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Semiclassical methods in solid state physics : two examples Jean Bellissard, Armelle Barelli To cite this version: Jean Bellissard, Armelle Barelli. Semiclassical methods in solid state physics : two examples. Journal de Physique I, EDP Sciences, 1993, 3 (2), pp.471-499. <10.1051/jp1:1993145>. <jpa00246736> HAL Id: jpa-00246736 https://hal.archives-ouvertes.fr/jpa-00246736 Submitted on 1 Jan 1993 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. j_ pJ1ys. I France (1993) 3 471-499 1993, FEBRUARY 471 PAGE Classification Physics Abs tracts 71.25.C 75.10 03.65F Semiclassical methods Bellissard Jean Laboratoire Physique de (Received 22 solid Armelle and Cedex, Toulouse in physics state examples two : Barelli Quantique Universit6 Paul 118, Sabatier Narbonne de route F-31062 France July 1992, accepted in final form August 1992) 9 de deux problbmes motiv6s l'6tude des modbles bidiest une revue par supraconductivitd I haute temp6rature critique. La premibre partie conpour l'6tude du des dlectrons Bloch de bidimensionnels soumis £ un spectre d'6nergie cerne pour champ magndtique uniforme. Une analyse semi-classique permet d'en comprendre les propr16t6s qualitatives et quantitatives. La deuxibme partie est un plaidoyer pour l'utilisation des m6thodes du "Chaos Quantique" dans l'6tude des systbmes de fermions fortement corr616s. La distribution des dcarts de niveaux d'un modble t J en deux dimensions, en fournit illustration. une Rdsumd. Ce mensionnels travail la Abstract. We here present a review of problems two superconductivity. The first part the concerns energy uniform magnetic field. A semiclassical analysis provides understanding of this spectrum. second In the part we fermion "Quantum Chaos" strongly correlated to systems. Tc distribution for the t J model in two a by motivated spectrum qualitative make It is the 2D of 2D as case illustrated models Bloch well for as the by the for electrons a high in a quantitative application of level spacing dimensions. the Paris meeting each other. Some Statistical Mechanics in January on similarity between Rammal's work on us dimensional almost studied the Sierpinski lattice [75] and periodic models by Bessis, some one why he wanted make Moussa and myself [15] was the acquaintance. to reason my During the short discussion we had together at this meeting, it became immediately clear had both got our degree in a French that we had two points in university, and common : we 1met R. Rammal 1983. Pierre both worked for the first Moussa introduced hard understand time at to of the Hofstadter structure spectrum [55]. hardship of being outsiders the French instituin research we dominated by the smart but arrogant "Grandes alumni. Ecoles" tions heavily both fascinated by the complexity and the importance of magnetic On the second, we were field effects in Solid State Physics. metal networks, the quantum Bloch electrons in a magnetic field, superconductor or normal Hall effect, electronic properties of quasicrystals, high Tc superconductors and quantum chaos On the first to point, shared the the PHYSIQUE DE JOURNAL 472 N°2 I dhcussions the years since then. Not of our frequent and continuing over professional and private problems as well. subjects daily life, politics of matters, to underlied his Scientific ambition, a strong need for affection and a wounded pride secretly after years of heartbreak discussion life. Rammal died on a Friday in the middle of a scientific medical for his country falling apart, and years of acute physical pain, despite the hope that technology gave him for the better. have the been focus the mention Bellksard Jean Toulouse, July 1992 electrons BlocI~ 1. motion Electronic in in a magnetic a field. crystal subject to magnetic a has field been one of the Physics. A magnetic field breaks the time reversal symmetry and the microscopic of a crystal. The Hall effect, the de Ham van Alphen structure diamagnetism in metals, the Meissner effect, flux quantization of vortices in oscillations disordered in slightly and recently the magnetoresistance more quantum Hall effect, are examples of the complexity a magnetic field introduces Solid State main reveals topics in details of effect, electronic superconductors metals, and the in homogeneous materials. first study of such problems goes back to Landau in 1930 [58] for the motion of electrons approximation. It was soon followed by the work of Peierls metal, in the effective mass The in a Bloch [72] for systematic electrons. translation operators [94] until took it But study of the problem. The generating are fifties, with a coming the main remark the effective by Luttinger [64] paper works of these out Hamiltonian is describing to start a that magnetic the electronic independent electron approximation. The case of two dimensional systems in perpendicular uniform magnetic field became increasingly important after the development a of electronic devices such as MOSFETS heterojunctions [2]. It is nowadays possible to get or metallic thin films 200 1 thick. Many of the afore mentioned effects actually enhanced in motion in the are dimensions. two recently, discovery high Tc superconductors in copper-oxides [12] led several Cooper pairs, lies probably in the planes of dimensions. Moreover, the flux two phases approach reduces the problem to the case of independent charged particles in a lattice and a magnetic field. Even though this last approximation is questioned nowadays, it led many physicists to go back to the question of the 2D electronic lattice motion in a uniform magnetic More experts to the that the propose the atoms, copper of leading reducing the problem to main mechanism to field. If denote we unitary by operators U V the and fulfilling the two magnetic following UV #/#o is the hle (h is quantum lo Here, o = ratio = To describe corresponding the to participate energy crossing the Fermi band. Let By Bloch electronic the between Planck's absence motion of the constant a describing translations commutation rules such a they system, e~~'"VU (I) = magnetic and e the are : flux # through the electron unit cell and the flux charge). in a crystal, let us consider magnetic field. Only those first the energy case levels for near which the a = 0 Fermi conduction, so that we can restrict ourselves to a finite set of bands restrict For simplicity, let us ourselves of only one such to the case E(k) (k (ki, k2) being the quasimomentum) be the corresponding band function. Peierls then proposed theory, E(k) is periodic with respect to the reciprocal lattice. in surface. = N°2 SEMICLASSICAL replace to k (ki, k2) by = dimensionless the K» METHODS operator ) I"h£ = SOLID IN K STATE (Ki, K~), = A») 473 where eA») (P» COnSt = PHYSICS , » expression the in nian to get the are the lattice rigorous justification The corrections always to the admits Peierls a hamiltonian. Here spacings in each A (Ai,A~) = of this Peierls Hamiltonian. substitution representation in the He~ ~j been has whatever But is the vector direction. e~~i and e~~? satisfy the commutation relation (I), the series in the convergent power operators U and V. as a that written be can = effective E 1, 2) of potential, a~ (p Remembering [IS, 85] in done corrections, the the Peierls Hamiltc- leads and to Hamiltonian effective form (2) v~~~~~~~~~~~ hmi,1112(°)U~~ " ' (~l~i,~1~~)~zz hm~,m~(al's where simplest The in terms (2). gives It smooth are case functions in rise the to analogy (I) between and at the This a. usually series lattice square a u~~ u + " jQ, pi ignore we namely [49] absolutely. the higher order : v~~ + v + commutation canonical converges which for "Harper model" sc-called HHarper The of looking consists (3) (where relations h h/2x), = (4) ;h = , main and thus We crystal 10 T we is that field and have realized with lattice get plays a fixed a varied be tunable a of 11, is which is operators, of once and for an if realized effective role we replace Planck all, whereas a is by e~~ U constant V , (I). in proportional l'he the to parameter. which system in order of of10~~, order the parameter like concrete a spacing of the a h is can P momentum 2xo difference magnetic a position Q and the by h. Therefore the between by e~Q and one can quite small the test justifies and methods. semiclassical magnetic realistic strong a field of of approach. semiclassical a In order artificially a network with lattice spacing of the order of lpm, in a magnetic indeed built by the Grenoble get o te I. Such networks group were we can by superconductors. The de Gennes Alexander of [68, 69] [I] theory for such [34] and mean networks Hamiltonian permits to relate in a simple way the groundstate energy of the Harper in the metal-superconductor transition temperature-magnetic field (3) to the normal curve phase space. In this problem however the electron charge e must be replaced by the Cooper pair charge 2e, in lo. However if field of few build we Gauss, ALGEBRAIC 1.I which o = representation where u and is unitaries are number. always can we v rational a such u~ Substituting matrix in which the U and U~ (I) V V by mean in considering commuting. In an consists are of q x q matrices the e~~iu for case irreducible and e'~?v that = v~ = I uu expression (2) for the effective numerically diagonalized on be can Then represent to and approach Another APPROACH. p/q e~~'P'~vu (5) = Hamiltonian, a computer, we get giving a rise k-dependent q x to a family of subbands. JOURNAL DE PHYSIQUE I f 3, N' 2, FEBRUARY lW3 17 q q ~ ',, PHYSIQUE DE JOURNAL 474 N°2 I .-.u-~~~~"--Zl-."""_ ~ .,, ", ""'"-~ '§ ~~~~~i~~' ~%O~~i'~ ~h&. 'W' W ~df ,Ai6Kl '"Ml' ~Q13~7 ~~4 _/@ ~ / /~j£fi~_ ~ .j If. ) '";:::, ::::.. :,: ~@if~' ~Jh~~ ~qy W 4m w ZdS :,;. & Y~4~ N~ ova ;,<Gd£S f /" / Jz7 ~4l V/l~~ _/.""' fi" ,1 /WS I las, ~3V / W' S .t, )/ A1 ~m ,©Wl l iW' ' '4 &f ~ j" N _/ ", /~ / ',_ ,_;.~" ,,' ,.--- ., /. :.. Spectrum for k~ I). As 0 = x/q or 1.1,:'. ', -1.0 diagonalization 2x. 0.5 0.0 -0.5 1.0 2.0 1.5 ., 3.0 3.5 edges are 2.5 (Hofstadter's butterfly). model mod .. .: -1.5 Harper's Harper's model, this For tained of .,,i>11" ,.,. -2.0 -3.5 , ~~ ~,. 'll., / Fig-I- It is especially simple rise gave famous the to for subband the one can see from A unit I is an element of A such C*-algebra has not necessarily enlarge a C*-algebra by adding two C*-algebras A and B, is a that aI = = one. involution a that structure if p is a is actually *-isomorphism very Ia such map from natural from A to B Banach a = , that (6) Va e A. Therefore from unit exists it is unique. A p(ab) *-homomorphism = then (6) ))I) One A p p(a)p(b), p(a*) points of view. purelj~ algebraic, is First of all, the topology generated by this norm concept). This (a of purely algebraic a*a spectral radius the is [[a[)~ This This lla*all any unit, but if some unit by brut force. a linear flux the = llall~ ob- butterfly [55] (see the magnetic flux varies, Hofstadter's figure I, the eigenvalue spectrum as of high exhibits a fractal complexity. structure difficulty, in this numerical approach, is to show that whenever The first mathematical becomes irrational, one can approximate the spectrum by mean of the rational o be done using C*-algebra techniques (see [71]). can with a bilinear product, and an antilinear A C*-algebra A is a complex vector space, A has A A b*a*. for which it is a* satisfying (ab)* Moreover E E a norm a -(namely converges) and satisfies each Cauchy space sequence Fig. / ~ ./6Y' ~3" 'NAG. lk W=.,_.)""' =.=W = -- can B = I. A always between p(a)*. two p is norm because is preserving. (6) confirmed Thus the tells that us by the fact norm is an N°2 algebraic antilinear of A be which can Quantum in It is the be can in generated by two algebra" subset compact the (2) form number a the hm~,m~'s I e Given The A the For in of show number continuous unit. a E a define to In such "c" of functions certain a C*-algebra AI a such C H that sense AI pa -- o a*a 0 < I and on AI Ao with has a cI. < vanish be can seen which numerical inverse calculation all 7 Many other trum, -- represent use This of is can is set : where with continuous is I respect compact a to a namely subset its gap of R. edges ~ I) + ~l(n + U and At last, V on one 1) + 2 also can be semiclassical done setting e'~~~+w~~~ x(U) in Namely space. and of space, a representation a *-homomorphism 2D them -I). = E~I(n) lattice, by on then, that see nothing but the to is (7) mean of L~(R) by e~w~i the discrete and e~W~? " useful 7 the on will we 2xo = practical for x(v) = )#i(n) a represent and Ill analysis namely follows as on forward To go ANALYSIS. the 2x(z cos f~(Z~) namely 2xa, 1(2 is the position operator representations do exist and are will we 0. the SpA(a) = SEMICLASSICAL 1.2 a = all C*-algebra of bounded operators on 7i; algebra several representations have physical meaning. For instance, Harper £~(Z), namely the Hilbert space of quantum states the following one 7i, chain, and the operator x(U) is the translation by one, whereas x(V) is the translations. where evaluating the ~l(n also finite a Actually in A. " can in consists multiplication by e~~'(~~°'~) if N is the position operator. It is not difficult Schr6dinger equation x(HHarper)~l E~I, where HHarper is given by (3) Harper equation magnetic for a. the We but Uo~ IAn. as spectrum element of AI, element is I, its unit a no represented by operators in a Hilbert 7i,, called the representation space be into rotation zI self adjoint a Sp A (pa(H)) = Hilbert a that the H* = E(a) can 1955 [49] used discrete lD a on been are positive elements, namely those elements A. This positivity property is essential and groundstates for instance. Then, energy some *-homomorphism functions of B(7i,) -- can justifying C*-algebra data one in a e z spectrum continuous x 475 unique largest C*-algebra Au (up to isomorphism) a U and V satisfying (I). Au contains a unit. It is called defined and studied by Rieflel [78]. Moreover, given any that there is a C*-algebra AI generated by elements of is unitaries had natural a Let i is the PHYSICS o. result the A x is at element Theorem are STATE define to a*a for negative there the hm~,m~ AI has always numbers main Then where an complex permits it non and R, I of there for that "abstract" them. of form that [78] the "'rotation in the smallest shown permits involution written Mechanics A, ))a II E a SOLID IN invariant. Moreover, the if METHODS SEMICLASSICAL we purposes. knowledge of the Hofstadter specdevelop an asymptotic calculus as the representation use e~~~?+w~?~ (8) = , and If k expand = the (ki, k~) gets for H an effective is chosen expression (3) Hamiltonian a maximum of the form as H = E(kc) + or in a powers of minimum @. k I(fif~~ )~vI(~ltv = + kc of the band function E(k), one (9) O(7~~~) , JOURNAL 47G PIIYSIQUE DE I N°2 Energy -2 ~ / / / ~~'~~ " o ' -2.5 -2.75 -3 -3.25 3 5 -3.75 Fig.2. between Comparison exact spectrum model; points are extracted from formulae given by Equation (11). semiclassical Harper the M~v where is a the effective (n EN) effective the (resp. minimum matrix mass maximum). near semiclassical numerical the formulae and extremum for (resp. + O(7) term is nothing but The corresponding eigenvalues : En E(kc) = + j ~~j+j~ + Landau the full spectrum, exact The approximation. mass and the ~~~ i,1 @,I»I o-fit, o.04 o.02 o curves -) easy in chosen for Hamiltonian in is Landau's are levels represent to namely get, o(72) (lo) , 1 where The m2 ml, O(7~) nomial can = 7(2" +1) + expression an indeed, using the the and n E N labels the corresponding levels. Landau (9) by perturbation theory. In particular if H is polyexpansion of En in (10) at every order in 7. In particular levels are given by (see Fig. 2) of M, obtained from V, one can get an equation, the Landau 4 Et Such eigenvalues be in U and Harper's for the are term can p/q of )~ ll u and Mq(C) © + (2~ + 1)~l fit"~ + (" + l)~l (7~)j (11) + generalized near each rational value p/q e Q of a. For eq.(5), and the generators Up pp(U) and Vp pp(V), v V' Ap generated by U' and is isomorphic © © Up Vp u v actually matrices sub-C*-algebra + 2 be in = = = = the effective Hamiltonian pa (H) can be Ap. In much the expansion get seen as a q x q same way we an near 6 2xfl te 0, if we replace Up and Vp by e~(~i+W~i) and e~(~?+W~?) and developing in power8 E(k) is replaced by a matrix valued "band function" difference is that for 6 0 the of li. The Ei(k), subbands Eq(k), which of q functions 7i(k). Diagonalizing 7i(k) gives a sequence Correspondingly we get k-dependent gives the spectrum of pp/q(H), whenever we vary k. I, eigenstates )I)k I q. Ao if to o = + fl. Therefore kp matrix with in this elements representation, in = = , = , , SEMICLASSICAL N°2 Such a T~ C~; k x "classical" describe To maximum the the or is the bundle near value of E;. The (I)k_k~, Q; eigenvalue E of kp (<lfip4 now in fl;(k) bundle. ((k, )I)~) the Here e subband. splits into The of the band rise to subbands. the for two Q;fiplo ~ in Mq(C). equation : jip(E) expansion of an (12) , matrix unit eigenvalue of = li gives of (I)kk(I(. (<lfipo; E; (kc) is then given by the near mass term + (I) (ii and Iq is the Iq = powers effective O(6) the However fiber a the i~~ ~~ E M is the in 477 formula = Expanding (12) motion to PHYSICS STATE edge, we pick up a band (namely we choose I e [I, q]), E;(kc) is the band edge we consider. Then kc is a order to reduce the problem to the case a -- 0, we replace Hamiltonian by mean of a standard by an effective band In = ii) SOLID which kp tip(z) where IN band a Hamiltonian to associated for of k minimum a complement Schur a valued matrix line spectrum kc be let and corresponds mechanics T~) e METHODS So we the Landau get form same sublevels at as eq.(9) each where band edge. pieces first term is proportional to (b( and gives rise to a discontinuity p/q (I.e. 6 edges (namely for n 0). Notice that 0) at o @@ is the density of states at the band edge. introduced of the i~~ bundle The other for the term is proportional to 6 and accounts curvature ~(k). It produces an asymmetry of the derivative of the band edges Hamiltonian by the matrix p/q (see Fig. 3). near o Rammal Formula (13) was written for the first time by Wilkinson [92] for the Harper model. fiE;,n(6) /fi6 the magnetization derivative Harper's model and related the rederived it for to [76] theory. The formula (13) was derived in full network, using Abrikosov's of a superconducting generality (namely for any model given by Eq.(7)) in [13] (see also [51, 77]) where it was called where = derivative of the = = = = the Wilkinson-Rammal 1.3 BANDTOUCHING. arated from of the conical jip(z) contact The bands touch at k them. between E;(kc) whenever is valid Only at a band edge sepsimple eigenvalue of ~(kc). On the formula Wilkinson-Rammal namely bands, other if two contrary, formula. is a kc, the asymptotic is = Whenever bands two different touch (see Fig. 4). Degenerate perturbation theory corresponding in (12) must be a 2 x 2 matrix 1<1(<1+ 1<'1("1 = each other, indicates the to depends and that the contact is eigenprojection order generically Hamiltonian effective the the upon : fl , if Ei(kc) One choice can of = E;>(kc), show that I' I and lowest the coordinates) by jip(z) being a = Dirac the order labels term of the in Vi and kc the is conical given (after a point. suitable namely operator, E; (kc + the touching bands, jip(z) expansion l~, '1 ~ ~~~ ~~ 2 ~'~~~ + O(d) (14) JOURNAL 478 PHYSIQUE I DE N°2 jliflfl W /$~flli~ A iw'l'wl J4' @ ~ ~,./ / ~ N ~~S~K ~~_~~, ..ll.I@#I,'I.," IS-I' ~~' 1J©K Jk~P'lK ,'. '.,~ .),b" ,"' "" / .~ ' I I lYl lift /~4~ IV I 4 r jP' @1 'W3©V' £ ~~K£ ...,:.i;.. /@~ # /~@$/l.,. IS 13Vl TALT W 1& £lP'k ~3L _,,~l'$~h~), ."~ ;÷;;;..;1::," .c' '. ;:I°,;;; b,Q, ....;,I:'... ., -1.2 -1.0 Fig-S- Asymmetry Thus Landau -0.8 -0.6 of the central sublevels are -0.2 replaced by Ez,~z where n o e Z is the Harper's For p/q, = what we can Ef~ generic touching Playing with such shown a 1/2 = had of a local permits the same in the in way, Dirac a subband principle to the near compute levels reveal produce a special packing understand [52]. It permits to may not near a 1/3. = O(6) + where vanish [88]. this the the Dirac the *) (l is gap no producing (15) levels 0. In particular if bandtouching. This is 2x(a 1/2)) given by (6 E at = conical a are = ~ O((6(~'~) + 2 n (16) > 0 in [9]. analysis permits system. subband. also spectrum studied semiclassical a extremum of curvature been there in [17] that gap is closed middle +2(2n(6()~'~ = underlying bands classical model namely 24@sgn(n) + ' Non Harper , was the q is even, in figure 4 at see the levels Dirac E; (kc) = for 1.0 label. Dirac model, it where edges band 0.6 0.4 A bunch of The slope of higher existence of permits of a conical of sublevels. This has in particular why the been of the Hausdorfl reveal to this the in At last power of In saddle and the of the existence function. Helfler dimension the measure level subband bandtouching. studied by topology the will levels derivatives order reveal to sublevels expansion these The extremum. actually Landau local of 6 much points Sj6strand Hofstadter METHODS SEMICLASSICAL N°2 IN SOLID STATE PHYSICS 479 L o r ki g ~ k2 r FigA. Conical between contact bands half in the previous analysis be produced by the at flux Harper model. for exponentially small not account tunneling effect. Actually considering E(k) to be the classical Hamiltonian where ki (resp. kz) represents classical momentum (resp. position) we may get tunneling between wells in the k (ki, k2) phase space. Because E(k) is periodic in k, any well is actually infinitely degenerate by mean of the translation in reciprocal space. It will produce a broadening of the Landau levels and sublevels which gives exponentially small correction as a or a p/q -- 0. This broadening has been described by an Wilkinson [92] and Helfler-Sj6strand [52]. TUNNELING 1.4 in terms a The EFFECT. (or p/q) a which may does = Before going into it, let us restrict ourselves degenerate wells in the cell of periods of the phenomenon for the following model appears the simpler case for which we have classically Such a reciprocal lattice, close to each other. studied by Wilkinson [92] and Barelli-Kreft [10] to Hw=U+U~~+V+V~~+t2(U~+U~~+V~+V~~), with12 > mod x 2x 1/4. Indeed, if12 which bifurcates symmetric by a fourfold classical degeneracy. Thus < at 12 rotation the (17) 1/4, Hw admits a unique classical minimum 1/4 into a local maximum and gives rise " ki around k2 " corresponding " Landau x. This sublevels to ki " four produces symmetry are at k2 " minima an exactly degenerate exact modulo O(tY"). To 4 x 4 describe effective the tunneling Hamiltonian effect for each between value these of the four Landau we replace quantum number wells, the n. Hamiltonian Using the by rotation a JOURNAL 480 matrix this symmetry, admits PHYSIQUE DE following form the ~"~7~~ t En (7) h the n~~ Landau techniques [50, 93], one where Sopp) neighbouring between where il ? r computed sublevel r ~ t WKB N°2 : 0 ~~~~~~~ I 0 before, as (resp. r) ~~~~ ' r f E = wells t in and t e R tunneling (resp, diagonally opposite wells) in the write can action8 of terms form C. Sn_n_ Using (re8p. : w corrections. aslov's It that out turns the Schr6dinger contrary to terms the in > (Imsopp( the case, E~ of level splitting +~de-lIm(sn_n )jH ~~~~-(lm(Sn n,)(/~ = ~ r that part of pression so (Imsnn.( real becomes 7 when ~~~ ~i~ to compare negligible varies. (~~(s~ /~ / ~ (fi~( s ~ i) ~ #) (~°) corresponding Landau level splits into four levels like a braid as 7 varies phenomenon has been also observed with Dirac sublevels [9]. In same sented the braiding of Dirac sublevels produced by the touching of two bands points with an exact fourfold symmetry around the fifth one. One can see on formulae fit quite well with the diagonalization WKB method. The The broadening The reciprocal symmetry replace the Hilbert each seen the the then Hamiltonian Since each acting on f~(Z~) by corresponding flux, one as scale invariance emphasized the role of the in [57]) precisely generates If H trix has fourfold the of Htunnei element Hamiltonian S' > this a effective of Ao> e proposed expansion of by using way same of the fourfold the rotation Htunnei acting in Hamiltonian given quantum translations. was fraction the instead (see Fig. 5). figure 6 is reprealong five conical figure 7 that the number associated n lattice, Ht~nn~j can be a However, when one computes 2D with for the a. The a' first lla = time Gauss mod by a' map instead I Azbel -- [6] who I lo (see expansion. action should original lattice, of the wells located between the two be accounted so sites at wells I and does and I'. This leads + Va> + VJ~) for. Htunn~j. Moreover the maI' is proportional to e~~"'/~ So that as 7 -- 0 only the to the following approximate : Htunnei This continuous much point in Htunn~j now spectrum symmetry tunneling neighbouring wells nearest of the between Sill is the where that see can magnetic of mean by labelled is in space from, by an eigenstates Landau well described k-phase the in started we the such be can symmetry generated by space well. [92, 52]. This with sublevels group used above. We to Landau of translation t. does n_n " 7~e~'~~'~'H (Uo> + UJ~ + O (e~~'/~) (21) , ImS. formula Hamiltonian, shows that properly the broadening renormalized; thus of each the Landau actually given by spectrum is again the sublevel corresponding is a Harper Hofstadter SEMICLASSICAL N°2 spectrum METHODS Square of SOLID IN Lattice PHYSICS , j ~,~ j , =.w.. -<' I 481 2.Neighbours(ml/2) with oi I STATE ,, , ~ ,, _,/ ~ _--.~'~ ,,' _.,,' , ,," , , -----' i , ., ,.'~. _>_:-' :.. .~ / ', / / ' 0 '> / $' l ] [ I o -2.S -3 -2.6 -2.4 Energy Fig-b- Braid Repeating this one. Landau sublevels in a Harper-like model with second -2 nearest neighbour fractal structure (from [io]). interaction of the for structure -2.2 construction again again gives and rise in principle to the spectrum. sublevels. previous analysis works only for well separated Landau Near the band have ideas and they shown that and Sj6strand have developed the kind of same renormalization analysis can be locally described by mean for Harper's model, each step of the standard forms, one of them being the Harper model, the others taking the possible of four However the Helfler centers, bandtouchings The or into The main Theorem result We (The remark Ten of Helfler > with an, performed by and 0, an Kerdelhud Sj6strand such that > C Vn, is if o then [56] for the contained admits the in the Harper following continuous a model on theorem fraction spectrum E(a) of Harper's a triangular [52] expansion of the equation has a , measure. however Martini irrational been is C There 2 form [al,a2, Lebesgve zero any account. analysis has same honeycomb lattice. a. that if C > I, the set of such a's of Kac [17]) is that E(a) problem has zero should Lebesgue have a zero measure. Lebesgue A conjecture measure for JOURNAL 482 Mf....~,"..' "~~=l)ill% , '_ ._ , "' , FJ0©'- ,~1~ I 'I I' ~o I I ~~ tin it,, ' id1 #.f "W& W f~'PO' I a f , / , Fig.6. + f w h ', ,, "'ii2/WJ/~lL -3 structure for -2 Dirac W I ' -1 sublevels a '~~~' ,' "_wwi h'_'W" lr IN i ii w~w< "I ?I fi7bs- W'.,' ' S 4 lilt AC'j twi- ii -I ?4@$" ww # I a ii ~' I m' _' / j ,O@ I ' i' I § I i ', i , / ,~'~ 2 f) in ~j; 4W' m y'~ J', W ~ I O~©' '~ J il jr A /J'f § J jj/ _.m1 / y' A '_ ~'k A3%W§ #' W I d /Wl j@I ' iw J I "W/i /lW ~ '~ , Braid 'W '#' HS@w" -4 NW K Ii j, lx. p I,a Ji v I 'F_~' M ~Sn WV- <,j# / ~X' w' 4')1' WWOOW'j' ,_/ W/ fi@)Kj Ill -5 Harper,like model ~ 4 with third neighbour nearest (from [9]). interaction 1.5 $& jR+@ ' '1 j< v £ '~@'~'j',"'~M'~ # ~ki M 16-1 j'j .AW.Yi j',=/ / ...I, / '.jm m _/ ,." / ,f~ my _/ '' I i~ w I ' q w%r j / / i 'p$/ '@ ( WV W ' C /J&2wHW3w¾w1 ~*d j%f I j 41'/ ~ m~t /4' Y £ ~ / -" ~ V ~ ( Hm~+'; ? wi ~/ ml / ~ «Ii hwh II J irik~ S~_~q,I' '~_J*O*£MM m_ .14W" till / '~_" ~ q '~fiQ _/ f-~ ""@_ * r/ / ( ~ id l _" _/ ~r~/ iV¾~' j,w WI _' ~' w &5JT +A ~W. ~ Rbb W M~' w+~ ~ r/ (&$£ _ Rl"1,' I^"i @~d ' i Rl~ojf i [l~/£ W@ it jfli_/ ' 1~ &+@ -m "ilk~~'~I.#ii.fli~ )" 1 1/W"', ','~W' '_ ill i' WI qy, O'f , I j at' " ~,~' j ~" '_ -.-ml ., ,w. / ~~y' .v ~ ~ i'W' li&' t'; &~ +£~' it ~z i j A'ii '~ A _' N°2 I ' ,/ l'~ci WW v/ '%k ii ii 0 hi N&, K~I~ i A~I ~< i m+ S , ' ' ' ~5w sw S, I PHYSIQUE DE FLUX probably PHASES been one reason is that it forces much are temperature. The discovery of important events THEORY. of the most provides probably a mechanism new superconductors in 1986 has Physics in years. The main pairing of electrons. The corresponding BCS theory, producing higher critical copper in Solid for oxide State stronger than the phonon coupling in mechanism is still not understood. Yet, this by theoreticians has permitted to the concentrate by of the that right accepted studies experts most on a now the key phenomena are to be found in a 2D model, representing the Fermi plectrons in the Copper orbitals, the Oxygen being there only to create an effective doping by holes in these Hubbard model in two dimensions, with a strong orbitals. One possible model is the sc-called electron-electron interaction, which can be approximated by the sc-called t J model as an the Nevertheless effective effort enormous small number of since models. then It is Hamiltonian. A great deal of work has been done in finding a good One of the approaches proposed was the sc-called model. idea in such an approach is that in the t J model, spin approximate ground state phase approximation. for flux variables can be considered such The as a basic slow rapid. Therefore a kind of Born-Oppenheimer approximation leads effective magnetic field acting on charges to a model in which the spins are frozen, creating an (here holes). In this adiabatic approximation, the charged particles become independent, and However the magnetic field is no longer similar to the Harper model. becomes Hamiltonian the whereas charge variables are N°2 SEMICLASSICAL METHODS SOLID IN STATE PHYSICS 483 / ~j 110÷x~ / ~ / ~ x x / / ' I h ow~÷~ ~ Q I x I , j ' i f~' Ax' "''°~ / <' _-~$ , i ', i "°°~"TI7~©f~i~~~T)~7k~jtS~llT ~ £ '% ~ ~j~w°~' ,, ~ , k ~ 'i~gncu,F'u, ~ ~/ I "~~j ,, , ~~_~ ~~ ~ t, , i I I I ~ i 'j ~ ~'~~~~ ~ t ~ x ', ' ' , ~ ~ ~ lX¾ll~ &l' , K Energy 0.6 o.4 0 ''~i 2 Flux ,'I 1-. l~~ -0.2 '" I -0.6 Fig.?. figure (from to crosses lower emiclassical figure expansions are : (from points from are exact extracted spectrum from whereas numerical exact full spectrum curves and are curves coming from correspond JOURNAL 484 PHYSIQUE DE I N°2 necessarily uniform, and it is still a dynamical variable in that it must be chosen to minimize concentration. Fermi energy, given the holes sea studies dealing with the ansatz of a this problem is already difficult, Because were many Hofstadter's uniform minimizing magnetic field. So that we are back to spectrum. that the miniOne of the main conjectures based upon this series of approximation, was #/#o should be equal to the concentration b of charges (b mizing dimensionless flux o calculations Hofsnumber Of charges per site). This conjecture was based upon numerical On in the semiclassical limit tadter's further proved to be correct spectrum (see in [59]) and was [77], namely b -+ 0 for the Hofstadter like models. This is what gave birth to anyons, a kind They can be reinterpreted in terms of of quasipartide with a magnetic flux attached to it. parastatistics, thanks to the fact that the indiscernability of identical particles in 2D leads to representation of the Braid Group [30]. a because of its fundamental character, it has Even though such an idea was very appealing been verified in experiments [39] and one future by now [3] not can say that it has hardly any of high Tc superconductors. at least in the context One of the main problems with such an approach is that the system should be driven spontaneously toward a state breaking the time reversal symmetry. mechanism Even though such a doubt that it can be broken in the t is a priori possible, there is some J model with spin 1/2 the = = [37]. problem is boson approach Another slave a [79]. However, that minimizing magnetic flux the uniform magnetic the shows that filled band is certainly field ansatz uniform. not gives rise to indeed For an unstable state Lieb be and One half and the case sea. This done was producing only the "anyon rule" a a that bipartite lattice, preliminary rigorous results by magnetic field ansatz with flux 1/2 mod I should of holes small should produce a slight amount a distribution. flux possible way to approach the periodic flux to use again a of for uniform suspect may we magnetic in the that indicate Nevertheless correct. disorder the at Loss [62] uniform non the in [8] to find out that the local correction to the ground global & = b was still valid as magnetic flux problem was to consider the approach for the stability of the Fermi disorder introduced by the periodicity, was by of a form factor, and state energy mean semiclassical b -+ 0. recognized that probably because of the the small field limit & -+ 0 and the long as period one do not commute, so-called "magnetic breakdown" caused by tunneling effect in phase space near separatrix [87]. Therefore, the understanding of the minimizing magnetic flux is still a pending question. This long discussion between at the heart of the exchanges Rammal and one of us (JB) was during the years 1990-1991. The very fact that the ground state of the Hubbard the t J model hard to compute or was so However led them 2. Quantum While one shown in [3] it has think that to of the usual been approaches were not suitable for a result. chaos. us (JB) kept talking for months of quantum chaos with Rammal on the basis phenomena should be important and universal, one day in April 1990, Rammal called him on the phone to claim that in the Hubbard model on a finite lattice, the eigenvalues avoided crossings as the coupling tend to exhibit varied [38]. That constant many was was an indication that chaotic behaviour, whatever that means, should in these models. some occur Still, a clear test of such an idea had to be done, and even with it the question of the mechanism unclear for the next producing this effect remained Apparently Rammal took this idea year. that such SEMICLASSICAL N°2 very seriously because May 1991 [80]. he METHODS produced several IN SOLID with works STATE his PHYSICS students on 485 chaos quantum before of "Quantum Chaos" several topics mainly developed since the name by a community of physicists who would probably disagree on a unified definition of the subject. however would say that it Most of them the quantum signature of classical chaos on concerns the dynamical properties of a system. As its stands, such a definition reflects a pragmatic point of view taking into the fact that most of the studies have concerned account systems with two degrees of freedom so far. It is very significant that the title of the Les Houches 1989 summer school proceeding was "Chaos and Quantum Physics". As explained by the organizers in the foreword, "the expression "Quantum Chaos" at first makes us think of a quasi-exponential proliferation of publications from the last decade hiding more or less legitimately behind a find We end scientific the under of the seventies ill-defined label". subject developed around three kinds of technical approaches (RMT), Semiclassical Theory (ST) and Dynamical Localization (DL). The Matrix Random Theory 2.I THE THEORY. First of all, the oldest RANDOM MATRIX approach is Random MatriX Theory (RMT). It was proposed in the fifties by Wigner [9 II as a convenient way of representing the Schr6dinger operator for a nucleus. The idea is that such a Hamiltonian is very complicated and can be approximated in many dimensional matrix HN, respects by a finite (but large) where N is the dimension, the elements of which are random variables. The simplest example consists in taking these variables to be gaussian, identically distributed and independent. This ensemble of matrices is called a GOE (gaussian orthogonal ensemble) or GUE (gaussian unitary ensemble) depending upon whether HN is real symmetric or complex hermitian. The main features (a) (b) the the One can Wigner these of ensembles are : density of states of such matrices is given by a semicircle law [91], level spacing distribution exhibits a repulsion of neighbouring levels. approximate the level spacing distribution by the one for 2 x 2 matrices surmise fl m Z~~sPe~'b~ the other hand follows if the random P(S) This is in Physics This eJlicient checked particular describe to has RMT tool in of the the disordered 1984, Bohigas systems by relevance The Porter the level spacing dis- [74] and [26] proposed (23) e~' = random giving developed by Dyson been calculation. sixties Anderson's in case = diagonal (or quasidiagonal) is matrix 2(GUE) namely [2 II law Poisson a I(GOE) = (22) fl tribution the : p(s) On called rise to and potential model Anderson Mehta [40] to theory for a eventually by Bohigas of such [70] used in Solid State localization. the point of becoming Physics started Nuclear et al. very a to be [48] in 1982. RMT as a statistical tool to investigate degrees of freedom. The first example was the Sinii billiard. Following a general idea of Berry [21], they established a phenomenological rule, namely that a "generic classically integrable system" should have a level spacing distribution p(s) given by a Poisson law implying some kind of level clustering, whereas a "generic" classically chaotic system should obey one of the Wigner distributions GOE or GUE depending the Hamiltonian. In such a statement, the word symmetry properties of the quantum upon In spectra of quantum et al. systems with small to use number the of JOURNAL 486 "generic" to say has more not precise meaning yet, a symmetries (such as negative of constant the harmonic [23] rigorous results of Sinii [83] permit that special systems having extra surface arithmetic geodesic motion on an or follow rule. the GUTZWILLER'S APPROACH N°2 recent fail to may I indicates it But oscillator [24, 81]) curvature SEMICLASSICAL THE 2.2 though even integrable systems. for least at PHYSIQUE DE The FORMULA. technical next tool approximation and first established used in upon a WKB a by Gutzwiller [46] in the context of impurity levels in Solid State Physics, and later by Balian and Bloch [7] in the case of eigenmodes of an Optical cavity. (q,p) E R~ x R~, classical phase space) be a classical Hamiltonian Let 7i(x) (x corresponding to a quantum one k. We consider the case of N degrees of freedom, and denote by h/2~ the Planck that k has a discrete h and h We spectrum and denote constant. assume their multiplicities. eigenvalues counted with by ED < El < En the corresponding < < "Quantum Chaos" formula is based = = If ~~~~ (it is a smooth d(E, e) by : approximation of the Dirac d(E, e) (E~ e2) measure) we ~j b<(E = ' the define smoothed density of states (24) En n Using a mately Feynman path integral, and written phase stationary a [47] d(E,e) method be can approxi- as d(E, e) m I(E) + jqq ~j A;(E)e~~~?(~)'~e j ~~ (25) ~ J In expression this d(E) is "Weyl the d(E) asymptotics" [90] or ~ / d~~xb(E = classical term, namely : 7i(x)) (26) Moreover, in (25) we sum up over the family of periodic orbits indexed by j, corresponding to E. Then, 2j(E) is the period of this orbit, S;(E) is its action £$_~ f. p~dq~, and energy the 7j a focal terms phase shift (the Maslov index of j) which points in the phase space. At last A;(E) of the Floquet matrix of orbit. formula large takes into is counting a account the factor (sometimes infinite) existence which oscillating can ~f caustics or be written in (as -+ 0) density of states from away its Weyl classical limit. different properties depending upon whether the This has very sum classical dynamics defined by 7i(x) on the energy shell 7i(x) E is quasi-integrable or chaotic. This is due to the exponential proliferation of classical periodic orbits in the latter case. This must be shows taken into that account a to number describe fluctuations the of of terms h the = 2.3 DYNAMICAL regime by the exhibit standard a Lo map Several CALIzATION. diffusive behaviour [32] given by in classical phase space. Hamiltonian The prototype systems in a strongly chaotic of such systems is provided : Pt+i = pt + Ii sin qt qt+i " qt + pt+i mod (27) 2~ , N°2 SEMICLASSICAL where a be can 1979 via a t~o ~ Even 89] on a diffusion a the For constant. is map a which be can ~~~~ ~~~~ mathematical map", only rough "sawtooth standard map has progress which made [18, 31, be ergodic, been happens estimates to finite at time have [36]. obtained been 487 ~~~ ~~~~ though this result is not rigorous, recent slightly different model, the sc-called has and PHYSICS for restricted described STATE SOLID time, and describing a rotator kicked periodically in time. Such describing the behaviour of a Hamiltonian system near a resonance to two degrees of freedom [63]. Chirikov [32] showed that for K > 4, the typical behaviour report, diffusive diffusion with (see Fig. constant 8) process : model his In IN discrete t is the generic METHODS This system evolution is quantized, and is unitary Floquet a be can given by known the as operator e~'lie~~+ F kicked " problem" [28]. The rotor quantum given by F ~"Q (~g) # , Q and Actually, position where P ih. here Q> P and h such as the period parameters, rotor, is led one the are usual are operators momentum two quantities. kicks, the effective Planck between T expression the to and dimensionless of the heft If on circle, satisfying [Q, PI introduce physical to a = wants one inertia of moment classical I of the constant ~) " (30) showing again that it can be tuned by varying the kicks frequency for instance. argued by Fishman, Grempel and Prange [42] that F should have the basis of an similar to the one leading to Anderson argument on nh (n E Z) and similar momentum space is quantized according to pn It has been spectrum the Then like in classical Anderson one, so that if we the As h -+ the compare that shows localization the diffusion of the quantum T The discrepancy interference between effects which ((pt PO )~) one for time constant system (like the insulator can theory diffusion transition be inital in state the classical and T. increase to related D is the to breaking time T and to system m ( G3 const (31) j~ h and behaviour is due to existence of quantum quantum quantum state in a finite region of phase space. classical diffusion local12ation has been and extended quantum to viewed works an chain to the It(n) a < t continues relationship between systems with a higher number of degrees of freedom. standard map in which It is actually a quasiperiodic localization over lD to converges classical trap The Such evolution quantum Beyond this point the quantum (see Fig. 8). Shepelyansky [33] supplemented by numerical calcu- agree classical length f the time average functions two fixed 0 and while the classical stay bounded argument by Chirikov, Izrailev and lations the model. mechanics, quantum average An localization. = the point pure a if = as a Itof(win, v-dimensional = to absolutely simple example of is then given by a time, namely ,wv_in) w2n; 2, all states v are increases exponentially constant) is expected leading A function lattice in localized fast with continuous momentum but K. now If v space, the > 3 spectrum for which localization an for Anderson large 1( the length metal- [29, 82]. JOURNAL 488 PHYSIQUE DE I N°2 E 20 20 Fig.8. Time straight the values WHAT 2.4 IS times been has a property the motion energy quantum other parameter is there long suitable always behaviour time time do to converge there However, as free a stationary behaviour states. of the complexity us Arnold first theorem (Ii, IN, Pi, , that means Therefore spectrum the these of of initial on in them [5] the asserts ,@N) E -+ all if the at consider as 0. case that R~ x curves = 4; different under time T~, not a -+ 0 and t -+ evolution that however has that classical systems. While at a finite the that an or plays the effective The constant. Planck semiclassical quantities quantum h constant ratio flux a a [86]. Planck's problem oo, -+ to see the number even not ? and classically integrable is a canonical change such Quantum by Berry " remarked conditions I do mechanics surprised be of notion as fixed at 0. definitely quantum system there h limits Notice of h as limits two the all, namely parameter, in would we h quantum K for deterministic, in the long run, This paradoxical situation appears. chaos extra counterpart between quantities appear They are unchanged Hamiltonian. of the Let "clash" a rotor [20]1 It do long time Where no is classical their regime chaotic system is when an First proportional to h. We have seen in section role of h and is no longer a universal In the kicked constant. rotor which be physically tuned desired. constant occurs can as notation let us denote by h this elsective To simplify Planck limit corresponds to h -+ 0. It is known that in most systems the some ~ loo the explicit more dependence" "sensitive a in map represent make us Hamiltonian more Of terms Let the classical becomes standard points and previously presented. characterizing a the 80 (from [16]). constant of mechanics In for energy classical technicalities the description explained in stochastic kinetic the to 60 QUANTUM CHAOS ? THEN behind [20], chaos is the of Planck effective lying Chaos" short evolution corresponds line of the 40 Hamiltonian systems effects simple as commute. Precisely when we they give rise to the of classical deal with spectrum in the chaos states, for h > 0, have degrees of freedom [44]. quantum of Hamiltonian of for 7i. The variables, the action be expressed as can Liouvilleand a angle function SEMICLASSICAL N°2 of the actions replacing I ,IN) only. IL>- = by njh each Ii METHODS where is nj The Efff smooth some (as cases into this have seen the account (ni, nN) and n, = 7i is now Still, expression as if two in levels not the is cross we consists then in : nNh) first can say if 7i 7i(1, @) procedure (32) , of term section exactly integrable, 489 get the eigenvalue spectrum I) exponentially of the tunneling effect due to h -+ 0, each eigenvalue is well occurence regions. classical If enough, we PHYSICS STATE quantization [41] to 7i(nib, = ~~ If 7i is EBK integer, an SOLID IN 7io(1) = terms exact classical them written ef(1, in powers be @) added degeneracies by the quantum independently. be + must labelled follow can expansion an small of h. to In take between numbers as (33) , (I, @), for e small, the KAM theorem (see [43]) asserts tori, I const, survive but a theorem by Poincard [73] that only a discrete set of periodic orbits of 7io survives the perturbation. asserts Perturbation theory in quantum mechanics can be used uniformly with respect to Planck's unstable periodic orbits of 7io corresponds in quantum meof constant [19]. The occurence chanics to level crossings producing (as h varies), after perturbation, kind of anticrossings some (like the Jahn-Teller effect of molecular physics). On the other hand, classical chaos precisely develops near unstable periodic orbits. So we are led to admit that this local "hyperbolicity" is associated with avoided crossings of levels [45]. the level splitphenomena occur which However in the fully chaotic regime, some new cause eigenstates is ting to increase. For indeed the tunneling effect between unperturbed resonant classical illustrated in [27]. In enhanced by the chaotic This has been beautifully transport. this latter orbits in the complex phase space, the usual WKB theory, requiring classical case does not provide the level splitting. Everything looks like if the transport of the wave correct barrier function through the dynamical efficient by using the classically chaotic mcmore was results in level splitting are of the tion than by using the usual tunneling elsect. These same order of magnitude as the spacing between levels. mean elsect on Rydberg Hydrogen A good example of such a phenomenon is given by the Zeeman number splits quantum atoms (see Fig. 9). If the magnetic field is small enough, each principal sublevels roughly linear in B. However, bne can see that rapidly these into a bunch of Zeeman they tend to avoid sublevels will start creating a lot of crossings. Due to higher order terms f is where that most each other. becomes function unperturbed number If the rapidly as larger very enough smooth a of the B of such increases. the if instead = crossings avoided Moreover classically in of invariant can one becomes that see the regime (namely chaotic B oscillate too high, each level will splitting at the avoided crossings large) than in the regular region [35]. remark We then h -+ that select 0 will we h consider quantum 0, and -+ numbers "n" -+ a given oo, and window we will arguments give an insight into what should be called assign quantum numbers the quasi-integrable regime, one can they exhibit sometimes avoided crossings. For indeed, in this small (typically of order e~"~ by WKB approximation) very These separation In the of chaotic fuzzy due to level spacing. in the chaotic order For case classical the energies [E-, E+], same situation. Quantum Chaos. First of all, in unambiguously to levels, even if latter case, the level splitting is compared to the mean spacing h. regime however, level of be in the quantum oscillations, and this (and this regular than reason in the the numbers level splitting is somewhat one. meaningless. The spectrum is of magnitude of the mean paradoxical) the spectrum is more rigid become is of the order JOURNAL 490 EE PHYSIQUE1 DE No2 io-~ -2 /f2 E Fig.9. magnetic the is in atom 2.5 LEVEL tics of level which of define (I) three a fl < illustrate to scales the of a variation that = = = point, let depending discrete Eo(h) set approximately compute us H upon Ei(h) < < statis~ the h, parameter some Ei+i(h) of < of h. for h h'- h if = O(b~) perturbation theory for individual levels convergent; is h'- (it) a short scale b,, such that in the eigenvalue distribution does not change, large number of times; a (iii) a macroscopic scale bM along which We the strong a (~ Hamiltonian a variation scale b~, such B/Bc is magnetic field B. ~ 2.3~ T). B corresponds The i to perturbative regime whereas here fl > 60, in units is i this consider us being given by sensitive microscopic Ei(h) Let atoms atomic regime (from [35]). chaotic To REPULSION. eigenvalues only We will a in ~~/(-2E)~. = strongly repulsion. spectrum expressed field is fl here parameter natural namely Rydberg Hydrogen for sublevels Zeeman dimensionless the the y ~~-9 2_ h range but each O(b,) = shape macroscopic the shape macroscopic the pair (Et Ei+i) of levels individual spectrum of the is of the oscillates modified. that assume 6p < b, < 6M Now let us consider [ho> fill of size from ho to hi, to the the level O(b~) along the main separation which variation subspace generated by H(hi) Ei(h) between Ei(h), Ei+i(h) of H(h) can be states H(ho) I and " + I. bH(h) m one fl~ by a 2 varies corresponding matrix 2 x interval an As h write can (ei Consider crossing. avoided one described Thus Ei+i(h). and exhibit + J fi) (34) , PWrS(I,I+1) where the the sum representation all concerns of 2 x 2 pairs of eigenvalues (I,I given by Pauli's matrices + I) one which J = anticross, (ai, a2> a3). and The we have constant used et METHODS SEMICLASSICAL N°2 the represents appreciable variation mean only value Ei+1)/2 + which we by hypothesis. So scale bM PHYSICS STATE neglect can et 0. * 491 here Here because is vi that if symmetric fi is real interval k=o L has et vi it an in R~. vector some component of E R~ (namely the second [h-, consider Let us h+] with size O(b,) and let us decompose now an O (b, /b~) subintervals Uf]o~[hk, hk+i] of size O(b~). variation of H(h) along this interval Hence the total then be written can as : Remark H(h) the on (Et of SOLID IN vanishes). into N = N-I H(h+) By hypothesis change, not variables, such on that so Therefore the interval an one whereas can fl~ = « (35) v)~~ large, but the macroscopic shape of v)~~'s (as k varies) as identically N is the the consider rapid the H(h-) make oscillations spectrum distributed does random independent. them sum N-1 £, ~(k) ~~ ~ i k=0 becomes The Gaussian a random spacing Ei+i level 2 x 2 with matrix zero mean N gets as large. Et is then given by s=Ei+i-Ei=2]Vi) namely distribution its p(s) p(s) have We here CHAOS 2.6 HA = of STRONGLY describing #-yj= Wigner V E R~ if V E R~ 2D a strongly real symmetric case complex hermitian explaining why argument an surmise if CORRELATED £ t = the Z~~se~'b~~ Z~~s~e~'~'~~ = beginning a IN model J t given by is fermion n~,_a) c$,,cy,a (I (I lattice ny,-a law Let SYSTEMS. FERMIONS correlated Wigner the case should be consider now us universal. the gas £ + J S~ (36) Sy , j~-yj=1,~,yEA i,~~yEA with periodic boundary conditions, c~,a is the fermion creator number of electrons of of electron of spin a, n~,, = cl,c~,, is the operator at the site z E A an spin a at z, and S~ is the spin operator for electrons at site I. HA acts on the Hilbert subspace "Gutzwiller number n~ = n~,t + n~,i < I at each site (the sc-called of states with occupation A is finite a lattice in Z~, taken projection" ). The the particular to its of number total No is number of that is N which value particles £~(n~,t holes. avoid To Since even. maximizes the 25(~) 0 or I depending upon thermodynamic limit namely (no magnetization here). model This speak t1 of describes prton any chaos in it has such -+ electron an whether N a any model? = Nt frustration total dimension oo, No is No IN gas with whether = unclear the + n~~ i) classical we spin is of the even -+ b + Ni will is fixed assume conserved we and equal to No where N bipartite and in 25(~) Ni Ni that A is will fix = corresponding sector. Since N is even, odd. physics will The correspond to the or 0 (b is the doping concentration), SIN -+ large number of quantum particles. It is quite a limit, the particles being fermions. So how can we JOURNAL 492 Even though we do not know yet eigenvalue spectrum is very sensitive distribution. spacing To do so, we phenomena liable to hide a possible the symmetry, Poisson a dilserent In the One physical must however question, that to answer to N°2 be we So parameters. one can enough, careful check can not whether the its level measure introduce to extra repulsion. For instance if our Hamiltonian admits a not be coupled together and the eigenvalues of H in two distribution then check that the level spacing will satisfy level will sectors independent. will be sectors PHYSIQUE I DE can law. model J t our it is important therefore restrict to H to of each sector a This symmetry. invariance, we N, No, the total spin S(~) Moreover, due to translation we indexed by k. Lastly the J term admits a spin choose a given quasi-momentum must sector rotation invariance which forces us to choose a spin sector. numerical point of view, the choice of the finite lattice A is done by defining a From the why is choose must £ of Z~ sublattice symmetry, generated by will we two where and p are identified q be can integers. two with the (q>P) " £ Thus this In way figure we 10. have can square 16 avoid To y Y (~q>P) or matrix sites 18 size too sides ~j ~ (we big " yZ xZ + = whose ~ in Z~. and x E order In to the preserve square choose X A vectors are A and and x = y Z~/£. and the number of sites becomes ~2 ~ ~2 ~ choose for N the to even spin frustrations) choose No memory, we avoid computer shown as " I (one hole). Now the compute break this difficulty lies in the spin symmetry. hard It is actually too each spin sector. calculation it is better To avoid such a long by modifying the J-term as technical main elements matrix symmetry to in £ (Ji (Sf)S)~) S(~)S)~)) + + Jjj S(~)S)~)) to (37) , jz-yj=1,~,YEA with At far mean Ji # Jjj. The limit Jjj -+ oo is actually the last, the level spacing must be normalized The normalization apart in the spectrum. level separation be one. This be done can Fig. II) and replacing approximation of p(E). The calculation In [65] it is also the Ising in has been shown that which allow to a obviously integrable. comparison between levels is usually chosen in such by calculating the density is way a of as states the local p(E) (see f~$ P(E)dE where p is some smooth Ei by the number fi operation is called "unfolding the spectrum" [25]. performed in [65] (see Fig. 12) and exhibits a GOE distribution. the level spacing distribution becomes Poisson. as Jjj -+ oo levels This model order = The previous analysis is certainly very preliminary. investigate whether system is integrable. Actually can a and still unpublished investigations of various models indicate that such a method is recent justified. For instance, a Heisenberg spin chain in lD, known to be integrable by mean of a DISCUSSION 2.7 it But Bethe In shows Ansatz holes One the way use such field, by computing a a shows should situation could be exhibits same calculation numerical SPECULATIONS. RMT [22] the much AND that a the be to Poisson lD t that distribution model J it satisfies to [4]. with a one hole should distribution Poisson also be integrable [60]. [84]. Presumably with A two [61]. dilserent method level used test spacing the integrability distribution on the of a 2D transfer Ising model matrix. with While magnetic for a zero N°2 METHODS SEMICLASSICAL 4 SOLID . . . . . . . . . . . . m m m m 3 2 j IN STATE PHYSICS 493 _i 3 2 / 4 ' ' / ' / ' / ' / m', ' I' m ' ' / ' / / m m ' m m / ' , / b ( , m / m m m / , / , / , / , " , " / , / , / , / , , -3 Fig.10. Lattice -2 sites for / / 2 -1 the t J model; upper figure : 16 3 sites; lower figure 18 sites. PHYSIQUE DE JOURNAL 494 I N°2 250 /h.~w~", q~,;w' ~ , . iso : ° ° a ° . loo ° 6 2 ~ -2 -4 0 4 E Fig-I Density I. of Jz of the t states 0.8 . model N for = 18 and Jz 0.25 = (from [65]). . 0.6 ~ i~ 0A 0.2 ~ 0.5 2 1.5 2.5 3 s Fig.12. and Poisson Level spacing distributions P(s) distribution are shown as full in the t lines Jz model (from [65]). for N = 18 and Jz = 0.25. Ideal GOE N°2 SEMICLASSICAL magnetic field [67] it is known to every magnetic field. Such a test The question next IN could integrable, give an indication. GOE character be the concerns METIIODS SOLID it is STATE unclear of the PHYSICS integrability whether for the t result 495 All model. J holds for matrix obviously real, so that this result should not be surprising. are the approach in terms of slave bosons, flux phases or However superconductivity suganyons should be broken. gests that, at least for low energy excitations, the time reversal symmetry We do not see anything like that here. It is however sufficient because this not a argument, breaking of the time reversal symmetry alsects only the ground state. For one hole, the Na~ elements of the gaoka Hamiltonian [66] requires theorem that the ground unique be state that so time symmetry reversal Moreover, the path integral approach used in quantum Monte Carlo calculations [54] shows that for spin 1/2, no time reversal symmetry breaking is required, namely there is no need to introduce complex field. Therefore, does not expect any of the a gauge one approximation like flux phases or anyons to work unless in trivial case, namely whenever the should be not broken. dimensionless flux is 0 or 1/2 mod I. Another important question raised by this result the validity of the quasiparticle concerns approximation in terms of a Fermi liquid theory. Even though quasiparticle peaks have been observed in a t J model or in the 2D Heisenberg ailtiferromagnet, if the theory made of were well defined independent quasiparticles we would expect a Poisson distribution How to occur. reconcile the two points of view ? Presumably, while level repulsion do occur to at high enough the density of states is large, whereas elementary at low energy, in the region where energy, excitations produce level clustering. However this has not been tested yet. If this is so a may liquid theory should be only a low energy approximation, and RMT should give the explanation for the If such that find to "scars" neither bosons Lastly, another [53] be may functions. in such relevant fermions, nor Green's in fails, namely if GOE explanation for the explanation an ought we background incoherent but scars still holds appearance would We case. conspiring to groundstate the near quasiparticles of then have produce a new stable a peaks. kind then energy, We suggest of excitations, excitation. important question the conductivity. that the linear We conjecture concerns behaviour of the resistivity with of the temperature ill] should be a respect to consequence random matrix approach. However, again, only low energy excitations relevant for such are calculations. Therefore if RMT holds at small and explains the linear law of resistivity, energy the quasiparticles approach becomes questionnable. an computation The operator. current basic Field Randomness two will requirements. conductivity involves required is structure Quantum in of This the requires the knowledge of the matrix locality properties of the Hamiltonian. It describe to the Hamiltonian in of terms random elements means for that matrices. the some Like Theory, translation invariance and locality are essential ingredients there. Hamiltonian which constrained by these only for the of the is part not occur investigating this question [14]. We are currently Acknowledgments. This with authors Bohigas, D. is paper R. summary a Rammal. cited A. Poilblanc, in Their the and a review solution reference could list. We of problems some not have want Connes, B. Dou~ot, P. Duclos, R. R. Seiler, C. Sire, J. 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