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Casimir effect around disclinations
28 August 1995 PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 204 ( 1995) 399-404
Casirnir eiTect around disclinations Fernando Moraes
’
Received 4 May 1995; accepted for publication 8
Communicated by L.J.
July 1995
Sham
Abstract This communication concerns the structure of the ele~tmmagnetic quantum vacuum in a disclinated insulator. It is shown that a nonzero vacuum energy density appears when the rotational symmetry of a continuous insulating elastic medium is broken by a disclination. An explicit expression is given for this Casimir energy density in terms of the parameter describing
the disclination.
1. Introduction
Space-time being isotropic and homogeneous implies that the vacuum state of a quantum field must be inv~iant under translations or rotations. The presence of external fields, a change in the geometry (curvature, torsion) or a change in the topology (defects, boundaries) of the space breaks the underlying symmetry of space-time, inducing changes in the vacuum state of the quantum field as compared to its field-free, unbound, Minkowski space counterpart. This is the celebrated Casimir effect [ 1,2]. Topological defects certainly offer an interesting (and quite unexplored) arena for the study of vacuum polarization phenomena in condensed matter systems. In previous publications f3,4] we studied the nonrelativistic quantum mechanics of electrons and holes in the presence of a disclination in an insulating elastic medium, with and without an external magnetic field. In the present work, I study the Casimir effect of the electromagnetic field in an infinite medium of constant dielectricity E and permeability p = 1 (a nonmagnetic medium), with a disclination at the origin of the coordinate system, at zero temperature. The electromagnetic Casimir effect is investigated since it is more likely to be experimentally observable in the solid state than the fermionic case [S]. This is due to the fact that massive fields, like the fermion field, give Casimir energies exponentially decreasing with their mass. An estimate [S] for electrons confined to regions of nanometric size, i.e. N 10m7 cm, gives an exponential damping factor for the Casimir energy of N 5 x 103. While the electromagnetic Casimir energy for light confined to a similar region is N 10 eV, the electronic effect is then roughly em5x lo’ weaker. Notice that while Ref. I.51 deals with the Casimir effect due to confinement to ’ On leave from:~p~a~nto
de Fisica, Unive~idade
Federal de ~mambuco,
0375~9601/95/$09.50 0 1995 Elsevier Science B.V. All rights resexved SSDI 0375.9601(95)00524-2
50670-901
Recife, PE, Brazil.
400
F. Moraes/
Physics Letters A 204 (1995)
399-404
mesoscopic regions, in this work the interest is in the effect outside a defect - no confinement involved. In any case, one should expect a much weaker electronic Casimir effect, then justifying the choice of presenting the electromagnetic effect here.
2. QED in conical space Quantum field theory has been extensively studied in the conical space-time of cosmic strings [6]. The large body of information available on this subject makes an excellent guide for the study of quantum field theoretical aspects of disclinations in solids, in the context of the theory of defects/three-dimensional gravity of Katanaev and Volovich [ 71. Disclinations, like cosmic strings, are associated with a conical metric [7] whose spatial part is given, in cylindrical coordinates, by ds2 = dz2 + dp2 + a2p2 dp2.
(1)
With (Y = 1 + A/27r, the disclination is obtained by either removing (A < 0) or inserting (h > 0) a wedge of material, of dihedral angle A, to the otherwise smooth elastic continuum. The resulting space, described by the above metric, has a null curvature tensor everywhere, except at the defect where it has a two-dimensional &function singularity given by [ 81 1-f-Y R’2 = R’1 = R2 = 27r ---42(P)~ 12 2
(2)
CY
where 82(p) is the two-dimensional delta function in flat space. It is clear that 0 < (Y < 1 corresponds to a positive-curvature disclination and cy > 1 to a negative-curvature disclination. (Y = I, of course, describes the Euclidean medium (absence of disclinations). The relevant physical quantity in the study of the Casimir effect is the vacuum expectation value of the energy-momentum tensor, or stress tensor, (fP “). Since this tensor diverges, in order to obtain physically meaningful results, one should subtract from it its also divergent flat space version 2. That is, one should calculate the renormalized vacuum expectation value (Tp “)ren = (pp
‘)discl
-
(fp
(3)
‘)flat.
In this work, I will be concerned only with the (pa ‘) component, since it is the one that gives the energy density. The electromagnetic field being a gauge field, care must be exercised in order that appropriate gauge-fixing be taken into account in the quantum theory. As it turns out [9] the convenient gauge here is V;A’ = V,,A” = 0 where form
A@
(i=O,
I; a=2,3), vector potential.
is the electromagnetic
(Pa O) = iti
lim V;V’G(x 1’i.X
-
’
In this gauge the energy density
x’),
where G(x - x’) is the scalar Green function, Ga(x - x’) = -1
(4) assumes
the very simple
(5) which in the absence of the defect is
I 4%-Z&(X - x’)2
In many cases an additional regularization is needed to ensure a finite result.
(6)
F. Moraes / Physics Letters A 204 (1995) 399-404
401
Imy
-
Fig.
x - x’ is the space-time
I.
Integration contour for IQ. (9).
interval between x and x’, its square being given by
(x - x1)2 =-C~(t-t’)~+(Z-~‘)~+~~+p’~-2pp’cos(qD-~’). The energy density is then
where G, (x - X’ ) is the scalar Green function in the medium with boundary conditions. Due to the disclination, the boundary condition in this problem is a periodicity of 27ra in the angular variable 40. Accordingly, imposition of this condition on the free space Green function can be done by reperiodising it via the contour integral [ lo]
G,(x -x’> = -k
’J
dy GO(P- rp’-Y)
exp(iyPa) sin(y/2a)
’
(9)
A
The singularities
in the integrand
(X - x’)~ = 2pp’[coshP
are poles that can be easily found by writing
- cos(rp - (o’) 1,
where cash p =
-c2(t
- t’)2 + (z - z’)2 + p2 + p’2 2PP’
Since coshp = cosi/?, the poles of Go(v--‘-y) are at y = fip+qP-_‘+2n7r. On the other hand, sin(y/2cY) contributes with poles at y = 2n7rcu. In both cases n E Z. As shown in Fig. 1, the contour A has two branches, one in the upper half y-plane from (r + cp - p’) + ioc to (-rr + 4p - 4p’) + ioo passing below the singularity y = i,L?+ Q - ‘p’ and the other in the lower half-plane from (-n- + p - 9’) - ioc to (T + (p - PO’)- ice passing above the singularity at y = -i/3 + (p - 40’. In the limit x’ = x the poles at y = &ip + p - 40’+ 2nr will be lying at y = 2n7r. So, one is left with poles at y = 2nr and y = 2naa. Considering this, there is a deformation [lo] of the contour A that will prove to be quite convenient. Just deform it into an anticlockwise loop B around the pole at y q O.and the two vertical lines, I’, y = b+iy and y = -b+iy, with -cc < y < 00 and b a constant (see Figs.,2 and 3). This can only be done if the poles lying on the Rey-axis, other than the one at y = 0, are outside the region limited by the lines,
402
F. Moraes / Phyks
Lotters A 204 f 1995) 399-404
-b
b
Rey
Fig. 2. Intermediate deformation of the contour. Fig. 3. Final form of the deformed contour.
i.e., b must be chosen to be less than the smaller of 2a and 29~~. As shown in the Appendix, the contribution from loop B turns out to be exactly GO{x - x’) , Since reno~ali~tion requires G, f x - x’) - Go( x - x’) , all one has to consider is the contribution from the two vertical lines IT. Eq. (8) becomes then
(10) It is easily found that
-$d:+d;
i
Go(v,-d-y)
*‘=x= 1677%p4sin4( y/2) ’
(11)
Furthermore, using the symmet~ of the contour r to replace the exponential by a cosine in Eq. ( 10) and then deforming the contour into a clockwise loop around the origin leads to
(12) This integral can be easily evaluated using calculus of residues, by expanding the integrand in powers of y such that the term of order y-’ gives $ dr,/r = -2rri. The expansion results in
cot(yl2ff) = 32& sin4(y/2)
+ ( +r - 8/3c~)y-~ + (gcr - 4/9ar - 2/45~z~)y-’ -t O(y’).
(13)
The energy density is then
(ToO)= 7205ep4 ( 11 - 10/a* - 1/ff4> 1
(14)
This result has been found by a variety of ways in the context of cosmic strings [9,11,12] and also for a metallic wedge [ 131. Notice that in the ftat space limit, LY= 1, (pi o)ren = 0 as it should. Also, although values of fy > 1 are probably not physically sensible for cosmic strings because this would correspond to negative linear mass densities, in condensed matter they describe negative-curvature disclinations [ 3,4]. The immediate consequence
F. Moraes / Physics Letters A 204 (&XV) 399404
403
for condensed matter systems with such defects is a dependence of the sign of the force exerted on the defect by the electromagnetic vacuum, on the sign of the defect curvature. This is certainly important for the study of the stability of the defects. The obvious implication is that a disc&nation dipole, or edge dislocation [ 71, is more stable than either a disclination standing by itself and that the interaction between the pair is attractive. This is a result of the vacuum polarization of the electromagnetic field only. The total interaction and overall stability depend on contributions from all the relevant fields.
3. Concluding remarks The purpose of this Letter is to draw attention to the importance of quantum field theoretical effects in condensed matter enclosing topological defects. A very simple case has been considered: the medium is infinite (e.g. no external boundaries were considered), nonmagnetic, has only one defect (disclination) , its dielectricity is fr~uency~inde~ndent and the temperature set to T = 0. Of course, these oversimpli~~ations give a distorted picture of the real phenomenon. Nevertheless, it is shown that a finite el~tromagneti~ Casimir energy density is present in the outer region of the disclination. It would be interesting to extend the present results to finite temperatures and to magnetic media. Moreover, use of a frequency-dependent complex dielectric constant would not only provide a more realistic calculation for the Casimir energy but also ene that takes into account absorption of energy inside the dielectric, something that happens in real materials.
Acknowledgement This work was partially supported by CNPq.
Appendix The anticlockwise loop integral --
i 4~ff f
dy
exp(iy/2a) Go(so-&Y) sin( y/2@)
B
can be evaluated in the same way as the integral in Eq. (12). Again, the symmetry of the contour permits the substitution of the exponential by a cosine. The resulting cot(y/2a) expanded around y = 0 is
2c+i Y -___ 6~y Y implying --
i 49Sraf
dyexp(iy/2fu) sin(y/2a)
G&Q-+y)
=-&
~Go((P-ip’-y)
B
References I 1 I V.M. Mostepanenko and N.N. Trunov, Sov. Phys. Usp. 31 (1988) 965. 12 I G. Plunien, B. Miiller and W. Greioer, Phys. Rep. 134 ( 1986) 87. 13 1 C. Fur&do and E Moraes, Phys. L&t. A I88 ( 1994) 394.
=Go(x-2’).
F. Moraes / Physics Letters A 204 f 1995) 399-404
404 I4
I C. Futtado. B.G.C. da Cunha. F. Moraes, E.R. Bezerra de Mello and V.B. Bezerra. Phys. Lert. A 151 I.H. Dur and M. Tomak, Phys. Len. A 176 (1993) 265. 161G.W. Gibbons, SW. Hawking and T. Vachaspati, eds., The formation and evolution of cosmic Cambridge, 17 I M.O.
18I
1990) part 4, and references therein.
Katanaev
D.D. Sokolov
191 VP
Frolov
and I.V. Volovich. and A.A.
and E.M.
Ann. Phys. (NY)
Starobinskii,
Serebriany,
2 16 ( 1992)
1.
Sov. Phys. Dokl. 22 ( 1977) 312.
Phys. Rev. D 35
I 101 J.S. Dowker,
Phys. Rev. D 36 (1987)
[ I 1I I 12 I 113I
M.R. Brown.
A.C. Ottewill
J.S. Dowker,
Phys. Rev. D 36 ( 1987) 3742.
( 1987)
3779.
3095.
and D.N. Page, Phys. Rev. D 33 ( 1986) 2840.
D. Deutsch and P Candelas. Phys. Rev. D 20 ( 1979) 3063.
195 ( 1994) 90. strings
(Cambridge
Univ.
Press
ELSEVIER
LETTERS
A
Physics Letters A 204 ( 1995) 399-404
Casirnir eiTect around disclinations Fernando Moraes
’
Received 4 May 1995; accepted for publication 8
Communicated by L.J.
July 1995
Sham
Abstract This communication concerns the structure of the ele~tmmagnetic quantum vacuum in a disclinated insulator. It is shown that a nonzero vacuum energy density appears when the rotational symmetry of a continuous insulating elastic medium is broken by a disclination. An explicit expression is given for this Casimir energy density in terms of the parameter describing
the disclination.
1. Introduction
Space-time being isotropic and homogeneous implies that the vacuum state of a quantum field must be inv~iant under translations or rotations. The presence of external fields, a change in the geometry (curvature, torsion) or a change in the topology (defects, boundaries) of the space breaks the underlying symmetry of space-time, inducing changes in the vacuum state of the quantum field as compared to its field-free, unbound, Minkowski space counterpart. This is the celebrated Casimir effect [ 1,2]. Topological defects certainly offer an interesting (and quite unexplored) arena for the study of vacuum polarization phenomena in condensed matter systems. In previous publications f3,4] we studied the nonrelativistic quantum mechanics of electrons and holes in the presence of a disclination in an insulating elastic medium, with and without an external magnetic field. In the present work, I study the Casimir effect of the electromagnetic field in an infinite medium of constant dielectricity E and permeability p = 1 (a nonmagnetic medium), with a disclination at the origin of the coordinate system, at zero temperature. The electromagnetic Casimir effect is investigated since it is more likely to be experimentally observable in the solid state than the fermionic case [S]. This is due to the fact that massive fields, like the fermion field, give Casimir energies exponentially decreasing with their mass. An estimate [S] for electrons confined to regions of nanometric size, i.e. N 10m7 cm, gives an exponential damping factor for the Casimir energy of N 5 x 103. While the electromagnetic Casimir energy for light confined to a similar region is N 10 eV, the electronic effect is then roughly em5x lo’ weaker. Notice that while Ref. I.51 deals with the Casimir effect due to confinement to ’ On leave from:~p~a~nto
de Fisica, Unive~idade
Federal de ~mambuco,
0375~9601/95/$09.50 0 1995 Elsevier Science B.V. All rights resexved SSDI 0375.9601(95)00524-2
50670-901
Recife, PE, Brazil.
400
F. Moraes/
Physics Letters A 204 (1995)
399-404
mesoscopic regions, in this work the interest is in the effect outside a defect - no confinement involved. In any case, one should expect a much weaker electronic Casimir effect, then justifying the choice of presenting the electromagnetic effect here.
2. QED in conical space Quantum field theory has been extensively studied in the conical space-time of cosmic strings [6]. The large body of information available on this subject makes an excellent guide for the study of quantum field theoretical aspects of disclinations in solids, in the context of the theory of defects/three-dimensional gravity of Katanaev and Volovich [ 71. Disclinations, like cosmic strings, are associated with a conical metric [7] whose spatial part is given, in cylindrical coordinates, by ds2 = dz2 + dp2 + a2p2 dp2.
(1)
With (Y = 1 + A/27r, the disclination is obtained by either removing (A < 0) or inserting (h > 0) a wedge of material, of dihedral angle A, to the otherwise smooth elastic continuum. The resulting space, described by the above metric, has a null curvature tensor everywhere, except at the defect where it has a two-dimensional &function singularity given by [ 81 1-f-Y R’2 = R’1 = R2 = 27r ---42(P)~ 12 2
(2)
CY
where 82(p) is the two-dimensional delta function in flat space. It is clear that 0 < (Y < 1 corresponds to a positive-curvature disclination and cy > 1 to a negative-curvature disclination. (Y = I, of course, describes the Euclidean medium (absence of disclinations). The relevant physical quantity in the study of the Casimir effect is the vacuum expectation value of the energy-momentum tensor, or stress tensor, (fP “). Since this tensor diverges, in order to obtain physically meaningful results, one should subtract from it its also divergent flat space version 2. That is, one should calculate the renormalized vacuum expectation value (Tp “)ren = (pp
‘)discl
-
(fp
(3)
‘)flat.
In this work, I will be concerned only with the (pa ‘) component, since it is the one that gives the energy density. The electromagnetic field being a gauge field, care must be exercised in order that appropriate gauge-fixing be taken into account in the quantum theory. As it turns out [9] the convenient gauge here is V;A’ = V,,A” = 0 where form
A@
(i=O,
I; a=2,3), vector potential.
is the electromagnetic
(Pa O) = iti
lim V;V’G(x 1’i.X
-
’
In this gauge the energy density
x’),
where G(x - x’) is the scalar Green function, Ga(x - x’) = -1
(4) assumes
the very simple
(5) which in the absence of the defect is
I 4%-Z&(X - x’)2
In many cases an additional regularization is needed to ensure a finite result.
(6)
F. Moraes / Physics Letters A 204 (1995) 399-404
401
Imy
-
Fig.
x - x’ is the space-time
I.
Integration contour for IQ. (9).
interval between x and x’, its square being given by
(x - x1)2 =-C~(t-t’)~+(Z-~‘)~+~~+p’~-2pp’cos(qD-~’). The energy density is then
where G, (x - X’ ) is the scalar Green function in the medium with boundary conditions. Due to the disclination, the boundary condition in this problem is a periodicity of 27ra in the angular variable 40. Accordingly, imposition of this condition on the free space Green function can be done by reperiodising it via the contour integral [ lo]
G,(x -x’> = -k
’J
dy GO(P- rp’-Y)
exp(iyPa) sin(y/2a)
’
(9)
A
The singularities
in the integrand
(X - x’)~ = 2pp’[coshP
are poles that can be easily found by writing
- cos(rp - (o’) 1,
where cash p =
-c2(t
- t’)2 + (z - z’)2 + p2 + p’2 2PP’
Since coshp = cosi/?, the poles of Go(v--‘-y) are at y = fip+qP-_‘+2n7r. On the other hand, sin(y/2cY) contributes with poles at y = 2n7rcu. In both cases n E Z. As shown in Fig. 1, the contour A has two branches, one in the upper half y-plane from (r + cp - p’) + ioc to (-rr + 4p - 4p’) + ioo passing below the singularity y = i,L?+ Q - ‘p’ and the other in the lower half-plane from (-n- + p - 9’) - ioc to (T + (p - PO’)- ice passing above the singularity at y = -i/3 + (p - 40’. In the limit x’ = x the poles at y = &ip + p - 40’+ 2nr will be lying at y = 2n7r. So, one is left with poles at y = 2nr and y = 2naa. Considering this, there is a deformation [lo] of the contour A that will prove to be quite convenient. Just deform it into an anticlockwise loop B around the pole at y q O.and the two vertical lines, I’, y = b+iy and y = -b+iy, with -cc < y < 00 and b a constant (see Figs.,2 and 3). This can only be done if the poles lying on the Rey-axis, other than the one at y = 0, are outside the region limited by the lines,
402
F. Moraes / Phyks
Lotters A 204 f 1995) 399-404
-b
b
Rey
Fig. 2. Intermediate deformation of the contour. Fig. 3. Final form of the deformed contour.
i.e., b must be chosen to be less than the smaller of 2a and 29~~. As shown in the Appendix, the contribution from loop B turns out to be exactly GO{x - x’) , Since reno~ali~tion requires G, f x - x’) - Go( x - x’) , all one has to consider is the contribution from the two vertical lines IT. Eq. (8) becomes then
(10) It is easily found that
-$d:+d;
i
Go(v,-d-y)
*‘=x= 1677%p4sin4( y/2) ’
(11)
Furthermore, using the symmet~ of the contour r to replace the exponential by a cosine in Eq. ( 10) and then deforming the contour into a clockwise loop around the origin leads to
(12) This integral can be easily evaluated using calculus of residues, by expanding the integrand in powers of y such that the term of order y-’ gives $ dr,/r = -2rri. The expansion results in
cot(yl2ff) = 32& sin4(y/2)
+ ( +r - 8/3c~)y-~ + (gcr - 4/9ar - 2/45~z~)y-’ -t O(y’).
(13)
The energy density is then
(ToO)= 7205ep4 ( 11 - 10/a* - 1/ff4> 1
(14)
This result has been found by a variety of ways in the context of cosmic strings [9,11,12] and also for a metallic wedge [ 131. Notice that in the ftat space limit, LY= 1, (pi o)ren = 0 as it should. Also, although values of fy > 1 are probably not physically sensible for cosmic strings because this would correspond to negative linear mass densities, in condensed matter they describe negative-curvature disclinations [ 3,4]. The immediate consequence
F. Moraes / Physics Letters A 204 (&XV) 399404
403
for condensed matter systems with such defects is a dependence of the sign of the force exerted on the defect by the electromagnetic vacuum, on the sign of the defect curvature. This is certainly important for the study of the stability of the defects. The obvious implication is that a disc&nation dipole, or edge dislocation [ 71, is more stable than either a disclination standing by itself and that the interaction between the pair is attractive. This is a result of the vacuum polarization of the electromagnetic field only. The total interaction and overall stability depend on contributions from all the relevant fields.
3. Concluding remarks The purpose of this Letter is to draw attention to the importance of quantum field theoretical effects in condensed matter enclosing topological defects. A very simple case has been considered: the medium is infinite (e.g. no external boundaries were considered), nonmagnetic, has only one defect (disclination) , its dielectricity is fr~uency~inde~ndent and the temperature set to T = 0. Of course, these oversimpli~~ations give a distorted picture of the real phenomenon. Nevertheless, it is shown that a finite el~tromagneti~ Casimir energy density is present in the outer region of the disclination. It would be interesting to extend the present results to finite temperatures and to magnetic media. Moreover, use of a frequency-dependent complex dielectric constant would not only provide a more realistic calculation for the Casimir energy but also ene that takes into account absorption of energy inside the dielectric, something that happens in real materials.
Acknowledgement This work was partially supported by CNPq.
Appendix The anticlockwise loop integral --
i 4~ff f
dy
exp(iy/2a) Go(so-&Y) sin( y/2@)
B
can be evaluated in the same way as the integral in Eq. (12). Again, the symmetry of the contour permits the substitution of the exponential by a cosine. The resulting cot(y/2a) expanded around y = 0 is
2c+i Y -___ 6~y Y implying --
i 49Sraf
dyexp(iy/2fu) sin(y/2a)
G&Q-+y)
=-&
~Go((P-ip’-y)
B
References I 1 I V.M. Mostepanenko and N.N. Trunov, Sov. Phys. Usp. 31 (1988) 965. 12 I G. Plunien, B. Miiller and W. Greioer, Phys. Rep. 134 ( 1986) 87. 13 1 C. Fur&do and E Moraes, Phys. L&t. A I88 ( 1994) 394.
=Go(x-2’).
F. Moraes / Physics Letters A 204 f 1995) 399-404
404 I4
I C. Futtado. B.G.C. da Cunha. F. Moraes, E.R. Bezerra de Mello and V.B. Bezerra. Phys. Lert. A 151 I.H. Dur and M. Tomak, Phys. Len. A 176 (1993) 265. 161G.W. Gibbons, SW. Hawking and T. Vachaspati, eds., The formation and evolution of cosmic Cambridge, 17 I M.O.
18I
1990) part 4, and references therein.
Katanaev
D.D. Sokolov
191 VP
Frolov
and I.V. Volovich. and A.A.
and E.M.
Ann. Phys. (NY)
Starobinskii,
Serebriany,
2 16 ( 1992)
1.
Sov. Phys. Dokl. 22 ( 1977) 312.
Phys. Rev. D 35
I 101 J.S. Dowker,
Phys. Rev. D 36 (1987)
[ I 1I I 12 I 113I
M.R. Brown.
A.C. Ottewill
J.S. Dowker,
Phys. Rev. D 36 ( 1987) 3742.
( 1987)
3779.
3095.
and D.N. Page, Phys. Rev. D 33 ( 1986) 2840.
D. Deutsch and P Candelas. Phys. Rev. D 20 ( 1979) 3063.
195 ( 1994) 90. strings
(Cambridge
Univ.
Press