There are no periodic wave-functions in the RDM but close to the critical
energies there exist periodic envelopes. These envelopes are given by the
non-disordered properties of the system.Comment: RevTex file, 1 page, Comment X. Huang, X. Wu and C. Gong, Phys. Rev.
  B 55, 11018 (1997

F. Domínguez-Adame

J. C. Flores

M. Hilke

Ph. Jacquod

X. Huang

English

arXiv

© 1998 The American Physical Society.
Part of this work was made possible due to the project FDI-Chile (J.C.F. and M.H.).In a recent report, Huang, Wu, and Gong [Phys. Rev. B 55, 11018 (1997)] studied numerically the electronic properties of a random dimer model (RDM) and found extended periodic wave functions near the critical energies. We argue that there are no periodic wave functions in the RDM but close to the critical energies there exist periodic envelopes. These envelopes are given by the nondisordered properties of the system.FDI-ChileDepto. de Física de MaterialesFac. de Ciencias FísicasTRUEpu

Hilke, M.

Flores, J. C.

Domínguez-Adame Acosta, Francisco

Docta Complutense

hedule asPHYSICAL REVIEW B 1 OCTOBER 1998-IVOLUME 58, NUMBER 13COMMENTSComments are short papers which criticize or correct papers of other authors previously published inPhysical Review B. EachComment should state clearly to which paper it refers and must be accompanied by a brief abstract. The same publication scfor regular articles is followed, and page proofs are sent to authors.Comment on ‘‘Periodic wave functions and number of extended statesin random dimer systems’’M. HilkeDepartment of Electrical Engineering, Princeton University, Princeton, New Jersey 08544J. C. FloresDepartamento de Fı´sica, Universidad de Tarapaca´, Casilla 7-D, Arica, ChileF. Domı́nguez-AdameDepartamento de Fı´sica de Materiales, Universidad Complutense, E-28040 Madrid, Spain~Received 12 November 1997!In a recent report, Huang, Wu, and Gong@Phys. Rev. B55, 11 018~1997!# studied numerically the elec-tronic properties of a random dimer model~RDM! and found extended periodic wave functions near the criticalenergies. We argue that there are no periodic wave functions in the RDM but close to the critical energies thereexist periodic envelopes. These envelopes are given by the nondisordered properties of the system.@S0163-1829~98!10337-5#ri-dethex.slasvrvncnsvtthrondri-ef.tal-enedIn a recent paper,1 Huang, Wu, and Gong studied numecally the electronic properties of a random dimer mo~RDM! and found extended periodic wave functions nearcritical energies. They also claimed that the number oftended states is proportional toAN, whereN is the size ofthe system, which is consistent with previous results.~i! There areno periodic wave functions in the RDMIndeed, even at the critical energies it was shown previou2that the wave functions have a constant norm but the phare random. Essentially the wave function isc i5eiu i, whereu i is random. This does, however, lead to a constant enlope. We define the envelope as the ensemble of cuwhich include all possible values of the random wave fution. In this point we present the analytical derivation agaithe main result presented in Ref. 1 and show that the enlope ~and not the wave function itself! is periodic near thecritical energies as long as the localization length exceedssize of the system.The RDM can be analyzed in terms of the product offollowing two transfer matrices3TA5S E2VA1 210 D 2, TB5S E2VB1 210 D 2. ~1!They reduce to the negative identity matrix whenE5VA orE5VB ~critical energies!, which means thatTA andTB com-mute. Close to the critical energy, i.e., whenDE5E2VA issmall, the commutator@TA ,TB#5O(DE), as opposed to thecase without the dimer condition, where the commutatoalways of order 1. The total system is described by a randPRB 580163-1829/98/58~13!/8837~2!/$15.00le-yese-es-te-heeismmixture of the products of the transfer matricesTA andTB ,thus T5P iTAniTBmi, whereni and mi are random. Thereforethe total product isT5TAnTBm1(N/4)O(DE), where n5( ini , m5( imi , and N52(n1m). This clearly demon-strates that the solution is equivalent to the system withT5~21!nT Bm , ~2!as long asN is smaller than 1/DE.Our main result is that the envelope of Eq.~2! reproducesall the numerical figures of Ref. 1. Indeed Huang, Wu, aGong studied numerically two cases:~a! VA521, VB51, andE5211DE. As the envelopecan be obtained from the solution ofTBm , the correspondingwave functions can be written asc i;cos(ki), whereE2VB52 cos(k). To first order inDE we obtaink.p2ADE/2.From this follows trivially that the general solution has peodic solutions for 2mADE/25pp, where p is an integer.This leads toDE52p2p2/(N/2)2, whereN/2 is the numberof dimers when they are equally distributed.~b! VA520.5, VB50.5, andE2VB5211DE. In thiscase we obtain k.2p/32)DE/2 and DE52pp/)(N/2).In both cases~a! and~b! the same dependences as in R1 are obtained.~ii ! The use of the inverse participation ratio~IPR! IPR5N( j 51N uc j u4, wherec j is the normalized wave function asite j, is very ambiguous. We can, indeed, construct delocized wave functions with arbitrary IPR. For instance, whconsidering the following periodic sparse and normaliz8837 © 1998 The American Physical Societyitthnolytatsdiceredect8838 PRB 58COMMENTSwave functionc j25T/N if j /T5 integer andc j250 other-wise, where the periodT is an integer (T,N), then IPR5T. In this way we have a set of delocalized states warbitrary IPR. Therefore, the relevant quantity to study isscaling of IPR withN, which gives consistent results evefor sparse states as, for example, in banded randmatrices.4 For a more general approach a multifractal anasis is proved useful.5~iii ! The number of extended states is proportionalAN,which means that the relative number of extended sthem-oestends to zero as 1/AN; therefore the delocalization propertieare important only in small systems.6In conclusion, there areno periodic wave functions in theRDM, but close to the critical energies there exist perioenvelopes. These envelopes are given by the nondisordproperties of the system.Part of this work was made possible due to the projFDI-Chile ~J.C.F. and M.H.!1X. Huang, X. Wu, and C. Gong, Phys. Rev. B55, 11 018~1997!.2J. C. Flores and M. Hilke, J. Phys. A26, L1255 ~1993!.3M. Hilke, J. Phys. A27, 4773~1994!.4Ph. Jacquod and D. L. Shepelyansky, Phys. Rev. Lett.75,3501 ~1995!.5F. Domı́nguez-Adame, E. Macia´, and A. Sa´nchez, Phys. Rev. B48, 6054~1993!.6M. Hilke and J. C. Flores, Phys. Rev. B55, 10 625~1997!.

Comment on "Periodic wave functions and number of extended states in random dimer systems"

Crossref

Comment on “Periodic wave functions and number of extended states in random dimer systems”

https://docta.ucm.es/bitstreams/74dc3f1e-2b26-441d-8ad0-1157eb49c26e/download

Comment on ``Periodic wave functions and number of extended states in random dimer systems'

Abstract

Similar works

Full text

Available Versions

Docta Complutense

Crossref