collective quantum tunneling of strongly correlated electrons in commensurate ...

Eur. Phys. J. B 20, 289–299 (2001) THE EUROPEANPHYSICAL JOURNAL Bc©

EDP SciencesSocieta Italiana di FisicaSpringer-Verlag 2001

Collective quantum tunneling of strongly correlated electronsin commensurate mesoscopic rings

I. Baldea1,2,a, H. Koppel2, and L.S. Cederbaum2

1 Physikalisches Institut, Universitat Karlsruhe (TH), 76129 Karlsruhe, Germany2 Theoretische Chemie, Physikalisch-Chemisches Institut, Universitat Heidelberg, INF 229, 69120 Heidelberg, Germany

Received 27 November 2000

Abstract. We present a new effect that is possible for strongly correlated electrons in commensuratemesoscopic rings: the collective tunneling of electrons between classically equivalent configurations, corre-sponding to ordered states possessing charge and spin density waves (CDW, SDW) and charge separation(CS). Within an extended Hubbard model at half filling studied by exact numerical diagonalization, wedemonstrate that the ground state phase diagram comprises, besides conventional critical lines separatingstates characterized by different orderings (e.g. CDW, SDW, CS), critical lines separating phases with thesame ordering (e.g. CDW-CDW) but with different symmetries. While the former also exist in infinitesystems, the latter are specific for mesoscopic systems and directly related to a collective tunnel effect. Weemphasize that, in order to construct correctly a phase diagram for mesoscopic rings, the examination ofCDW, SDW and CS correlation functions alone is not sufficient, and one should also consider the sym-metry of the wave function that cannot be broken. We present examples demonstrating that the jumps inrelevant physical properties at the conventional and new critical lines are of comparable magnitude. Thesetransitions could be studied experimentally e.g. by optical absorption in mesoscopic systems. Possiblecandidates are cyclic molecules and ring-like nanostructures of quantum dots.

PACS. 71.45.Lr Charge-density-wave systems – 75.30.Fv Spin-density waves – 72.15.Nj Collective modes(e.g., in one-dimensional conductors – 73.20.Mf Collective excitations (including excitons, polarons, plas-mons and other charge-density excitations)

1 Introduction

Exact or almost exact numerical methods (exact diag-onalization, Monte Carlo, density matrix renormaliza-tion group) [1] are extensively used to study models forstrongly correlated systems that cannot be solved ana-lytically. Usually, such numerical studies are ultimatelydevoted to infinite systems. Although inherently carriedout for finite systems, they investigated the size (N) de-pendence only to deduce, via finite-size scaling analysis,results for N →∞. In view of recent advances in the fieldsof nanostructures and synthesis of larger molecules, the N -dependence can also be considered a problem of interestby itself. Phenomena that do not occur or are irrelevant forinfinite systems can be interesting for finite (mesoscopic)ones. This was confirmed, e.g., by the results of severalrecent studies on mesoscopic Peierls rings [2,3]. Thesestudies dealt with weakly correlated systems whose prop-erties are dominated by electron-phonon coupling. Withthe present paper, we aim to demonstrate that stronglycorrelated mesoscopic systems of interacting electrons areeven more exciting topics.

a Permanent address: Institute for Space Sciences, NationalInstitute for Lasers, Plasmas and Radiation Physics, Bucha-rest-Magurele, Romaniae-mail: [email protected]

Hubbard models have been often used to de-scribe strongly correlated electrons in numerous low-dimensional compounds ranging from conducting poly-mers to cuprates [4]. The one-dimensional extendedHubbard model is interesting because its ground statedisplays various types of orderings. In the case of a half-filled band two ordered phases are most relevant for quasione-dimensional compounds: the charge- and spin-density-waves (CDW, SDW). The previous studies devoted to thiscase have paid much attention to the CDW-SDW criticalline [5–8]. Another possible ordering is the charge separa-tion (CS) [8,10]. The interest in this phase was recently re-vived in the context of strip formation in certain cuprates.A common feature of all these studies is the interest in infi-nite systems. Excepting in part for reference [7], even whenactually dealing with (small) finite systems, they exam-ined averaged properties (energies, correlation functions)such that a finite-size scaling analysis could ultimatelyprovide information relevant for infinite systems. An im-portant qualitative difference between finite and infinitecases is that, contrary to the latter, no symmetry break-ing is possible in the former case. To illustrate the basicidea, let us refer to a dimerized system like polyacety-lene, where single and double bonds alternate. In an infi-nite system, there are two degenerate ground states |Φa,b〉,say · · ·ABAB · · · and · · ·BABA · · · , of broken symmetry.

290 The European Physical Journal B

In a finite system, the ground state is in general nondegen-erate. The two eigenstates of lowest energy are compatiblewith the original symmetry, i.e. they are either even orodd; approximately, they are represented by the symmet-ric and antisymmetric linear combinations |Φa〉±|Φb〉 [13].The physics behind this is the quantum tunneling betweenthe two configurations that are classically equivalent. Thisis similar to the well known NH3-molecule. We shall showbelow that a similar phenomenon also occurs in a sys-tem of strongly correlated electrons. It dramatically af-fects physical properties which could be investigated ex-perimentally.

The remaining part of this paper is organized as fol-lows. In Section 2 we formulate the model used in thiswork. Section 3 includes a qualitative discussion of theground states. The critical lines and the ground statephase diagram are presented in Section 4. Section 5 is de-voted to correlation functions characterizing the variousorderings as well as to optical properties. Some conclu-sions and remarks make the object of the final Section 6.

2 Model

Let us consider N (supposed even throughout) electronson N sites (half-filling case) described by an extendedHubbard Hamiltonian:

H = −t0N−1∑l=0

∑σ=↑,↓

(c†l,σcl+1,σ + c†l+1,σcl,σ

)

+N−1∑l=0

(Unl,↑nl,↓ + V nlnl+1

). (1)

Here, c (c†) denote creation (annihilation) operators forelectrons, nl,σ ≡ c†l,σcl,σ, nl ≡ nl,↑+nl,↓, t0 is the hoppingintegral between nearest neighbors, U and V are on-siteand nearest-neighbor potentials, respectively. An infinitesystem of noninteracting electrons possesses a divergentdensity of states at the Fermi level, and this plays a keyrole for the occurrence of various types of ordering spe-cific for the physics in one dimension: Peierls dimerization,CDW, SDW, etc. In a finite system, such orderings are fa-vored by small values of the energy difference between thelowest unoccupied orbital and the highest occupied or-bital (the so called HOMO-LUMO gap [12]) [2,8,13]. TheHOMO-LUMO gap vanishes in the thermodynamic limit,but it is difficult in general to make it much smaller thanother energies for sizes (N ∼ 10) for which exact diagonal-ization can be applied. The HOMO-LUMO gap vanishesidentically only in two cases: for sizes N = 4n by impos-ing periodic boundary conditions (PBC, cN+l,σ ≡ cl,σ)as well as for N = 4n + 2 and antiperiodic (Mobius)boundary conditions (ABC, cN+l,σ ≡ −cl,σ) (n is an in-teger) [2,8,13]. Below, we shall always use these combina-tions of N -values and boundaries (open shell case). There-fore, the results reported in this paper are relevant for fi-nite (mesoscopic) rings. Although our primary interest is

in mesoscopic rings, one should remember that such fi-nite clusters are exactly equivalent to extended periodicsystems where the sampling of the Brillouin zone is re-stricted to N specific points and includes the Fermi points(so called small-crystal approach [7,11]).

As well known, a CDW-SDW transition occurs inthe extended Hubbard model (1) with a critical lineU . 2V [6,8,9]; this line is slightly pushed towardsthe CDW region as compared to the Hartree-Fock crit-ical line (U = 2V ) [5]. However, excepting in part forreferences [7,9], these studies paid little attention tosymmetry. The Hamiltonian (1) remains invariant un-der particle-hole transformation P [cl,σ → (−1)lc†l,σ] andspin flip F (cl,σ → cl,−σ). Besides, the previous studies(e.g. [9]) on chains have mentioned the elementary trans-lation T (cl,σ → cl+1,σ) and the space inversion mv(0)(cl,σ → cN−l,σ), The elementary symmetry operationscan be used to generate all elements of the symmetrygroup of the Hamiltonian (1). One can easily show thatthe spatial symmetry group is CN ,v (N = N for PBCand N = 2N for ABC) comprising rotations Tk ≡ Tk

(cl,σ → cl+k,σ) around a principal axis CN as well as twodistinct classes of reflection planes perpendicular to CN ,mv(j) (cl,σ → cN+2j−l,σ) and md(j) (cl,σ → cN+1+2j−l,σ)(0 ≤ k ≤ N − 1, 0 ≤ j ≤ N/2− 1). No symmetry break-ing is possible in finite rings, unlike in infinite ones. Con-sequently, under the aforementioned transformations, thenondegenerate eigenstates |Ψα〉 of (1) should be either ofeven (|Ψα〉 → +|Ψα〉) or of odd (|Ψα〉 → −|Ψα〉) parity. Thecorresponding eigenvalues (±1) of the operators T, P, F,mv(0) and md(0) will be denoted by T , P , F , mv andmd, respectively. We note that a full classification alongthese lines has not been given before in the literature.

We have studied small clusters by means of numericalexact (Lanczos) diagonalization and found no significantchanges in the phase diagram if N varies from 6 to 12. Weshall present below results on ground state properties aswell as lowest excitations (parities, energies and correla-tion functions). Based on them, we shall demonstrate thatthe phase diagram of the investigated model is richer thanpreviously claimed [6–10]. Namely, we shall present exam-ples showing new quantum phase transitions: a CDW-to-CDW transition in the region previously assigned as theCDW phase, and an SDW-to-SDW transition in the re-gion previously assigned as the SDW phase. We shall showthat such quantum phase transitions represent the man-ifestation of a collective tunneling occurring in stronglycorrelated electron systems. To illustrate that the phys-ical properties of a mesoscopic system are dramaticallyaffected by these transitions, we present results on thematrix elements of relevant optical transitions.

3 Types of ground states of finite rings

The nature of the ground state and its assignment in thephase diagram results from the competition of the threeterms entering equation (1) that prefer a uniform state(t0), favoring (U < 0) or preventing (U > 0) doubly

I. Baldea et al.: Collective tunneling of strongly correlated electrons in mesoscopic rings 291

occupied sites, with preference for (V < 0) or against(V > 0) occupation of adjacent sites, respectively. Threewell-established types of orderings that are possible inmodel (1) can easily be understood by examining the(classical) limit t0 → 0. Then, for V > 0 and U < 2V ,the lowest energy corresponds to doubly occupied sitesand unoccupied sites alternating periodically (the bipola-ronic limit of a CDW). In this case, there are two equiv-alent multielectronic configurations expressed in occupa-tion number representation as |CDW1〉 = | · · · 0202 · · · 〉,|CDW2〉 = | · · · 2020 · · · 〉. For U > 0 and U > 2V , thelowest energy is obtained if each site is occupied exactlyby one electron with either up or down spin; two possibleconfigurations correspond to alternating spin orientationsbetween adjacent sites, i.e. |SDW1〉 = | · · · ↑↓↑↓ · · · 〉 and|SDW2〉 = | · · · ↓↑↓↑ · · · 〉 (the antiferromagnetic limit ofan SDW). Finally, for U < 0 and V < 0, the lowest energyis achieved in a CS state (i.e. half of the sites are adja-cently doubly occupied, while the other half is empty).There are N such limiting configurations that are equiv-alent: |CS1〉 = |0 · · · 02 · · · 2〉, |CS2〉 = |0 · · · 02 · · · 20〉, · · ·|CSN/2+1〉 = |2 · · · 20 · · ·0〉, · · · |CSN 〉 = |20 · · · 02 · · · 2〉.

All the aforementioned configurations correspond tostates with broken symmetry; therefore, they cannot beground states (or, in general, eigenstates) of a finite quan-tum mechanical system. However, for each type of order-ing, they can be used to construct pairs of states thatpreserve the symmetry of the problem:

|CDW±〉 = (|CDW1〉 ± |CDW2〉) /√

2,

|SDW±〉 = (|SDW1〉 ± |SDW2〉) /√

2, (2)

|CS±〉 = N−1/2N−1∑r=0

(±1)r |CSr〉 .

4 Phase diagram and critical lines

Before discussing phase diagrams for finite rings, a com-ment is in order. In infinite systems with strong correla-tions, it is possible to change the type of the ground stateby varying e.g. pressure (and hence interaction strengths)at zero temperature (T = 0). Such quantum phase tran-sitions have investigated recently in variety of systems(quantum spin systems, two dimensional electron gas,heavy fermions, etc.) [16]. Traditional (T 6= 0) phase tran-sitions result from the competition between ordering andthermal fluctuations. In quantum phase transitions, therole of the latter is played by quantum fluctuations. Tradi-tional and quantum phase transitions are qualitatively dif-ferent phenomena. A strictly one dimensional system withshort range interaction cannot undergo traditional phasetransitions, but quantum phase transitions are possible.The phase diagram represents an important property ofinfinite systems. There, critical lines separate the spaceof interaction strengths in different (normal or ordered)phases. Ordered phase are characterized by ground statewave functions with certain broken symmetries. However,what one can directly extract (even in principle) from ex-periments are often not the related order parameters (e.g.,

BCS’s, Peierls distortion amplitude, staggered magneti-zation) but rather some correlation functions. Therefore,a critical point of a phase transition should realisticallybe defined by a discontinuous (or at least non-smooth)change of certain correlation functions. Within such apragmatic standpoint, no formal objections exist againstquantum phase transitions in finite systems where a sym-metry breaking is impossible [15]. Critical lines obtainedmonitoring correlation functions in finite systems are use-ful if one wants to deduce, via conformal field theory andfinite-size analysis, extrapolations for infinite systems.

More or less abrupt variations of certain correlationfunctions may serve to define more or less approximatelycritical points. We stress, however, that the phase dia-gram of a finite (mesoscopic) system constructed in thisway would miss an important property: the symmetry ofthe ground state. Unlike those deduced by means of corre-lation functions, the critical points where the symmetry ofthe ground state does change can be determined exactly.Therefore, we shall define and present below the criticallines for finite rings that separate the phase diagram inphases characterized by ground states of a certain welldefined symmetry.

Each of the states of equation (2) corresponding to theideal orderings discussed in Section 3 transforms accordingto one of the four one-dimensional irreducible representa-tions (A1,2, B1,2) of the group CN ,v (N = N for PBCand N = 2N for ABC). Their complete symmetry is givenin Table 1. In the presence of a finite hopping (t0 6= 0),the exact ground states contain, besides the terms enter-ing equation (2), extra terms that change e.g. the averageelectron numbers per site from the values 0, 1 or 2 enter-ing the states with ideal orderings. However, in the wholerange of parameters, the ground states found by exact nu-merical diagonalization always transform according to theone-dimensional irreducible representations (A1,2, B1,2) ofthe group CN ,v. Two or three energy curves correspondingto the lowest energy states of symmetries A1,2 and B1,2

can cross by varying the interaction strengths U/t0 andV/t0 and this defines, as already anticipated, the criticallines or tricritical points, respectively. The phase diagramsobtained for 6-, 8-, and 10-site rings are shown in Figure 1.Besides the critical lines, we indicate there which physicalordering (CDW, SDW, CS) dominates in a certain regionof the parameter space, i.e. which of the multielectronicconfigurations entering the states |CDW±〉, |SDW±〉 and|CS±〉 of equations (2) gives the largest contribution tothe ground state. To distinguish the exact ground statesfrom the ideal states |CDW±〉, |SDW±〉 and |CS±〉, wedenote the former by |c±〉, |s±〉 and |cs±〉, respectively.In addition to the ordered phases c±, s± and cs±, thephase diagram comprises, as previously pointed out [8],an intermediate phase denoted by im in Figure 1. Thephase diagrams of Figure 1 confirm a number of interest-ing results reported previously on the transitions betweenvarious types of orderings [6–9]. However, we shall mainlyemphasize below the new findings of the present study.

The total spin is a good quantum number. The ex-act ground states we found numerically are either singlet


0

-2

-4

-6-3 0 3

V/t0

U/t0

c+ c+

c-

cs+

cs- im+,t

s+,t s-

(a)

0

-2

-4

-3 0 3 6

V/t0

U/t0

c+ c+

c-

cs+ cs- im-,t

s-,t s+

(b)

2

0

-2

-4

-3 0 3 6

V/t0

U/t0

c+

c+

c-

cs+ cs- im+,t

s+,t s-

(c)

Fig. 1. Phase diagrams for 6-, 8-, and 10-site rings (a, b, and c, respectively). The critical lines separate the parameter space inphases of well defined symmetries. Solid lines are between phases with the same ordering but with different symmetries, whiledashed ones separate phases with both different orderings and different symmetries. By their locations, the symbols c±, s±,and cs± indicate the types of main orderings CDW, SDW, and CS, respectively; im± correspond to intermediate orderings.The total spin of the ground state is zero, S = 0 (singlet), excepting for the regions specified by the subscript t, where S = 1(triplet). In the small closed region at U, V . 0 the ground state is of s− (s+) type for N = 6, 10 (N = 8). The full symmetryof each phase is indicated in Table 1.

(S = 0) or triplet (S = 1) states [17]. To our knowl-edge, previous investigators of the model (1) have alwaysreported singlet ground states. Therefore, to indicate ex-plicitly that the ground state is not a singlet, we shallintroduce the index t for ground states with S = 1. Unlikethe total spin operator, the total spin projection Sz is notinvariant under charge conjugation P (Sz

P→ −Sz). TheP-symmetry given in Table 1 as well as the correlationfunctions Kc,s,b of Section 5 for triplet cases correspondto the state with Sz = 0.

Perhaps the most interesting new finding of the presentpaper is the existence of the new critical lines c+-c−,s+-s−, cs+-cs−, separating ground states with the sameordering (CDW, SDW, CS) but possessing different sym-metries. In Figure 1, they are shown as solid lines. In ad-dition to these, the phase diagrams comprise critical linesbetween phases characterized by both different orderingsand different symmetries. This is the case of the lines ofthe CDW-SDW and CS-SDW transitions investigated pre-viously [6–8,10]. Such critical lines (dashed lines in Fig. 1)will be called old, although the fact that the phases sep-

arated by them possess not only different orderings, butalso different symmetries has not been emphasized previ-ously. More precisely, these are transitions c+ s± andcs± s±, respectively; see Figure 1. The dashed lines ofFigure 1 roughly correspond to the critical lines obtainedfor N = 6 and N = 8 in reference [8]. Old critical linesalso exist in infinite systems. For the present purpose, thenew critical lines are most interesting because they arespecific for finite (mesoscopic) systems.

Intuitively speaking, the coherent tunneling of allelectrons e.g. between the configurations |CDW1〉 and|CDW2〉 (cf. Sect. 3) yields two approximate eigenstates|CDW+〉 ' |c+〉 and |CDW−〉 ' |c−〉 with different sym-metry. Depending on the model parameter values, one oranother of the states |c+〉 and |c−〉 has the lowest en-ergy. The system undergoes a new type of transition c+-c− because, for certain interaction strengths (defining thec+-c− critical line), the energy levels corresponding to theeigenstates |c+〉 and |c−〉 do cross. The traditional normal-superconductor transition is driven by electron pairing.Similarly, the phenomena occurring at the new critical


Table 1. Total spin (S) and symmetries of the ideal orderedstates (i.s.) and the lowest exact eigenstates (e.s.) character-izing the various orderings in open shell systems classified ac-cording to the irreducible representations of the point groupsspecified in parentheses. Each of these e.s. is the ground statein certain regions of the phase diagram (cf. Fig. 1). (Noticethat, in all cases, Pmd = +1 and Tmdmv = +1.)

N = 4n (PBC)

i.s. e.s. irrep (CN,v) T mv md P F S

CDW+ c+1A1 +1 +1 +1 +1 +1 0

CDW− c−1B1 −1 +1 −1 −1 +1 0

SDW+ s+1B2 −1 −1 +1 +1 +1 0

SDW− s−,t3A2 +1 −1 −1 −1 −1 1

CS+ cs+1A1 +1 +1 +1 +1 +1 0

CS− cs−1B2 −1 −1 +1 +1 +1 0

im−,t3A2 +1 −1 −1 −1 −1 1

N = 4n+ 2 (ABC)

i.s. e.s. irrep (C2N,v) T mv md P F S

CDW+ c+1A1 +1 +1 +1 +1 −1 0

CDW− c−1B1 −1 +1 −1 −1 −1 0

SDW+ s+,t3A2 +1 −1 −1 −1 +1 1

SDW− s−1B2 −1 −1 +1 +1 −1 0

CS+ cs+1A1 +1 +1 +1 +1 −1 0

CS− cs−1B1 −1 +1 −1 −1 −1 0

im+,t3A2 +1 −1 −1 −1 +1 1

lines may be termed quantum phase transitions driven bythe collective tunneling of electrons. The property thatsuddenly changes at the critical lines c+-c−, cs+-cs−, ands+-s− is neither CDW, nor SDW or CS correlation func-tion but the symmetry of the ground state wave function.Such tunneling-driven quantum phase transitions repre-sent a new effect of strong electron correlations and arespecific for finite (mesoscopic) systems.

A further support to assign such phenomena to a col-lective electron tunneling is given by the behavior of thelowest excitation energy ε. To illustrate, let us considerthe c−-sector. There, ε ≡ 〈c+|H|c+〉−〈c−|H|c−〉 obtainedin our numerical calculations is considerably smaller thanall other excitation energies and, as shown below, scalesas expected for a tunnel splitting energy. For small t0,the classically equivalent configurations |CDW1,2〉 havethe energy Emin ≈ UN/2. The switching between theseconfigurations (| · · · 0202 · · · 〉 | · · · 2020 · · · 〉) can be re-garded as a deformation of the electron distribution. Inthis process of reorganization of the entire system, thehighest energy Emax ≈ UN/4+V N is reached in the (nor-mal) state where the electron density of either spin is uni-form (〈nl,σ〉 = 1/2). Because Emax > Emin (in the CDWregime, 2V > U), the transitions |CDW1〉 |CDW2〉cannot occur classically, but are possible quantum me-

chanically by tunneling. Semiclassically, the tunnel split-ting energy ε should be proportional to the transmis-sion coefficient τ ∼ exp(−a

√2EBM/~) over a distance

of the order of the lattice constant a (| · · · 0202 · · · 〉 | · · · 2020 · · · 〉) through a barrier of height EB = Emax −Emin ≈ (V − U/2)N of an object of mass M ≈ Nm(m ≈ ~2/(2t0a2) being the electron mass). This yieldslog ε ∼ −N

√(V − U/2)/t0, a dependence confirmed by

the exact numerical results presented in Figure 2. In par-ticular, the size dependence log ε ∼ −N (already visiblein Fig. 2b in spite of the small N -values used there) is aclear indication of the collective nature (M ∝ N) of thistunneling effect through a barrier created self-consistentlyby all electrons (EB ∝ N).

A given symmetry of the ground state does not nec-essarily specify uniquely the type of ordering. Transitionsfrom one type of ordering to another can also occur grad-ually. It has been mentioned previously [8] that the CDW-CS transition is smooth. This is possible because the states|c+〉 and |cs+〉 have the same symmetry (cf. Tab. 1). Asshown in Figure 1, the smooth CDW-CS transition doesoccur for all investigated values of N . The same holdsfor the states |s+〉 and |cs−〉 for N = 8. As shown in Ta-ble 1, there also exists another pair of states with the samesymmetry: |s±,t〉 and |im±,t〉. Interestingly, a smooth s+,t

- im+,t transition does occur for N = 6, while the regionss−,t and im−,t (s+,t and im+,t) are separated in the phasediagram for N = 8 (N = 10). In Figure 1a, in agreementwith those aforementioned, the s+,t - s− transition is in-dicated by a solid line, while a dashed line is used forthe im+,t - s− transition. Of course, the point where theline style changes can be defined only by some arbitraryconvention.

For the investigated open shell rings, we have foundthat the exact ground state always transforms accordingto one of the irreducible representations A1,2 and B1,2

of the symmetry group CN ,v. Interestingly, not all afore-mentioned symmetries given are independent: as seen inTable 1, Pmd = +1 and Tmdmv = +1. We have alsoinvestigated closed shell rings (i.e. N = 4n and ABC aswell as N = 4n + 2 and PBC). The relations Pmd = +1and Tmdmv = +1 also hold for the lowest eigenstatesof closed shell rings with A1,2- and B1,2-symmetries. Theresult Pmd = +1, valid for CDW-, SDW-, and CS-typeeigenstates, and both for open and closed shell rings, rep-resents the generalization of a recent finding. This rela-tionship, and not the incorrect one Pmv = +1 [19] statedin the first reference [9], has been reported in the secondof references [9], where only closed shell rings near theCDW-SDW transition 0 < U ≈ 2V and without consider-ing CS-states have been investigated. Roughly, the CDW,SDW and CS orderings in the ground state of closed shellrings occur in the same regions of the (U/t0, V/t0)-planeas found for open shell rings. This agrees with the expecta-tion that, for large enough sizes and/or strong enough cou-plings, averaged physical properties should become insen-sitive to the boundary conditions (thermodynamic limit).We point out that the relationships Tmdmv = +1 andPmd = +1 are valid for the eigenstates which transform


10-4

10-5

10-6

1.6 1.8 2.0

ε/t0

w (a)

10-3

10-4

10-5

10-6

4 6 8 10

ε/t0

N (b)

Fig. 2. CDW tunnel splitting energy ε ≡ 〈c+|H|c+〉−〈c−|H|c−〉 for U/t0 = 3.0: (a) versus w ≡p

(V − U/2)/t0 for N = 6; 8; 10(solid, dashed and dotted lines, respectively); (b) versus N for V/t0 = 4.4; 5.0; 5.6 (circles, crosses and stars, respectively).

according to one of the one-dimensional irreducible rep-resentations A1,2- and B1,2, but they do not necessarilyhold for low energy eigenstates of symmetries described bytwo-dimensional irreducible representations of the groupCN ,v. Without entering into details, we note that the lat-ter are also physically relevant [18].

However, as far as finite (mesoscopic) rings are con-cerned, important differences exist between the open andclosed shell cases. The phase diagram for the closed shellcase is trivial: there are no critical lines (as defined above).In the whole (U/t0, V/t0)-plane, the ground state is an 1A1

state and remains separated from the next eigenstate bya nonvanishing energy. In a recent attempt to clarify thephase diagram of the model (1) for the infinite case, nu-merical diagonalization results deduced by the level cross-ing method have been extrapolated for N →∞ [9]. To geta nontrivial phase diagram for the 4n (4n + 2)-site ringswith ABC (PBC) (closed shells in the ground state) inves-tigated there, the crossing of levels corresponding to lowestexcited eigenstates have been monitored. Intuitively, onemight expect similarities between (i) the ground statespossible for 4n (4n + 2)-site rings with PBC (ABC) and(ii) the lowest excited states for the reversed combina-tions, i.e. 4n (4n + 2)-site rings with ABC (PBC), be-cause both correspond to open shell situations. While (i)and (ii) yield insignificant differences for the CDW-SDWtransition lines 0 < U ≈ 2V , the corresponding phase di-agrams are qualitatively different. In the case (ii), thereare no tunneling driven quantum phase transitions; thestates |CDW+〉, |SDW+〉 and |CS+〉 as well as their exactcounterparts (|c+〉, |s+〉, and |cs+〉, respectively) are ofthe same symmetry: 1A1. P = +1, and F = (−1)N/2. De-pending on the values of U/t0 and V/t0, the exact groundstate is either |c+〉, or |s+〉, or |cs+〉, i.e. the dominant cor-relation is either CDW, or SDW or CS. Level crossings be-tween e.g. |c+〉 and |c−〉 states never occur for N = 4n+2(N = 4n) and PBC (ABC).

It is worth noting that the phase transitions of newtype and only these exhibit a reentrant behavior. As seenin Figure 1, by increasing U starting from large negativeU -values at fixed V > 0, the phase c+ disappears but

reappears later again. s− (s+) is another phase displayinga reentrant behavior for N = 6, 10 (N = 8); this occurse.g. by decreasing U starting from large positive valuesat V/t0 ≈ −0.5. A variety of traditional phase transitionscan display a reentrant behavior: by monotonic changesof temperature or composition, a certain phase can dis-appear and reappear again. So, our results show that thereentrant behavior is a common feature of traditional and(tunneling driven) quantum phase transitions.

5 Correlation functions and optical properties

Within a semiclassical (mean-field) picture, a certain or-dered state is characterized by an order parameter, i.e.a nonvanishing average of an appropriate operator. For aCDW, one can use the amplitude of the modulated elec-tron density, i.e. the nonvanishing site-independent aver-age (−1)l 〈(nl,↑ − 1/2) + (nl,↓ − 1/2)〉. This means that,e.g. the charge excess at each even-numbered site is equalto the charge deficit at each odd-numbered site. Sincethere is another equivalent configuration (obtained by in-terchanging the words even and odd above), the afore-mentioned average vanishes in the ground state computedquantum mechanically because of the tunneling betweentwo equivalent configurations. Rather than such averages,CDW, SDW and BOW (bond order wave) orderings in acertain eigenstate |Ψ〉 (〈Ψ |Ψ〉 = 1 ) of a mesoscopic sys-tem can be characterized by nonvanishing 2kF-correlationfunctions Kc,s,b, respectively [14]:

Kc,s =N−1∑l=0

(−1)l 〈Ψ | (nl,↑ ± nl,↓) (n0,↑ ± n0,↓) |Ψ〉

(3)

Kb =N−1∑σ;l=0

(−1)l⟨Ψ |(c†l,σcl+1,σ + h.c.

)×(c†0,σc1,σ + h.c.

)|Ψ⟩.


The quantity Kcs will be used as a measure of the CSordering:

Kcs =N−1∑k=0

| 〈CSk|Ψ〉 |2. (4)

Calculations show that, by changing the model parame-ters, the CDW, SDW and CS correlations behave in ac-cord with what one expects from the qualitative analysisof Section 3. For illustration, we present in Figure 3 curvesfor the correlation functions computed for 10-site rings byvarying U/t0 at fixed V/t0 and vice versa. To facilitate theunderstanding of the results presented in this section, wegive below the values of the correlation functions of equa-tions (3, 4) computed for the states with ideal orderingsof equation (2):

|CDW±〉 : Kc = N ;Ks = 0;Kb = 2;Kcs = 0;|SDW±〉 : Kc = 0;Ks = N ;Kb = 2;Kcs = 0; (5)

|CS±〉 : Kc = 2[1− (−1)N/2]/N ;Ks = 0;Kb = 4/N ;Kcs = 1.

The numerical results confirm that, sufficiently away fromthe lines of separating both CDW-SDW orderings (U =2V > 0) and CDW-CS orderings (V = 0, U < 0). theCDW correlations become larger, if one moves into the c±sector, This happens, e.g. by increasing V at fixed neg-ative U (Fig. 3a) or by decreasing U at fixed positive V(Fig. 3b). Deep enough inside the CDW region, the groundstate wave function |c±〉 is dominated by the symmetrizedsuperposition |CDW±〉 of equation (2). Consequently, thecorrelation functions in Figures 3a and 3b tend towardsthe limiting values of first of equation (5). In particular,this means that a CDW state can evolve towards the per-fect (bipolaronic) limit.

The charge segregation can also become perfect. Weillustrate this by presenting curves for the correlation func-tions along two lines passing through the cs± sector. Fig-ure 3c (3a) shows that the correlation function Kcs in-creases and finally saturates to 1 by decreasing U (V ) atnegative V (U), while the other correlation functions aresmall and approach the limiting values of the third equa-tion (5). In the CS region, the weight of the symmetrizedsuperposition |CS±〉 of equation (2) in the expansion ofthe ground state |cs±〉 is by far the largest. Although aCS-CDW crossover is clearly visible in Figure 3c, noneof the correlation functions Kc,ss,b exhibits a jump; thesmooth CS-CDW transition is a manifestation of the factthat the states |c+〉 and |cs+〉 possess the same symmetries(cf. Sect. 4 and Tab. 1).

Similar to the c±- and cs±-phases, the im+,t-phase isalso dominated by a few symmetric superpositions cor-responding to smeared CS’s. Sufficiently far from thes−-phase, the dominant contribution comes from the sym-metrized superposition |X1〉 of 20 (in general, 2N) termsof the form | ↑ 2222 ↓ 0000〉, the remaining weight be-ing practically exhausted by the symmetrized superpo-sition |X2〉 of 80 (in general, 8N) terms of the form| ↑ 02222 ↓ 000〉 and |2 ↑ 222 ↓ 000〉; typical weights

are ∼ 90% and ∼ 9%, respectively. Only very close to thes− border, another superposition |X3〉 of 40 (in general,4N) terms of the type | ↑ 222 ↓↑↓ 000〉 (a hybrid be-tween CS and SDW ordering) becomes more significant.A few values are illustrative. For (U/t0 = 4.5, V/t0 =−3.47); (U/t0 = 6.0, V/t0 = −4.36), the correspondingweights are wX1 = 80.58; 90.03%, wX2 = 10.02; 6.21%,and wX3 = 4.56; 2.07%. In view of this behavior, it seemsto us unlikely to assign this im-phase as a metallic phase.Ending the discussion on the im-phase we note that theexistence of an intermediate state between the CS andSDW phases can also be qualitatively understood withina classical analysis (t0 → 0). Although, to our knowledge,not mentioned by any previous study, this fact can be eas-ily seen. For t0 → 0, the energy (U + 4V )(N/2− 1) of thestate |X1〉 defined above is lower than the energies of theideal states |CSk〉 (1 ≤ k ≤ N) and |SDW1,2〉 (equal toUN/2 + 2V (N − 1) and V N , respectively) in the region0 < U < −2V (1− 4/N)/(1− 2/N).

Straightforward numerical analysis of the weights ofthe different multielectronic configuration contributing tothe ground state has indicated an important differencebetween the s± sectors and all other sectors of the phasediagram. In the latter case, 2; N ; 2N appropriately sym-metrized terms dominate or (almost) exhaust the expan-sion the ground state wave function in the c±; cs±; im±sectors, respectively; as discussed above, these terms cor-respond to classically equivalent configurations. In the s±sectors, the number of classically equivalent multielec-tronic configurations of lowest energy (NV ) is consider-ably larger: there are

(NN/2

)degenerate states where each

site is occupied by one electron of either spin up or down;the states |SDW1,2〉 (cf. Sect. 3) are only two out of these.When the hopping is switched on (t0 6= 0), it lifts the de-generacy of the states that are classically equivalent in allsectors. This has two important consequences.

First, the higher degeneracy lowers the energy morein the s±-sectors than in the c±- and cs±-sectors. There-fore, the classical critical lines separating an SDW-typeground state from states of other types are pushed towardsthe latter. In particular, the critical line 0 < U . 2Vis slightly pushed towards the CDW region with respectto the classical line U = 2V , as already mentioned [6].Second, the CDW and SDW correlation functions behavedifferent. Deep inside the CDW regions, Kc approachesthe value N corresponding to ideal ordering (cf. firstEq. (5)). By contrast, in the SDW regions, Ks saturatesto a value significantly smaller than the ideal value N (cf.second Eq. (5)); see Figures 3c and 3d. In other words,a CDW-type ground state can evolve towards the bipola-ronic limit (|c±〉 → |CDW±〉), but an SDW-type groundstate can never reach the perfect antiferromagnetic order-ing (|s±〉 6→ |SDW±〉).

The finite hopping yields a certain difference betweenthe weights (w) in the ground states |s±〉 of the various(NN/2

)states mentioned above. However, although some-

how preferred, the weights of the states |SDW1,2〉 re-main small, as illustrate by several examples, where thecontributions of the states |SDW1〉, | ↓↑↓↑↓↑↓↓↑↑〉, and


8

6

4

2

0-4 -2 0 2

K

V/t0

c

cs s

b

U/t0= - 1.6

(a)

9

6

3

0-3 0 3 6

K

U/t0

c

s

b

V/t0=2.0

(b)

3

2

1

0-3 0 3 6

K

U/t0

s

cs

c

b

V/t0= - 3.5

(c)

3

2

1

0-3 0 3 6

K

U/t0

s

cs

c

b V/t0= - 1.0

(d)

Fig. 3. CDW, SDW, BOW and CS correlation functions K (subscripts c, s, b, and cs, respectively) for 10-site rings. (ForV/t0 = −3.5, the Kb-curve is multiplied by 0.75). For U/t0 = −1.6 (a), one can observe a smooth CS-CDW transition aroundV/t0 ≈ −0.9. Jumps in Kc,s,cs,b are visible at the critical points: (b) U/t0 = 2.66 (c−-c+); 3.86 (c+-s−). (c) U/t0 = 0.74(cs−-im+,t); 4.57 (im+,t-s−); (d) U/t0 = −0.76 (cs+-s+,t); 1.17 (s+,t-s−); The small discontinuities in Kc,s,cs,b cannot be seenwithin the drawing accuracy at the following critical points: (a) V/t0 = −3.84 (cs−-cs+); 1.50 (c+-c−); (b) U/t0 = 2.05 (c+-c−);(c) U/t0 = 1.41 (cs+-cs−).

| ↓↓↑↓↑↓↑↑↓↑〉 with the largest weights (w) are given. At(V/t0 = −3.5, U/t0 = 4.58), in the s− region very close tothe im+,t border, the values are w = 6.88; 1.12; 0.67%, re-spectively. Deep inside the s−-region, at (V/t0 = 2, U/t0 =6), the weights are 7.36%, 12.32%, and 7.54%, respec-tively. In the s+,t-region, at (V/t0 = −1, U/t0 = 1),w = 0.70; 0.13; 0.08%, respectively.

Actually, there is a more general reason why a perfectantiferromagnetic ordering can never be reached withinthe model (1). Straightforward calculations show that, un-like |CDW±〉 and |CS±〉, the states |SDW±〉 are not eigen-states of the total spin operator S. Since [S2,H] = 0, |s±〉are exact eigenstates of both operators H and S. There-fore, whatever the parameter values (in particular, evenfor U/t0 � 2V/t0 � 1), the expansions of |s±〉 alwayscontain nonvanishing contributions from multielectronicconfigurations different from |SDW±〉. which diminish Ks

from the ideal value N (cf. Eqs. (5)).

At the CDW-SDW and CS-SDW transitions, the cor-responding correlation functions display significant dis-

continuities: see e.g. Kc,s at U/t0 = 3.86 in Figure 3band Kcs,s at U/t0 = −0.76 in Figure 3d, respectively. Weshall not discuss in detail such transitions between varioustype of orderings, which also occur in infinite systems [20].We only make one more remark on the interplay betweenorderings. Figures 3a and 3b reveal that the Kb-values arelarger around the c±-s± critical lines, that is the BOW or-dering is stabilized by the CDW-SDW competition. Thissupports the similar conclusion of references [9].

The jumps of the CDW, CS, and SDW correlationfunctions at the c+-c−, cs+-cs−, and s+-s− transitions areextremely small, practically invisible within the drawingaccuracy scala of Figure 3. The reason for this is that,e.g., similarly to the approximate states |CDW+〉 and|CDW−〉, the CDW correlations in the lowest eigenstates|c+〉 and |c−〉 do not differ appreciably for a given param-eter set. From the almost equal values of Kc,cs,s at eitherside of the tunneling-driven quantum phase transitions,one might be attempted to conclude that the correspond-ing phenomena are not of physical interest. However, the


3

2

1

0-4 -2 0 2 4

J/t0

G/t0

U/t0

V/t0=1.0

(a)

3

2

1

0-4 -2 0 2

J/t0

G/t0

U/t0

V/t0= - 1.0

(b)

Fig. 4. Results on the first transition allowed optically from the ground state in 10-site rings. Shown are the absorption frequencyG (solid line) and the matrix element of the current operator J (dashed line). The jumps in G and J at the tunneling-driven phasetransitions occur in (a), at U = −1.14 and U = 1.33 (c+-c−); in (b), at U = 1.17 (s+,t-s−). They are of comparable magnitudeto those occurring at the quantum transitions between states with different orderings – in (a), at U = 1.94 (CDW-SDW); in(b), at U = −0.76 (CS-SDW).

contrary is true, as demonstrated below by means of twoexamples.

First, the BOW correlation function Kb doessignificantly jump when crossing the aforementionedcritical lines, as clearly seen in Figure 3: Kb is muchmore sensitive to such changes in the ground state sym-metry than Kc,s because of the different symmetry prop-erties of the operators entering these functions: the op-erators Oc,s ≡

∑N−1l=0 (−1)l(nl,↑ ± nl,↓) transforms ac-

cording to the irreducible representation B1, while Ob ≡∑N−1σ;l=0(−1)l(c†l,σcl+1,σ + h.c.) transforms as B2 [21].Most interesting for experiments, the optical prop-

erties are dramatically affected at all the quantumphase transitions, irrespective whether they occur betweenstates of different physical orderings (e.g., CDW-SDW)or between states of different symmetries but character-ized by the same ordering (e.g., CDW-to-CDW). Sud-den changes in such properties at CDW-to-CDW andSDW-to-SDW transitions can be as large as those occur-ring at CDW-SDW transitions. This is clearly illustratedby the curves of Figure 4 which present results on the spec-tral absorption line of lowest frequency from the groundstate: absorption frequency (optical gap)G and the ampli-tude of the corresponding matrix element J of the currentoperator. Details on the relevance of these results for ex-periments are given in Section 6.

6 Summary and outlook

Basically, quantum fluctuations affect the phase diagramobtained classically in two ways. On one side, they changethe positions of the critical lines; this effect is more pro-nounced for values of t0 comparable to those of U and V .As discussed in Section 5, the hopping tends to enlarge theSDW sector, because of the higher degeneracy of the mul-

tielectronic configurations that are classically equivalent.More important, we have demonstrated in this paper thatthe ground state can change the symmetry when crossingcertain critical lines even inside a parameter region char-acterized, classically speaking, by a definite (i.e. eitherCDW or SDW or CS) ordering. We termed such phenom-ena tunneling-driven quantum phase transitions, becausethey are caused by the collective tunneling of stronglycorrelated electrons between two configurations which areclassically equivalent. The modifications of relevant phys-ical properties (cf. Figs. 3 and 4) at such critical linesare as drastic as those at other, more conventional (e.g.CDW-to-SDW) phase transitions.

We have emphasized the important role played infinite systems of strongly correlated electrons by thequantum tunneling between two configurations that areclassically equivalent. Similar to other cases – like theNH3-molecule or systems dimerized due to the coupling ofelectrons to intramolecular (Holstein) [13] or intermolec-ular (Su-Schrieffer-Heeger) [18] phonons –, the symmetryof the Hamiltonian, broken within semiclassical (mean-field-type) approximations, is restored by quantum tun-neling. However, there are important differences betweenthe aforementioned systems and that presently consid-ered, which reflect the important role of the strong elec-tronic correlations. In the former, the tunneling alwaysyields pairs of nearly degenerate states of opposite parities.Therefore, the optical absorption in the ground state |Ψ0〉cannot be practically distinguished from that in the lowestexcited state |Ψ1〉 if the tunneling splitting ε = E1 − E0

(Eλ ≡ 〈Ψλ|H|Ψλ〉) is sufficiently small [18]. Pairs of ex-cited states |Φα〉 and |Φβ〉 close in energy (|Eβ −Eα| ≈ ε)exist, such that the transitions |Ψ0〉 → |Φα〉 and |Ψ1〉 →|Φβ〉 are allowed, while the transitions |Ψ0〉 → |Φβ〉 and|Ψ1〉 → |Φα〉 are forbidden. In the presence of strong corre-lations, positions and intensities of spectral lines for tran-sitions allowed from the state |Ψ0〉 can significantly differ


from those allowed from |Ψ1〉. In this case, the frequency(ω = Eα − E0) of a photon causing a certain transition|Ψ0〉 → |Φα〉 can be well separated from the frequencies(ω′ = Eγ − E1) of all transitions |Ψ1〉 → |Φγ〉 allowedoptically (|ω′ − ω| � ε).

This feature can be exploited to reveal the quantumtunneling experimentally. On the basis of the above con-siderations, a strong temperature (T ) dependent absorp-tion can be expected. For T � ε, only transitions from|Ψ0〉 are possible, while for T ∼ ε the transitions from|Ψ1〉 also contribute to the spectrum. The absorption inthe state |Ψ1〉 could be extracted by comparing the twospectra. Of course, this is practically realizable only ε issufficiently large. ε strongly depends on the model pa-rameters, and a quantitative estimation of ε for largersizes is difficult. In accord with the collective nature ofthe electron tunneling, one may assume log ε ∝ −N (cf.Sect. 4). Then, by setting V/t0 = 1, the following val-ues could be estimated for U/t0 = 1.8 (CDW regime),ε/t0 = 0.065; 0.00004; 4.5× 10−8 at N = 40; 100; 200. Fort0 ∼ 1 eV, even for the last of these values it is possible topopulate only the ground states at temperatures of a fewmK. In the SDW regime, the ε-values are generally largerthan in the CDW regime and, hence, the experimental re-quirements are easier to fulfill. For U/t0 = 2.5, we get:ε/t0 = 0.095; 0.03; 0.004; 10−5 at N = 40; 100; 200; 500.Actually, even values N ∼ 10 are not very far from ex-perimental conditions: what matters e.g. in conjugatedpolymers is not the length of a macroscopic sample, butthe conjugation length, and the latter could be of the or-der of several tens of units [23]. Nevertheless, the scal-ing log ε ∝ −N presumably remains a severe experimen-tal challenge to observe a collective electron tunnelingin larger systems. A possible way out of this difficultyis to study the photon-assisted tunneling. Resonant pho-tons of energy ω can bring the system in an excited state;the corresponding decrease in the effective barrier height(EB → EB−ω) makes the tunneling more probable. Suchan attempt was successful recently to reveal a coherenttunneling in a SQUID [22].

A direct experimental investigation of a collective elec-tron tunneling in strongly correlated systems will be pos-sible once the interaction strengths can be changed. Tech-niques used for conventional solids (e.g. applied pressure)are difficult to apply to mesoscopic systems. Attempts tofabricate e.g. ring-like nanostructures consisting of quan-tum dots appear most promising, since this will permit thedirect modification of the interaction strengths. Recentachievements of nanometer-scale site-control techniquesfor the growths of individual quantum dots [24] and quan-tum rings [25] on semiconductor surfaces are encouragingin this direction.

Demonstrating that interesting phenomena can bedriven by quantum tunneling, the present theoreticalstudy represents a motivation for such efforts in the field ofnanostructures. Besides, more common systems (interact-ing spins, cyclic molecules, commensurability other thantwo) are also interesting, since similar effects can also beexpected for similar reasons.

I.B. thanks Professors E. Dormann and H. von Lohneysen forvaluable discussions. The work at Karlsruhe has been finan-cially supported by the Sonderforschungsbereich 195 and thatat Heidelberg, in part, by the Fonds der Chemischen Industrie.

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14. Alternative (short-range) correlation functions were em-ployed in references [7,8]. As already pointed out [8], theirbehavior is similar to those used here.

15. Finite nuclei are other examples where phase transitionsbeyond the conventional sense are discussed; see, e.g., F.Iachello et al., Phys. Rev. Lett. 81, 1191 (1998), R.F.Casten et al., Phys. Rev. Lett. 82, 5000 (1999) and Phys.Rev. C 57, R1553 (1998).

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17. The fact that there exist certain parameter regions wherethe total spin is S = 1 and, consequently a threefold de-generate ground state, is beyond doubt. While, for larger


systems, one might suspect that the Lanczos algorithmmight converge to an eigenstate different from the trueground state, this can be definitely ruled out for 6-siterings, where we have computed accurately and tested alleigenvectors by diagonalizing directly (924 × 924) ma-trices. For triplet ground states, the SDW correlationfunctions computed for the states with spin projectionsSz = +1 and Sz = −1 are equal, but they differ fromthat corresponding to Sz = 0. In this case, the curves forKs presented in Figure 3 correspond to the state withSz = 0.

18. I. Baldea, (unpublished).19. In the notation of the first reference [9], the incorrect

relation reads CP = +1, where C stands for particle-hole (charge conjugation) symmetry and P for inter-changing of left- and right-moving fermions (R L).One can straightforwardly check that the interchangeR L, amounting to sign reversal of electron momen-tum (p→ −p), represents the transformation mv(0) andnot md(0).

20. For this reason, we shall not address here the question onthe exact location of a critical point (U ∼ 3 − 6) wherethe CDW-SDW phase transition changes from second tofirst order [6,9].

21. In view of the translation invariance, one can alternativelyexpress the correlation functions as Kν = 〈OνOν〉/N(ν = c, s, b). In the main text, we have preferred to givethe forms (3), because they are more convenient numeri-cally.

22. J.B. Friedman, V. Patel, W. Chen, S.K. Tolpygo, J.E.Lukens, Nature 406, 43 (2000).

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25. R.J. Warburton, C. Schaflein, D. Haft, F. Bickel, A.Lorke, K. Karai, J.M. Garcia, W. Schoenfeld, P.M.Petroff, Nature 405, 927 (2000).

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