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Submitted on 1 Jan 1985

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Landau level spectrum of Bloch electrons in ahoneycomb lattice

R. Rammal

To cite this version:R. Rammal. Landau level spectrum of Bloch electrons in a honeycomb lattice. Journal de Physique,1985, 46 (8), pp.1345-1354. �10.1051/jphys:019850046080134500�. �jpa-00210078�

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Landau level spectrum of Bloch electrons in a honeycomb lattice

R. Rammal

Centre de Recherches sur les Très Basses Températures, CNRS, B.P. 166X, 38042 Grenoble Cedex, France,and Department of Physics, University of Pennsylvania, Philadelphia, PA 19104-3859, U.S.A.

(Reçu le 25 juin 1984, accepté le 2 avril 1985)

Résumé. 2014 On analyse le spectre d’énergie du modèle des liaisons fortes, sur un réseau en nid d’abeilles, en présenced’un champ magnétique uniforme. Le graphe du spectre est montré pour différents flux réduits rationnels à traversles cellules élémentaires. On montre que ce spectre possède des propriétés récursives analogues à celles du réseaucarré et du réseau triangulaire. Les propriétés spécifiques (bandes interdites, sous-bandes, etc.) obtenues sontattribuées à une propriété de frustration et sont comparées avec celles des réseaux de Bravais. Nos résultats ontune implication directe pour les mesures récentes du champ critique supérieur d’un réseau supraconducteur ennid d’abeilles. Nous comparons aussi la structure du bord du spectre dans les trois réseaux : carré, triangulaire etnid d’abeilles.

Abstract. 2014 The energy spectrum of a tight-binding honeycomb lattice in the presence of a uniform magneticfield is analysed. The graph of the spectrum over a wide range of rational reduced flux 03A6/03A60 through elementaryhexagonal cells is plotted. The energy spectrum is found to have recursive properties similar to those discussedpreviously on the square and triangular lattices. New features of the spectrum are also obtained. Specific properties(gaps, subbands, etc.) are shown to be a direct consequence of frustration and are compared with the spectrum ofBravais lattices. Our results are shown to be relevant for the recent measurements of the upper critical field of a

superconducting honeycomb network. A comparison of the structure of the edge of the spectrum on square,triangular and honeycomb lattices is also outlined.

J. Physique 46 (1985) 1345-1354 AOÛT 1985,

Classification

Physics Abstracts71.10 - 71.25 - 74.10

1. Introduction.

Recently, it has been shown that [1, 2] the magneticproperties of regular superconducting networks arecontrolled, within the framework of mean field theory,by the so-called Harper’s equation [3]. More precisely,the edge of the spectrum of the tight-binding model,in the presence of a magnetic field [4] was identifiedwith the upper critical line of the superconductingnetworks. This connection has been achieved experi-mentally [5] by a direct measurement of the criticaltemperature of a regular two dimensional super-conducting network. The observed well-defined struc-tures of the critical line reflect the expected features ofthis line.

In this paper we propose to investigate the energyspectrum of a tight-binding model on a honeycomblattice (H) in the presence of a uniform magnetic field.Our motivations are as follows :

a) Up to now, only Bravais lattice have beenstudied : square (S) lattice [4, 6] and triangular lat-tice [7]. Previous work has shown that the spectrum

have various symmetry properties some of whichmay be traced to the point group of the lattice. Others,such as recursiveness are general features of the modelHamiltonian used and the difference between rationaland irrational numbers [4, 8]. What happens on a non-Bravais lattice such as the honeycomb lattice ?On the other hand, the topological structure of the

lattice plays an important role in the band structure,already in zero magnetic field. The occurrence of oddrings in the lattice induces some particular features ofthe density of states which may be traced to frustra-tion [9] phenomena. What are the consequences of theparticular band structure at zero field on the Landaulevel spectrum ?

Finally, the rich structure of the spectrum of Landaulevels results from the coexistence of two incommen-surate periods. The first is given by the lattice structureand the second is fixed by the magnetic field. Therelevant parameter in the band structure is the ratiof = 4>14>0’ where 00 denotes the quantum flux and 0the flux through an elementary cell of the lattice. Inthis respect, the Landau level spectrum can also be

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019850046080134500

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viewed as a frustration problem [10]. To what extentdoes the band structure (closure of the gaps, bandtouching, etc.) result from geometrical considerationsalone ?

b) In view of the experimental measurement of theedge of the spectrum, using superconducting net-

works [5], it is useful to compare this edge on variouslattices. In particular, the reduction of the critical

temperature Teat different rationals f = 0/00 issensitive to the topology of the network. This is clearlyvisible when we compare the square and triangularlattices. What is the corresponding situation on thehoneycomb lattice ?This paper is organized as follows. In section 2, a

general formulation of the problem is given, as well asa coherent set of notations. Section 3 is devoted to thecalculation of the band structure at zero and low

magnetic fields. Section 4 is actually the main part ofthis paper. The spectrum in calculated at rational fluxand the density of states is expressed in each subband.Various symmetry and recursiveness properties arealso discussed as well as the measure of the spectrum.The structure of the edge spectrum is discussed inrelation with superconducting networks in section 5.In particular, a comparison between square, triangularand honeycomb lattices is outlined in this section.In the last section we give some concluding remarksand discuss some other illustrations of the resuts.

2. Notations and general formulation.

The honeycomb lattice is made up of two dimensionalarray of hexagonal unit cells, of side a, with atoms atthe vertices. Such a structure is encountered in solidstate physics in some strongly anisotropic crystals [11].The unit cell is a rhombus of side a/, with angles2 7T/3 and n/3 at its vertices. We take the x-axis parallelto the long diagonal and the y-axis parallel to the shortdiagonal of the rhombus. There are two atoms

(Fig. la) in each unit cell, A and B. In the reciprocalspace k = (kX’ ky), the first Brillouin zone is the

hexagon formed by the appropriate perpendicularbisectors. Particular points of relative importance,lying at the edge of the first Brillouin zone, are shown

on figure lb : F(2 n/3a, 0) and

respectively.As a model for the honeycomb lattice, we consider

an stight-binding model with nearest-neighbour over-lap only. If cp(r) is used to denote the wave functionamplitude on site r, then the Schrodinger equationreduces to a finite-difference equation

where c denotes the associated eigenvalue, r’ denotesnearest neighbours of r and V the strength of the over-lap integral. In the following, we take V = 1 for

simplicity.

Fig. 1. - (a) The honeycomb lattice made up of hexagonalcells of side a. A(x - a, y) and B(x, y) denote two atoms inthe unit cell. (b) The first Brillouin zone in the reciprocal spacek = (kX’ ky). Particular points lying at the edge of the zoneare shown : F(2 n/3a, 0) and K(2 nl3a, ± 2 7r/3 a -VI-3).

In the presence of a uniform magnetic field H, per-pendicular to the planar lattice, the above equation ismodified by phase factors depending on the magneticfield strength. Let Yap be the phase factor betweenatoms a and fl :

A being the vector potential, equation (2. 1) becomes :

As shown in section 4, the whole spectrum of equa-tion (2. 3) depends only on the reduced flux f = 0/00,giving the ratio of the magnetic flux through an elemen-tary hexagonal cell I(o = 6 Ha’,13-14) to the flux

quantum (00 = hcle for electrons).

3. Spectrum at zero and low magnetic fields.

3.1 ZERO FIELD SPECTRUM. - In the absence of the

magnetic field, the dispersion relation is given by :

where k = (k, ky) is the wave vector associated to theperiodic solutions CPa = tfa exp(ikx. x + iky.Y) of equa-tion (2.1), (a = A, B).The spectrum (two bands) is confined in the interval

[ - 3, + 3] and the edges are reached for k = 0.Close to this point we have two parabolic bands andthe curves of constant energy are circular.As can be seen the adjacent band extrema are at the

comers of the first Brillouin zone (points K). Thebottom of the valence band (- in Eq. (3.1)) and thetop of the conduction band (+ in Eq. (3.1)) lie atk=0.

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Eigenstates corresponding to e = 0 (point K) areshown on figure 2. Similarly the associated eigen-states to point F are shown on figure 3. Note that,close to points K, the dispersion relation becomeslinear instead of quadratic. The density of statesn(s) associated with equation (3.1) is given [12] bythe following expression

Here is the

argument of the complete elliptic integral K(k) offirst kind.Note that in addition to the Van Hove’s singulari-

ties (at s = ± 1, ± 3), which are typical of twodimensional lattices, the density of states n(E) vanishesat B = 0 where a non-analytical singularity appears

The origin of this singularity at 8 = 0 may be tracedback to the linear dispersion relation close to points K,resulting from the degeneracy of the two bands atthese points.

3.2 Low MAGNETIC FIELD LIMIT. - The calculationof the energy spectrum at low magnetic field can becarried out using a semiclassical theory (Onsagerquantization scheme [13]) or a continuum approxi-mation [14] for the tight-binding equations (Eq. (2.3))in the presence of a small magnetic field. Note that[15] the semiclassical method may break down nearsaddle points. In the present case, the Onsager’sscheme can still be applied near the Van Hove singu-larities, but it will be shown to break down near thenon-analytic singularity s = 0.

3.2.1 Onsager’s quantization scheme. - Near theband edges s = ± 3, the quantization rule gives thefollowing Landau levels :

where y = 2 noloo and n is a positive integer (n > 0).As shown in figure 4, equation (3.4) reproducesquantitatively the Landau level structure near theband edges.

Fig. 2. - Degenerate eigenmodes associated with the energylevel e = 0 at zero magnetic field. Here a and b denote twoarbitrary numbers and w is the complex conjugate ofm = exp(2 in/3).

Fig. 3. - Eigenmodes associated with s = + 1 and - 1 at

zero magnetic field.

Similarly, near the singular point s = 0, one obtains

This result, implying a square-root departure of g as afunction of the magnetic field must be contrasted withthe usual linear behaviour given by equation (3.4).However, as shown below, the result of equation (3..5)is not entirely correct : n + 1 2 must be replaced by n.When this correction is taken into account, equa-tion (3 . 5) gives accurate results, as shown in figure 4.3. 2. 2 Continuum approximation. - In this approach,the tight-binding equations (Eq. (2.3)) are linearizedin the presence of a vanishing magnetic field. In thefollowing, we choose for convenience the Landau

gauge A = H(O, x, 0).Let us consider first the case of the band edge B = 3.

Taking into account the translation symmetry in

y direction, solutions of equations (2.3) can bechosen to be of the form cp(x, y) = tJI(x) exp(iky. y).One obtains a system of two coupled equations

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Close to e = 3, one can choose ky = 0 and look at solutions of equation (3.6) where t/J A = t/JB = 0. Puttingu = x - a/2, a more symmetric equation is obtained by adding the above two equations :

Now, the two sides of equation (3. 7) are expanded in ula and y. This leads to :

Fig. 4. - Energy spectrum for broadened Landau levels.Only low index levels are shown and the fine structure corres-ponds to just a few rational values of the reduced flux 0/00.Solid lines correspond to Landau levels starting at zero fieldfrom s = 3 and 8 = 0 respectively (Eqs. (3.4) and (3. 10)).

which is simply the Schrodinger equation for a simpleharmonic oscillator. The energy levels associated with

equation (3.8) reproduce the result of equation (3.4).In order to perform the continuum approximation

for the degenerate level s = 0, we use equation (3.6),

written in variable u = x - al2 and for kycorresponding to the point (kx = 0, ky) of the firstBrillouin zone. In the limit *y 1, we expand the sys-tem as above and this yields

The solution of this set of equations can be expressedin terms of normalized harmonic oscillator functions.In particular, the corresponding eigenvalues are

given by

where n takes on all positive integer values.The (partial) breakdown of Onsager’s rule is simply

caused by the band degeneracy at e = 0. This is a

consequence of the lattice structure partially neglectedin the semi-classical approach.

4. Landau level subband structure.

Some time ago, Hofstadter [4] studied the energyspectrum of a single tight binding square lattice. Thesame study was extended later for the triangularlattice [7] and generalized [6] square lattice structures.In all these cases, it was shown that the relevant para-meter in the band structure is the ratio f = 0/00.For rational /(= p/q), the tight-binding band is

split up into q non-overlapping subbands, each

containing an equal number of states. For irrationalvalues of f the number of bands is infinite, but thegaps that exist at rational values of f persist oversome finite range of f, through a succession of closuresand openings. The graph of the energy spectrum as afunction of magnetic field exhibits a recursive structurein the plot whereby copies of the entire spectrum arecontained within the spectrum itself. Although arigorous proof of this property is not available, onecan see its plausibility by looking at the spectrum as acomposite of broadened and split Landau levels.More general properties were also discovered :

i) periodicity in f along the field axis, with period 1due to gauge invariance, ii) the closures of the gaps.For instance on the square lattice, at f = 1/2, thespectrum comprises two bands which touch eachother. This means that at this energy the density of.states has an isolated zero with a non-analytic beha-viour. Other such touchings of bands can be observedon the central horizontal axis. The case of the triangu-lar lattice exhibits similar behaviour. In this case, a

gap closure due to touching of two bands is observedwhen f = 1/6 and 5/6.

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As was shown in section 2, a band touching occursalready in the honeycomb lattice in the absence of amagnetic field, at s = 0. A natural question arises :to what extent do the above properties of band touch-ings result from geometrical considerations alone ?In this section we shall investigate this question, bystudyingttie band structure in the presence of a

magnetic field. For this, we focus our attention tothese properties instead of recursiveness, which isalso present in this spectrum.

4.1 REDUCTION TO ONE DIMENSION PROBLEM. - Aswas noticed in section 2, the honeycomb lattice is anon-Bravais lattice. Each unit cell contains two atomsA and B. We choose as above the Landau gaugeA = H(O, x, 0) ,for convenience. Writing the tight-binding equations (Eq. (2. 3)) for sites B(x;y), A(x - a, y)and A(x + a/2, y ± a v’3/2), one obtains a systemof four eigenvalue equations. Eliminating the A sublat-tice sites, the following equation for B site is obtained

In equation (4.1), Fi(l i 6) denote phase factors, resulting from the elimination of A sites. For the choosengauge, they are given by the following expressions

It is reasonable to assume plane wave behaviour in the y direction, since the coefficients in the above equationonly involve x. Therefore we write

and then deduce the one dimensional eigenvalue equation

After some rearrangements and the use of equation (4. 2), this leads to the finite difference equation

where t/lm = OB(x) at x = m(3 a/2), A = B2 - 3 and K = ky a,J3/2. Here m denotes an integer, and y = 2n4>/4>ois the magnetic field parameter. Only the ratio f = 0/0 0 is involved in equation (4. 5) as expected.

It is instructive to compare equation (4.5) to similar equations obtained on square [4] and triangular [7]lattices

and

Equation (4.5) is a second-order difference equationwhose solutions determine the broadening and fine

structure of the Landau levels for a given magneticfield. The spectrum is confined to values of A between

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- 6 and + 6, i.e. - 3 6 .- + 3. A close inspectionof equation (4.5) shows that the spectrum is unchangedunder reflections E H - s for a given value of y. Inaddition, the spectrum is invariant under translations :f - f + n, n integer and exhibits a reflection symme-try about half integer values of f. All these symmetryproperties are also present on the square lattice

spectrum, reflecting translation and gauge inva-riance.

4.2 RATIONAL FIELDS. - Because of the symmetriesin the spectrum, we shall limit our discussion to

0 f 1/2. For rational values p/q of f (p, q inte-gers prime relative to each other), the system (Eq. (4. 5))becomes closed after translation by q periods thusleaving 2 q separate equations. As was noticed byseveral authors [7, 16], not all these equations areindependent, however. For f = p/q, equation (4.5)becomes

’

Making the substitution :

one obtains a system of q equations for amplitudes g(u), u = 1, 2, ..., q.

where 01 = 2 x and y = 2 nplq, Indeed, one can easily verify that g(u) and g(q + u) obey the same equation.If we use Floquet’s theorem to construct these two functions through the relation

a Hermitian system of q equations for q unknowns results. In our notations, the q x q secular determinant hasperiod 2 7r/q in 81, and the indices 81 and 82 appear in the constant term only. The non-zero matrix elementsof the secular determinant are given by

Expanding this secular determinant in terms of

products of its elements, we know from its periodicity2 nlq with respect to 01 that all non-constant termscontaining less than q factors exp(i01) cancel. Theonly terms depending on 01 and 02 are thereforeeasy to obtain.The constant term involving 02 is given by

Similarly, the term involving (Jl only is given by

Therefore the equation for A that results has thefollowing form

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where Pq(À) is a polynomial of degree q in Å notcontaining 01 and 02- W(01, 82) is given by

When 01 and 02 cover their range, W «(J l’ 02) variesbetween W 1 = - 3 and W2 = + 6. Intercepts ofthe polynomial between these two values definetherefore the subbands for the rational field chosen.

In general, it is easy to verify the following form

of the secular equation

Furthermore, A = - 3 is a zero for this equation forall p and q. This solution corresponds to a bandtouching at s = 0 in the spectrum. More generally,the q bands in variable A transform into 2 q bandswith a reflection symmetry about c = 0.Some examples of polynomials P,(A) are given in

what follows, for simple values of f.

Intercepts of P,,(A) with W 1 and W2 give the subbandedges. More precisely the edges are fixed by the fol-lowing two conditions

and

For instance, if p = 1 and q = 2, equation (4.1b)yields the following four symmetrical bands :

,/3 8 ! [ /6, and 0 s [ % /3 touching at

s = 0 and e = +..,/3-. For small values of q, the cal-culations can be achieved analytically up to q = 5.For other values of q, we have calculated (using thealgorithm of Eq. (4.16)) the band edges for p/q givenby the first elements of the Farey series. We stopped thecalculations at q = 20 because the band widths becameas narrow as 10-9. Figure 5 shows the spectrumobtained in this manner. Broadening of the Landaulevels (Fig. 4) increases with f and a fine structure ispresent between any two rational values of f. Onfigure 4 are shown the Landau levels calculated insection 3 in the portion of the spectrum (0 , s , 3,0 f 1/2) shown in figure 5. These levels near8=3 and c = 0 are clearly recognizable, their fielddependence in the limit of low field matching exactlythe results of section 3 (indicated by full lines). Largegaps separate the low index levels, and two majorgaps are obtained for 0 f - 1/2. A detailed dis-cussion of the nesting hypothesis (recursiveness pro-perty) will be found in reference [7] and therefore notdiscussed here. Instead we shall discuss the densityof states in each subband.

Fig. 5. - Spectrum of a tight-binding model on a honey-comb lattice in a perpendicular magnetic field. The eigen-value energy a, ranging between - 3 and + 3, is the shownvertical variable and 0/00 the reduced magnetic flux throughone lattice hexagonal cell, is the horizontal variable, rangingfrom 0 to 1. Only rational values p/q of the field variable0/00 are represented with q 20.

4.3 DENSITY OF STATES. - The multiplicity of thesolutions of equation (4.5) can be obtained in astraight-forward manner since the phases 01 and 02appear in the constant term only. For this we follow

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the method proposed in references [7] and [17]. Themain idea is to transform the counting over an energyrange to a counting over a range of the polynomialPq(A), thus yielding a form of the density of statesappropriate for an arbitrary rational value of the fieldparameter f Let Às(Ol’ (2) be a root of equation (4.13)for a given 01 and O2 and a fixed value of the field. Asthese phases cover their range the root will scan anentire subband (in A). The phases have constant weightsince they are phase variables and the density of states,in variable A is then given by the expression

The sum is over all subbands and the total number ofstates has been normalized to one. The 5 function canbe integrated out by transforming the integral over 02to an integral over A,, with the aid of equation (4.15).In this way, P is treated as a variable, and the remain-ing integration can be performed in terms of completeelliptic integrals. Using A = E2 - 3, one gets

where

and

Here P is the value of the polynomial at A = s2 - 3and K(x) is the complete elliptic integral of the firstkind. Within a subband, only this latter factor variessignificantly giving a A-shaped logarithmic singula-rity to the density of states. At subband edges, thedensity of states drops discontinuously to zero as agap is encountered. Non-analytic singularities comingfrom the touchings of neighbouring subbands appearalso as in zero field. In addition it is easy to recover the

expression of n(s) obtained in section 3 at f = 0. Inthis case, P(A) = - À, v === ! I s I1/2 and u =F [ e 11/2

coincides with argument k of the elliptic integral givenin equation (3.2).4.4 MEASURE OF THE SPECTRUM. - As a by-productof the previous calculation, we have considered themeasure of the spectrum for rational f. Recently, thetotal measure of the spectrum has been studied [18]for different lattices. In particular, it was suggestedthat for a given f = p/q the total sum of bandwidths(measure of the spectrum) has the asymptotic forma/q for isotopic square and triangular lattices. Here adenotes a constant of proportionality which asympto-tically approaches 9.3300 on the square lattice, formost values of the numerator p. This very accurateresult (one part in 105) suggests the vanishing of thespectral measure, when q goes to infinity. The value 1of the exponent of q was attributed to the self-simi-larity of the diagram of energy bands as a functionof f.

Let us denote S(p/q) the sum of bandwidths for agiven f = p/q. We have deduced the values of theproduct qs(plq) from our numerical results. Althoughthe calculations were performed for q , 20 someconclusions can be stated. A selection of the values of

qS(p/q) is shown in table I. For q a 10, qS(p/q) takesits value inside the range from 7.2 to 8.6 independentof the way in which p varies. It is clear that the generalform of the result is in agreement with what is expec-ted, but it does not appear that data can be extrapo-lated in a reliable manner. The sequence p = 2,q = 9, 11, 13, 15, 17, 19 suggests the limiting valuea/2 = 3.95, while the sequence q even, p = q/2 - 1is too short to show convergence. Other sequencesare scattered and do not follow an obvious pattern.Therefore, it is likely that the sum of bandwidths istending to zero as fast as q - ’ is this case. However,more extensive data are needed to extract the precisevalue of the constant a.

5. Application to superconducting networks.

The theory involved in the computation of the uppercritical field of a superconducting network (neglectingfluctuations) consists of solving the linearized Ginz-burg-Landau equation, which is essentially equivalentto the Schrodinger equation. For superconductingapplication, one is only interested in the lowest eigen-value (edge of the spectrum). Note that the same equa-tions are involved in the mean field theory of Josephson-junction arrays [19]. The basic concepts for treatingsuperconducting networks near the second order

phase boundary have been worked in detail by variousauthors. The linearized Ginzburg-Landau equations

Table I. - Values of qs(plq) for selected values ofplq for the honeycomb lattice.

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lead in general to an eigenvalue problem which isbest expressed in terms of the order parameter valuesat the nodes. If node a is linked to n nodes via strandsof length L,,,# (# = 1 to n), the basic equation, atnode a may be written as

where A# is the value of the order parameter at nodesand y,,,p is the circulation of the vector potential alongthe strand (Eq. (2.2)) linking a and jS. Çs is the super-conducting coherence length.For a regular network, L,,,, = a is the same for all

strands and equation (5.1) reduces to

In equation (5.2), z denotes the coordination numberof the lattice. Thus the problem reduces to the Landaulevel spectrum of a tight-binding model in the samegeometry. The energy of the corresponding Bloch elec-trons (Eq. (2. 3)) is given by : c = z. cos (a/çs).The upper critical field of the superconducting

network is given by the edge of the spectrum calcu-lated in section 4. For a honeycomb lattice, z = 3and the results are given in table II.The results obtained in the previous section have a

direct significance for the magnetic behaviour of thesuperconducting network. In particular, at low field H,

Table II. - Edge (s) of the spectrum for rational valuesof the reduced flux f( = 0/00) = p/q with p and q inte-gers prime to each other (q , 20).

the continuum approximation (Eq. (3.4)) reprodu-ces [2] the linear behaviour in temperature

More generally, the variation of the lowest eigenvalueas function of f = 41/410 (Fig. 6) determines the effectof the frustration, induced by the magnetic field, on thetransition temperature Tc (or ground state energy) ofthe superconducting transition, treated in mean fieldtheory. When the flux through one lattice cell is half aflux quantum, f = 1/2 the lattice is effectively fullyfrustrated [10, 19]. The above study leads to the pre-diction of visible dips on the superconducting criticalline, actually observed experimentally [5], at rationalvalues of f. More pronounced depressions are atf = 1/3. Noticeable dips of smaller amplitudes appearsalso at f = 2/5 and 1/2.We conclude by comparing the relative reduction

of the critical temperature Tr on different lattices. Forthis we consider only the fully frustrated case f = 1/2.In this case, the total bandwidth is : 4,.,,/-2, 9 and 2 J6respectively for square (S), triangular (T) and honey-comb (H). These numbers must be compared with thecorresponding widths at f = 0 : 8, 9 and 6. A directcomparison is given by the value of s at the edge forf = 1/2 : 2V2, 3 and V6 for S, T and H. This leads tocos (a/g) = V2/2, 1/2 and J6/2 respectively on thesethree lattices. However AT, - l/ç;, therefore

ATe ,(T) > AT,,(S) >, AT,(H) for superconductingnetworks made with the same material and same a.The frustration is therefore more efficient on triangu-lar lattice than on the two other lattices, as expectedfrom intuitive expectations. In comparing the squareto the honeycomb lattice, the large number of sides(6 against 4) in the elementary loop shows that frus-

Fig. ’6. - Edge of the spectrum shown in figure 5, for0 % 0/00 1/2.

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tration has less dramatic effect on the honeycomblattice. Further comparison between different super-conducting networks as well as experimental resultswill be discussed elsewhere [20].

6. Conclusion.

The results presented in this paper for an electrondescribed by a tight-binding honeycomb lattice in thepresence of a magnetic field complete those previouslyobtained on square and triangular lattices. Ourresults show that the touching of neighbouring sub-bands, already present in zero field is still present atfinite rational field. The recursive property is also

present. New features of the spectrum were alsoobtained. Specific properties (gaps, subbands, ...)are shown to be a direct consequence of frustrationand are compared with the other lattices. Howevertwo important questions have not been discussed indetail. The first concerns the gaps and the definitionof a gap index. The value of this index is defined by theintegrated density of states below a gap and is ofdirect interest for the quantized Hall effect [21, 22].The second is related to the nesting hypothesis andthe recursive property of the spectrum. Both of thesequestions have been discussed in great detail in theliterature and the analysis can be carried over step bystep from known cases to ours.

Let us conclude by noting that the question ofirrationality is no longer artificial in real materials,as was pointed out in reference [4] some years ago.

The observation of the predicted features on figures 5and 6 requires, of course, an enormous magnetic fieldof about 106 kOe to let f - 1 for a typical latticespacing of the order of 2 A. Nowadays ,experimentallimitations do not allow for magnetic fields higher than103 kOe, which implies that the influence of the

magnetic field on the motion of the electrons is verysmall and the rich spectrum is not seen experimentallyin this regime. As shown in this paper and in refe-rences [5 and 20], the superconducting networksprovide a powerful and unique example, where someof the main features of the spectrum can be observedin realistic conditions. Furthermore, in these systems,the frustration induced by the magnetic field entersalone, and in a controllable way. Continuous variationof the applied field allows for a fine tuning of thefrustration, that is not easy to achieve otherwise.To our knowledge, this is the first available systemwhere a small modification of an external parameter(y = 0/00) may lead to a qualitatively different statein a predictable and adjustable way.

Acknowledgments.I would like to thank Dr G. Toulouse for useful and

friendly conversations. I would also like to expressmy dept to my collaborator Dr J. C. Angles D’Auriacfor valuable help in various ways. Finally, I wouldlike to thank Professor T. C. Lubensky at the Univer-sity of Pennsylvania for the hospitality in the PhysicsDepartment. This work was supported in part by theNSF under grand number DMR 82-19216.

References

[1] ALEXANDER, S., Phys. Rev. B 27 (1983) 1541.ALEXANDER, S. and HALEVI, E., J. Physique 44 (1983)

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