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Hall 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 corresponding 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. 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