend on α and, consequently, the divergences are cancelled in the evaluationof the first term in the right-hand side of (2.22). This term vanishes in the limit s → e z /q . This contribution does not depend on the opening angle of the cone and on themagnetic flux. It coincides with the corresponding quantity in the Minkowski spacetime in theabsence of the magnetic flux. Subtracting the Minkowskian part and taking the limit s →
0, forthe renormalized FC we find: h ¯ ψψ i , ren = − qm π ) / r Z ∞ dx x − / e − m r / x − x X δ = ± J ( q, δα , x ) . (2.27)Note that in the case q = 2 p the contribution of the additional term (2.26) to the renormalizedFC vanishes.By taking into account Eq. (2.24), the integration over x in Eq. (2.27) is performed explicitlyand one finds the following formula h ¯ ψψ i , ren = m πr n − p X l =1 ( − l cot( πl/q ) e mr sin( πl/q ) cos(2 πlα )+ q π Z ∞ dy e − mr cosh( y/ cosh( y/ P δ = ± f ( q, δα , y )cosh( qy ) − cos( qπ ) o , (2.28)where p is an integer defined by 2 p q < p + 2. Note that the sum in the integrand may bewritten in the form X δ = ± f ( q, δα , y ) = − y/ X δ = ± cos [ qπ (1 / δα )] sinh[ q (1 / − δα ) y ] . (2.29)For integer q and for the parameter α given by the special value (2.14), from (2.28) we obtain theresult (2.17). At distances larger than the Compton wavelength of the spinor particle, mr ≫ e − mr for 1 q e − mr sin( π/q ) for q > h ¯ ψψ i , ren ≈ m cos(2 πα )2 πr cot( π/q ) e mr sin( π/q ) , mr ≫ . (2.30)In the special case when the magnetic flux is absent we have α = 0 and the general formula(2.28) simplifies to h ¯ ψψ i , ren = − m πr n p X l =1 ( − l cot( πl/q ) e mr sin( πl/q ) + 2 qπ cos( qπ/ Z ∞ dx sinh ( qx ) tanh( x ) e − mr cosh x cosh(2 qx ) − cos( qπ ) o . (2.31)In this case the FC is only a consequence of the conical structure of the space. For odd valuesof the parameter q the second term in the figure braces vanishes and for the FC we have thesimple formula h | ¯ ψψ | i , ren = − m πr ( q − / X l =1 ( − l cot( πl/q ) e mr sin( πl/q ) . (2.32)Another limiting case corresponds to the magnetic flux in background of Minkowski spacetime.In this case, taking q = 1, from Eq. (2.28) we find h ¯ ψψ i , ren = − m sin( πα )2 π r Z ∞ dx sinh x cosh x sinh (2 α x ) e mr cosh x , (2.33)and the FC is negative for α = 0.In Fig. 2, the fermionic condensate is plotted as a function of the magnetic flux for a massivefermionic field in conical spaces with φ = π (left plot) and φ = π/ q = 2 the first term in figure braces of (2.28) vanishes and the second term contains the factorcos (2 πα ). Consequently, in this case the FC vanishes at α = π/ mr = = - - - - - - Α r < ΨΨ > , r e n mr = = - - - - Α r < ΨΨ > , r e n Figure 2: The FC as a function of the magnetic flux for a massive fermionic field in boundary-freeconical spaces with q = 2 (left plot) and q = 4 (right plot).8n alternative expression for the FC is obtained by using the formula [13] I ( q, α , x ) = A ( q, α , x ) + 2 q Z ∞ dz I z ( x ) − πq Z ∞ dz Re (cid:20) sinh( zπ ) K iz ( x ) e π ( z + i | qα − / | ) /q + 1 (cid:21) , (2.34)with A ( q, α , x ) = 0 for | α − / q | /
2, and A ( q, α , x ) = 2 π sin[ π ( | qα − / | − q/ K | qα − / |− q/ ( x ) , (2.35)for 1 / < | α − / q | <
1. Substituting the representation (2.34) into the expression (2.22) forthe regularized FC, we see that the part with the second term on the right-hand side of (2.34)does not depend on the opening angle of the cone and on the magnetic flux. It is the same as inthe Minkowski bulk in the absence of the magnetic flux and, hence, it should be subtracted inthe renormalization procedure. Subtracting the part corresponding to q = 1 and α = 0, in theremaining part the limit s → h ¯ ψψ i , ren = − m (2 π ) / r Z ∞ dx x − / e − m r / x − x × h qB ( q ( | α | − /
2) + 1 / , x ) − Z ∞ dz K iz ( x ) h ( q, α , z ) i . (2.36)In this formula we have used the notations h ( q, α , z ) = X δ = ± Re (cid:20) sinh( zπ ) e π ( z + i | qδα − / | ) /q + 1 + sinh( zπ ) e πz − (cid:21) . (2.37)and B ( y, x ) = (cid:26) , y , sin( πy ) K y ( x ) y > . (2.38)The representation (2.36) is valid for conical spaces with q <
4. For special values q = 2 and α = 1 /
4, by taking into account that h (2 , / , z ) = 0, we see that the FC defined by (2.36)vanishes.In the case of a magnetic flux in background of the Minkowski spacetime ( q = 1) we find h ¯ ψψ i , ren = − m sin( π | α | )(2 π ) / r Z ∞ dx x − / e − m r / x − x × (cid:20) K α ( x ) − π | α | ) Z ∞ dz K iz ( x ) cosh( πz )cosh(2 πz ) − cos(2 πα ) (cid:21) . (2.39)For a conical space in the absence of the magnetic flux the general formula reduces to h ¯ ψψ i , ren = 4 m (2 π ) / r Z ∞ dx x − / e − m r / x − x × Z ∞ dz K iz ( x ) cosh( πz ) cos( π/q ) + cosh[ πz (2 /q − πz/q ) + cos( π/q ) . (2.40)For q = 2 the integral over z is evaluated explicitly (see, for instance, [23]) and we get a simpleexpression h ¯ ψψ i , ren = ( m/π ) R ∞ dt K (2 mrt ) /t . Recall that for odd values of q we have the9imple formula (2.32). For the second representation of the Clifford algebra the renormalizedFC in a boundary-free conical space changes the sign.We can generalize the results given above for a more general situation where the spinor field ψ obeys quasiperiodic boundary condition along the azimuthal direction ψ ( t, r, φ + φ ) = e πiχ ψ ( t, r, φ ) , (2.41)with a constant parameter χ , | χ | /
2. With this condition, the exponential factor in theexpression for the eigenspinors (2.6) has the form e − iq ( n + χ ) φ + iEt . The corresponding expressionfor the eigenfunctions is obtained from that given above with the parameter α defined by α = χ − e Φ / π. (2.42)The same replacement generalizes the expression of the FC for the case of a field with periodicitycondition (2.41).In general, the fermionic modes in background of the magnetic vortex are divided into twoclasses, regular and irregular (square integrable) ones. In the problem under consideration, forgiven q and α , the irregular mode corresponds to the value of j for which q | j + α | < / α in the form (2.19), then the irregular mode is present if | α | > (1 − /q ) /
2. This mode corresponds to j = − n − sgn( α ) /
2. Note that, in a conical space, underthe condition | α | (1 − /q ) /
2, there are no square integrable irregular modes. As we havealready mentioned, there is a one-parameter family of allowed boundary conditions for irregularmodes. These modes are parametrized by the angle θ , 0 θ < π (see Ref. [22]). For | α | < / θ = 3 π/
2. If α isa half-integer, the irregular mode corresponds to j = − α and for the corresponding boundarycondition one has θ = 0. Note that in both cases there are no bound states. In this section we consider the change in the FC induced by a circular boundary concentric withthe apex of the cone. We assume that the field obeys the MIT bag boundary condition on thecircle with radius a : (1 + in µ γ µ ) ψ (cid:12)(cid:12) r = a = 0 , (3.1)where n µ is the outward oriented normal (with respect to the region under consideration) to theboundary. For the interior region n µ = δ µ . In this region the negative-energy eigenspinors aregiven by the expression [13] ψ ( − ) γj = ϕ e − iqjφ + iEt ǫ j γe − iqφ/ E + m J β j + ǫ j ( γr ) e iqφ/ J β j ( γr ) ! , (3.2)with the same notations as in Eq. (2.6). From the boundary condition at r = a we find thatthe eigenvalues of γ are solutions of the equation J β j ( γa ) − γǫ j J β j + ǫ j ( γa ) p γ + m + m = 0 . (3.3)For a given β j , Eq. (3.3) has an infinite number of solutions which we denote by γa = γ β j ,l , l = 1 , , . . . . The normalization coefficient in Eq. (3.2) is given by the expression ϕ = yT β j ( y )2 φ a µ + p y + µ p y + µ , (3.4)10ith the notations µ = ma and T β j ( y ) = yJ β j ( y ) h y + ( µ − ǫ j β j ) (cid:16) µ + p y + µ (cid:17) − y p y + µ i − . (3.5)Substituting the eigenspinors (3.2) into the mode-sum formula h ¯ ψψ i = X j ∞ X l =1 ¯ ψ ( − ) γj ψ ( − ) γj , (3.6)for the FC we find h ¯ ψψ i = q πa X j ∞ X l =1 yT β j ( y ) h(cid:16) − µ p y + µ (cid:17) J β j + ǫ j ( yr/a ) − (cid:16) µ p y + µ (cid:17) J β j ( yr/a ) i , (3.7)with y = γ β j ,l . Here we assume that a cutoff function is introduced without explicitly writingit. The specific form of this function is not important for the discussion below.For the summation of the series over l in Eq. (3.7) we use the summation formula (see[25, 26]) ∞ X l =1 f ( γ β j ,l ) T β ( γ β j ,l ) = Z ∞ dx f ( x ) − π Z ∞ dx × (cid:20) e − β j πi f ( xe πi/ ) K (+) β j ( x ) I (+) β j ( x ) + e β j πi f ( xe − πi/ ) K (+) ∗ β j ( x ) I (+) ∗ β j ( x ) (cid:21) , (3.8)where the asterisk means complex conjugate. In this formula, for a given function F ( x ), we usethe notation F (+) ( x ) = ( xF ′ ( x ) + ( µ + p µ − x − ǫ j β j ) F ( x ) , x < µ,xF ′ ( x ) + (cid:16) µ + i p x − µ − ǫ j β j (cid:17) F ( x ) , x > µ. (3.9)Note that for x < µ one has F (+) ∗ ( x ) = F (+) ( x ). The ratio of the combinations of the modifiedBessel functions in Eq. (3.8) may be presented in the form K (+) β j ( x ) I (+) β j ( x ) = W (+) β j ,β j + ǫ j ( x ) + i p − µ /x x [ I β j ( x ) + I β j + ǫ j ( x )] + 2 µI β j ( x ) I β j + ǫ j ( x ) , (3.10)with the notation defined by W ( ± ) β j ,β j + ǫ j ( x ) = x (cid:2) I β j ( x ) K β j ( x ) − I β j + ǫ j ( x ) K β j + ǫ j ( x ) (cid:3) ± µ (cid:2) I β j + ǫ j ( x ) K β j ( x ) − I β j ( x ) K β j + ǫ j ( x ) (cid:3) . (3.11)The notation with the lower sign will be used below.Applying to the series over l in Eq. (3.7) the summation formula and comparing with Eq.(2.10), we see that the term in the FC corresponding to the first integral in the right-hand sideof Eq. (3.8) coincides with the condensate in a boundary-free conical space. As a result, the FCis presented in the decomposed form h ¯ ψψ i = h ¯ ψψ i , ren + h ¯ ψψ i b , (3.12)11here h ¯ ψψ i b is the part induced by the circular boundary. For the function f ( x ) correspondingto Eq. (3.7), in the second term on the right-hand side of Eq. (3.8), the part of the integral overthe region (0 , µ ) vanishes. Consequently, the boundary-induced contribution for the FC in theregion inside the circle is given by the expression h ¯ ψψ i b = q π X j Z ∞ m dx x × n m I β j ( xr ) − I β j + ǫ j ( xr ) √ x − m Re[ K (+) β j ( xa ) /I (+) β j ( xa )] − [ I β j ( xr ) + I β j + ǫ j ( xr )]Im[ K (+) β j ( xa ) /I (+) β j ( xa )] o . (3.13)The real and imaginary parts appearing in this equation are easily obtained from Eq. (3.10).Note that under the change α → − α , j → − j , we have β j → β j + ǫ j , β j + ǫ j → β j . From hereit follows that the real/imaginary part in Eq. (3.13) is an odd/even function under this change.Now, from Eq. (3.13) we see that the boundary-induced part in the FC is an even functionof α . For points away from the circular boundary and the cone apex, the boundary-inducedcontribution is finite and the renormalization is reduced to that for the boundary-free geometry.This contribution is a periodic function of the parameter α with the period equal to 1. So, if wepresent this parameter in the form (2.19) with n being an integer, then the FC depends on α alone.In the case of a massless field the expressions for the boundary-induced part in the FC takesthe form h ¯ ψψ i b = − q π a X j Z ∞ dz I β j ( zr/a ) + I β j + ǫ j ( zr/a ) I β j ( z ) + I β j + ǫ j ( z ) . (3.14)As it is seen, this part is always negative. We would like to point out that the boundary-inducedFC does not vanish for a massless filed. The corresponding boundary-free part vanishes and,hence, for a massless field h ¯ ψψ i = h ¯ ψψ i b .Various special cases of general formula (3.13) can be considered. In the absence of themagnetic flux one has α = 0 and the contributions of the negative and positive values of j tothe FC coincide. The corresponding formulas are obtained from (3.13) and (3.14) making thereplacements X j → X j =1 / , / ,... , β j → qj − / , β j + ǫ j → qj + 1 / . (3.15)In the case q = 1, we obtain the FC induced by the magnetic flux and a circular boundary in theMinkowski spacetime. And finally, in the simplest case α = 0 and q = 1 one has h ¯ ψψ i , ren = 0,and the expression (3.13) gives the FC induced by a circular boundary in the Minkowski bulk: h ¯ ψψ i = 1 π a ∞ X n =0 Z ∞ µ dxI n ( x ) + I n +1 ( x ) + 2 µI n ( x ) I n +1 ( x ) /x × n µ W (+) n,n +1 ( x ) p x − µ (cid:2) I n ( xr/a ) − I n +1 ( xr/a ) (cid:3) − p − µ /x (cid:2) I n ( xr/a ) + I n +1 ( xr/a ) (cid:3) o , (3.16)where the function W (+) n,n +1 ( x ) is defined by Eq. (3.10).Now we turn to the investigation of the FC in asymptotic regions of the parameters. Forlarge values of the circle radius, we replace the modified Bessel functions in Eq. (3.13), with xa
12n their arguments, by asymptotic expansions for large values of the argument. In the case of amassive field the dominant contribution to the integral comes from the integration range nearthe lower limit. In the leading order one has h ¯ ψψ i b ≈ qm e − ma √ π ( ma ) / X j ǫ j h β j I β j + ǫ j ( mr ) − ( β j + ǫ j ) I β j ( mr ) i , (3.17)and for a fixed value of the radial coordinate, the boundary-induced FC is exponentially small.For a massless field, assuming r/a ≪
1, we expand the modified Bessel function in thenumerator of integrand in Eq. (3.14) in powers of r/a . The dominant contribution comes fromthe term j = 1 / α < j = − / α >
0. To the leading order wefind h ¯ ψψ i b ≈ − q π a ( r/ a ) q α − Γ ( q α + 1 / Z ∞ dz z q α − I q α +1 / ( z ) + I q α − / ( z ) , (3.18)where q α is defined by the relation q α = q (1 / − | α | ) . (3.19)Hence, for a massless field the FC decays as a − (2 q α +1) .For points near the apex of the cone, r →
0, we use the expansion of the modified Besselfunction for small values of the argument. The leading term in the boundary-induced FC takesthe form h ¯ ψψ i b ≈ q π a ( r/ a ) q α − Γ ( q α + 1 / Z ∞ µ dz z q α p z − µ × µW (+) q α − / ,q α +1 / ( z ) − ( z − µ ) /zz [ I q α − / ( z ) + I q α +1 / ( z )] + 2 µI q α − / ( z ) I q α +1 / ( z ) . (3.20)Note that for a massless field this expression reduces to Eq. (3.18). As it is seen, in thelimit r → | α | < / − / (2 q ) and diverges for | α | > / − / (2 q ). Notice that in the former case the irregular mode is absent and thedivergence in the latter case comes from the irregular mode. For the magnetic vortex in thebackground Minkowski spacetime, the boundary-induced contribution diverges as r − | α | . In thecase | α | = 1 / − / (2 q ), corresponding to q α = 1 /
2, the boundary-induced FC tends to a finitelimiting value.The boundary-induced part in the FC diverges on the circle. For points near the circle themain contribution to Eq. (3.14) comes from large values of j . Introducing a new integrationvariable y = z/β j , we use the uniform asymptotic expansion for the modified Bessel functionfor large values of the order. To the leading order in the expansion over (1 − r/a ) one finds thebehavior h ¯ ψψ i b ≈ − π ( a − r ) . (3.21)This leading term does not depend on the opening angle of the cone and on the magnetic flux. Itcoincides with the corresponding term for the FC in the geometry of a circle in (2+1)-dimensionalMinkowski spacetime. This asymptotic behavior is well seen in Fig. 3 where the dependence ofthe FC on the radial coordinate is presented for a massless fermionic field for various values ofthe parameter q . The left/right plot corresponds to the value of the parameter α = 0/ α = 0 . α = 0 . q <
5, vanishes for q > q = 5. In particular,13 = =
15 100.0 0.2 0.4 0.6 0.8 - - - - (cid:144) a a < ΨΨ > b q = Α = - - - - (cid:144) a a < ΨΨ > b Figure 3: The FC inside a circular boundary as a function on the radial coordinate for a masslessfermionic field.for q = 10 one has h ¯ ψψ i ∝ r in the limit r →
0. These properties are well seen from the rightplot of Fig. 3.In Fig. 4, we present the condensate for a massless fermionic field inside a circular boundaryas a function of the magnetic flux. The graphs are plotted for r/a = 0 . q = - - - - - - - Α a < ΨΨ > b Figure 4: The FC for a massless field inside a circular boundary as a function of α . In the region outside a circular boundary the negative-energy eigenspinors, obeying the boundarycondition (3.1) with n µ = − δ µ , have the form [13] ψ ( − ) γj ( x ) = c e − iqjφ + iEt γǫ j e − iqφ/ E + m g β j ,β j + ǫ j ( γa, γr ) g β j ,β j ( γa, γr ) e iqφ/ ! , (4.1)14ith the function g ν,ρ ( x, y ) = ¯ Y ( − ) ν ( x ) J ρ ( y ) − ¯ J ( − ) ν ( x ) Y ρ ( y ) , (4.2)and Y ν ( x ) being the Neumann function. The barred notation in Eq. (4.2) is defined by therelation ¯ F ( − ) β j ( z ) = − ǫ j zF β j + ǫ j ( z ) − ( p z + µ + µ ) F β j ( z ) , (4.3)with F = J, Y and µ = ma . The normalization coefficient is given by the expression c = 2 Eγφ ( E + m ) [ ¯ J ( − )2 β j ( γa ) + ¯ Y ( − )2 β j ( γa )] − . (4.4)The positive-energy eigenspinors are found with the help of the relation ψ (+) γn = σ ψ ( − ) ∗ γn . Notethat for the region under consideration the conical singularity is excluded by the boundary andall modes described by eigenspinors (4.1) are regular.Substituting the eigenspinors into the mode-sum formula (2.5), the FC is written in the form h ¯ ψψ i = q π X j Z ∞ dγ γE ( E − m ) g β j ,β j + ǫ j ( γa, γr ) − ( E + m ) g β j ,β j ( γa, γr )¯ J ( − )2 β j ( γa ) + ¯ Y ( − )2 β j ( γa ) . (4.5)As before, we assume the presence of a cutoff function which makes the expression on the right-hand side of Eq. (4.5) finite. Similar to the interior region, the FC outside a circular boundarymay be written in the decomposed form (3.12).In order to find an explicit expression for the boundary-induced part, we note that theboundary-free part is given by Eq. (2.10). For the evaluation of the difference between the totalFC and the boundary-free part, we use the identity g β j ,λ ( x, y )¯ J ( − )2 β j ( x ) + ¯ Y ( − )2 β j ( x ) − J λ ( y ) = − X l =1 , ¯ J ( − ) β j ( x )¯ H ( − ,l ) β j ( x ) H ( l )2 λ ( y ) , (4.6)with λ = β j , β j + ǫ j , and with H ( l ) ν ( x ) being the Hankel function. For the boundary-inducedpart in the FC we find the expression h ¯ ψψ i b = − q π X j X l =1 , Z ∞ dγ γE ¯ J ( − ) β j ( γa )¯ H ( − ,l ) β j ( γa ) × h ( E − m ) H ( l )2 β j + ǫ j ( γr ) − ( E + m ) H ( l )2 β j ( γr ) i . (4.7)In the complex plane γ , the integrand of the term with l = 1 ( l = 2) decays exponentially in thelimit Im( γ ) → ∞ [Im( γ ) → −∞ ] for r > a . By using these properties, we rotate the integrationcontour in the complex plane γ by the angle π/ l = 1 and by the angle − π/ l = 2. The integrals over the segments (0 , im ) and (0 , − im ) of the imaginaryaxis cancel each other. Introducing the modified Bessel functions, the boundary-induced partin the FC is presented in the form h ¯ ψψ i b = q π X j Z ∞ m dz z × n m K β j ( zr ) − K β j + ǫ j ( zr ) √ z − m Re[ I ( − ) β j ( za ) /K ( − ) β j ( za )] − [ K β j ( zr ) + K β j + ǫ j ( zr )]Im[ I ( − ) β j ( za ) /K ( − ) β j ( za )] o , (4.8)15here F ( − ) ( z ) = zF ′ ( z ) − ( µ + i p z − µ + ǫ j β j ) F ( z ) . (4.9)By using the definition (4.9), the ratio in the integrand of Eq. (4.8) can be written in theform I ( − ) β j ( x ) K ( − ) β j ( x ) = W ( − ) β j ,β j + ǫ j ( x ) + i p − µ /x x [ K β j ( x ) + K β j + ǫ j ( x )] + 2 µK β j ( x ) K β j + ǫ j ( x ) , (4.10)with the notation W ( − ) β j ,β j + ǫ j ( x ) defined by Eq. (3.11). Now the real and imaginary parts ap-pearing in Eq. (4.8) are easily obtained from Eq. (4.10). By taking into account that underthe change α → − α , j → − j , one has β j → β j + ǫ j , β j + ǫ j → β j , we conclude that thereal/imaginary part in Eq. (4.10) is an odd/even function under this change. Now, from Eq.(4.8) it follows that the boundary-induced part in the FC is an even function of α . This functionis periodic with the period equal to 1.For a massless field the expression for the boundary-induced part in the FC simplifies to h ¯ ψψ i b = − q π a X j Z ∞ dz K β j ( zr/a ) + K β j + ǫ j ( zr/a ) K β j ( z ) + K β j + ǫ j ( z ) . (4.11)As in the case of the interior region, the boundary-induced FC does not vanish for a massless filed.The corresponding boundary-free part vanishes and, hence, in this case we have h ¯ ψψ i = h ¯ ψψ i b .When the magnetic flux is absent, α = 0, the corresponding expression for the boundary-inducedpart is obtained from Eq. (4.8) by the replacements (3.15). In particular, for the circle in theMinkowski bulk the formula for the fermionic condensate is obtained from Eq. (3.16) by theinterchange I ⇄ K , replacing W (+) n,n +1 ( x ) → W ( − ) n,n +1 ( x ).Now let us consider the behavior of the boundary-induced part in the FC in the asymptoticregions of the parameters. First we consider the limit a →
0, for fixed values of r . By takinginto account the asymptotics of the modified Bessel functions for small values of the arguments,to the leading order we find the expression h ¯ ψψ i b ≈ q ( a/ r ) q α π r Γ ( q α + 1 / Z ∞ mr dz z q α √ z − m r × [ (cid:0) m r − z (cid:1) K q α − / ( z ) − z K q α +1 / ( z )] , (4.12)with the notation (3.19). For a massless field the integral in (4.12) is evaluated in terms of thegamma function and one has h ¯ ψψ i b ≈ − q Γ( q α + 1)Γ(2 q α + 1 / πr Γ ( q α + 1 / (cid:16) a r (cid:17) q α . (4.13)Hence, in the limit a → r , the boundary-induced part in FC vanishesas a q α .For a massive field, at large distances from the boundary, under the condition mr ≫
1, themain contribution to the integral in Eq. (4.8) comes from the region near the lower limit of theintegration. In the leading order we find h ¯ ψψ i b ≈ − qe − mr πr X j Im[ I (+) β j ( ma ) /K (+) β j ( ma )] . (4.14)and the boundary-induced FC is exponentially suppressed. For a massless field, the asymptoticat large distances is given by Eq. (4.13) and the boundary-induced condensate decays as r − q α − .16or points near the circle the main contribution to (4.11) comes from large values of j . By usingthe uniform asymptotic expansion for the Macdonald function for large values of the order,to the leading order one finds h ¯ ψψ i b ≈ − [8 π ( r − a ) ] − . The leading term in the asymptoticexpansion does not depend on the opening angle of the cone and on the magnetic flux. Thedependence of the FC outside a circular boundary on the radial coordinate is presented in Fig.5 for a massless field for various values of the parameter q . The left/right plot corresponds tothe value of the parameter α = 0/ α = 0 . Α = = - - - - (cid:144) a a < ΨΨ > b Α = =
10 511.5 2.0 2.5 3.0 - - - - - - - (cid:144) a a < ΨΨ > b Figure 5: The FC outside a circular boundary as a function on the radial coordinate for amassless fermionic field.In Fig. 6, the fermionic condensate is plotted for a massless field outside a circular boundaryas a function of the magnetic flux. The graphs are plotted for r/a = 1 . α at half-integer values. In particular, its derivative vanishesat these points. Note that this is not the the case for the interior region. q = - - - - - - - Α a < ΨΨ > b Figure 6: The FC outside a circular boundary as a function of the magnetic flux.17
Half-integer values of the parameter α In this section we consider the FC for half-integer values of the parameter α . In this case forthe boundary-free geometry the eigenspinors with j = − α are still given by Eqs. (2.6). For theeigenspinor corresponding to the special mode with j = − α one has [13] ψ ( − )(0) γ, − α ( x ) = (cid:18) E + mπφ rE (cid:19) / e iqαφ + iEt γe − iqφ/ E + m sin( γr − γ ) e iqφ/ cos( γr − γ ) ! , (5.1)where γ = arccos[ p ( E − m ) / E ]. As we have noted, for half-integer values of α the mode with j = − α corresponds to the irregular mode. The contribution of the modes with j = − α to theFC is the same as before. Special consideration is needed for the mode with j = − α only. Forthe contribution of this mode to the FC one has h ¯ ψψ i ,j = − α = Z ∞ dγ ¯ ψ ( − )(0) γ, − α ψ ( − )(0) γ, − α = − q π r Z ∞ dγ m + γ sin(2 γr ) − m cos(2 γr ) p γ + m . (5.2)The part with the last term in the numerator is finite, whereas the part with the first two termsis divergent. As before, in order to deal with this divergence we introduce the cutoff function e − sγ . The integral in the right-hand side of Eq. (5.2) is expressed in terms of the Macdonaldfunction.For half-integer values of α , it can be easily seen that for the series in the contribution ofthe modes with j = − α one has X j = − α I β j ( x ) = X j = − α I β j + ǫ j ( x ) = ∞ X n =1 (cid:2) I qn − / ( x ) + I qn +1 / ( x ) (cid:3) . (5.3)Summing the contributions from the mode with j = − α and from the modes j = − α , for theregularized FC we find the expression h ¯ ψψ i , reg = − qme m s (2 π ) / ∞ X n =1 Z r / s dx x − / e − m r / x − x √ r − xs (cid:2) I qn − / ( x ) + I qn +1 / ( x ) (cid:3) − qm π r [ e m s/ K ( m s/
2) + 2 K (2 mr ) − K (2 mr )] . (5.4)After the summation over n by using the formula given in Sect. 2, we find the following repre-sentation h ¯ ψψ i , reg = − m π ( e m s √ π Z r / s dx x − / e − m r / x √ r − xs + 1 r p X l =1 cot( πl/q ) e mr sin( πl/q ) + q πr Z ∞ dy sinh( y/
2) sinh( qy )cosh( qy ) − cos( qπ ) e − mr cosh( y/ cosh( y/ ) − qm π r [ K (2 mr ) − K (2 mr )] + o ( s ) , (5.5)where 2 p q < p + 2. The first term in the figure braces of this expression corresponds to thecontribution coming from the Minkowski spacetime part. It is subtracted in the renormalization18rocedure and for the renormalized FC in a boundary-free conical space one finds h ¯ ψψ i , ren = − qm π r Z ∞ dy sinh( y/
2) sinh( qy )cosh( qy ) − cos( qπ ) e − mr cosh( y/ cosh( y/ − m πr p X l =1 cos( πl/q )sin( πl/q ) e − mr sin( πl/q ) − qm π r [ K (2 mr ) − K (2 mr )] . (5.6)As before, the FC is a periodic function of α with the period 1. Note that, in the case underconsideration the renormalized FC in a boundary-free conical space does not vanish for a masslessfield: h ¯ ψψ i , ren = − q π r , m = 0 . (5.7)This corresponds to the contribution of the irregular mode.Now we consider the region inside a circle with radius a . The contribution of the modes with j = − α is given by Eq. (3.13) where now the summation goes over j = − α . For the evaluationof the contribution coming from the mode with j = − α , we note that the negative-energyeigenspinor for this mode has the form [13] ψ ( − ) γ, − α ( x ) = b √ r e iqαφ + iEt γe − iqφ/ E + m sin( γr − γ ) e iqφ/ cos( γr − γ ) ! , (5.8)where γ is defined after Eq. (5.1). From boundary condition (3.1) it follows that the eigenvaluesof γ are solutions of the equation m sin( γa ) + γ cos( γa ) = 0 . (5.9)The positive roots of this equation we denote by γ l = γa , l = 1 , , . . . . From the normalizationcondition, for the coefficient in Eq. (5.8) one has b = E + maEφ [1 − sin(2 γa ) / (2 γa )] − . (5.10)Using Eq. (5.8), for the contribution of the mode under consideration to the FC we find: h ¯ ψψ i j = − α = − aφ r ∞ X l =1 µ + γ l sin(2 γ l r/a ) − µ cos(2 γ l r/a ) q γ l + µ [1 − sin(2 γ l ) / (2 γ l )] , (5.11)where µ = ma and the presence of a cutoff function is assumed. For the summation of the seriesin Eq. (5.11), we use the Abel-Plana-type formula ∞ X l =1 πf ( γ l )1 − sin(2 γ l ) / (2 γ l ) = − πf (0) / /µ + 1 + Z ∞ dz f ( z ) − i Z ∞ dz f ( iz ) − f ( − iz ) z + µz − µ e z + 1 . (5.12)The latter is obtained from the summation formula given in [27] (see also [26]) taking b = 0and b = − /µ . For the functions f ( z ) corresponding to Eq. (5.11) one has f (0) = 0. Thesecond term on the right-hand side of (5.12) gives the part corresponding to the boundary-freegeometry. As a result, the FC is presented in the form h ¯ ψψ i j = − α = h ¯ ψψ i ,j = − α + h ¯ ψψ i b ,j = − α , (5.13)19here the boundary-induced part is given by the expression h ¯ ψψ i b ,j = − α = qπ r Z ∞ m dx m − x sinh(2 xr ) − m cosh(2 xr ) √ x − m (cid:16) x + mx − m e ax + 1 (cid:17) . (5.14)The contribution of the modes j = − α remains the same and is obtained from the correspondingexpressions given above for non-half-integer values of α by the direct substitution α = 1 / h ¯ ψψ i b ,j = − α = − q π r (cid:20) πr/a sin( πr/a ) − (cid:21) . (5.15)Note that this part is finite at the circle center. By taking into account Eq. (5.7) and addingthe contribution coming from the modes with j = − α , for the total FC one finds h ¯ ψψ i = − q πar sin( πr/a ) − qπ a ∞ X n =1 Z ∞ dz I qn − / ( zr/a ) + I qn +1 / ( zr/a ) I qn − / ( z ) + I qn +1 / ( z ) . (5.16)The expression on the right-hand side is always negative. The first term dominates near the coneapex. Near the boundary this term behaves as (1 − r/a ) − , whereas the second term behaveslike (1 − r/a ) − . Hence, the latter dominates near the circle.In the region outside a circular boundary there are no irregular modes and the FC is acontinuous function of the parameter α at half-integer values. The corresponding expression isobtained taking the limit α → / h ¯ ψψ i = lim α → / [ h ¯ ψψ i , ren + h ¯ ψψ i b ], where the separateterms are given by expressions (2.28) and (4.8). However, note that the limiting values of theseparate terms h ¯ ψψ i , ren and h ¯ ψψ i b , defined by these expressions, do not coincide with theboundary-free and boundary-induced parts of the FC at half-integer values of α . In this paper we have investigated the FC in a (2+1)-dimensional conical spacetime with acircular boundary in the presence of a magnetic flux. The case of massive fermionic field isconsidered with the MIT bag boundary condition on the circle. As the first step we haveconsidered a conical space without boundaries and with a special case of boundary conditionsat the cone apex, when the MIT bag boundary condition is imposed at a finite radius, which isthen taken to zero. For the evaluation of the FC the direct summation over the modes is usedwith the spinorial eigenfunctions (2.6). If the ratio of the magnetic flux to the flux quantum isnot a half-integer number, the regularized FC with the exponential cutoff function is given byexpression (2.13). A simple expression for the renormalized FC, Eq. (2.17), is obtained in thespecial case when the parameter q is an integer and is related to the parameter α by Eq. (2.14).In this special case the renormalized FC vanishes for a massless field and for a massive field ina conical space with q = 2 and is negative for other cases.For the general case of the parameters q and α , a convenient expression for the regularizedFC is obtained by using the integral representation (2.23) for the series involving the modifiedBessel function. This formula allows us to extract explicitly the part in FC corresponding tothe Minkowski spacetime in the absence of the magnetic flux. Subtracting this part, for therenormalized FC we derived formula (2.28). At distances larger than the Compton wavelengthof the spinor particle, mr ≫
1, the FC is suppressed by the factor e − mr for 1 q < e − mr sin( π/q ) for q >
2. In the special case when the magnetic flux is absent thegeneral formula simplifies to Eq. (2.31). Another limiting case corresponds to the magnetic20ux in background of Minkowski spacetime with the renormalized FC given by Eq. (2.33).An alternative expression for the FC is obtained by using the integral representation (2.34)for the series involving the modified Bessel function. This leads to the expression (2.36) forthe renormalized FC. In the special cases of a magnetic flux in background of the Minkowskispacetime and for a conical space in the absence of the magnetic flux the general formula reducesto Eqs. (2.39) and (2.40), respectively.In Section 3 we have considered the FC inside a circular boundary concentric with theapex of the cone. The corresponding eigenspinors are given by the expression (3.2) and theeigenvalues of the quantum number γ are solutions of Eq. (3.3). The mode-sum for the FCcontains series over these solutions. For the summation of this series we have used the Abel-Plana-type formula (3.8). This allows us to decompose the FC into the boundary-free andboundary-induced parts, Eq. (3.12), with the boundary-induced part given by Eq. (3.13). Theasymptotic near the cone apex is given by Eq. (3.20). In this limit the boundary-induced partvanishes when | α | < / − / (2 q ) and diverges for | α | > / − / (2 q ). In the former case theirregular mode is absent and the divergence in the latter case comes from the irregular mode.The boundary-induced FC diverges on the circle. The leading term in the asymptotic expansionover the distance from the boundary is given by Eq. (3.21). This term does not depend on theopening angle of the cone and on the magnetic flux and coincides with the corresponding termfor the FC in the geometry of a circle in (2+1)-dimensional Minkowski spacetime.The region outside a circular boundary is considered in Section 4. The boundary-inducedpart of the FC in this region is given by Eq. (4.8). This expression is obtained from thecorresponding formula for the interior region by the interchange of the modified Bessel functions I and K . For a massless field the general formula is simplified to Eq. (4.11) and the boundary-induced part is negative. In the limit when the circle radius tends to zero, a →
0, and for afixed value of r , the boundary-induced part in FC vanishes as a q α . At large distances from theboundary, for a massive field, the asymptotic behavior is given by Eq. (4.14) and the boundary-induced FC is exponentially suppressed. For a massless field, the asymptotic at large distancesis given by Eq. (4.13) and the boundary-induced condensate decays as r − q α − .The special case of the magnetic flux corresponding to half-integer values of the parameter α is discussed in Section 5. For this case the contribution of the mode with j = − α shouldbe considered separately. The renormalized FC in the boundary-free geometry is given by Eq.(5.6) and does not vanish in the massless limit. In the region inside a circular boundary thecontribution of the special mode with j = − α to the FC is given by Eq. (5.14) and is finite atthe circle center. For a massless fermionic field the total FC inside a circular boundary is givenby Eq. (5.16) and is negative. In the region outside a circular boundary the FC is a continuousfunction of the parameter α at half-integer values and the corresponding expression is obtainedfrom that in Section 4 taking the limit α → / Acknowledgments
E.R.B.M. thanks Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq) forpartial financial support. A.A.S. would like to acknowledge the hospitality of the INFN Labo-ratori Nazionali di Frascati, Frascati, Italy.
References [1] S. Deser, R. Jackiw and S. Templeton, Ann. Phys. , 372 (1982); A.J. Niemi and G.W.Semenoff, Phys. Rev. Lett. , 2077 (1983); R. Jackiw, Phys. Rev. D , 2375 (1984); A.N.21edlich, Phys. Rev. D , 2366 (1984); M.B. Paranjape, Phys. Rev. Lett. , 2390 (1985);D. Boyanovsky and R. Blankenbecler, Phys. Rev. D , 3234 (1985); R. Blankenbecler andD. Boyanovsky, Phys. Rev. D , 612 (1986).[2] T. Jaroszewicz, Phys. Rev. D , 3128 (1986).[3] E.G. Flekkøy and J.M. Leinaas, Int. J. Mod. Phys. A , 5327 (1991).[4] H. Li, D.A. Coker, and A.S. Goldhaber, Phys. Rev. D , 694 (1993).[5] V.P. Gusynin, V.A. Miransky and L.A. Shovkovy, Phys. Rev. D , 4718 (1995); R.R.Parwani, Phys. Lett. B , 101 (1995).[6] Yu.A. Sitenko, Phys. At. Nucl. , 2102 (1997); Yu.A. Sitenko, Phys. Rev. D , 125017(1999).[7] G.V. Dunne, Topological Aspects of Low Dimensional Systems (Springer, Berlin, 1999).[8] A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, and A.K. Geim, Rev. Mod.Phys. , 109 (2009).[9] S. Bellucci and A.A. Saharian, Phys. Rev. D , 085019 (2009); S. Bellucci and A.A. Sa-harian, Phys. Rev. D , 105003 (2009); S. Bellucci, A.A. Saharian, and V.M. Bardeghyan,Phys. Rev. D , 065011 (2010).[10] Yu.A. Sitenko and N.D. Vlasii, Low Temp. Phys. , 826 (2008).[11] S. Leseduarte and A. Romeo, Commun. Math. Phys. , 317 (1998).[12] C.G. Beneventano, M. De Francia, K. Kirsten, and E.M. Santangelo, Phys. Rev. D ,085019 (2000); M. De Francia and K. Kirsten, Phys. Rev. D , 065021 (2001).[13] E.R. Bezerra de Mello, V.B. Bezerra, A.A. Saharian, and V.M. Bardeghyan, Phys. Rev. D , 085033 (2010).[14] I.L. Buchbinder and E.N. Kirillova, Int. J. Mod. Phys. A , 143 (1989); E. Elizalde, S.D.Odintsov, and Yu.I. Shil’nov, Mod. Phys. Lett. A , 931 (1994); E. Elizalde, S. Leseduarte,and S.D. Odintsov, Phys. Rev. D , 5551 (1994); E. Elizalde, S. Leseduarte, and S.D.Odintsov, Phys. Lett. B , 33 (1995); D.K. Kim and G. Koh, Phys. Rev. D , 4573(1995); E. Elizalde and S.D. Odintsov, Phys. Rev. D , 5990 (1995); E. Elizalde, S.Leseduarte, S.D. Odintsov, and Yu.I. Shil’nov, Phys. Rev. D , 1917 (1996).[15] I. Brevik and T. Toverud, Class. Quantum Grav. , 1229 (1995).[16] E.R. Bezerra de Mello, V.B. Bezerra, A.A. Saharian, and A.S. Tarloyan, Phys. Rev. D ,025017 (2006).[17] E.R. Bezerra de Mello, V.B. Bezerra, and A.A. Saharian, Phys. Lett. B , 245 (2007).[18] E.R. Bezerra de Mello, V.B. Bezerra, A.A. Saharian, and A.S. Tarloyan, Phys. Rev. D ,105007 (2008).[19] A.A. Saharian, Classical Quantum Gravity , 165012 (2008); E.R. Bezerra de Mello andA. A. Saharian, J. High Energy Phys. (2008) 081.2220] P.E. Lammert and V.H. Crespi, Phys. Rev. Lett. , 5190 (2000); A. Cortijo and M.A.H.Vozmediano, Nucl. Phys. B , 293 (2007); Yu.A. Sitenko and N.D. Vlasii, Nucl. Phys.B , 241 (2007); C. Furtado, F. Moraes, and A.M.M. Carvalho, Phys. Lett. A ,5368 (2008); A. Jorio, G. Dresselhaus and M.S. Dresselhaus, Carbon Nanotubes: AdvancedTopics in the Synthesis, Structure, Properties and Applications (Springer, Berlin, 2008).[21] A. Krishnan, et al, Nature , 451 (1997); S.N. Naess, A. Elgsaeter, G. Helgesen and K.D.Knudsen, Sci. Technol. Adv. Mater. , 065002 (2009).[22] P. de Sousa Gerbert and R. Jackiw, Commun. Math. Phys. , 229 (1989); P. de SousaGerbert, Phys. Rev. D , 1346 (1989); Yu.A. Sitenko, Ann. Phys. , 167 (2000).[23] A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series (Gordon andBreach, New York, 1986), Vol. 2.[24] E. R. Bezerra de Mello, Classical Quantum Gravity , 095017 (2010).[25] A.A. Saharian and E.R. Bezerra de Mello, J. Phys. A: Math. Gen. , 3543 (2004).[26] A.A. Saharian, The Generalized Abel-Plana Formula with Applications to Bessel Functionsand Casimir Effect (Yerevan State University Publishing House, Yerevan, 2008); ReportNo. ICTP/2007/082; arXiv:0708.1187.[27] A. Romeo and A.A. Saharian, J. Phys. A35