Table Of ContentHall conductivity in the cosmic defect and dislocation space-time
∗
Kai Ma and Jian-hua Wang
School of Physics Science, Shaanxi University of Technology, Hanzhong 723000, Shaanxi, P. R. China
Huan-Xiong Yang
Interdisciplinary Center for Theoretical Study, University of Science and Technology of China, Hefei 200026, P. R. China
6
Hua-wei Fan
1
0 School of Physics and Information Technology, Shaanxi Normal University, Xian 710000, Shaanxi, P. R. China
2
Influences of topological defect and dislocation on conductivity behavior of charge carries in
g
external electromagnetic fields are studied. Particularly the quantum Hall effect is investigated
u
in detail. It is found that the nontrivial deformations of spacetime due to topological defect and
A
dislocationproduceanelectriccurrentattheleadingorderofperturbationtheory. Thiscurrentthen
induces a deformation on the Hall conductivity. The corrections on the Hall conductivity depend
8
on the externalelectric fields, thesize of the sample and the momentum of the particle.
2
PACSnumbers: 04.62.+v,11.27.+d,12.20.Ds
]
l
l
a
h Topologicaldefects, whichencode the globalgeometry of charge carries in electromagnetic fields. It is shown
- structure of the background space-time, can be realized thattheHallconductivityreceivesacorrectionwhichde-
s
e invariousphysicalsystemsrangingfromthemacroscopic pends linearly on the scale parametersof the topological
m scaletothemicroscopicscale,andthereforesimulatedex- defects, and are in the same order with the corrections
tensivestudiesonitsphysicaleffects[1–36]. Forinstance, on the spin-Hall conductivity[22, 23]. However, we can
.
at the cosmic string[1–4], global monopole[5], and domain expect that very steady and precise quantized Hall con-
m wall[6–11] can be formed at the phase transitions of the ductivitycanprovidebetterexperimentalsensitivityand
early universe from symmetric to asymmetric configura- can bound on parameters of the models.
-
d tions. Even though topological defects have been com- We consider the quantum Hall effect in the presence
n monlyobservedincondensedmattersystems,sofarthere of a topological defect. The dynamics of a relativistic
o
isnodirectexperimentalevidenceofthecosmologicalde- Diracparticleinthecurvedspacetimeisexpressedinthe
c
fectsduetotheirextremelyhighenergies. Thereforeitis generalized covariant form of the Dirac equation [35],
[
still attractive to explore the nontrivial geometry struc-
2 tures at the fundamental level. [γ˜µ(x)(pµ−qAµ(x)−Γµ(x))+mc2]ψ(x)=0, (1)
v
6 Γ (x) is the spinor connection, and γ˜µ(x) are the
Variousstudiesontheelectromagneticdynamicsofthe µ
8 elements of coordinate dependent Clifford algebra
1 magnetic and electric dipole moments shown that both
in the curved spacetime and satisfy the relation
0 their global and local physical properties can be sensi-
{γ˜µ(x),γ˜ν(x)} = 2gµν(x), with gµν(x) being the matric
0 tive to non-trivial geometries[12–30]. However, it is still
. difficult to observe them experimentally. On the other of the spacetime in the presence of a topological defect.
2
In the formalism of vierbein (or tetrad), which allows
0 hand, it is naturally expected that the energy spectra
us to define the spinors in the curved spacetime, the
6 of the charged carries in various quantum systems can
metric has the form [35], g (x) = ea (x)eb (x)η and
1 be deformed by the topological defects, and hence can µν µ ν ab
: be used to detect the topological properties of the back- the coordinate dependent Dirac matrix reads γ˜µ(x) =
v γ ea (x). The inverse of vierbein is defined by the rela-
i ground spacetime. The effects of nontrivial geometries a µ
X tions, ea (x)eµ(x)=δa and eµ (x)ea (x)=δµ .
survive in the non-relativistic limit, and have been stud- µ b b a ν ν
r ied in Refs. [12, 13]. Nevertheless, the electromagnetic The non-relativistic electromagnetic dynamics for a
a charged Dirac particle in the presence of topological de-
dynamics of charge carries in the presence of topological
fects has been discussed in Ref.[22] by using the Foldy-
defects were less discussed. It was pointed out that the
Wouthuysen transformation[37]. It was shown that at
infinite degeneracyofthe Landaulevelsisliftedbytopo-
the non-relativistic limit the Scho¨rdinger equation be-
logical disclinations and defects[31, 32]. In this letter we
comes ( here we have neglected all the terms involving
focus onconductivity behaviorof the chargercarriesun-
spin)
der the influences of topological defects and dislocation
in the non-relativistic limit. We consider the quantum
2
1
Halleffectwhichisoneofthemostprofoundphenomena H = p~−qA~(~x)−qK~(p~,~x) +qV(~x) , (2)
2m(cid:18) (cid:19)
where K~ behaving like an effective vector potential is
∗Electronicaddress: [email protected] the correction to the ordinary minimal coupling, but it
2
is as functions of both position and momentum. In the by using the leading order wave-functions which are ob-
convention of this work it is tained by neglecting the χ-dependent terms in the full
Hamiltonian. The leading order results can be found in
K~ = 1Ω~~ · p~−qA~ , (3) the standard textbooks, e.g., in Ref. [38].
q (cid:18) (cid:19) The electric current can be obtained by using the
Heisenberg equation. For the Hamiltonian (2) we obtain
wherethe matrixΩis the correctiontothe ordinaryver-
bena, px qBx2y ylz+lzy
J = qρ +χ −χ , (10)
x c(cid:18)m mρ2 mρ2 (cid:19)
Ωaµ(x)=eaµ(x)−δaµ, Ωµa(x)=eµa(x)−δµa. (4) p qB qBx3 xl +l x
y z z
J = qρ − x−χ +χ ,(11)
y c(cid:18)m m mρ2 mρ2 (cid:19)
Next, we focus on the influences of the topologicalde-
p
fect with line element given by J = qρ z . (12)
z c
m
2 2 2 2 2 2 2 2
ds =c dt −dρ −η ρ dϕ −dz . (5) Thenattheleadingordertheexpectationvaluesofthese
currentoperatorsareJ =hn,ℓ,k |J |n,ℓ,k iwhichhave
where η = 1−4λG/c2 is the deficit angle and λ is the i z i z
the following forms,
linear mass density of the cosmic string. In general, the
deficit angle can assume η >1, which corresponds to an px χ 2qBx2y ylz
J = qρ + −2ℜ , (13)
anti-conical spacetime with negative curvature. The ge- x c(cid:20)m m(cid:18) ρ2 ρ2 (cid:19)(cid:21)
ometry(5)correspondstoaconicalsingularitydescribed
by the curvature tensor Rρρ,,ϕϕ = 14−ηηδ2(~r). The vierbein Jy = qρc(cid:20)pmy − qmBx−χ(cid:18)2mqBρx23 −2ℜmxlρz2(cid:19)(cid:21) ,(14)
for this metric reads
p
z
J = qρ . (15)
1 0 0 0 z cm
ea =0 cosϕ −ηρsinϕ 0 (6) Theexpectationvalueoftheoperatorx2y/ρ2 canbeeas-
µ 0 sinϕ ηρcosϕ 0
ily calculated by using the symmetry as follows.
0 0 0 1
x2y 1 a x2y 1 a x2y
hk | |k i= dy =− dy =0
and y ρ2 y Ly Z−a ρ2 Ly Z−a ρ2
(16)
1 0 0 0
wherea=L /2,andL isthelengthofthesamplealong
eµa =00 −cossinϕϕ scionsϕϕ 00. (7) the yˆ-directioyn. Thus, ythis term does not contribute the
ηρ ηρ electric current along the xˆ-direction. The expectation
0 0 0 1 value of the operator ylz/ρ2 is slightly complicated. In-
tegrating out the variable y we obtain
The flat spacetime can be recovered for η = 1. In the
system of rectangular coordinates, the corresponding Ω yl 2x L
z y
=hn| arctan( )−1 p |ni , (17)
matrix, ρ2 (cid:18)L 2x (cid:19) x
y
0 0 0 0 Noticing that px|ni is purely image[38], then the expec-
Ωaµ =(1−η) 00 si−nφsinco2sφφ s−incφocso2sφφ 00 . (8) titatdioonesvnaolutealosfotghiivseoapneyractoonrtrisibaultsioonp.uTreheimeaxgpee.ctaTthiouns
0 0 0 0 value of the operator x3/ρ2 is more complicated. Inte-
grating our the variable y we obtain
Inserting this into (3) we can obtain the explicit expres- x3 2x2 L
y
sion of the effective vector potential, =hn| arctan( )|ni . (18)
ρ2 L 2x
y
2
K~ = χ qBx − lz ~e , (9) Duetothefactthatthequantumfluctuationoftheposi-
q(cid:18) ρ ρ(cid:19) φ tionoperatorisverysmallforharmonicoscillator,wecan
neglect these effects in the order of χ. Then we obtain
where χ = 1−η. This correction is proportional to the the approximated expectation value,
angularmomentumoftheparticleandtheexternalmag-
x3 2x2 L
netic field. Further, the topological defect affects only ≈ c arctan( y ) . (19)
thephysicsontheazimuthaldirection. Thisisduetothe ρ2 Ly 2xc
scalingoftheazimuthalangleinthelineelement(5). We
Using the same approximation we can obtain the expec-
will use the perturbation method to calculate the influ- tation value of the operator xl /ρ2,
z
encesofthis effectivevectorpotentialonthe expectation
valuesoftheelectriccurrents. Inthisapproximation,the xlz ¯hkyx2 2xc Ly
=hn| |ni≈¯hk arctan( ) . (20)
expectation values of the various currents are calculated ρ2 ρ2 y L 2x
y c
3
Then collecting all these results we obtain the expecta- Thecorrectionagaindependsontheangularmomentum
tion values of the electric currents and external magnetic field. Furthermore the compo-
nentsarethe sameasinlastsection. Theonlydifference
J¯ = 0 , (21)
x isthatthe correctionisalongthezˆ-directionratherthan
J¯ = qρ E 1+2χM(κ) , (22) φˆ-direction. This is due to the couplingbetween φ andz
y c
B in the line element (26). Again we use the perturbation
(cid:0) (cid:1)
J¯ = qρ ¯hkz (23) theory to calculate the expectation values of the elec-
z c m tric current operators in this case. Using the Heisenberg
equation we obtain
Here we define a dimensionless function
M(κ)=κarctan(κ1) , κ= 2Lxc (24) Jx = qρc(cid:18)pmx +ξρy2pmz(cid:19) , (31)
y
p qB x p
It is aas functions of the momentum in the yˆ-direction J = qρ y − x−ξ z , (32)
k , the external fields E and B, center of the harmonic y c(cid:18)m mc ρ2 m(cid:19)
y
oscillator x as well as the size of the sample L . In qρ l −qBx2
c y c z
J = p −ξ . (33)
terms of the mass density of the topological defect we z m (cid:18) z ρ2 (cid:19)
have χ=4λG/c2. Then corrected Hall conductivity is
Then at the leading order the expectation values of the
8λG current operators are
σ˜ =σ 1+ M(κ) (25)
H H(cid:18) c2 (cid:19)
ξ¯hk y
z
J = qρ hn,k | |n,k i , (34)
We study the Hall effect in the presence of topological x c m y ρ2 y
dislocation. The line element is
¯hk qB ξ¯hk x
y z
J = qρ −hn,k | x+ |n,k i ,(35)
ds2 =c2dt2−dρ2−ρ2dϕ2−(dz+ξdϕ)2. (26) y c(cid:18) m y mc m ρ2 y (cid:19)
¯hk l −qBx2
where ξ is the torsion of the topological dislocation[33, J = qρ z −ξhn,k | z |n,k i . (36)
z c(cid:18) m y mρ2 y (cid:19)
34]. Topologicaldislocationsaremuchmorerealisticline
defects. They can modify the energy spectrum of elec-
Wehavecalculatedtheexpectationvalueoftheoperator
trons moving in a uniform magnetic field. Landau levels y/ρ2, it is zero at the leading order. By using the same
in the presence of dislocations have been investigated.
approximation, we obtain the expectation values of the
The torsion can be identified with the surface density of 2 2 2 2
operators x/ρ , x /ρ and l /ρ ,
z
the Burgers vector in the classical theory of elasticity.
The vierbein in this case reads x 2 L
y
≈ arctan( ) , (37)
ρ2 L 2x
1 0 0 0 y c
0 cosϕ −ρsinϕ 0 l ¯hk x 2h¯k L
eaµ(x)= 0 sinϕ ρcosϕ 0, (27) ρz2 = ρy2 ≈ L y arctan(2xy ) . (38)
y c
0 0 ξ 1
Then collecting all these results we obtain the expecta-
and, tion values of the electric current operators,
1 0 0 0
J = 0 , (39)
x
0 cosϕ sinϕ 0
eµa(x)= 0 −sinϕ cosϕ 0 . (28) J = −qρ E 1− 2ξ ¯hkz B 1M(κ) , (40)
0 ξ sinρϕ −ξ cρosϕ 1 y cB(cid:18) Ly m Eκ (cid:19)
ρ ρ ¯hk 2ξ ¯hk 1
z y
J = qρ + M(κ) . (41)
The flat spacetime can be recovered for ξ = 0. In the z c(cid:18) m L m κ (cid:19)
y
reference frame of the rectangular coordinates, the cor-
responding Ω matrix is, The corrected Hall conductance is
2ξ ¯hk B 1
0 0 0 0 σ˜ =σ 1− z M(κ) . (42)
H H
Ωa = 0 0 0 0 . (29) (cid:18) Ly m Eκ (cid:19)
µ 0 0 0 0
0 −ρξ sinφ ρξ cosφ 0 CthoemcpoarrreecdtetodtHhaelrlecsounltdsuicntatnhceecadseepoenfdtospoonlotghiceamldoemfeecnt-,
tum in the zˆ-direction.
Inserting this into (3) we can obtain the explicit expres-
In summary, the influences of topological defect and
sion of the effective vector potential,
dislocation on quantum Hall effects have been studied.
ξ l qBx2 For the topologicaldefect with line element (5), the vec-
K~ = z − ~e . (30)
q(cid:18)ρ2 ρ2 (cid:19) z tor potential A~ receives corrections proportional to the
4
lineardensityofthecosmicstring,angularmomentuml tion along the zˆ-direction, this is a distinct feature from
z
of the particle and itself. These additional potentials af- the last case. Even though J is affected, the effect will
z
fectonlythedynamicsalongtheazimuthaldirection. By be negligible due to the smallness of the Raaga vector
using the perturbationmethodwe obtainedthe expecta- compared with the sample size L . However the Hall
y
tion values of the charge current Ji as well as the Hall current Jy receives significant correction due to the en-
conductanceσH. ThechargecurrentJx doesnotreceive hancement factor B/E which is of order 108 typically.
correction due to the harmonic motion along this direc- Thecorrectionalsodependsonthemomentumalongthe
tion, and Jz does not also influence due to the fact that zˆdirection, andmaybe positive andnegativedepending
the line element does not deform the motion of particle on the expectation value of the coordinate x = x , with
c
along this direction. However, the Hall current Jy re- −Lx/2 < xc < Lx/2. For a Ragga vector of order 1nm,
ceivescorrectionproportionaltothelineardensityofthe the correction is of order 10−3 that is measurable.
cosmic string. The correction is always positive and de-
pends on the expectation value of the coordinate x=x
c
as well as the sample size L . The dependence is repre-
y
sented by a function M(κ)=κarctan(1/κ),κ=2x /L . Acknowledgments: K. M. is supported by the China
c y
For a light cosmic string with χ = 4λG/c2 ∼ 10−9, it is Scholarship Council under Grant No. 201207010002,
verydifficulttoobservetheeffects. Howeverforacosmic and the Hanjiang Scholar Project of Shaanxi University
string with heaviermass density, the correctioncould be of Technology. J. H. W. is supported by the National
observable. Natural Science Foundation of China under Grant No.
For the topological dislocation with line element (26), 11147181and the Scientific Research Project in Shaanxi
the canonicalmomentumis deformedandreceivesa cor- Province under Grant No. 2009K01-54 and Grant No.
rection proportional to the Ragga vector ξ~ and the an- 12JK0960. H.-X. Y. is supported in part by the Startup
gular momentum l of the charge particle. The charge Foundation of the University of Science and Technology
z
current J does not receive correction due to the har- of China and the Project of Knowledge Innovation Pro-
x
monic motion along this direction. There is a correc- gram (PKIP) of the Chinese Academy of Sciences.
[1] KibbleT W B 1976 J. Phys. A:Math. Gen. 19 1387 [21] Shen J Q 2004 Phys.Rev.D70 067501
[2] Muniz C R, Bezerra V B and Cunha M S 2014 Ann. [22] WangJ-H,MaKandLiK2013Phys.Rev.A87032107
Phys.350 105 [23] Wang J-H, Ma K, Li K and Fan H-W 2015 Ann. Phys.
[3] Hiscock W A 1985 Phys.Rev.D31 3288 362 327
[4] Vilenkin A 1985 Phys. Rep.121 263 [24] Yang H-Xand Ma H-L 2008 JCAP 8 24
[5] BarriolaMandVilenkinA1989Phys.Rev.Lett.63341 [25] Ma K and Dulat S 2011 Phys. Rev.,A84 012104
[6] Vilenkin A 1983 Phys. Lett. B 133 177 [26] Ma K,WangJ-H and YangH-X2016 Phys.Lett. B756
[7] Kleinert H 1989 World Scientific (Singapore) vol 4 chap 221
2 p 1362 [27] Chowdhury Dand Basu B2014 Phys.Rev.D90125014
[8] Katanaev M O and Volovich I V 1999 Ann. Phys. 271 [28] Chowdhury D and Basu B 2013 Ann.Phys. 329 166
203 [29] Basu B and Chowdhury D 2013 Ann.Phys. 335 47
[9] Katanaev M O and Volovich I V 1992 Ann. Phys. (NY) [30] Basu B, Chowdhury D and Ghosh S 2013 Phys. Lett.
216 1 A377 1661
[10] FurtadoC and Moraes F1994 Phys.Lett. A188 392 [31] Furtado C, Cunha B G C da and Moraes F et. al. 1994
[11] Assis J G de, Furtado C and Bezerra V B 2000 Phys. Phys. Lett. A195 90
Rev.D62 045003 [32] Marques G A, Furtado C, Bezerra V B and Moraes F
[12] Bakke K, Ribeiro L R, Furtado C and Nascimento J R 2001 J. Phys. A:Math. Gen. 34 5945
2009 Phys.Rev. D79024008 [33] Gal’tsovDVandLetelierPS1993Phys.Rev.D474273
[13] BakkeK 2014 Ann.Phys. (NY) 346 51 [34] LorenciVADeandMoreiraJrES2002Phys.Rev.D65
[14] Eugenio R Bezerra de Mello 2004 JHEP 06 016 085013
[15] Mota H F and BakkeK 2014 Phys.Rev.D89 027702 [35] Nakahara M 1998 (Institute of Physics Publishing,
[16] BakkeK 2013 Gen. Relativ. Gravit. 45 1847 Bristo)
[17] BakkeK,NascimentoJRandFurtadoC2008Phys.Rev. [36] Hindmarsh M B and Kibble T W B 1995 Rep. Prog.
D78064012 Phys. 58 477
[18] BakkeK and FurtadoC 2013 Ann.Phys.(NY)336 489 [37] Foldy L L and WouthuysenS 1950 Phys.Rev.78 29
[19] Belich H and BakkeK 2014 Phys.Rev.D 90 025026 [38] Landau L D and Lifshitz L M 1981 Elsevier(Singapore)
[20] MonteroMandMartin-MartinezE2011Phys.Rev.A84 Pte Ltd vol 3 chap 3 p 67
012337
