Casimir effect for fermion condensate in conical rings
aa r X i v : . [ h e p - t h ] F e b Casimir effect for fermion condensate in conical rings
A. A. Saharian , T. A. Petrosyan , A. A. Hovhannisyan Department of Physics, Yerevan State University,1 Alex Manoogian Street, 0025 Yerevan, Armenia Institute of Applied Problems of Physics NAS RA,25 Nersessian Street, 0014 Yerevan, Armenia
Abstract
The fermion condensate (FC) is investigated for a (2+1)-dimensional massive fermionic fieldconfined on a truncated cone with an arbitrary planar angle deficit and threaded by a magneticflux. Different combinations of the boundary conditions are imposed on the edges of the cone.They include the bag boundary condition as a special case. By using the generalized Abel-Plana-type summation formula for the series over the eigenvalues of the radial quantum number, theedge-induced contributions in the FC are explicitly extracted. The FC is an even periodic func-tion of the magnetic flux with the period equal to the flux quantum. Depending on the boundaryconditions, the condensate can be either positive or negative. For a massless field the FC in theboundary-free conical geometry vanishes and the nonzero contributions are purely edge-inducedeffects. This provides a mechanism for time-reversal symmetry breaking in the absence of magneticfields. Combining the results for the fields corresponding to two inequivalent irreducible represen-tations of the Clifford algebra, the FC is investigated in the parity and time-reversal symmetricfermionic models and applications are discussed for graphitic cones.
Keywords : fermion condensate; Casimir effect; conical geometry; graphene
Field theoretical fermionic models in (2+1)-dimensional spacetime appear as long-wavelength effectivetheories describing a relatively large class of condensed matter systems, including graphene family ma-terials, topological insulators, Weyl semimetals, high-temperature superconductors, ultracold atomsconfined by lattice potentials, and nano-patterned 2D electron gases [1, 2]. In the low-energy approx-imation, the corresponding dynamics of charge carriers is governed with fairly good accuracy by theDirac equation, where the velocity of light is replaced by the Fermi velocity [3]-[5]. The latter is muchless than the velocity of light, and this presents a unique possibility for studying relativistic effects.Among the most interesting topics in quantum field theory is the dependence of the properties ofthe vacuum state on the geometry of the background spacetime. The emergence of Dirac fermions inthe abovementioned condensed matter systems and availability of a number of mechanisms to controlthe corresponding effective geometry provide an important opportunity to observe different kinds offield-theoretical effects induced by the spatial geometry and topology. In particular, it is of specialinterest to investigate the influence of boundaries on the physical characteristics of the ground state.This influence can be described by imposing appropriate boundary conditions on the field operator.Those conditions modify the spectrum of vacuum fluctuations and, as a consequence, the vacuumexpectation values of physical observables are shifted by an amount that depends on the bulk and1oundary geometries and on the boundary conditions. The general class of those effects is knownunder the name of the Casimir effect (for reviews see [6]-[10]). In recent years, the Casimir effectfor the electromagnetic field in physical systems with graphene structures as boundaries has beenwidely discussed in the literature (see references [11]-[34] and references [35]-[37] for reviews). Byusing external fields, different electronic phases can be realized in Dirac materials. The magnitudeand the scaling law of the corresponding Casimir forces are essentially different for those phases [38].New interesting features arise in interacting fermionic systems [39]-[44].Graphene family materials also offer a unique opportunity to investigate the boundary-inducedand topological Casimir effects for a fermionic field. On the edges of graphene nanoribbons boundaryconditions are imposed on the effective fermionic field that ensure the zero flux of the quasiparticles.Those conditions are sources for the Casimir-type contributions to the expectation values of physicalcharacteristics of the ground state. Similarly, the periodicity conditions along compact dimensionsimposed on the fermionic field in graphene nanotubes and nanorings give rise to the topologicalCasimir effect for those characteristics. As such characteristics, in [45, 46, 47] the fermion condensate,the expectation values of the current density and of the energy-momentum tensor have been studied.The edge-induced Casimir contributions in finite length carbon nanotubes were discussed in [48]-[50].Tubes with more complicated curved geometries have been considered in [51]-[54]. These geometriesprovide exactly solvable examples to model the combined influence of gravity and topology on theproperties of quantum matter. Note that various mechanisms have been considered in the literaturethat allow to control the effective geometry in graphene type materials [55]-[58].As background geometry, in the present paper we consider a 2-dimensional conical space with twocircular boundaries (conical ring). The corresponding spacetime is flat and is a (2+1)-dimensionalanalog of the cosmic string geometry. We investigate the influence of the edges and of the magneticflux, threading the ring, on the fermion condensate (FC). The corresponding vacuum expectationvalues of the fermionic charge and current densities have been recently studied in [59]. Among theinteresting applications of the setup under consideration are the graphitic cones. They are obtainedfrom a graphene sheet by cutting a sector with the angle πn c / n c = 1 , , . . . ,
5, and then appropriatelygluing the edges of the remaining sector. The opening angle of the cone, obtained in this way, is givenby φ = 2 π (1 − n c / φ for all the values corresponding to n c = 1 , , . . . ,
5, have been observed experimentally [60]-[62]. The corresponding electronic propertieswere studied in references [63]-[70]. Our main interest here is the investigation of the Casimir-typecontributions to the FC induced by the edges of a conical ring for general values of the opening angle.The ground state fermionic expectation values for limiting cases of the geometry under consideration,corresponding to boundary-free cones and to cones with a single circular edge, have been examined inreferences [71]-[76]. In particular, the FC has been discussed in [74]. The effects of finite temperatureon the FC were investigated in [77, 78]. The vacuum expectation values for the charge and currentdensities on planar rings have been studied in [79].The paper is organized as follows. In the next section we describe the geometry and present thecomplete set of fermionic modes. Based on those modes, the FC is evaluated in Section 3. Variousrepresentations are provided for the edge-induced contributions and numerical results are presented. InSection 4, by combining the results for the fields realizing two inequivalent irreducible representationsof the Clifford algebra, we consider the FC in parity and time-reversal symmetric models. Applicationsto graphitic cones are discussed. The main results are summarized in Section 5.
We consider a charged fermionic field in (2+1)-dimensional conical spacetime described by the coor-dinates x = t , x = r , x = φ , with r >
0, 0 φ φ . The corresponding metric tensor is givenby g µν = diag(1 , − , − r ) . (1)2or φ = 2 π this metric tensor corresponds to (2+1)-dimensional Minkowski spacetime. For φ < π one has a planar angle deficit 2 π − φ and the spacetime is flat everywhere except at the apex r = 0 where it has a delta type curvature singularity. In (2+1)-dimensional spacetime there aretwo inequivalent irreducible representations of the Clifford algebra with the 2 × γ µ ( s ) = ( γ , γ , γ s ) ), where s = ± γ = diag(1 , −
1) and γ = i (cid:18) e − iqφ e iqφ (cid:19) , γ s ) = sr (cid:18) e − iqφ − e iqφ (cid:19) , (2)where q = 2 π/φ . Note that one has the relation γ s ) = − isγ γ /r .Let ψ ( s ) , s = ±
1, be two-component spinor fields corresponding to two inequivalent irreduciblerepresentations of the Clifford algebra. In the presence of an external gauge field A µ , the correspondingLagrangian density has the form L ( s ) = ¯ ψ ( s ) ( iγ µ ( s ) D ( s ) µ − m ( s ) ) ψ ( s ) (3)with the covariant derivative operator D ( s ) µ = ∂ µ + Γ ( s ) µ + ieA µ , the spin connection Γ ( s ) µ and theDirac adjoint ¯ ψ ( s ) = ψ † ( s ) γ . We are interested in the effects of two circular boundaries r = a and r = b , a < b , on the fermion condensate (FC) h | ¯ ψ ( s ) ψ ( s ) | i ≡ h ¯ ψ ( s ) ψ ( s ) i , (4)where | i corresponds to the vacuum state. On the edges the boundary conditions (cid:16) iλ ( s ) r n ( r ) µ γ µ ( s ) (cid:17) ψ ( s ) = 0 , r = a, b, (5)will be imposed with λ ( s ) r = ± n ( r ) µ being the inward pointing unit vector normal tothe corresponding boundary. We can pass to the new set of fields ψ ′ ( s ) defined as ψ ′ (+1) = ψ (+1) , ψ ′ ( − = γ γ ψ ( − . The corresponding Lagrangian density is presented as L ( s ) = ¯ ψ ′ ( s ) ( iγ µ D µ − sm ( s ) ) ψ ′ ( s ) , where γ µ = γ µ (+1) and D µ = D (+1) µ . The boundary conditions are transformed to (cid:16) iλ ( s ) ′ r n ( r ) µ γ µ (cid:17) ψ ′ ( s ) = 0, with λ ( s ) ′ r = sλ ( s ) r and r = a, b . By taking into account that ψ ( − = γ γ ψ ′ ( − , for the FC we get h ¯ ψ ( s ) ψ ( s ) i = h ¯ ψ ′ ( s ) ψ ′ ( s ) i . The boundary condition (5) with λ ( s ) r = 1 hasbeen used in MIT bag models to confine the quarks inside hadrons (for a review see [80]). In condensedmatter applications it is known as infinite mass or hard wall boundary condition [81]. As it has beenmentioned in [81], another possibility to confine the fermions corresponds to the condition (5) with λ ( s ) r = −
1. Note that one has (cid:16) in ( r ) µ γ µ ( s ) (cid:17) = 1 and for the eigenvalues of the matrix in ( r ) µ γ µ ( s ) we get ±
1. Here, the upper and lower signs correspond to the boundary conditions (5) with λ ( s ) r = − λ ( s ) r = 1, respectively. More general boundary conditions for the confinement of fermions, containingadditional parameters, have been discussed in [82]-[87].In the discussion below, the investigation for the FC will be presented in terms of the fields ψ ′ ( s ) = ψ ,omitting the prime and the index. So, we consider a two-component fermionic field ψ ( x ) obeying theDirac equation ( iγ µ D µ − sm ) ψ ( x ) = 0 , (6)and the boundary conditions (cid:16) iλ r n ( r ) µ γ µ (cid:17) ψ ( x ) = 0 , r = a, b, (7)3here λ r = sλ ( s ) r take the values ±
1. We will consider the FC in the region a r b where n ( u ) µ = n u δ µ , with n a = − n b = 1. Note that the topology of the conical ring is nontrivial andthe periodicity condition on the field should be specified along the φ -direction as well. Here we imposea quasiperiodicity condition with a phase 2 πχ : ψ ( t, r, φ + φ ) = e πiχ ψ ( t, r, φ ) . (8)The special cases include the untwisted and twisted fermionic fields with χ = 0 and χ = 1 /
2, respec-tively. For the external gauge field we assume a simple form with the covariant components A µ = Aδ µ .This corresponds to a magnetic flux threading the ring and localized in the region r < a . The phys-ical azimuthal component of the vector potential is given as A φ = − A/r and it corresponds to themagnetic flux Φ = − φ A . The effect of this flux on the properties of the fermionic vacuum is purelytopological and the results given below do not depend on the profile of the magnetic field sourcing theflux. The spatial geometry of the problem under consideration with the magnetic flux is presented inFigure 1. Figure 1: The geometry of a conical ring threaded by a magnetic flux.The ground state FC is expressed in terms of the fermion two-point function S (1) ( x, x ′ ) as (cid:10) ¯ ψψ (cid:11) = − lim x ′ → x Tr( S (1) ( x, x ′ )) . (9)The two-point function describes the correlations of the vacuum fluctuations and is defined as the VEV S (1) ik ( x, x ′ ) = h | [ ψ i ( x ) , ¯ ψ k ( x ′ )] | i with the spinor indices i and k . The trace in (9) is taken over thoseindices. The FC plays an important role in discussions of chiral symmetry breaking and dynamicalmass generation for fermionic fields. Expanding the fermionic operator in terms of a complete setof the positive and negative energy mode functions ψ (+) σ and ψ ( − ) σ , obeying the conditions (7), (8),and using the anticommutation relations for the fermionic annihilation and creation operators, thefollowing mode sum is obtained for the FC (cid:10) ¯ ψψ (cid:11) = − X σ X κ = − , + κ ¯ ψ ( κ ) σ ψ ( κ ) σ . (10)Here, σ stands for the complete set of quantum numbers specifying the solutions of the equation (6), P σ is understood as a summation for discrete components and as an integration for continuous ones.In the problem under consideration for the mode functions in the region a r b one has [59] ψ ( κ ) σ = C κ e iq ( j + χ ) φ − κiEt g β j ,β j ( γa, γr ) e − iqφ/ ǫ j γe iqφ/ κE + sm g β j ,β j + ǫ j ( γa, γr ) ! , (11)4here E = p γ + m is the energy, j = ± / , ± / , . . . , ǫ j = 1 for j > − α and ǫ j = − j < − α , β j = q | j + α | − ǫ j / . (12)Here and in what follows α = χ + eA/q = χ − e Φ / (2 π ) . (13)The radial functions in (11) are given by the expression g β j ,ν ( γa, γr ) = Y ( a ) β j ( γa ) J ν ( γr ) − J ( a ) β j ( γa ) Y ν ( γr ) , (14)with the Bessel and Neumann functions J ν ( x ), Y ν ( γr ), and with the notation f ( u ) β j ( x ) = λ u n u ( κ p x + m u + sm u ) f β j ( x ) − ǫ j xf β j + ǫ j ( x ) , (15)for f = J, Y , u = a, b , and m u = mu .The mode functions (11) obey the boundary condition on the edge r = a . The eigenvalues of theradial quantum number γ are determined by the boundary condition on the edge r = b . They aresolutions of the equation C β j ( b/a, γa ) ≡ J ( a ) β j ( γa ) Y ( b ) β j ( γb ) − J ( b ) β j ( γb ) Y ( a ) β j ( γa ) = 0 . (16)We will denote by γ = γ l , l = 1 , , . . . , the positive roots of this equation. The eigenvalues of γ areexpressed as γ = γ l = z l /a . Note that under the change ( α, j ) → ( − α, − j ) one has β j → β j + ǫ j and β j + ǫ j → β j . From here we can see that under the change( κ, α, j ) → ( − κ, − α, − j ) (17)we get f ( u ) β j ( uγ ) → − ǫ j ( λ u n u /u ) ( κE + sm ) f ( u ) β j ( uγ ) , (18)and, hence, the roots γ l are invariant under the transformation (17).The normalization coefficient is given by | C κ | = πqz a E + κsmE T abβ j ( z ) , (19)where z = z l = γ l a , E = p z /a + m , and we have defined the function T abβ j ( z ) = zE + κsm B b J ( a )2 β j ( z ) J ( b )2 β j ( zb/a ) − B a − , (20)with B u = u (cid:20) E − κλ u n u u (cid:18) E − κsm E + ǫ j β j (cid:19)(cid:21) . (21)Note that the parameters χ and Φ enter in the expression of the mode functions in the gauge-invariantcombination α . This shows that the phase χ in the quasiperiodicity condition (8) is equivalent to amagnetic flux − πχ/e threading the ring and vice versa.In addition to an infinite number of modes with γ = γ l , depending on the boundary conditions,one can have a mode with γa = iη , η >
0. As it has been shown in [59], for that mode η ma and,hence, E >
0. This means that under the boundary conditions (7) the vacuum state is always stable.For half-integer values of the parameter α , in addition to the modes with j = − α and discussedabove, a special mode with j = − α is present. The upper and lower components of the corresponding5ode functions are expressed in terms of the trigonometric functions. These mode functions and theequation determining the eigenvalues of the radial quantum number are given in [59]. For j = − α and for boundary conditions with λ b = − λ a , one has also a zero energy mode with γ = im . In a waysimilar to that discussed in [59] for the vacuum expectation values of the charge and current densities,it can be seen that the special mode with j = − α and E = 0 does not contribute to the FC. Thelatter is a consequence of the cancellation of the contributions coming from the positive and negativeenergy modes. The contribution of the zero energy mode to the FC is zero as well. Note that thelatter is not the case for the expectation values of the charge and current densities. Given the complete set of fermionic modes, the FC on the conical ring is obtained by using the modesum formula (10). First let us consider the case when all the roots of the eigenvalue equation are real.Substituting the mode functions (11), the FC in the region a r b is presented in the form (cid:10) ¯ ψψ (cid:11) = − πq a X j X κ = ± ∞ X l =1 T abβ j ( z ) zE h ( sm + κE ) g β j ,β j ( z, zr/a ) + ( sm − κE ) g β j ,β j + ǫ j ( z, zr/a ) i z = z l , (22)where E = p z /a + m and the summation goes over j = ± / , ± / , . . . . The operators ¯ ψ and ψ in the left-hand side of (22) are taken at the same spacetime point and the expression on the right-hand side is divergent. Various regularization schemes can be used to make the expression finite. Tobe specific, we will assume that the regularization is done by introducing a cutoff function withoutwriting it explicitly. The final result for the renormalized FC does not depend on the specific form ofthat function. By taking into account that the roots z l are invariant under the transformation (17)and by using the transformation rule (18) we can see that the FC is an even periodic function of theparameter α , defined by (13), with the period 1. In particular, we have periodicity with respect to theenclosed magnetic flux with the period equal to the flux quantum 2 π/e . If we present the parameter α in the form α = n + α , with | α | / n being an integer, then the FC will depend on thefractional part α only. Note that the vacuum expectation values of the charge and current densitiesare odd periodic functions of the magnetic flux with the same period.An alternative representation of the FC is obtained from (22) by using the Abel-Plana-type formula[88, 89] ∞ X l =1 w ( z l ) T abβ j ( z l ) = 4 π Z ∞ dx w ( x ) J ( a )2 β j ( x ) + Y ( a )2 β j ( x ) − π Res z =0 w ( z ) H (1 b ) β j ( zb/a ) C β j ( b/a, z ) H (1 a ) β j ( z ) − π Z ∞ dx X p =+ , − w ( xe piπ/ ) K ( bp ) β j ( xb/a ) /K ( ap ) β j ( x ) K ( ap ) β j ( x ) I ( bp ) β j ( xb/a ) − I ( ap ) β j ( x ) K ( bp ) β j ( xb/a ) . (23)for the function w ( z ) analytic in the half-plane Re z > z . Here and below H ( l ) ν ( x ), with l = 1 ,
2, are the Hankel functions and the notation H ( lu ) β j ( x ) is defined in accordancewith (15). In the second integral on the right-hand side of (23), for the modified Bessel functions f ν ( x ) = I ν ( x ) , K ν ( x ), the notations f ( up ) β j ( x ) = δ f xf β j + ǫ j ( x ) + λ u n u (cid:20) κ q(cid:0) xe pπi/ (cid:1) + m u + sm u (cid:21) f β j ( x ) , (24)are introduced with p = + , − , u = a, b , and δ I = 1 , δ K = − . (25)6dditional conditions on the function w ( z ) are given in [89]. By taking into account that q(cid:0) xe pπi/ (cid:1) + m u = (cid:26) p m u − x , x < m u ,pi p x − m u , x > m u , (26)for x >
0, we see that f ( u +) β j ( x ) = f ( u − ) β j ( x ) in the range x ∈ [0 , m u ]. In addition, for the function w ( x ) corresponding to the series over l in (22) one has w ( xe − iπ/ ) = − w ( xe iπ/ ) for x ∈ [0 , m a ]. Fromthese properties it follows that for the FC the integrand of the last integral in (23) vanishes in theintegration range x ∈ [0 , m a ]. It can also be seen that for the FC the residue in (23) is zero.As a result, applying the formula (23) for the series over l in (22), the FC in the region a r b is decomposed into two contributions. The first one, denoted below as (cid:10) ¯ ψψ (cid:11) a , comes from the firstterm in the right-hand side of (23) and is presented in the form (cid:10) ¯ ψψ (cid:11) a = − q πa X j X κ = ± Z ∞ dz zE ( sm + κE ) g β j ,β j ( z, zr/a ) + ( sm − κE ) g β j ,β j + ǫ j ( z, zr/a ) J ( a )2 β j ( z ) + Y ( a )2 β j ( z ) . (27)The second contribution comes from the last term in (23). Introducing the modified Bessel functions,we get the following representation of the FC: (cid:10) ¯ ψψ (cid:11) = (cid:10) ¯ ψψ (cid:11) a + q π ∞ X n =0 X p = ± Z ∞ m dx x √ x − m Re ( K ( b ) n p ( bx ) /K ( a ) n p ( ax ) G ( ab ) n p ( ax, bx ) × h ( sm + i p x − m ) G ( a )2 n p ,n p ( ax, rx ) − ( sm − i p x − m ) G ( a )2 n p ,n p +1 ( ax, rx ) io , (28)where instead of the summation over j we have introduced the summation over n with n p = q ( n + 1 / pα ) − / . (29)Here and in what follows, for the functions f ν ( z ) = I ν ( z ) , K ν ( z ) we use the notation f ( u ) n p ( z ) = δ f zf n p +1 ( z ) + λ u n u ( i p z − m u + sm u ) f n p ( z ) , (30)with u = a, b , and the functions in the right-hand side of (28) are defined by G ( u ) n p ,ν ( x, y ) = K ( u ) n p ( x ) I ν ( y ) − ( − ν − n p I ( u ) n p ( x ) K ν ( y ) ,G ( ab ) n p ( x, y ) = K ( a ) n p ( x ) I ( b ) n p ( y ) − I ( a ) n p ( x ) K ( b ) n p ( y ) . (31)Similar to the case of the charge and current densities, discussed in [59], it can be shown that theexpression (28) is valid in the presence of bound states as well. Under the replacements λ u → − λ u , s → − s we have f ( u ) n p ( z ) → h f ( u ) n p ( z ) i ∗ and the last term in (28) changes the sign.The term (cid:10) ¯ ψψ (cid:11) a in (28) does not depend on b . By using the asymptotic formulas for the modifiedBessel functions (see, for example, [90]), it can be seen that in the limit b → ∞ the last term in (28)behaves as e − mb for a massive field and like ( a/b ) q (1 − | α | )+1 for a massless field. From here it followsthat (cid:10) ¯ ψψ (cid:11) a = lim b →∞ (cid:10) ¯ ψψ (cid:11) and the contribution (cid:10) ¯ ψψ (cid:11) a presents the FC in the region a r < ∞ of (2+1)-dimensional conical spacetime for a fermionic field obeying the boundary condition (7) at r = a . Hence, the last term in (28) is interpreted as the contribution induced by the second boundaryat r = b when we add it to the conical geometry with a single edge at r = a . In order to furtherextract the edge-induced contribution in (cid:10) ¯ ψψ (cid:11) a we use the relation g β j ,ν ( z, y ) J ( a )2 β j ( z ) + Y ( a )2 β j ( z ) = J ν ( y ) − X l =1 , J ( a ) β j ( z ) H ( l )2 ν ( y )2 H ( la ) β j ( z ) . (32)7here ν = β j , β j + ǫ j . This relation is easily obtained by taking into account that J ( a )2 β j ( z ) + Y ( a )2 β j ( z ) = H (1 a ) β j ( z ) H (2 a ) β j ( z ). Applying (32) for separate terms in (27), we can see that the part in the FC comingfrom the first term in the right-hand side of (32), denoted here by (cid:10) ¯ ψψ (cid:11) , does not depend on a andis presented as (cid:10) ¯ ψψ (cid:11) = − qsm π X j Z ∞ dx x J β j ( xr ) + J β j + ǫ j ( xr ) √ x + m . (33)This part corresponds to the FC in a boundary-free conical space and has been investigated in [74]for the case s = 1. The corresponding renormalized value is given by the expression h ¯ ψψ i , ren = − sm πr n [ q/ X l =1 ( − l cot( πl/q ) e mr sin( πl/q ) cos(2 πlα )+ qπ X δ = ± cos [ qπ (1 / δα )] Z ∞ dy tanh ye mr cosh y sinh[ q (1 − δα ) y ]cosh(2 qy ) − cos( qπ ) o , (34)where [ q/
2] is the integer part of q/
2. Note that for points away from the edges of the conical ringthe boundary-induced contribution in the FC is finite and the renormalization is required for theboundary-free part only.The contribution to the FC (cid:10) ¯ ψψ (cid:11) a coming from the last term in (32) is induced by the edge at r = a in the region a r < ∞ . That contribution is further transformed by rotating the integrationcontour over z by the angle π/ H (1) β j ( zr/a ), H (1) β j + ǫ j ( zr/a ),and by the angle − π/ H (2) β j ( zr/a ), H (2) β j + ǫ j ( zr/a ). The parts of theintegrals over the intervals [0 , ima ] and [0 , − ima ] cancel each other. Introducing in the remainingintegrals the modified Bessel functions the FC (cid:10) ¯ ψψ (cid:11) a is presented in the form (cid:10) ¯ ψψ (cid:11) a = h ¯ ψψ i , ren + q π ∞ X n =0 X p = ± Z ∞ m dx x √ x − m Re ( I ( a ) n p ( ax ) K ( a ) n p ( ax ) × h(cid:16) sm + i p x − m (cid:17) K n p ( rx ) − (cid:16) sm − i p x − m (cid:17) K n p +1 ( rx ) io , (35)where n p is defined by (29). It can be seen that in the special case s = 1, λ a = 1 this expression coin-cides with the result from [74]. The condensate given by (35) changes the sign under the replacement( s, λ a ) → ( − s, − λ a ). Combining this property with the corresponding behaviour of the last term in(28), we conclude that the FC (cid:10) ¯ ψψ (cid:11) in the region a r b changes the sign under the transformation( s, λ a , λ b ) → ( − s, − λ a , − λ b ) . (36)For a massless field the FC in the boundary-free geometry vanishes and the single edge inducedcontribution is simplified to (see also [74] for the boundary condition with λ a = 1) (cid:10) ¯ ψψ (cid:11) a = − λ a q π a ∞ X n =0 X p = ± Z ∞ dx K n p ( xr/a ) + K n p +1 ( xr/a ) K n p +1 ( x ) + K n p ( x ) . (37)In this special case the total FC on a conical ring takes the form (cid:10) ¯ ψψ (cid:11) = (cid:10) ¯ ψψ (cid:11) a − q π ∞ X n =0 X p = ± Z ∞ dx x Im " K ( b ) n p ( bx ) K ( a ) n p ( ax ) × G ( a )2 n p ,n p ( ax, rx ) + G ( a )2 n p ,n p +1 ( ax, rx ) G ( ab ) n p ( ax, bx ) , (38)8here now f ( u ) n p ( z ) = δ f zf n p +1 ( z ) + iλ u n u zf n p ( z ) . (39)Of course, for a massless field the FC does not depend on the parameter s . The zero FC for amassless field, realizing one of the irreducible representations of the Clifford algebra and propagatingon a conical space without boundary, is related to the time-reversal ( T -)symmetry of the model. Thepresence of the edges gives rise to nonzero FC and, hence, breaks the T -symmetry. This mechanismof T -symmetry breaking for planar fermionic systems have been discussed in [81]. The symmetrybreaking was interpreted semiclassically in terms of the phases accumulated by the waves travellingalong closed geodesics inside a bounded region and reflected from the boundary.In the representation (28) for the FC on a conical ring with edges r = a and r = b , the partcorresponding to a cut cone with a r < ∞ is explicitly separated. An alternative representation,where the part corresponding to a cone with finite radius b is extracted, is obtained from (28) usingthe identity I ( a ) n p ( ax ) K ( a ) n p ( ax ) K ν ( y ) + K ( b ) n p ( bx ) K ( a ) n p ( ax ) G ( a )2 n p ,ν ( ax, y ) G ( ab ) n p ( ax, bx )= K ( b ) n p ( bx ) I ( b ) n p ( bx ) I ν ( y ) + I ( a ) n p ( ax ) I ( b ) n p ( bx ) G ( b )2 n p ,ν ( bx, y ) G ( ab ) n p ( ax, bx ) , (40)with ν = n p , n p + 1. Separating the contributions coming from the first term in the right-hand side,the FC in the region a r b is presented in the form (cid:10) ¯ ψψ (cid:11) = (cid:10) ¯ ψψ (cid:11) b + q π ∞ X n =0 X p = ± Z ∞ m dx x √ x − m Re ( I ( a ) n p ( ax ) /I ( b ) n p ( bx ) G ( ab ) n p ( ax, bx ) × h(cid:16) sm + i p x − m (cid:17) G ( b )2 n p ,n p ( bx, rx ) − (cid:16) sm − i p x − m (cid:17) G ( b )2 n p ,n p +1 ( bx, rx ) io , (41)where (cid:10) ¯ ψψ (cid:11) b = h ¯ ψψ i , ren + q π ∞ X n =0 X p = ± Z ∞ m dx x √ x − m Re ( K ( b ) n p ( bx ) I ( b ) n p ( bx ) × h ( sm + i p x − m ) I n p ( rx ) − ( sm − i p x − m ) I n p +1 ( rx ) io . (42)In the limit a → | α | < / a q (1 − | α | ) whereas the first term does not depend on a . This allows to interpret the part (cid:10) ¯ ψψ (cid:11) b as the FC on a cone 0 r b for a field obeying the boundary condition (7) on a single circularboundary at r = b . With this interpretation, the last term in (41) corresponds to the contributionwhen we additionally add the boundary at r = a with the respective boundary condition from (7). Inthe special case s = 1, λ b = 1 the FC (42) coincides with the result derived in [74].For a massless field, from (41) we get the following alternative representation for the FC: (cid:10) ¯ ψψ (cid:11) = (cid:10) ¯ ψψ (cid:11) b − q π ∞ X n =0 X p = ± Z ∞ m dx x Im " I ( a ) n p ( ax ) I ( b ) n p ( bx ) × G ( b )2 n p ,n p ( bx, rx ) + G ( b )2 n p ,n p +1 ( bx, rx ) G ( ab ) n p ( ax, bx ) , (43)with the single edge contribution (cid:10) ¯ ψψ (cid:11) b = − λ b q π b ∞ X n =0 X p = ± Z ∞ dx I n p ( xr/b ) + I n p +1 ( xr/b ) I n p ( x ) + I n p +1 ( x ) . (44)9he latter is negative for the boundary condition with λ b = 1 and positive for the condition with λ b = − r = a, b . The divergence at r = a comes from the singleboundary part (cid:10) ¯ ψψ (cid:11) a in the representation (28) and the divergence on the edge r = b comes from theterm (cid:10) ¯ ψψ (cid:11) b in (41). In order to find the leading term in the asymptotic expansion over the distancefrom the edge at r = u , u = a, b , we note that for | r/u − | ≪ (cid:10) ¯ ψψ (cid:11) u (the last terms in (35) and (42)) come from large values of x and n . Byusing the uniform asymptotic expansions for the modified Bessel functions, to the leading order weget (cid:10) ¯ ψψ (cid:11) ≈ − λ u π ( r − u ) . (45)In deriving this result we have additionally assumed that m | r − u | ≪
1. Near the edges the leading termdoes not depend on the mass, on the magnetic flux and on the angle deficit of the conical geometry.It is of interest to note that the vacuum expectation values of the charge and current densities arefinite on the ring edges [59].In Figure 2 we display the FC for a massless fermionic field on a conical ring as a function of theradial coordinate. The graphs are plotted for b/a = 8, q = 1 . α = 1 /
4. The curves I andII correspond to the boundary conditions on the edges with ( λ a , λ b ) = (1 ,
1) and ( λ a , λ b ) = (1 , − λ a , λ b ) are obtained by takinginto account the property that for a massless field the FC changes the sign under the replacement( λ a , λ b ) → ( − λ a , − λ b ). In the case I the FC is negative everywhere. For the case II the condensateis negative near the edge r = a and positive near r = b . This behavior is in accordance with theasymptotic estimate (45). III - - - r / a a 〈 ψψ 〉 Figure 2: The radial dependence of the FC for a massless field on a conical ring with the parameters b/a = 8, q = 1 . α = 1 /
4. The graphs I and II correspond to the sets ( λ a , λ b ) = (1 ,
1) and( λ a , λ b ) = (1 , − α is depicted in Figure 3 for a massless field andfor the parameters b/a = 8 and q = 1 .
5. The full and dashed curves correspond to r/a = 3 and r/a = 5. As in Figure 2, the graphs I and II are for the sets ( λ a , λ b ) = (1 ,
1) and ( λ a , λ b ) = (1 , − λ a , λ b ) = (1 ,
1) is continuous as well. For the10oundary conditions with ( λ a , λ b ) = (1 , −
1) the derivative of the FC with respect to the magnetic fluxis discontinuous for half-integer values of the ratio of the magnetic flux to the flux quantum.
I III II - - - - α a 〈 ψψ 〉 Figure 3: The FC versus the parameter α for a massless field. The graphs are plotted for b/a = 8, q = 1 . r/a = 3 (full curves) and r/a = 5 (dashed curves).Figure 4 displays the FC as a function of the parameter q , determining the planar angle deficitfor conical geometry. The graphs are plotted for a massless field and for the values of the parameters b/a = 8, α = 1 / r/a = 3 (full curves) and r/a = 5 (dashed curves). As before, the curves I and IIcorrespond to ( λ a , λ b ) = (1 ,
1) and ( λ a , λ b ) = (1 , − IIIIII - - a 〈 ψψ 〉 Figure 4: The dependence of the FC on the planar angle deficit of the conical space for α = 1 /
4. Thevalues of the remaining parameters are the same as those for Figure 3.All the graphs above were plotted for a massless field. In order to see the effects of finite mass, inFigure 5 we depicted the dependence of the FC on the dimensionless parameter ma for s = 1, b/a = 8, q = 1 . r/a = 2 and α = 1 /
4. The curves I,II,III,IV correspond to the sets of discrete parameters( λ a , λ b ) = (1 , , − − ,
1) and ( − , − s = − IIIIII IV - - a 〈 ψψ 〉 Figure 5: The FC as a function of the mass for s = 1 and for fixed values b/a = 8, q = 1 . r/a = 2, α = 1 /
4. The separate graphs correspond to different combinations of the boundary conditions onthe ring edges.
For a fermion field ψ ( x ) in two spatial dimensions, realizing one of the irreducible representations ofthe Clifford algebra, the term m ¯ ψψ in the corresponding Lagrangian density is not invariant withrespect to the parity ( P ) and time-reversal ( T ) transformations. The P - and T -symmetries can berestored considering models involving two fields ψ (+1) and ψ ( − realizing inequivalent irreduciblerepresentations and having the same mass. The corresponding Lagrangian density is given by L = P s = ± L ( s ) with the separate terms from (3). We assume that the fields obey the boundary conditions(5) on the ring edges. The total FC is presented in two equivalent forms, P s = ± h ¯ ψ ( s ) ψ ( s ) i and P s = ± h ¯ ψ ′ ( s ) ψ ′ ( s ) i . An equivalent representation of the model is obtained combining the two-componentfields in a single 4-component spinor Ψ = ( ψ (+1) , ψ ( − ) T with the Lagrangian density L = ¯Ψ( iγ µ (4) D µ − m )Ψ , (46)where the 4 × γ µ (4) = I ⊗ γ µ for µ = 0 ,
1, and γ = σ ⊗ γ with σ being the Pauli matrix. For the corresponding FC one has the standard expression h ¯Ψ( x )Ψ( x ) i . Theboundary conditions on the edges r = a, b are rewritten as (cid:16) i Λ r n µ γ µ (4) (cid:17) Ψ( x ) = 0 , (47)with Λ r = diag( λ (+1) r , λ ( − r ). Alternatively, we can introduce the spinor Ψ ′ = ( ψ ′ (+1) , ψ ′ ( − ) T andthe set of gamma matrices γ ′ µ (4) = σ ⊗ γ µ . For the corresponding Lagrangian density one gets L = ¯Ψ ′ ( iγ ′ µ (4) D µ − m )Ψ ′ and for the FC h ¯Ψ ′ ( x )Ψ( x ) ′ i . Now the boundary conditions take the form12 i Λ r n µ γ ′ µ (4) (cid:17) Ψ ′ ( x ) = 0. The latter has the same form as (47), though with different representationof the gamma matrices.Let us consider different combinations of the boundary conditions for the fields ψ (+1) and ψ ( − .First we assume that λ (+1) u = λ ( − u , u = a, b . For the coefficients in the boundary conditions forthe fields ψ ′ (+1) and ψ ′ ( − one gets λ ( − ′ u = − λ (+1) ′ u . From here we conclude that the condensates h ¯ ψ (+1) ψ (+1) i and h ¯ ψ ( − ψ ( − i are obtained from the formulas in the previous sections taking s = 1, λ u = λ (+1) u and s = − λ u = − λ (+1) u , respectively. If the parameter χ in the condition (8) andthe charges e are the same for the fields ψ (+1) and ψ ( − , then the parameter α is the same as well.Now, recalling that the FC discussed in the previous section, changes the sign under the replacement( s, λ u ) → ( − s, − λ u ), we see that the total fermionic condensate vanishes. This means that in themodel at hand with two fields and with the parameters in the boundary conditions λ (+1) u = λ ( − u theCasimir contributions induced by the edges do not break the parity and time-reversal symmetries. Inthe second case with λ (+1) u = − λ ( − u , the fields ψ (+1) and ψ ( − obey different boundary conditions,whereas for the fields ψ ′ (+1) and ψ ′ ( − the boundary conditions are the same. In this case the total FCis nonzero and the parity and time-reversal symmetries are broken by the boundary conditions. Notethat, the nonzero FC may appear in the first case as well if the masses or the phases χ for separatefields are different. Hence, the edge-induced effects provide a mechanism for time-reversal symmetrybreaking in the absence of magnetic fields.Among the interesting condensed matter realizations of fermionic models in (2+1)-dimensionalspacetime is graphene. For a given spin degree of freedom, the effective description of the long-wavelength properties of the electronic subsystem is formulated in terms of 4-component fermionicfield Ψ = ( ψ + ,A , ψ + ,B , ψ − ,A , ψ − ,B ) T . (48)Two 2-component spinors ψ + = ( ψ + ,AS , ψ + ,BS ) and ψ − = ( ψ − ,AS , ψ − ,BS ) correspond to two inequiv-alent points K + and K − at the corners of the hexagonal Brillouin zone for the graphene lattice. Thecomponents ψ ± ,A and ψ ± ,B present the amplitude of the electron wave function on the triangularsublattices A and B . The Lagrangian density for the field Ψ is given as (in standard units) L g = ¯Ψ[ i ~ γ ∂ t + i ~ v F X l =1 , γ l (4) ( ∇ l + ieA l / ~ c ) − ∆]Ψ , (49)where c is the speed of light, v F ≈ . × cm/s is the Fermi velocity, and ∆ is the energy gap inthe spectrum. The spatial components of the covariant derivative are expressed as D l = ∇ l + ieA l / ~ c with e being the electron charge. Various mechanisms for the generation of the gap, with the range1 meV . ∆ . m = ∆ /v F and a C = ~ v F / ∆. The characteristic energy scale ingraphene made structures is given by ~ v F /a ≈ .
51 eV, where a is the inter-atomic distance for thegraphene lattice. The fields ψ + and ψ − correspond to the fields ψ (+1) and ψ ( − in our considerationabove and the Lagrangian density (49) is the analog of (46). Hence, the parameter s corresponds tothe valley-indices + and − in graphene physics.For graphitic cones the allowed values of the opening angle are given by φ = 2 π (1 − n c / n c = 1 , , . . . ,
5. The transformation properties of the spinor fields under the rotation by the angle φ about the cone axis are studied in [63, 65, 67, 70]. For odd values of n c the condition that relatesthe spinors with the arguments φ + φ and φ mixes the valley indices by the matrix e − iπn c τ / withthe Pauli matrix τ acting on those indices. One can diagonalize the corresponding quasiperiodicitycondition by a unitary transformation. For graphitic cones with even n c the components with differentvalues of the valley-index are not mixed by the quasiperiodicity condition. The latter corresponds to(8) with the inequivalent values χ = ± / χ . In accordance with the considerationgiven above, if the boundary conditions and the masses for the fields corresponding to different valleys13re the same, the contributions to the FC coming from those fields cancel each other and the totalFC vanishes. However, some mechanisms for the gap generation in the spectrum break the valleysymmetry (an example is the chemical doping) and the corresponding Dirac masses for the fields ψ + and ψ − differ. In this case one has no cancellation and a nonzero total FC is formed. As it hasbeen mentioned above, the nonzero FC is also generated by imposing different boundary conditionson the edges of the ring for the fields corresponding to different valleys. In these cases the expressionof the FC for a given spin degree of freedom is obtained by combining the formulas given above forseparate contributions coming from different valleys. In the corresponding expressions it is convenientto introduce the Compton wavelengths a C + and a C − instead of the Dirac masses m + and m − throughthe replacements m ± u → u/a C ± for u = a, b, r .The boundary conditions for fermions in the effective description of graphene structures with edges(graphene nanoribbons) depend on the atomic terminations. For special cases of zigzag and armchairedges those conditions have been discussed in [83] (for a generalization see [87]). The equivalencebetween the boundary conditions considered in [81] and [83] has been discussed in [87]. The boundaryconditions for more general types of the atomic terminations in graphene sheets were studied in[82, 84, 87]. The general boundary conditions contain four parameters. The FC is an important characteristic of fermionic fields that plays an important role in discussionsof chiral symmetry breaking and dynamical generation of mass. It appears as an order parameterfor the confinement-deconfinement phase transitions. In the present paper we have investigated theFC for a (2+1)-dimensional fermionic field localized on a conical ring with a general value of theplanar angle deficit. The consideration is presented for both inequivalent irreducible representationsof the Clifford algebra. The boundary conditions on the edges of the ring are taken in the form (7)with discrete parameters λ a and λ b . As a special case they include the boundary condition used inMIT bag model of hadrons for confinement of quarks. The mode-sum for the FC contains summationover the eigenvalues of the radial quantum number γ . The latter are determined from the boundaryconditions on the ring edges and are roots of the transcendental equation (16). Depending on thevalues of the discrete parameters ( s, λ a , λ b ), one can have modes with purely imaginary values of γ .For those modes, corresponding to bound states, we have γ + m >
0. This shows that for boundaryconditions under consideration the fermionic vacuum state is always stable.For an equivalent representation of the FC, we have applied the generalized Abel-Plana-type for-mula (23) to the series over the eigenvalues of γ . That allowed to extract explicitly the part in the FCcorresponding to the region a r < ∞ of a conical space with a single adge and to present the partinduced by the second edge in the form of the integral that is well adapted for numerical evaluations(last term in (28)). The first contribution, corresponding to the conical region a r < ∞ (the secondedge at r = b is absent), is further decomposed in the form of the sum of the boundary-free andedge-induced terms (formula (35)). An alternative representation of the FC on a conical ring, givenby (41), is obtained by using the identity (40) for the modified Bessel functions. In that representationthe part in the FC is extracted which corresponds to a finite radius cone (with the radius b ) and thelast term in (41) is induced by the second edge at r = a , added to that geometry. For a masslessfield the boundary-free contribution in the FC vanishes and the nonzero FC is entirely due to thepresence of boundaries (due to the Casimir effect). In this case the expressions for the edge-inducedcontributions to the FC are simplified to (38) and (43) with the single-edge geometry parts (37) and(44). The latter are positive for the boundary condition with λ u < λ u > r = u , u = a, b , the leading term in the asymptotic expansion over the distanceis given by the simple expression (45). The leading term does not depend on the mass and on the14agnetic flux and is positive (negative) for the boundary condition with λ u < λ u > T -symmetry breaking in the absenceof magnetic fields.For a fermionic field realizing one of the irreducible representations of the Clifford algebra, the massterm in the Lagrangian density is not invariant under the parity and time-reversal transformations.Invariant fermionic models are constructed combining two fields corresponding to two inequivalentirreducible representations. In those models, the total FC is obtained by summing the contributionscoming from the separate fields. The latter are obtained based on the results presented in section 3.If the parameters ( χ, λ u ) and the masses for the separate fields are the same, then the correspondingcontributions cancel each other and the total FC is zero. In this case the Casimir-type contributionsdo not break the parity and time-reversal symmetries of the model. If at least one of the parameters( χ, λ u , m ) is different for the fields in the combined Lagrangian, the total FC is nonzero and thesymmetries are broken. The results obtained in the paper can be applied for the investigation of theFC in graphitic cones with circular edges. The opening angle of the latter can be used as an additionalparameter to control the electronic properties. In the long-wavelength approximation these propertiesare well described by the Dirac model with appropriate periodicity conditions with respect to therotations around the cone axis. Acknowledgments
A.A.S. was supported by the grant No. 20RF-059 of the Committee of Science of the Ministry ofEducation, Science, Culture and Sport RA, and by the ”Faculty Research Funding Program” (PMIScience and Enterprise Incubator Foundation). T.A.P. was supported by the grant No. 20AA-1C005of the Committee of Science of the Ministry of Education, Science, Culture and Sport RA, and by the”Faculty Research Funding Program” (PMI Science and Enterprise Incubator Foundation).
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