Electronic excitations and correlations in quantum bars

The spectrum of boson fields and two-point correlators are analyzed in a quantum bar system (a superlattice formed by two crossed interacting arrays of quantum wires), with a short-range interwire interaction. The standard bosonization procedure is shown to be valid, within the two-wave approximatio...

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Дата:	2002
Автори:	Kuzmenko, I., Gredeskul, S., Kikoin, K., Avishai, Y.
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Мова:	English
Опубліковано:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2002
Назва видання:	Физика низких температур
Теми:	Сильно коррелированные системы и высокотемпературная сверхпроводимость
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/130232
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Цитувати:	Electronic excitations and correlations in quantum bars / I.Kuzmenko, S.Gredeskul, K.Kikoin, Y.Avishai // Физика низких температур. — 2002. — Т. 28, № 7. — С. 752-762. — Бібліогр.: 17 назв. — англ.

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spelling	irk-123456789-1302322018-02-10T03:04:16Z Electronic excitations and correlations in quantum bars Kuzmenko, I. Gredeskul, S. Kikoin, K. Avishai, Y. Сильно коррелированные системы и высокотемпературная сверхпроводимость The spectrum of boson fields and two-point correlators are analyzed in a quantum bar system (a superlattice formed by two crossed interacting arrays of quantum wires), with a short-range interwire interaction. The standard bosonization procedure is shown to be valid, within the two-wave approximation. The system behaves as a sliding Luttinger liquid in the vicinity of the Γ point, but its spectral and correlation characteristics have either 1D or 2D nature depending on the direction of the wave vector in the rest of the Brillouin zone. Due to the interwire interaction, unperturbed states propagating along the two arrays of wires are always mixed, and the transverse components of the correlation functions do not vanish. This mixing is especially strong around the diagonals of the Brillouin zone, where the transverse correlators have the same order of magnitude as the longitudinal ones. 2002 Article Electronic excitations and correlations in quantum bars / I.Kuzmenko, S.Gredeskul, K.Kikoin, Y.Avishai // Физика низких температур. — 2002. — Т. 28, № 7. — С. 752-762. — Бібліогр.: 17 назв. — англ. 0132-6414 PACS: 73.23.-b, 73.40.-c http://dspace.nbuv.gov.ua/handle/123456789/130232 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Сильно коррелированные системы и высокотемпературная сверхпроводимость Сильно коррелированные системы и высокотемпературная сверхпроводимость
spellingShingle	Сильно коррелированные системы и высокотемпературная сверхпроводимость Сильно коррелированные системы и высокотемпературная сверхпроводимость Kuzmenko, I. Gredeskul, S. Kikoin, K. Avishai, Y. Electronic excitations and correlations in quantum bars Физика низких температур
description	The spectrum of boson fields and two-point correlators are analyzed in a quantum bar system (a superlattice formed by two crossed interacting arrays of quantum wires), with a short-range interwire interaction. The standard bosonization procedure is shown to be valid, within the two-wave approximation. The system behaves as a sliding Luttinger liquid in the vicinity of the Γ point, but its spectral and correlation characteristics have either 1D or 2D nature depending on the direction of the wave vector in the rest of the Brillouin zone. Due to the interwire interaction, unperturbed states propagating along the two arrays of wires are always mixed, and the transverse components of the correlation functions do not vanish. This mixing is especially strong around the diagonals of the Brillouin zone, where the transverse correlators have the same order of magnitude as the longitudinal ones.
format	Article
author	Kuzmenko, I. Gredeskul, S. Kikoin, K. Avishai, Y.
author_facet	Kuzmenko, I. Gredeskul, S. Kikoin, K. Avishai, Y.
author_sort	Kuzmenko, I.
title	Electronic excitations and correlations in quantum bars
title_short	Electronic excitations and correlations in quantum bars
title_full	Electronic excitations and correlations in quantum bars
title_fullStr	Electronic excitations and correlations in quantum bars
title_full_unstemmed	Electronic excitations and correlations in quantum bars
title_sort	electronic excitations and correlations in quantum bars
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2002
topic_facet	Сильно коррелированные системы и высокотемпературная сверхпроводимость
url	http://dspace.nbuv.gov.ua/handle/123456789/130232
citation_txt	Electronic excitations and correlations in quantum bars / I.Kuzmenko, S.Gredeskul, K.Kikoin, Y.Avishai // Физика низких температур. — 2002. — Т. 28, № 7. — С. 752-762. — Бібліогр.: 17 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT kuzmenkoi electronicexcitationsandcorrelationsinquantumbars AT gredeskuls electronicexcitationsandcorrelationsinquantumbars AT kikoink electronicexcitationsandcorrelationsinquantumbars AT avishaiy electronicexcitationsandcorrelationsinquantumbars
first_indexed	2025-07-09T13:06:44Z
last_indexed	2025-07-09T13:06:44Z
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fulltext	Fizika Nizkikh Temperatur, 2002, v. 28, No. 7 , p. 752–762 Electronic excitations and correlations in quantum bars I. Kuzmenko, S. Gredeskul, K. Kikoin, and Y. Avishai Department of Physics, Ben-Gurion University of the Negev, P.O.Box 84105 Beer-Sheva, Israel E-mail: sergeyg@bgumail.bgu.ac.il Received February 1, 2002 The spectrum of boson fields and two-point correlators are analyzed in a quantum bar sys- tem (a superlattice formed by two crossed interacting arrays of quantum wires), with a short-range interwire interaction. The standard bosonization procedure is shown to be valid, within the two-wave approximation. The system behaves as a sliding Luttinger liquid in the vicinity of the � point, but its spectral and correlation characteristics have either 1D or 2D nature depending on the direction of the wave vector in the rest of the Brillouin zone. Due to the interwire interaction, unperturbed states propagating along the two arrays of wires are al- ways mixed, and the transverse components of the correlation functions do not vanish. This mixing is especially strong around the diagonals of the Brillouin zone, where the transverse correlators have the same order of magnitude as the longitudinal ones. PACS: 73.23.–b, 73.40.–c 1. Introduction Diverse D � 1 dimensional objects embedded in D-dimensional structures were recently investi- gated experimentally and analysed theoretically. Rubbers and various percolation networks are ex- amples of such disordered D � 1 objects, whereas self-organized stripes in oxicuprates, manganites, nanotube ropes, and quantum Hall systems are ex- amples of ordered (periodic) structures of this kind. In some cases, the effective dimensions of such structures may be intermediate, e.g., between D � 1 and D � 2. They are especially promising can- didates for studying novel electronic correlation properties, which, in particular, are relevant for the search of Luttinger liquid (LL) fingerprints in two dimensions. This challenging idea is motivated by noticing some unusual properties of electrons in Cu–O planes in high-Tc materials [1]. However, the Fermi liquid state seems to be rather robust in two dimensions. In this respect, a 2D system of weakly coupled 1D quantum wires [2–4] looks promising. Indeed, a theoretical analysis of stable LL phases was recently presented for a system con- sisting of coupled parallel quantum wires [5–7] and for 3D stacks of sheets of such wires in parallel and crossed orientations [8]. In most of these cases, the interaction between the parallel quantum wires is assumed to be perfect along the wire [8], whereas the interaction between the modes generated in dif- ferent wires depends only on the interwire distance. Along these lines, generalization of the LL theory for quasi-2D (and even 3D) systems is reported in Ref. 8, where the interaction between two crossed arrays of parallel quantum wires forming some kind of a network depends on the distance between points belonging to different arrays. As a result, the grid of crossed arrays retains its LL properties for propagation along both subsets of parallel wi- res, whereas the cross-correlations remain nonsin- gular. This LL structure can be interpreted as a quantum analog of a classical sliding phases of cou- pled XY chains [9]. A special case of 2D grid where the crossed wires are coupled by tunneling interaction is considered in Refs. 4, 10. In the present paper, a different course is elabo- rated. We ask the question of whether it is possible to encode both 1D and 2D electron liquid regimes in the same system within the same energy scale. In order to unravel the pertinent physics we con- sider a grid with a short-range capacitive interwire interaction. This approximation might look shaky if applied for crossed stripe arrays in the cuprates. On the other hand, it seems natural for 2D grids of © I. Kuzmenko, S. Gredeskul, K. Kikoin, and Y. Avishai, 2002 nanotubes [11,12] or artificially fabricated bars of quantum wires with grid periods a12, which exceed the lattice spacing of a single wire or the diameter of a nanotube. It will be shown below that this in- teraction can be made effectively weak. Therefore, such a quantum bars (QB) retain the 1D LL cha- racter for the motion along the wires, similarly to the case considered in Ref. 8. At the same time, however, boson mode propagation along the diago- nals of the grid is also feasible. This process is es- sentially a two-dimensional one, as are the shape of the Brillouin zone and equipotential surfaces in the reciprocal QB lattice. Before developing the formalism, a few words about the main assumptions are in order. Our at- tention here is mainly focused on charge modes, so it is assumed that there is a gap for spin excitations. Next, we are mainly interested in electronic proper- ties of QBs which are not related to simple charge instabilities like commensurate CDWs, so that the (for simplicity equal) periods a a a1 2� � are sup- posed to be incommensurate with the lattice spa- cing. The Brillouin zone (BZ) of the QB super- lattice is two-dimensional, and the nature of excitations propagating in this BZ is determined by Bragg interference of modes with the superlattice wave vector. This interference (umklapp processes) is, of course destructive for LL excitations with both wave vector components close to multiple in- tegers of Q a� 2� . However, in case of weak scat- tering, only two-wave interference processes near the boundaries of the BZ are significant. One can then hope that the harmonic boson modes survive in the major part of the BZ, and that the Hamil- tonian of the QB might still be diagonalized with- out losing the main characteristic features of the LL physics. 2. Quantum bars: basic notions Quantum bars may be defined as a 2D periodic grid, i.e., two crossed periodic arrays of 1D quan- tum wires with a period a = (a , a )1 2 . In fact these arrays are placed on two parallel planes sepa- rated by an interplane distance [12], but in this section we consider QB as a genuine 2D system. The arrays are oriented along the unit vectors e12, with an angle � between them. Here we consider a square grid (� � � 2) formed by identical wires of length L with basis vectors a j ja� e , j � 1 2, (Fig. 1). The interaction between the excitati- ons in different wires is assumed to be concentra- ted near the crossing points with coordinates n n n a n a1 2 1 2a a1 2� � ( , ). The integers n j enumerate the wires within the jth array. Such interaction im- poses a superperiodicity on the energy spectrum of initially one-dimensional quantum wires, and the eigenstates of this superlattice are characterized by a 2D quasimomentum q g g= 1 2q q q q1 2 1 2� � ( , ). Here g1,2 are the unit vectors of the reciprocal superlattice satisfying the standard orthogonality relations ( )e gi j ij� � . The corresponding basis vectors of the reciprocal superlattice have the form (m Q m Q1 2, ), where Q a� 2� and m12, are integers. In conventional 2D systems, forbidden states in reciprocal space arise due to Bragg diffraction in a periodic potential, whereas the whole plane is al- lowed for wave propagation in real space, at least till the periodic potential is weak enough. In sharp- ly anisotropic QBs most of the real space is forbid- den for electron and plasmon propagation, whereas the Bragg conditions for the wave vectors are still the same as in conventional 2D plane modulated by a periodic potential. The excitation motion in QBs is one-dimensional in the major part of the 2D plane, and the anisotropy in real space imposes re- strictions on the possible values of the 2D coordi- nates x � ( , )x x1 2 . At least one of them, e.g., x2 should be an integer multiple of the interwire dis- tance a, so that the coordinate x � ( , )x n a1 2 cha- racterizes the point with the 1D coordinate x1 lying at the n2-th wire of the first array. The 2D Brillouin zone of a QB is constructed as an extension of the 1D Brillouin zones of two crossed arrays and subsequent folding of this BZ in accordance with the square superstructure. How- ever, one cannot resort to the standard basis of 2D plane waves when constructing the eigenstate with a given wave vector k in the BZ because of the ki- nematic restrictions mentioned above. Even in non- interacting arrays of quantum wires the 2D basis is formed as a superposition of two sets of 1D waves. Fizika Nizkikh Temperatur, 2002, v. 28, No. 7 753 Electronic excitations and correlations in quantum bars Fig. 1. Square quantum bars formed by two interacting arrays of parallel quantum wires. e1 , e2 are the unit vectors of the superlattice. The first of them is a set of 1D excitations propa- gating along each wire of the first array character- ized by a unit vector k1 1e with a phase shift ak2 be- tween adjacent wires. The second set is the similar manifold of excitations propagating along the wires of the second array with a wave vector k2 2e and phase shift ak2 . The states of equal energy ob- tained by means of this procedure form straight lines in the 2D BZ. For example, the QB Fermi surface developed from the points kF for an indi- vidual quantum wire consists of two sets of lines \| \|,k kF12 � . Accordingly, the Fermi sea is not a cir- cle with radius kF like in the case of free 2D gas, but a cross in the k plane bounded by these four lines [4] (see Fig. 2). Of course, these equipo- tential lines describe the 1D excitations in the BZ of a QB. Due to weakness of the interwire interaction, the excitations in the 2D BZ depicted in Fig. 3 ac- quire genuine two-dimensionality characterized by the quasimomentum q = (q , q )1 2 . However, in ca- se of weak interaction the 2D waves constructed from the 1D plane waves in accordance with the above procedure form an appropriate basis for the description of elementary excitations in QB in close analogy with the nearly free electron approxima- tion in conventional crystalline lattices. It is easily foreknown that the interwire interaction does not completely destroy the above quasimomentum clas- sification of eigenstates, and the 2D reconstruction of the spectrum may be described in terms of wave mixing similarly to the standard Bragg diffraction in a weak periodic potential. Moreover, the classi- fication of eigenstates of noninteracting crossed ar- rays of 1D wires («empty superlattice») may be ef- fectively used for the classification of energy bands in a real QB superlattice. Our next step is to con- struct a complete 2D basis for this empty super- lattice. Due to LL nature of the excitations in a gi- ven wire, they are described as plane waves L ikx�1 2 exp( ) with wave number k and initial dis- persion law �0( ) \| \|k v k� . Each excitation in a cor- responding «empty superchain» is described by its quasi-wavenumber q and the band number p ( , , ...)p � 1 2 . Its wave function has the Bloch-type structure � p q p qx L u x, ,( ) ( )� 1 e iqx (1) with the Bloch amplitude u xp q n nn , ( ) sin � �� cos [( ) ] exp2 1 4p i q Qn n� � , � � n x a n� �� 2 . The corresponding dispersion law �p q( ) has the form ( ) ( )vQ q p p � � � �1 2 1 4 � 754 Fizika Nizkikh Temperatur, 2002, v. 28, No. 7 I. Kuzmenko, S. Gredeskul, K. Kikoin, and Y. Avishai Fig. 2. Fermi surface of square metallic quantum bars in the absence of charge transfer between wires. g1 , g2 are the unit vectors of the reciprocal superlattice. Fig. 3. Two dimensional Brillouin zone of a square QB. The three directions along which the dispersion of Bose excitations is calculated in Sec. 5 are marked by the in- dices 1, 2, 3. � � � � � � � 2 1 2 1 2 2 1 2 2 1 ( ) ( ) cos ( )p n n n Q� � . Within the first BZ of a superchain, \| \|q Q� 2, ex- pressions for the Bloch amplitude and dispersion law are substantially simplified: u x iQx p qp q p , ( ) exp ( ) sgn� � � �� ! " # �1 2 1 , (2) �p pq vQ p q Q ( ) ( ) \| \| � � �� 2 1 1 . (3) The 2D basis of periodic Bloch functions for an empty superlattice is constructed in terms of the 1D Bloch functions (1), (2): $p p p q p qx x, , , ,( ) ( ) ( )% %�q r � � 1 21 2 . (4) Here p p, , , ...% � 1 2 , are the band numbers and the 2D quasimomentum q � ( , )q q1 2 belongs to the first BZ, \| \|q Qj � 2. The corresponding eigenfrequ- encies are � � �pp p pq q% %� �( ) ( ) ( )q 1 2 . (5) We will use this basis in the next Section when constructing the excitation spectrum of QBs within the reduced band scheme. 3. Hamiltonian The full Hamiltonian of the QBs is H H H H� � �1 2 int , (6) where Hj describes the 1D boson field in the jth array H v dx g x n a L L n 1 1 1 2 1 2 2 2 2 2 � �� &�� ( , ) � ! " # 1 1 1 1 2 2 g x n ax( ( , ))' ( , H v dx g n a x L L n 2 2 2 2 1 2 2 2 2 1 � �� &�� ( , ) � ! " # 1 2 2 1 2 2 g n a xx( ( , ))' ( , and ( , )( �j j are the conventional canonically con- jugate boson fields (see, e.g., Ref. 13). The Fermi velocities v v12, � and the LL parameters g g12, � are taken to be the same for both arrays. Genera- lization to the case of different parameters vj , g j , aj is straightforward. The interwire interaction results from a short- range contact capacitive coupling in the crosses of the bar, H dx dx V x n a n a x L L n n int , ( , )� � � � &� 1 2 1 1 2 2 2 2 1 2 ) )1 1 2 2 1 2( , ) ( , )x n a n a x . Here )i ( )r are density operators, and V( )r r1 2� is a short-range interwire interaction. Physically, it represents the Coulomb interaction between charge fluctuations e x r( )0 , ( )0 1� around the points r1 1 2� ( , )x n a and r2 1 2� ( , )n a x . The size of these fluctuations is determined by the screening radius r0 within the wire. One may neglect the interwire tunneling and restrict oneself to the capacitive in- teraction only, provided the vertical distance be- tween the wires d is substantially larger than the screening radius r0 . Therefore the interaction has the form V V x r x r ( ) ,r � � � �� 0 1 0 2 02 + , where the function + ( , )� �1 2 is + ( , ) ( ) ) ( ) � � * � ,� � � 1 2 1 2 0 2 1 2 2 21 � � � � � � � � � r d . (7) It is seen from these equations that + ( , )� �1 2 is an even function of its arguments; it vanishes for \| \|,�12 1- and is normalized by condition +( , ) .0 0 1� The effective coupling strength is V e d0 22 � . In terms of boson field operators ( i , the interac- tion is written as H V dx dx x n a r n a x r L L n n int , ,� � �� &0 1 2 1 1 0 2 2 02 2 1 + 2 � ' ( ' (x xx n a n a x 1 21 1 2 2 2 2( , ) ( , ) . In the quasimomentum representation (4), (1), (2) the full Hamiltonian (6) acquires the form H vg a jp jp pj � �� 2 1 2 � �q q q � % % % % %% � �� 2 1 2 vga Wjpj p jp + j p ppjj q q q q ( ( (8) with matrix elements for interwire coupling given by Fizika Nizkikh Temperatur, 2002, v. 28, No. 7 755 Electronic excitations and correlations in quantum bars Wjpj p jp j p jj pp jpj p jj% % % % % % % % %� �q q q q� � � . [ ( )]1 . Here � �jp p j p jq v p Q qq � � � �� ( ) ( ) \| \| 2 1 1 (9) are eigenfrequencies (3) of the «unperturbed» 1D mode pertaining to an array j, band p, and quasimomentum q j jg . The coefficients . .jpj p p p jpj pq q% % � % % %� �q q( ) sgn ( )1 1 2 + . � gV r va 0 0 2 � (10) are proportional to the dimensionless Fourier com- ponent of the interaction strengths, + +1 2 1 2 1 2 0 1 1 2 2 p p ir q qd d% � �� &q � � � � � �( , ) ( )e �% %u r u rp q p q p p, , ( ) ( ) 1 20 1 0 2 2 1� � + q . (11) The Hamiltonian (8) describes a system of cou- pled harmonic oscillators, which can be exactly diagonalized with the help of a certain canonical linear transformation (note that it is already diago- nal with respect to the quasimomentum q). The diagonalization procedure is, nevertheless, rather cumbersome due to the mixing of states belonging to different bands and arrays. However, it will be shown below that the dimensionless interaction pa- rameter . (10) is effectively weak, and a perturba- tion approach is applicable. 4. Main approximations As was mentioned in the Introduction, we con- sider rarefied QBs with a short-range capacitive in- teraction. In the case of QBs formed by nanotubes, this is a Coulomb interaction screened at a dis- tance of the order of the nanotube radius [14] R0 , and therefore r R0 0~ . The minimal radius of a single-wall carbon nanotube is about R0 � � 0.35–0.4 nm (see Ref. 15). The intertube vertical distance d in artificially produced nanotube net- works is estimated as d / 2 nm [12]. Therefore the dimensionless interaction . (10) can be estimated as . 0~ R a 0 , (12) where 0 � 2 0 2R d ge v� . (13) The first factor is about 0.35. The second one, which is nothing but the «fine structure» constant for the nanotube QBs, can be estimated as 0.9 (we used the values d � 1 3 and v � �8 107 cm/s [11]). Therefore 0 is approximately equal to 0.1. The modulus of the matrix element (11) with the expo- nential form of � �( ) exp( \| \| )� � does not exceed unity, so that the interaction (11) is really weak. The smallness of dimensionless interaction . en- ables one to apply perturbation theory. In this limit, the systematics of unperturbed levels and states is grossly conserved, at least in the low-en- ergy region corresponding to the first few bands. This means that they should be described by the same quantum numbers (array number, band num- ber, and quasimomentum). Indeed, as follows from the unperturbed dispersion law (9), the interband mixing is significant only along the high-symmetry directions in the first BZ (BZ boundaries and lines gi � 0). In the rest of the BZ this mixing can be taken into account perturbatively. The interarray mixing within the same energy band is strong for waves with quasimomenta close to the diagonal of the BZ. Away from the diagonal, it can also be cal- culated perturbatively. For quasimomenta far from the BZ diagonals and high-symmetry directions, and in second order of perturbation theory, the above-mentioned cano- nical transformation results in the following renormalized field operators for the first array: ~( 1 ( . ( 1 1 1 2 1 2p p p pp p p q q q q q� �� % % % � , (14) where . . � � � � � pp p p p p p p % % % � % q q q q q q 1 2 1 2 2 2 1 2 and 1 .p pp p q q 2� % % � . (15) Below, the specific values of these coefficients . � .p pq q1 , (16) . � . � .q q q1 11 , (17) % � % % 2 �1 .p pp p p q q 2 (18) will also be used. The renormalized eigenfrequ- encies for the first array are 756 Fizika Nizkikh Temperatur, 2002, v. 28, No. 7 I. Kuzmenko, S. Gredeskul, K. Kikoin, and Y. Avishai ~� � 31 1 1p p pp p q 2 q 2 q� � � � � � � � � �% % � , (19) where 3 . � � � � 4 pp p p p p p % % % % �q q q q q 1 2 2 2 1 2 22 . Corresponding formulas for the second array are obtained by replacing 1 2p p5 , and 1 2% 5 %p p . Consider now the frequency correction in Eq. (19) more attentively. In the case under consi- deration, all terms in Eq. (19) are nonsingular. Then, away from the BZ boundary (\| \|q 66 Q 2) the following estimate is valid: � � � � jp jp j j O q pQ q q 2 q 2 2 1 2 2 1 % % / � � � � � � � � � � �( ) , ( , ) ( , ),( , );j j % � 1 2 2 1 p 7 1. Therefore, the correction term can be estimated ap- proximately as �11 2 q qS , with S R a p p p p q q q� �� . 0 +112 2 0 2 2 112 2 . (20) Due to the short-range character of the interac- tion, the matrix elements +112pq vary slowly with band number, being of the order of unity for p p a R6 max ~ 0, and decrease rapidly for p p7 max. Therefore, the right-hand side in Eq. (20) can roughly be estimated as S R a R aq ~ .0 0 001 1� 66 . (21) One should also remember that the energy spec- trum of nanotube remains one-dimensional only for frequences smaller than some �m . Therefore, an ex- ternal cutoff arises at p akm� , where k vm m~ � . As a result, one gets the estimate S R a k Rmq ~ 0 0 0 . (22) Hence, one could hope to gain here an additional power of the small interaction radius, but for na- notubes, km is of the order of 1 0R (see Refs. 16, 11) and the two estimates coincide. For quasimomenta close to the BZ center, the coefficient Sq can be calculated exactly. Due to smoothness of the matrix elements .112pq with re- spect to the band number p, the sum over p in Eq. (20) can be replaced by an integral over the ex- tended BZ with wave vector k whose components are k q q p Qj j j p j j� � � � sgn( )( ) [ ]1 2 1 . (23) For \| \|q � Q 2 one gets \| \|Sq \| \| S0� , where S R a dk k0 0 2 2 2 � �� & 0 � + ( ) and + +, 8( ) ,k d d ir k� & � � � � � 1 2 1 2 0 2e– . Finally, one obtains S R a0 0 2 0� 0+ , (24) where the constant + + +0 2 1 2 1 2� & d d d� � � � � � �( , ) ( , ) for the exponential form of , 8* � �( ) exp \| \|9 � is / 15. . As a result, instead of the preliminary esti- mate (21), we have S R a0 0014� . . Thus the correction term in Eq. (19) is in fact small. 5. Energy spectrum In the major part of the BZ, for quasimomenta q lying far from the diagonal, the spectrum is de- scribed by Eqs. (14), (19). Here each eigenstate (14) mostly conserves its initial systematics, i.e., belongs to a given array, and mostly depends on a given quasimomentum component. The correspon- ding dispersion laws (19) remain linear, being slightly modified near the BZ boundaries only. The main changes are therefore reduced to a renorma- lization of the plasmon velocity. In Fig. 4 the dis- persion curves, corresponding to quasi-momenta varying along line 1 of Fig. 3, are plotted in com- parison with those for noninteracting arrays. (In all figures within this Section we use the units � � � �Q v 1.) In what follows we use the (j p, ) no- tation for the unperturbed boson propagating along the jth array in the pth band. Then the lowest curve in Fig. 4 is, in fact, the slightly renormalized dispersion of a (2,1) boson, the middle curve de- scribes a (1,1) boson, and the upper curve is the dispersion of a (1,2) boson. The fourth frequency, corresponding to a (2,2) boson, is far above and is not displayed in the figure. It is seen that the dis- Fizika Nizkikh Temperatur, 2002, v. 28, No. 7 757 Electronic excitations and correlations in quantum bars persion remains linear along the whole line 1 except in the nearest vicinity of the BZ boundary (see in- set in Fig. 4). The interband hybridization gap for the bosons propagating along the first array can be estimated as :� 012 0~ vQ r a . Similar gaps exist near the boundary of the BZ for each odd and next even energy band, as well as for each even and next odd band near the lines q1 0� or q2 0� . The energy gap between the pth and ( )p � 1 -th bands is estimated as :� 0p p vQ r a o p, ~ ( )� � 1 0 1 . For large enough band number p interaction is effec- tively suppressed, .1 2 0p p% 5 , and the gaps vanish. Dispersion curves corresponding to quasi-mo- menta lying at the BZ boundary q Q1 2� , 0 22� �q Q (line 2 in Fig. 3), are displayed in Fig. 5. Again, the dispersion laws are nearly linear, and deviations from linearity are observed only near the corner of the BZ. The lowest curve de- scribes the dispersion of the (2,1) wave. Its coun- terpart in the second band (2,2) is described by the highest curve in the figure. In the zero the approxi- mation, two modes (j,1) propagating along the first array are degenerate with unperturbed fre- quency � � 0.5. The interaction lifts the degene- racy. The lowest of two middle curves corresponds to the (1,u) boson, and upper of them describes the (1,g) boson. Here the indices g, u denote a boson parity with respect to the transposition of the band numbers. Note that the (1,g) boson exactly con- serves its unperturbed frequency � � 0.5. The latter fact is related to the square symmetry of the QBs. The points a, b, c in Figs. 4 and 5 are the same. Now consider the dispersion of modes with qua- si-momenta on line 3 in Fig. 3. We start with q not too close to the BZ corner q q Q1 2 2� � . In this case, the initial frequencies of modes belonging to the same band coincide, � � �1 2p p pq q q� � . There- fore the modes are strongly mixed: ~ ( )( 1 ( (gp p p pq q q q� �� 1 2 1 1 2 2 1 � �% % % % % 2 �1 2 1 2( ). ( . (p p p pp p p p q q q q , (25) ~ ( )( 1 ( (up p p pq q q q� �� 1 2 1 1 2 2 1 � �% % % % % 2 �1 2 1 2( ). ( . (p p p pp p p p q q q q . (26) The corresponding eigenfrequencies are shifted from their bare values 758 Fizika Nizkikh Temperatur, 2002, v. 28, No. 7 I. Kuzmenko, S. Gredeskul, K. Kikoin, and Y. Avishai Fig. 4. The energy spectrum of QB (solid lines) and noninteracting arrays (dashed lines) for quasi-momenta at line 1 of Fig. 3 (q q2 102� . ). Points a, b, c on the figure correspond to the point A at the BZ boundary. Inset: Zoomed vicinity of the point q Q1 05� . ; � � 05. . Fig. 5. Upper panel: The energy spectrum of QB (solid lines) and noninteracting arrays (dashed lines) for qua- si-momenta at the BZ boundary (line 2 in the Fig. 3). Points d, e, f in the figure correspond to the corner B of the BZ. Lower panel: Zoomed vicinity of the line � � 05. . ~� � . 3gp p p p pp p p q q q q 2 2 1 21� � � � � � � � � � � % % 2 � , (27) ~� � . 3up p p p pp p p q q q q 2 2 1 21� � � � � � � � � � � % % 2 � , (28) In zeroth order of perturbation theory, the modes (25), (26) have a definite j-parity with respect to transposition of array numbers j � 1, 2. Due to the repulsive character of the interaction, the odd modes ( , )u p , p � 1, 2, (26) correspond to lower fre- quencies (28) and the even modes (g,p) (25) corre- spond to the higher ones (27). The dispersion curves at the BZ diagonal are displayed in Fig. 6. The points d, e, and f in Fig. 6 are the same as in the Fig. 5. At the BZ corner q q Q1 2 2� � all four initial modes in the zeroth approximation are degenerate and have also a definite p-parity with respect to transposition of band numbers p � 1, 2. The inter- wire interaction partially lifts the degeneracy. In zeroth-order approximation, the lowest frequency corresponds to a (g,u) boson, symmetric with re- spect to transposition of the array numbers, but antisymmetric with respect to the transposition of band numbers. The upper curve describes a ( , )u u boson with odd j-parity and p-parity. Degeneracy of two middle modes with even band parity, (g,g) and (u,g) bosons, is provided by the symmetry of interaction in a square superlattice (7). Note once more that their frequency equals to its unperturbed value � � 0.5. All these results show that the quantum states of the 2D quantum bar conserve the quasi-1D charac- ter of the Luttinger-like liquid in the major part of momentum space, and that the 2D effects can be successfully calculated within the framework of perturbation theory. However, bosons with quasi- momenta close to the diagonal of the first BZ are strongly mixed bare 1D bosons. These excitations are essentially two-dimensional, and therefore the lines of equal energy in this part of BZ are modified by the 2D interaction (see Fig. 7). It is clearly seen that deviations from linearity occur only in a small part of the BZ. The crossover from LL to FL beha- vior around isolated points of the BZ due to a sin- gle-particle hybridization (tunneling) for Fermi ex- citations was noticed in Refs. 4 and 10, where a mesh of horizontal and vertical stripes in supercon- ducting cuprates was studied. 6. Correlations and observables The structure of the energy spectrum analyzed above predetermines optical and transport proper- ties of the QBs. Let us consider an ac conductivity whose spectral properties are given by a cur- rent—current correlator ; � ; � ; �jj jj jji% % %� % � %% �( , ) ( , ) ( , )q q q � � %& 1 0 0 1 1� �dt J t Jt j je i [ ( ), ( )]q q † . Here J vgjp jpq q� 2 � is a current operator of the jth array. For simplicity we restrict ourselves to the first band. For noninteracting wires, the current— current correlator is reduced to the conventional LL expression [17], [ ( ), ( )] sin ( )J t J ivg tj j j j jj1 1 0 1 10 2q q q q% %� �† � � with metallic Drude peak Fizika Nizkikh Temperatur, 2002, v. 28, No. 7 759 Electronic excitations and correlations in quantum bars Fig. 6. The energy spectrum of QB (solid lines) and noninteracting arrays (dashed lines) for quasi-momenta on the diagonal of the BZ (line 3 in Fig. 3). Points d, e, f in the figure correspond to corner B of the BZ. Fig. 7. The lines of equal frequency for QB (solid lines) and noninteracting arrays (dashed lines). Lines 1, 2, 3 correspond to the frequencies �1 � 0.1, �2 � 0.25, �3 � 0.4. % 7 � �% %; � � � � jj j jjvg( , ) ( )q q0 1 . (29) The corresponding imaginary part contains a single pole at the resonance frequency � j1q , % 7 � � % %; � � � � � jj jp jp jj vg ( , ) ( ) q q 2 q 2 0 2 2 . For interacting wires, where . jpj p% % 2q 0, the cor- relators may easily be calculated after diagonaliza- tion of the Hamiltonian (8) by the transformations (14) for q off the diagonal of the first BZ, or by the transformations (25) and (26) for q lying on the di- agonal of the BZ. Consider first the quasi-momenta q lying far from the diagonal of the first BZ. In this case, the transformations for the field momenta can be ob- tained in a similar manner to the transformations (14) for the field coordinates. As a result, one has: [ ( ), ( )]J t J11 11 0q q † � � � � �2 1 1 11 11ivg t( )~ sin (~ )1 � �q q q � �2 2 2 2ivg tp p p p. � �q q q ~ sin (~ ) , [ ( ), ( )]J t J ivg11 21 0 2q q q † � � . �[~ sin (~ ) ~ sin (~ )]� � � �11 11 21 21q q q qt t , where 1 pq , .pq and .q are defined in Eqs. (15), (17), (18). Then, for the optical absorbtion %; one obtains % � � � �; � � 1 � �11 1 111( , ) ( ) ( ~ )q q qvg � �� . � �vg p p pq q 2 2( ~ ) , (30) % � � � �; � � . � � � �12 11 21( , ) [ ( ~ ) ( ~ )]q q q qvg . (31) The longitudinal optical absorbtion (30) (i.e. the conductivity within a given set of wires) has its main peak at frequency ~ \| \|�11 1q / v q , corresponding to the first band of the pertinent array, and an ad- ditional weak peak at the frequency ~ \| \|,�21 2q / v q corresponding to the first band of a complementary array. It contains also a set of weak peaks at fre- quencies ~ [ ]�2 2p p vQq / (p � 2, 3, …) correspond- ing to the contribution from the higher bands of the complementary array. At the same time, a second observable becomes relevant, namely, the transver- se optical conductivity (31). It is proportional to the interaction strength and has two peaks at frequ- encies ~�11q and ~�21q in the first bands of both sets of wires. For \| \|q 5 0 Eq. (30) reduces to that for an array of noninteracting wires (29), and the trans- verse optical conductivity (31) vanishes. In the case when the quasi-momenta q belong to the diagonal of the first BZ, the transformations for the field momenta are similar in form to Eqs. (25) and (26). The current–current correlation func- tions have the form [ ( ), ( )] ( )J t J ivg11 11 10 1q q q † � � � % 1 � �[~ sin (~ ) ~ sin (~ )]� � � �g g u ut t1 1 1 1q q q q � 2 �ivg tp p p p. � �q q q 2 1 ~ sin (~ ) , [ ( ), ( )]J t J1 2 0q q † � � � �ivg t tg g u u[~ sin (~ ) ~ sin (~ )]� � � �1 1 1 1q q q q with �pq and %1 pq defined by Eqs. (3), (18), and the optical conductivity is estimated as % � � % ; � 7 < � 111 12 1( , ) ( )q q vg � � � �[ ( ~ ) ( ~ )] � � � �q u1 1q q � � 2 �� . � �vg p p pq q 2 1 ( ) , (32) % � � � �; � 7 < � � � � �12 1 12 ( , ) [ ( ~ ) ( ~ )]q q q vg q u . (33) The longitudinal optical conductivity (32) has a split double peak at frequencies ~�11q and ~�21q , in- stead of a single peak. Again, a series of weak peaks occurs at frequencies �pq corresponding to contri- butions from higher bands p � 2, 3, 4,... The trans- verse optical conductivity (33), similarly to the nondiagonal case (31), has a split double peak at frequencies ~�11q and ~�21q . The imaginary part of the ac conductivity %% %; �jj ( , )q is calculated within the same approach. Its longitudinal component for q far from the BZ diagonal equals %% � � � � � �; � � � 1 � � . � � 11 11 2 1 2 11 2 2 2 2 2 2 1 ( , ) ~ ~ q q q q q q vg p pp � � � � � � � � . Beside the standard pole at zero frequency and the main pole at the resonance frequency �11q , the real part has an additional series of high-band satellites. The corresponding expression for %%; �22( , )q can be obtained after the replacement 1 2= . For quasi- 760 Fizika Nizkikh Temperatur, 2002, v. 28, No. 7 I. Kuzmenko, S. Gredeskul, K. Kikoin, and Y. Avishai momenta at the diagonal, the longitudinal optical absorption is %% � � � � � ; � 1 � � � � � � � 11 1 1 2 2 1 2 1 2 2 1 1( , ) ( ) ~ ~ ~ ~ q q q q qvg g g u u q 2 � � � � � � � � � � �� 2 1 2 2 2 2 1 vg p pp � � . � � q q q . It contains two main poles similar to two peaks in the optical conductivity, and the series of poles contributed by the higher bands. The transverse component of the imaginary part of the ac conduc- tivity has the same form for any q: %% � � � � � � � � � � � � ; � . � � � � � 12 2 1 2 2 1 2 2 1 1 ( , ) ~ ~ q q q q vg u g . It always contains two poles and vanishes for non- interacting wires. One of the main effects specific for a QB is the appearance of a nonzero transverse momentum—mo- mentum correlation function. In space–time coordi- nates (x,t) its representation reads G t x t x12 1 1 2 20 0 0( , ) [ ( , ; ), ( , ; )]x � � � . (34) This function describes the momentum response at the point ( , )0 2x of the second array at the moment t caused by initial (t � 0) perturbation at the point ( , )x1 0 of the first array. Standard calculations si- milar to those described above lead to the following expression: G t i V r v dk dk k k12 0 0 2 2 1 2 4 1 2 ( , )x � � �� & � � + � � sin ( ) sin ( ) sin ( ) sin ( ) k x k x k k vt k k vt k k 1 1 2 2 2 2 1 1 2 2 1 2 . (35) Here +k k1 2 is defined by Eq. (11) with p p� 1 , % �p p2 , and k12, from Eq. (23). This correlator is shown in Fig. 8. Here the non- zero response corresponds to the line determined by the obvious kinematic condition \| \| \| \|x x vt1 2� � . The finiteness of the interaction radius slightly spreads this peak and changes its profile. 7. Conclusion In conclusion, we have demonstrated that the energy spectrum of QBs shows the characteristic properties of LL at \| \|,q �5 0, but at finite q, the density and momentum waves may have either 1D or 2D character depending on the direction of the wave vector. Due to an interwire interaction, un- perturbed states propagating along the two arrays are always mixed, and the transverse components of the correlation functions do not vanish. For quasi- momentum lying on the diagonal of the BZ, such mixing is strong, and the transverse correlators have the same order of magnitude as the longitudi- nal ones. Acknowledgement S. G. and K. K. are indebted to L. Gorelik, M. Jonson, I. Krive, and R. Shekhter for helpful dis- cussions. They also thank Chalmers Technical Uni- versity, where this work was started, for hospita- lity and support. This research is supported in part by grants from the Israel Science foundations, the DIP German–Israel cooperation program, and the USA–Israel BSF program. S. G. is happy to see this paper published in the special issue devoted to the Jubilee of his old friend and colleague, Acade- mician Victor Eremenko. 1. P. W. Anderson, Science 235, 1196 (1987). 2. J. E. Avron, A. Raveh, and B. Zur, Rev. Mod. Phys. 60, 873 (1988). 3. Y. Avishai and J. M. Luck, Phys. Rev. B45, 1074 (1992). 4. F. Guinea and G. Zimanyi, Phys. Rev. B47, 501 (1993). 5. V. Emery, E. Fradkin, S. A. Kivelson, and T. C. Lubensky, Phys. Rev. Lett. 85, 2160 (2000). 6. A. Vishwanath and D. Carpentier, Phys. Rev. Lett. 86, 676 (2001). Fizika Nizkikh Temperatur, 2002, v. 28, No. 7 761 Electronic excitations and correlations in quantum bars Fig. 8. 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Electronic excitations and correlations in quantum bars

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