We show that the pitfalls encountered in earlier calculations of the RKKY
range function for a non interacting one dimensional electron gas at zero
temperature can be unraveled and successfully dealt with through a proper
handling of the impurity potential.Comment: to appear in Phys. Re

C. Kittel

G. F. Giuliani

Gabriele F. Giuliani

Giovanni Vignale

I. Gradshteyn

Trinanjan Datta

English

arXiv

URL:http://link.aps.org/doi/10.1103/PhysRevB.72.033411
DOI:10.1103/PhysRevB.72.033411We show that the pitfalls encountered in earlier calculations of the RKKY range function for a noninteracting one- dimensional electron gas at zero temperature can be unraveled and successfully dealt with through a proper handling of the impurity potential

Giuliani, Gabriele, 1953-

Vignale, Giovanni, 1957-

Datta, Trinanjan

University of Missouri: MOspace

RKKY range function of a one-dimensional noninteracting electron gasGabriele F. Giuliani,1,* Giovanni Vignale,2,† and Trinanjan Datta11Department of Physics, Purdue University, West Lafayette, Indiana 47907, USA2Department of Physics, University of Missouri, Columbia, Missouri 65211, USAReceived 15 February 2005; published 19 July 2005We show that the pitfalls encountered in earlier calculations of the RKKY range function for a noninteract-ing one-dimensional electron gas at zero temperature can be unraveled and successfully dealt with through aproper handling of the impurity potential.DOI: 10.1103/PhysRevB.72.033411 PACS numbers: 71.10.Ca, 72.10.d, 71.55.i, 75.30.HxThe apparently straightforward evaluation of theRuderman-Kittel-Kasuya-Yosida RKKY range function, ormore generally, of the linear density modulation nx in-duced at zero temperature in a noninteracting one-dimensional electron gas by a localized static impurity mod-eled with a  function potential has proven surprisinglytroublesome. Although the original calculation gave an incor-rect answer,1 more recent investigations appear to suggestthat only a certain procedure, known to lead to a physicallysensible answer, should be employed.2The original and most popular procedure is based on thestandard theory of linear response.3 If the impurity potentialis assumed to be of the form Ux= 2u /2mx where u isa suitable wave vector, then one hasnx =2u2mLq 0q,0eiqx, 1where L is the system length and 0q ,0 is the staticLindhard response function in one dimension given by0q,0 =2LPknk − nk+qk+q − k, 2where the notation P stipulates that the principal part of thesum must be taken. Substituting 2 into 1 and using thezero temperature occupation numbers immediately leads tothe formulanx =u22−dqeiqx−kFkFdk 12kq + q2 − 12kq − q2 .3The interpretation and handling of this expression rest at theorigin of the problem at hand.If one simply proceeds to explicitly evaluate the integralover k in Eq. 3, or, which is the same, that of 2, the resultis well known and is given by0q,0 =2m2qln2kF + q2kF − q . 4At this point the second integral over q can be readilyevaluated leading tonx = −usi2kFx , 5where the sine integral function appears.4 This result pro-vides the correct answer to the problem. In particular, thelarge distance behavior turns out to benx  −ucos2kFxx, 6an expression displaying the expected decay and Friedeloscillations.5 One should note at this point that, in view ofthe singular behavior of the integrand at the origin, howevertempting, the order of the k and q integrations in Eq. 3cannot be freely exchanged.2 Doing so leads to the mani-festly unphysical answer1nx = −usi2kFx + 2  not! . 7This pitfall is unique to the one-dimensional case for in twoand three dimensions, formulas equivalent to 3 can be de-rived and safely evaluated by exchanging at will the order ofthe intervening integrations.An alternative, appealing and equally physically validprocedure to obtain nx is in the case offered by extractingthe leading linear term of the formulanx − n0 = 2 	k	kF„		kx	2 − 		k0x	2… , 8valid for single Slater determinant states. Now, if the usualfirst-order perturbation theory result for 	kx is used in 8,one is immediately led to an equation that differs from 3merely for the exchange of the order of the quadratures.1Since, as observed above, such an expression does lead to anunphysical result, it has been suggested that this route isunphysical and should be avoided.2 We find that this conclu-sion is unwarranted for, as we show next, the difficulty lieshere with the use of perturbation theory, which is invalid, andnot with Eq. 8 per se.To prove our assertion we observe that the exact delocal-ized eigenstates of the Schrödinger equationPHYSICAL REVIEW B 72, 033411 20051098-0121/2005/723/0334112/$23.00 ©2005 The American Physical Society033411-122m− 2x2 + ux	x = E	x , 9can be written as a superposition of the two following nor-malized scattering states	k+x = eikxL+u2ik − ue−ikxL, x 02ik2ik − ueikxL, x 0 10and	k−x = 2ik2ik − ue−ikxL, x 0e−ikxL+u2ik − ueikxL, x 0, 11where k is limited to positive values only. By adding themodulus squares of 	k+x and 	k−x and by summing overk0 in 8, one obtains the following exact expression forthe electronic density nx:nx =20kFdk1 + 2uk sin2k	x	4k2 + u2 − u2 cos2k	x	4k2 + u2  .12At this point we are left with extracting the linear term inu. This must be done with some care. In particular, one mustresist the temptation to simply drop the last term in Eq. 12and, at the same time, to neglect the u2 in the denominator ofthe second one. To do so coincides with making use of first-order perturbation theory and leads exactly to the originalunphysical result of Eq. 7. Although it is safe to handle thesecond term of 12 as just described, the third term doescontribute a first-order term that can be readily extracted bymaking use of the relationlimu→0u4k2 + u2=2k , 13leading tonx 2kF+u0kFdksin2k	x	k−2  , 14a result that can be readily seen to coincide with the correctanswer of Eq. 5.We have therefore shown that proper handling of the im-purity potential allows one to correctly carry out the calcu-lation of the linear density modulation via either the responsefunction method of Eq. 1 or the alternative direct procedureoffered by 8.6We conclude by remarking that a proper treatment ofthe problem of the effects of a static localized impurity inan interacting one-dimensional electron liquid can beachieved by means of the Luttinger liquid model. Theproblem is highly nontrivial as, although Friedel-like oscil-lations with an amplitude decay generally ruled by theinteraction strength exist at intermediate distances,3 for largedistances the physics of the phenomenon is nonlinear inthe impurity potential.7 Finally, a recent discussion of theeffects of the Coulomb interaction on the Friedel oscillationsof an electron liquid in two and three dimensions can befound in Ref. 8.The authors wish to thank A. W. Overhauser and GeorgeE. Simion for useful conversations.*Electronic address: gfg@physics.purdue.edu,tdatta@physics.purdue.edu†Electronic address: vignaleg@missouri.edu1 C. Kittel, in Solid State Physics, vol. 22, edited by F. Seitz, D.Turnbull, and H. Ehreinreich Academic Press, New York, 1968see also erratum ibidem.2 Y. Yafet, Phys. Rev. B 36, 3948 1987.3 G. F. Giuliani and G. Vignale, Quantum Theory of the ElectronLiquid Cambridge University Press, Cambridge, England,2005.4 I. Gradshteyn and I. Ryzhik, Table of Integrals, Series and Prod-ucts Academic Press, San Diego, 1965.5 J. Friedel, Adv. Phys. 3, 446 1954.6 Of course the correct answer can also be arrived at by calculatingnx at finite temperature and then taking the T→0 as done forinstance in V. I. Litvinov and V. K. Dugaev, Phys. Rev. B 58,3584 1998. This sheds no light onto the problem at hand.7 R. Egger and H. Grabert, Phys. Rev. Lett. 75, 3505 1995.8 G. Simion and G. F. Giuliani Phys. Rev. B 72, 045127 2005.BRIEF REPORTS PHYSICAL REVIEW B 72, 033411 2005033411-2

Physical Review B

RKKY range function of a one-dimensional noninteracting electron gas

Gabriele F.  Giuliani

Crossref

We show that the pitfalls encountered in earlier calculations of the RKKY range function for a noninteracting one-dimensional electron gas at zero temperature can be unraveled and successfully dealt with through a proper handling of the impurity potential

Giuliani, G. F.

Vignale, G.

Datta, T.

Purdue E-Pubs

PhysicsPhysics Research PublicationsPurdue University Year RKKY range function of aone-dimensional noninteracting electrongasG. F. Giuliani G. VignaleT. DattaThis paper is posted at Purdue e-Pubs.http://docs.lib.purdue.edu/physics articles/159RKKY range function of a one-dimensional noninteracting electron gasGabriele F. Giuliani,1,* Giovanni Vignale,2,† and Trinanjan Datta11Department of Physics, Purdue University, West Lafayette, Indiana 47907, USA2Department of Physics, University of Missouri, Columbia, Missouri 65211, USAReceived 15 February 2005; published 19 July 2005We show that the pitfalls encountered in earlier calculations of the RKKY range function for a noninteract-ing one-dimensional electron gas at zero temperature can be unraveled and successfully dealt with through aproper handling of the impurity potential.DOI: 10.1103/PhysRevB.72.033411 PACS numbers: 71.10.Ca, 72.10.d, 71.55.i, 75.30.HxThe apparently straightforward evaluation of theRuderman-Kittel-Kasuya-Yosida RKKY range function, ormore generally, of the linear density modulation nx in-duced at zero temperature in a noninteracting one-dimensional electron gas by a localized static impurity mod-eled with a  function potential has proven surprisinglytroublesome. Although the original calculation gave an incor-rect answer,1 more recent investigations appear to suggestthat only a certain procedure, known to lead to a physicallysensible answer, should be employed.2The original and most popular procedure is based on thestandard theory of linear response.3 If the impurity potentialis assumed to be of the form Ux= 2u /2mx where u isa suitable wave vector, then one hasnx =2u2mLq 0q,0eiqx, 1where L is the system length and 0q ,0 is the staticLindhard response function in one dimension given by0q,0 =2LPknk − nk+qk+q − k, 2where the notation P stipulates that the principal part of thesum must be taken. Substituting 2 into 1 and using thezero temperature occupation numbers immediately leads tothe formulanx =u22−dqeiqx−kFkFdk 12kq + q2−12kq − q2 .3The interpretation and handling of this expression rest at theorigin of the problem at hand.If one simply proceeds to explicitly evaluate the integralover k in Eq. 3, or, which is the same, that of 2, the resultis well known and is given by0q,0 =2m2qln2kF + q2kF − q . 4At this point the second integral over q can be readilyevaluated leading tonx = −usi2kFx , 5where the sine integral function appears.4 This result pro-vides the correct answer to the problem. In particular, thelarge distance behavior turns out to benx  −ucos2kFxx, 6an expression displaying the expected decay and Friedeloscillations.5 One should note at this point that, in view ofthe singular behavior of the integrand at the origin, howevertempting, the order of the k and q integrations in Eq. 3cannot be freely exchanged.2 Doing so leads to the mani-festly unphysical answer1nx = −usi2kFx + 2  not! . 7This pitfall is unique to the one-dimensional case for in twoand three dimensions, formulas equivalent to 3 can be de-rived and safely evaluated by exchanging at will the order ofthe intervening integrations.An alternative, appealing and equally physically validprocedure to obtain nx is in the case offered by extractingthe leading linear term of the formulanx − n0 = 2 	k	kF„		kx	2 − 		k0x	2… , 8valid for single Slater determinant states. Now, if the usualfirst-order perturbation theory result for 	kx is used in 8,one is immediately led to an equation that differs from 3merely for the exchange of the order of the quadratures.1Since, as observed above, such an expression does lead to anunphysical result, it has been suggested that this route isunphysical and should be avoided.2 We find that this conclu-sion is unwarranted for, as we show next, the difficulty lieshere with the use of perturbation theory, which is invalid, andnot with Eq. 8 per se.To prove our assertion we observe that the exact delocal-ized eigenstates of the Schrödinger equationPHYSICAL REVIEW B 72, 033411 20051098-0121/2005/723/0334112/$23.00 ©2005 The American Physical Society033411-122m− 2x2+ ux	x = E	x , 9can be written as a superposition of the two following nor-malized scattering states	k+x = eikxL+u2ik − ue−ikxL, x  02ik2ik − ueikxL, x  0 10and	k−x = 2ik2ik − ue−ikxL, x  0e−ikxL+u2ik − ueikxL, x  0, 11where k is limited to positive values only. By adding themodulus squares of 	k+x and 	k−x and by summing overk0 in 8, one obtains the following exact expression forthe electronic density nx:nx =20kFdk1 + 2uk sin2k	x	4k2 + u2−u2 cos2k	x	4k2 + u2 .12At this point we are left with extracting the linear term inu. This must be done with some care. In particular, one mustresist the temptation to simply drop the last term in Eq. 12and, at the same time, to neglect the u2 in the denominator ofthe second one. To do so coincides with making use of first-order perturbation theory and leads exactly to the originalunphysical result of Eq. 7. Although it is safe to handle thesecond term of 12 as just described, the third term doescontribute a first-order term that can be readily extracted bymaking use of the relationlimu→0u4k2 + u2=2k , 13leading tonx 2kF+u0kFdksin2k	x	k−2  , 14a result that can be readily seen to coincide with the correctanswer of Eq. 5.We have therefore shown that proper handling of the im-purity potential allows one to correctly carry out the calcu-lation of the linear density modulation via either the responsefunction method of Eq. 1 or the alternative direct procedureoffered by 8.6We conclude by remarking that a proper treatment ofthe problem of the effects of a static localized impurity inan interacting one-dimensional electron liquid can beachieved by means of the Luttinger liquid model. Theproblem is highly nontrivial as, although Friedel-like oscil-lations with an amplitude decay generally ruled by theinteraction strength exist at intermediate distances,3 for largedistances the physics of the phenomenon is nonlinear inthe impurity potential.7 Finally, a recent discussion of theeffects of the Coulomb interaction on the Friedel oscillationsof an electron liquid in two and three dimensions can befound in Ref. 8.The authors wish to thank A. W. Overhauser and GeorgeE. Simion for useful conversations.*Electronic address: gfg@physics.purdue.edu,tdatta@physics.purdue.edu†Electronic address: vignaleg@missouri.edu1 C. Kittel, in Solid State Physics, vol. 22, edited by F. Seitz, D.Turnbull, and H. Ehreinreich Academic Press, New York, 1968see also erratum ibidem.2 Y. Yafet, Phys. Rev. B 36, 3948 1987.3 G. F. Giuliani and G. Vignale, Quantum Theory of the ElectronLiquid Cambridge University Press, Cambridge, England,2005.4 I. Gradshteyn and I. Ryzhik, Table of Integrals, Series and Prod-ucts Academic Press, San Diego, 1965.5 J. Friedel, Adv. Phys. 3, 446 1954.6 Of course the correct answer can also be arrived at by calculatingnx at finite temperature and then taking the T→0 as done forinstance in V. I. Litvinov and V. K. Dugaev, Phys. Rev. B 58,3584 1998. This sheds no light onto the problem at hand.7 R. Egger and H. Grabert, Phys. Rev. Lett. 75, 3505 1995.8 G. Simion and G. F. Giuliani Phys. Rev. B 72, 045127 2005.BRIEF REPORTS PHYSICAL REVIEW B 72, 033411 2005033411-2

We show that the pitfalls encountered in earlier calculations of the RKKY range function for a noninteracting one- dimensional electron gas at zero temperature can be unraveled and successfully dealt with through a proper handling of the impurity potential

On the RKKY range function of a one dimensional non interacting electron gas

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