Two-dimensional Bloch electrons in perpendicular magnetic fields: an exact calculation of the Hofstadter butterfly spectrum
TTwo-dimensional Bloch electrons in perpendicular magnetic fields: an exactcalculation of the Hofstadter butterfly spectrum
S. Janecek ∗ ,
1, 2, 3, 4
M. Aichinger,
3, 5 and E. R. Hern´andez
1, 2 Instituto de Ciencia de Materiales de Madrid (ICMM–CSIC), Campus de Cantoblanco, 28047 Madrid, Spain Institut de Ciencia de Materials de Barcelona (ICMAB–CSIC), Campus de Bellaterra, 08193 Barcelona, Spain Johann Radon Institute for Computational and Applied Mathematics (RICAM),Austrian Academy of Sciences, Altenberger Strasse 69, A-4040 Linz, Austria MathConsult GmbH, Altenberger Strasse 69, A-4040 Linz, Austria Uni Software Plus GmbH, Kreuzstrasse 15a, A-4040 Linz, Austria (Dated: March 11, 2013)The problem of two-dimensional, independent electrons subject to a periodic potential and auniform perpendicular magnetic field unveils surprisingly rich physics, as epitomized by the fractalenergy spectrum known as Hofstadter’s Butterfly. It has hitherto been addressed using variousapproximations rooted in either the strong potential or the strong field limiting cases. Here wereport calculations of the full spectrum of the single-particle Schr¨odinger equation without furtherapproximations. Our method is exact, up to numerical precision, for any combination of potentialand uniform field strength. We first study a situation that corresponds to the strong potential limit,and compare the exact results to the predictions of a Hofstadter-like model. We then go on toanalyze the evolution of the fractal spectrum from a Landau-like nearly-free electron system to theHofstadter tight-binding limit by tuning the amplitude of the modulation potential.
The motion of electrons in a crystalline solid subjectto a magnetic field has been considered since the earlydays of quantum mechanics [1]. The field splits the crys-tal’s electronic bands into sub-bands and internal mini-gaps; the energetic arrangement of these sub-bands formsa fractal structure reminiscent of a butterfly when plot-ted as a function of the field [8]. Significant experimen-tal effort has been devoted to detecting signatures ofthis energy spectrum in two-dimensional electron gases(2DEGs) [2, 3]. For independent electrons, the situationis described by the single-particle Schr¨odinger equation, Hψ ( r ) ≡ (cid:20) m Π + V ( r ) (cid:21) ψ ( r ) = Eψ ( r ) , (1)where H is the Hamiltonian, ψ ( r ) is an eigenstate withenergy E , Π = p + e A ( r ) is the dynamical momentumoperator and A ( r ) is the vector potential correspondingto the magnetic field, B = ∇ × A . We take the field tobe uniform and oriented along the z -direction, B = B e z .The electrons are restricted to the two-dimensional (2D) xy -plane, and the external potential V ( r ) is periodic ona Bravais lattice defined by vectors R n = j a + k b , n = ( j, k ) ∈ Z . (2)To date, this problem has been chiefly approached byapproximations starting from two complementary limits,considering either the influence of a weak magnetic fieldon the band structure resulting from a strongly varyingpotential V ( r ) [4, 5], or the influence of a small mod-ulation potential on the Landau-quantized electrons ina strong field [5, 6]. In the strong potential limit, onetypically starts with a tight-binding (TB) approxima-tion for a single band of the zero-field ( A = 0) prob-lem, E ( k ), where k is the crystal momentum. Then, an effective Hamiltonian for the magnetic field problemis generated from E ( k ) through the Peierls substitution k → ( p + e A ) / (cid:126) [1, 7]. This procedure was used, amongothers, by Hofstadter in his seminal article for a nearest-neighbor (NN) TB model of the 2D square lattice [8].There are a number of simplifications inherent to this ap-proach: (i) the TB approximation of the zero-field bandstructure, (ii) the restriction to electrons in a single band,(iii) the neglect of the diamagnetic energy of the TB or-bitals and the field dependence of the TB hopping in-tegrals. This has been shown to lead to quantitative aswell as qualitative errors for both nearly free and tightlybound two-dimensional electrons [9–12]. Generalizationof the effective Hamiltonian approach has turned out tobe difficult, see, e.g., Ref. [13]. Surprisingly, the strongfield approach is closely related to the strong potentialone: if potential-induced coupling between different Lan-dau levels is neglected, the same secular equation results,but with the magnetic field replaced by its inverse [5].Including such coupling has a profound effect on the cal-culated energy spectrum [14, 15], even for weak couplingstrength. The resulting rearrangement of the Hofstadterbutterfly has recently been confirmed by experiments [3];we take this as a strong indication that experiments canonly be fully understood by going beyond the approxi-mate schemes described above. Method.
In the absence of magnetic field, Bloch’s the-orem, which results from the commutation of the latticetranslations with the Hamiltonian, allows to restrict thecalculation of the eigenfunctions ψ ( r ) to one primitivecell of the lattice in Eq. (2). As these translations do notleave the vector potential A ( r ) invariant, they no longercommute with H when a magnetic field is present; con-sequently, the eigenfunctions ψ ( r ) are not Bloch waves . a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r By combining a lattice translation with a suitable gaugetransformation to counteract its effect upon the vectorpotential, one can define magnetic translation operators , T A ( R n ), that do commute with H , see Refs. [16–18]. Theoperators T A ( R n ) do not, however, form a group. For arational field, where the number of magnetic flux quantaper unit cell, α = π e (cid:126) ( a × b ) B , is a rational number, α = p/q with p and q relative prime, one can choose alarger magnetic lattice that has an integer number of p flux quanta passing through each cell, e.g., S n = j a + k ( q b ) , n = ( j, k ) ∈ Z . (3)On this lattice, a generalized version of the Bloch theo-rem holds [18]: for an orthorhombic lattice and Landaugauge, A ( r ) = Bx e y , the solutions of Eq. (1) can bechosen of the form φ ( r ) ≡ e i θ r u θ ( r ), where u θ ( r ) obeys u θ ( r + S ) = exp (cid:104) − i e (cid:126) BS x y (cid:105) u θ ( r ) , (4)and the magnetic crystal momentum θ is restricted tothe first Brillouin zone of (3). This condition allows torestrict the calculation of u θ ( r ) to one primitive cell ofthe magnetic lattice. The functions u θ ( r ) exhibit a pe-culiar topology in a space where their y -component isexpanded in a Fourier series, see Ref. [19]. This allows totake into account the boundary condition (4) in a natu-ral way when the diffusion method is used for solving theSchr¨odinger equation [20, 21]. We have summarized thesteps leading to Eq. (4) and the technical details of thediffusion method in the Supplementary Information.At a fixed rational field, α = p/q , we obtain the bands E j ( θ ) by numerically solving Eq. (1) for a grid of θ -valuesspanning the magnetic Brillouin zone (MBZ) correspond-ing to this field. The density of states (DOS) as a functionof the field, ρ ( B, E ), can then be calculated by integrat-ing the bands over the MBZ and repeating the processfor different fields. The rational field can only be tunedin discrete steps, α P = P/Q , where
P, Q ∈ Z . The size ofthe magnetic unit cell depends on the reduction of P/Q to a quotient of relatively prime integers p/q , and is q times as large as the zero-field (“geometric”) unit cell.At the field α P , one zero-field band is thus expected tosplit into q magnetic bands, which are known to clusterin groups of p bands [8]. The pattern of distinct primefactors of P and Q as α P is swept across a range of fieldsgives rise to the self-similar, fractal pattern of gaps in theDOS that has become known as the Hofstadter butterfly.When the Fermi energy of the system lies in a gap, i.e.,a region where ρ ( B, E F ) is zero, the Hall conductance σ xy assumes a quantized value, σ gap xy = ne /h, n ∈ Z . Thefundamental topological reason for this quantization wasrevealed by Thouless et al. [22], who showed that bothin the strong field and strong potential limits the Kubo-Greenwood formula for σ xy is related to the Chern num-ber of the U (1) bundle over the magnetic Brillouin zone.This was later argued [23] to be a direct consequence of the magnetic translation symmetry, Eq. (4). Sweeping ei-ther the magnetic field or the Fermi energy through thefractal pattern of minigaps inside a broadened band orLandau level results in a peculiar, non-monotonous Halleffect [22]. Indications of this behavior have been foundin experiments [2]. In this work we have used an alterna-tive approach, introduced by Stˇreda [24], to obtain σ gap xy from the numerically calculated DOS, σ gap xy ( B, E F ) = e ∂ρ ( B, E (cid:48) ) ∂B (cid:12)(cid:12)(cid:12) E (cid:48) = E F . (5)In our numerical calculations, we have studied a sim-ple system consisting of a two-dimensional square latticeof potential wells with the symmetrized Fermi functionform [25], V ( r ) = U coth (cid:16) r d (cid:17) sinh (cid:0) r d (cid:1) cosh (cid:0) rd (cid:1) + cosh (cid:0) r d (cid:1) , (6)with parameters r = 39 . , d = 1 .
59 nm , a = 100 nm , (7)where a is the lattice spacing. The potential is illustratedin Fig. (1a); the parameters are set to loosely reproducethe conditions of earlier experimental studies [2, 3]. Strong Potential Regime.
We first compare the spec-trum generated by the full Schr¨odinger equation (1) tothe results of a TB approximation similar to the one usedby Hofstadter. We thus choose a fairly deep modulationpotential, U = − V ≡ − . . (8)We obtained the band structure and DOS at zero mag-netic field [see Fig. (1c)] by numerically solving the corre-sponding single-particle Schr¨odinger equation. We thenfitted a TB model to the lowest three bands [black dashedlines in Fig. (1c)], and employed the Peierls substitutionto obtain the fractal energy spectra shown in Fig. (1d),see the Supplementary Information for details. The low-est band [ n = 1 in Fig. (1c)] required only nearest-neighbor hopping integrals in the TB band model for anadequate fit, and thus yields a spectrum corresponding tothat obtained by Hofstadter [8] [left panel of Fig. (1d)].The second and third bands required up to 3 rd -nearest-neighbor hoppings, which lead to significantly distortedversions of Hofstadter’s butterfly, shown for the secondband on the right panel of Fig. (1d). The spectrum ofthe third band is similar and is not shown. Our findingsqualitatively agree with the results of Ref. [26].In Fig. (2a) we show the energy spectrum of the sixlowest bands, as obtained by numerically solving the fullmagnetic eigenvalue problem, Eq. (1), using the schemeoutlined above. A maximum magnetic unit cell size of Q = 32 was employed in the calculation. For compari-son, the spectrum of an isolated potential well is plotted E U [ V ] n = 1 n = 2 Flux quanta per unit cell ( ↵ ) M X M | k | E n e r gy E U [ V ] ⇢ ( E ) ⇡/a a) b) XM ⇡/a ⇡/a ⇡/a k y c)d) n =1 n =2 n =3 U = V = . k x FIG. 1: (color online). a) Schematic plot of the periodicFermi well potential. b) First Brillouin zone of the recip-rocal lattice. c)
Left panel : Zero-field band structure of thepotential with parameters (7) and U = − V . Solid red linesshow bands obtained by numerically solving the Schr¨odingerequation, dashed black lines are TB bands fitted to these ex-act bands (see text). Right panel : DOS ρ ( E ) obtained fromthe full Schr¨odinger equation. d) Hofstadter butterflies forthe two lowest bands. Areas with non-zero DOS are printedblack, and the gaps are colored according to the correspond-ing quantized Hall conductance σ gap xy in units of e /h . Whiteindicates zero Hall conductance, warm (cold) colors indicatepositive (negative) Hall conductance. A number of larger gapsare labeled with the corresponding Hall conductance for ref-erence. The butterflies are periodic in the flux, one periodbeing shown in each case. in Fig. (2c). It bears a strong resemblance to the Fock-Darwin (FD) spectrum [27] of a parabolic well, we willthus refer to these states as “FD states” in the following.It can be seen in Fig. (2a) that in the periodic systemthe FD states of the isolated well are broadened intobands with a fractal internal structure that is qualita-tively well described by the Hofstadter butterfly. In theexact result the periodicity of the Hofstadter spectrumis superimposed onto the field dependence of the corre-sponding FD state; this field-dependence is not taken intoaccount in the TB model with constant hopping integrals.In general, higher energy levels, having more extendedwave functions, undergo larger broadening at a given fluxvalue. Conversely, bands become narrower with increas-ing field, as their wave functions become more spatially E n e r gy E U [ V ] (1)(2) (3) (4)(5) (6) (7)(8) (9)(10)(11) E U [ V ] . . . . . Flux quanta per unit cell ( ↵ )a)b) c) U = V = .
12 -1 -20 12 -1 0
FIG. 2: (color online). Magnetic energy spectrum (DOS andHall conductance σ gap xy ) for the lowest six bands of the of thesquare Fermi well lattice with parameters (7) and U = − V ,calculated by numerically solving Eq. (1). The color codingand labeling is as in Fig. (1). Panel ( b ) shows a magnified por-tion of the lowest band indicated by the dashed box in panel( a ). Panel ( c ) shows the energy spectrum of an isolated Fermiwell potential, Eq. (6). It strongly resembles the spectrum ofa parabolic potential (the Fock-Darwin spectrum [27]). localized, tending to Landau levels in the limit of highfield intensities. In the TB model, the main effect of in-creasing the second- and third-nearest neighbor hoppingsis a distortion of the butterfly that opens a gap at fluxstrengths of α j = j + 1 /
2, with j ∈ Z , indicated by thevertical (red) dotted lines in the butterflies in Fig. (1d).This behavior is also present in the exact results, compareregions (4)–(5) or (7)–(8) in Fig. (2a). The gap widensfor higher-energy bands, consistent with the TB approx-imation, where such bands need to be modeled by largerhopping integrals to more distant neighbors. The gapdecreases again with increasing field due to the strongerlocalization of the wave functions, an effect that is notincluded in the TB description with B -independent hop-ping. Band crossings, which are not described withinthe single-band Peierls approximation, are an interestingsubject for future studies: the narrow third FD bandseems to disrupt the butterfly pattern of the broaderbands it crosses in regions (3) and (6), but does not seemto exert any noticeable effect in region (9). At higher en-ergies [see region (10)], where multiple bands cross, theresulting fractal spectrum can assume a form that is verydifferent from the original Hofstadter butterfly. Intermediate Potential Regime.
We now explore theevolution of the spectrum as the potential is changedfrom nearly flat (strong field limit) to highly modulated(strong potential limit). The intermediate stages of thisevolution are not accessible to the approximate method-ologies hitherto employed. We use the square lattice po-tential with the parameters given in Eq. (7), but changethe well depth U . We start with a very shallow potential( U = − . V ), which corresponds to a Landau-like sys-tem with nearly-free electrons in a magnetic field. Theresulting spectrum is illustrated in Fig. (3a); it exhibitsthe typical “Landau fan” form, with slightly broadenedLandau levels (LLs) that display an internal fractal struc-ture of minigaps (most evident in the lowest level, L ).The Hall conductance between LLs increases monotoni-cally in steps of e /h , consistently with the integer quan-tum Hall effect. Furthermore, an emerging white gap( σ gap xy = 0) can be observed at low field ( α = 0 − E ≈ . V ). When the potential modulation is in-creased to U = − . V [Fig. (3b)] this gap is seen towiden further, practically cutting off a low-energy trian-gular section (marked “T” in the figure) from all LLs L n with n >
0. At α = 1 the upper-right tip of thistriangular cluster of states retains a tenuous link to thebroadened L Landau band, at the position indicated bythe blue arrow. At the same time, the minigaps at α ≥ L have broadened further, tothe point where the band is only held together by a nar-row sub-band at α = 1, indicated by the red arrow. TheHall conductance in the gap between L and L at α = 1is still σ gap xy = 1. But upon further increasing the modu-lation strength, the gap first closes and then reopens withthe above links reversed: at U = − . V [Fig. (3c)], thetip of the triangular cluster is now connected to the low-est miniband of L (blue arrow), while the rest of the L miniband is connected to L (red arrow). At the sametime, the gap has changed its character to σ gap xy = 0, re-sulting in the first clearly discernible FD-like band F ,separated from all higher energy states by an unbrokenwhite gap. Most of F is formed from the lowest mini-band of the LL L , except for the triangular cluster ofstates in α <
1, which originates from higher LLs. Theprocess of formation of the first FD band ( F ) that isincipient at α = 1 in Fig. (3b) can be seen to repeatitself at α = 2 for the second FD band ( F ) [Fig. (3c),green and purple arrows] and at α = 3 for the third FDband ( F ) [Fig. (3d)]. The gradual transformation fromLLs to FD-like bands thus proceeds by the re-connectionof minibands from one broadened LL to those of neigh-boring LLs. This rearrangement process fragments thelarge triangular gaps between LLs in the weak potentiallimit into minigaps encapsulated by the emerging FD-likebands. As a result, the FD states F , F , F in Fig. 3dare composed of a low-field section that originates fromhigher LLs (triangle T in case of F , one Hofstadter but-terfly segment plus a triangular cluster in case of F , etc.), and a high-field section that is one miniband of thelowest LL L .The above results show that with the method pre-sented here it is possible to perform exact calculationsof the spectrum of independent electrons in a 2D pe-riodic potential and constant perpendicular magneticfield. The same technique is readily applicable to solvethe Kohn-Sham equations of density functional the-ory (DFT) which are expected to provide a reasonable de-scription at least for weakly correlated electron systems.Experimental techniques have recently become availableto directly probe the local DOS in 2DEGs on surfacesin a perpendicular magnetic field using scanning tunnel-ing spectroscopy, making possible the detection of spa-tial features of some LLs [32], and even measuring theirresponse to a 1D periodic potential due to surface buck-ling [33]. Such developments may presently bring aboutadditional possibilities of experimentally determining thespectral features of the 2DEG subject simultaneously toa periodic potential and a perpendicular magnetic field,contrasting them with the predictions reported herein.The Hofstadter butterfly is not unique to the system dis-cussed here. The basic principle underlying the fractalenergy spectrum is the presence of two competing sym-metries (in the case studied here, the periodicity of thelattice and the symmetry of the Landau orbits, which isgoverned by the area required by one magnetic flux quan-tum). Similar patterns have been observed or predictedto occur in a variety of very different systems, such asmicrowaves transmitted through a waveguide with a pe-riodic arrangement of scatterers [28], the electronic [29]and vibrational [30] spectra of incommensurate crystals,ultracold atoms in optical lattices [31], and photonic crys-tals [34, 35]. The topological protection of the quantumHall phase has been shown to improve the performanceof optical delay lines and to overcome limitations relatedto disorder in photonic technologies [35]. We hope thatthe method presented here will also prove beneficial inthese related fields.We wish to thank A. Garc´ıa and E. Krotscheck forhelpful discussions. SJ was funded by the Austrian Sci-ence fund FWF under project no. J2936-N, E.R.H. bythe Spanish Ministry of Science and Innovation throughproject FIS2009-12721-C04-03. SJ would like to thankICMAB and ICMM for their hospitality during his stay.We acknowledge CESGA and the Johannes Kepler Uni-versity Linz for the use of their computer facilities, wherethe results reported here were obtained. [1] R. Peierls, Z. Physik , 763 (1933).[2] C. Albrecht, J. Smet, K. von Klitzing, D. Weiss,V. Umansky, and H. Schweizer, Phys Rev Lett , 147(2001).[3] M. Geisler, J. Smet, V. Umansky, K. von Klitzing, E n e r gy E U [ V ] a) U = . V U = . V U = . V U = . V Flux quanta per unit cell ( ↵ ) L L L -1 L L L L L -1 F L F F F T T 123 1
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