Are there quantum oscillations in an incommensurate charge density wave?
AAre there quantum oscillations in an incommensurate charge density wave?
Yi Zhang, Akash V. Maharaj, and Steven Kivelson
Department of Physics, Stanford University, Stanford, California 94305, USA (Dated: June 27, 2018)Because a material with an incommensurate charge density wave (ICDW) is only quasi-periodic, Bloch’stheorem does not apply and there is no sharply defined Fermi surface. We will show that, as a consequence,there are no quantum oscillations which are truly periodic functions of 1 / B (where B is the magnitude of anapplied magnetic field). For a weak ICDW, there exist broad ranges of 1 / B in which approximately periodicvariations occur, but with frequencies that vary inexorably in an unending cascade with increasing 1 / B . For astrong ICDW, e.g. in a quasi-crystal, no quantum oscillations survive at all. Rational and irrational numbersreally are di ff erent. I. INTRODUCTION
Quantum oscillations (QOs) provide powerful experimentalavenues to precisely characterize the Fermi surface structurein an electron system . At zero temperature and in the limitof small magnetic field B (where the semiclassical approxi-mation is asymptotically exact), measurable quantities suchas the density of states (DOS) and various transport proper-ties oscillate as a function of 1 / B with a periodicity that isinversely proportional to the cross-sectional area of the Fermisurface. In the presence of a commensurate charge densitywave (CDW), the QOs can be simply understood by foldingthe momentum space Brillouin zone in accord with the en-largement of the crystal unit cell. In the case of an incommen-surate charge density wave(ICDW), however, the translationalsymmetry along the direction of the CDW wave vector (orvectors) is completely broken; Bloch’s theorem no longer ap-plies, and hence the notion of a Fermi surface is at best an ap-proximate concept. Nonetheless, it has been widely acceptedthat new oscillation periods arise from the reconstructed elec-tron and hole pockets that result from the perturbative foldingof the Fermi surface in a manner similar to that of the com-mensurate case .We have extensively studied the problem of QOs in a two-dimensional system in the presence of a unidirectional ICDW.We obtain numerical solutions on systems with a linear di-mension up to ∼ O (cid:16) (cid:17) sites using an e ffi cient recursiveGreen’s function method. For weak ICDW potentials V (cid:28) W where W is the bandwidth, we obtain a perturbative under-standing of these results, while larger values of V / W canbe understood by introducing an exact duality between thetwo-dimensional system with a unidirectional ICDW, and athree-dimensional system without the ICDW but with an ad-ditional magnetic flux. For the two-dimensional system withan ICDW of ordering vector Q and amplitude V , the dualthree-dimensional system has a hopping matrix element V / z ) direction, and a “tilted” magnetic field (cid:126) B e f f = B ˆ z + ( Q / π )ˆ y where ˆ x and ˆ y are the in-plane direc-tions parallel and perpendicular to the charge ordering vector(where magnetic fields are measured in units of a flux quantaper unit cell). This mapping is readily generalized to other di-mensions, as well as to the case of bidirectional ICDW order.We find that there is no limit in which perfectly periodicQOs occur in the presence of an ICDW. In the strong coupling limit, for V / W greater than a critical value of order 1, thereare no well defined QOs at all. In the dual system, the crit-ical V / W is identified with the point at which the underlyingthree-dimensional Fermi surface transforms from a quasi-two-dimensional cylinder to a closed three-dimensional surface.This limit likely applies to the case of quasi-crystals in whichthe incommensurate potential is not small in any sense. Bycontrast, in the small V / W limit, oscillations that approximatethe QOs of a true crystal are observed over large but finite in-tervals of 1 / B , but the observed period varies depending onthe range of B studied.The nature of the QOs for small V / W , and the manner inwhich they deviate from strict periodicity can be understoodintuitively in terms of the conventional theory of magneticbreakdown as a perturbative e ff ect. Specifically, there is a hi-erarchy of gaps on the Fermi surface, ∆ n ∼ W ( V / W ) n , whosemagnitudes are governed by the lowest order in perturbationtheory at which they appear. Depending on the position ofthe unperturbed Fermi surface and its relation to the CDWordering vector, only some of these gaps open on the Fermisurface, and so it is only these that matter. The hierarchyof gaps on the Fermi surface terminates at n max ≤ q for a q th order commensurate CDW, but it continues to arbitrarilyhigh n in the incommensurate case. In a given range of B ,magnetic breakdown e ff ectively eliminates the e ff ect on theelectron dynamics of all gaps that are su ffi ciently small that ∆ / W (cid:28) ω c , where ω c ∼ W B is the e ff ective cyclotron en-ergy, while all gaps that are large compared to √ ω c W arerespected by the semiclassical dynamics. If n = N and n = N + M are two sequential terms in this hierarchy, thenfor small V / W there exists a parametrically broad range offields, ( ∆ N / W ) (cid:29) B (cid:29) ( ∆ N + M / W ) , in which all gaps oforder n ≥ N + M can be neglected, but the Fermi surface ise ff ectively reconstructed by all gaps with n ≤ N . However,inevitably, a further reconstruction of the Fermi surface mustoccur in some order of perturbation theory, giving a fractalcharacter to the QO spectrum.The rest of this paper is organized as follows. Sec. II re-views the canonical understanding of QOs in the presence andabsence of CDWs. In Sec. III we introduce the tight bindingmodel which forms the basis of our numerical analysis, andalso discuss its duality properties. Numerical results are pre-sented in Sec. IV, where we show the field dependence ofQO periods for a commensurate CDW and a small amplitudeICDW, along with the spectacular absence of QOs for a large a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b amplitude ICDW. We also explain the eventual breakdown ofthe perturbative picture for the ICDW QOs. In Sec. V we nu-merically verify the exactness of the duality picture, using thisview to uncover several novel properties of ICDW in magneticfields. Finally, Sec. VI discusses further applications of thismodel to localization problems in one dimension as well asthe case of bidirectional ICDW order. II. CHARGE DENSITY WAVES IN A MAGNETIC FIELDA. Quantum oscillations in ordinary metals
Quantum oscillatory phenomena can be generally under-stood to arise from the semi-classical quantization of elec-tronic energies in an applied magnetic field – which is asymp-totically exact in the B → (cid:126) d (cid:126) kdt = − e (cid:104) (cid:126) v ( (cid:126) k ) × (cid:126) B (cid:105) = e (cid:126) (cid:126) B × dE ( (cid:126) k ) d (cid:126) k , (1)where (cid:126) v ( (cid:126) k ) = (1 / (cid:126) ) dE / d (cid:126) k is the electron group velocity. Thisimplies (cid:126) k · (cid:126) B = dE ( (cid:126) k ) / dt =
0, i.e. the electrons move inorbits of constant energy in planes perpendicular to the mag-netic field.In the absence of any Berry phase, the quantum phase theelectrons accrue on each orbit around the Fermi surface is (cid:126) S k / eB , where S k is the area of Fermi surface cross sec-tion perpendicular to the applied magnetic field. Thus themaximum and minimum values of S k govern the interferencebetween multiple trajectories. Maxima in the semiclassicalDOS (corresponding to the point at which a Landau level justcrosses the Fermi energy) occur whenever1 B = (cid:32) n + (cid:33) π Φ S k (2)for integer n ∈ Z where Φ = h / e is the magnetic flux quan-tum. For fixed Fermi energy (and hence constant S k ) this leadsto perfectly periodic QOs with period ∆ (1 / B ) = π e / (cid:126) S k . B. Commensurate CDWs and magnetic breakdown
The above arguments can be straightforwardly applied tosystems in which the translation symmetry of the crystal isspontaneously broken by a commensurate CDW with wave-vector (cid:126) Q = Q ˆ x where, in units in which the lattice constant is a = Q = π p / q with p , q ∈ Z relatively prime. This simplydefines a new crystal structure with q times as large a unit celland q times as many bands in a folded Brillouin zone which is q times smaller in the ˆ x direction. Generically, the CDWcauses a “reconstruction” of the Fermi surface due to gaps thatopen at the new Brillouin zone-boundary, resulting in smallerelectron and hole pockets (plus open Fermi surfaces that do ~Q~Q~Q ~Q FIG. 1. The Fermi surface reconstruction with a momentum transferof ± Q by scattering from a CDW is shown as the yellow-coloredregion (both the dark and light yellow regions). In addition, a higherorder scattering with a momentum transfer of ± Q (cid:48) may lead to asmaller pocket (dark yellow region) and so on. Consequently, theremay be a hierarchy of Fermi surface pockets protected by gaps ofdi ff erent sizes. not contribute to the QOs), thus giving rise to new periodici-ties in the QOs. The reconstruction can be viewed as arisingfrom processes in which the electrons on the Fermi surface arescattered by the CDW with momentum transfer ± nQ ˆ x modulo2 π ˆ x . In the case of a weak sinusoidal density wave, V / W (cid:28) n , as such processes arise first in n th order per-turbation theory, see Fig. 1.In the notation used in the introduction, gaps in-dexed in this way depend parametrically on V as ∆ n = α W ( V / W ) n (cid:104) + O ( V / W ) (cid:105) where α does not depend on V .Depending on the relation between the Fermi surface locationand the ordering vector Q , gaps with a given index n may notinvolve states at the Fermi surface, in which case they can beneglected for present purposes. For instance, if Q (cid:48) in Fig. 1were slightly longer, it would fail to span the indicated Fermisurface, and so would not produce any Fermi surface recon-struction. Notice that the area of the reconstructed Fermi sur-face pockets is purely geometric in the limit of vanishinglysmall V / W , determined solely by the structure of the underly-ing Fermi surface and the specified CDW ordering vector, butfor finite V the opening of gaps causes a non-zero displace-ment of the Fermi surface which can produce changes in thearea of the various pockets of order ∆ n / W . This e ff ect be-comes qualitatively significant if V / W is not small, in whichcase the perturbative approach must be abandoned.For finite field strength B corrections to semiclassical quan-tization arise due to ‘magnetic breakdown’, associated withtransitions between semiclassical trajectories. It is easy toshow that magnetic breakdown is negligible so long as (cid:96)δ k (cid:29)
1, where (cid:96) = √ Φ / π B is the magnetic length and δ k is the distance of closest approach in reciprocal space be-tween two semi-classical trajectories. (The precise criteriondepends on the local curvature of the trajectories as well .)Alternatively, for weak CDW order, this criterion can be ex-pressed as ω c (cid:28) ∆ / W , where ω c is the cyclotron energy and ∆ is the relevant gap induced by the CDW. When there exists aseries of Fermi surface pockets produced by gaps of di ff erentsizes from multiple orders of reconstructions, this results in acorresponding hierarchy of breakdown magnetic fields. C. Incommensurate CDW in a magnetic field
When the period of charge modulation is not a rational mul-tiple of the underlying lattice spacing, i.e. when Q = πα where α is an irrational number, the CDW is referred to asincommensurate. While many properties of ICDW materialscan be understood by treating the CDW potential perturba-tively, or better still by approximating α by a nearby com-mensurate approximant, p / q ≈ α , we will show in this paperthat an exact treatment of the problem reveals a number offundamental properties that are not captured by this type ofapproximation. III. THE MODEL
To begin with, we define a general model on a d -dimensional hyper-cubic lattice in the presence of a uniformmagnetic field and a CDW potential: H = − (cid:88) (cid:104) (cid:126) r ,(cid:126) r (cid:48) (cid:105) (cid:104) t (cid:126) r − (cid:126) r (cid:48) exp (cid:2) iA ( (cid:126) r ,(cid:126) r (cid:48) ) (cid:3) c † (cid:126) r c (cid:126) r (cid:48) + h . c . (cid:105) − (cid:88) (cid:126) r U ( (cid:126) r ) c † (cid:126) r c (cid:126) r (3)where c † (cid:126) r creates an electron on site (cid:126) r , (cid:104) (cid:126) r , (cid:126) r (cid:48) (cid:105) designatesnearest-neighbor sites, t ν ≡ t ( ± ˆ e ν ) is assumed real and pos-itive, but can depend on direction ˆ e ν with ν running from 1to d (with the convention ˆ e = ˆ x , ˆ e = ˆ y , etc.), the magneticflux through any plaquette in units in which the flux quan-tum is Φ = π is given by the sum of A around the plaque-tte, and by definition A ( (cid:126) r ,(cid:126) r (cid:48) ) = − A ( (cid:126) r (cid:48) ,(cid:126) r ). In all cases, wewill assume the flux is spatially uniform and penetrates onlyplaquettes that are parallel to ˆ x . We are thus free to chose agauge that preserves translational symmetry in all but the ˆ x direction, A ( (cid:126) r ,(cid:126) r + ˆ e ν ) = Φ ν (cid:126) r · ˆ x = Φ ν x , where, for example, Φ ≡ Φ y = Φ is the flux through each plaquette in the x − y plane, and Φ ≡ Φ x =
0. For the present, we will also con-sider a unidirectional CDW in the ˆ x direction which consistsof a small number m of distinct Fourier components, U ( (cid:126) r ) = m (cid:88) j = V j cos[ Q j x − θ j ] . (4)Here θ j is the relative phase between a given component of theCDW and the underlying lattice, and x j ≡ θ j / Q j can be inter-preted as the location of the minimum of the correspondingCDW potential in Eq. 3 in continuum space.Because the CDW is unidirectional and we have chosenthe appropriate gauge, we can exploit translation invarianceto Bloch diagonalize the Hamiltonian by Fourier transform perpendicular to ˆ x . We define a x ,(cid:126) k ≡ ( N ⊥ ) − d / (cid:88) (cid:126) r ⊥ e i (cid:126) k · (cid:126) r ⊥ c (cid:126) r ⊥ + x ˆ x (5)where (cid:126) r ⊥ ≡ (cid:126) r − x ˆ x and (cid:126) k is the corresponding d − H = (cid:80) (cid:126) k H (cid:126) k with H (cid:126) k = − (cid:88) x t x (cid:18) a † x ,(cid:126) k a x + ,(cid:126) k + h . c . (cid:19) (6) + d (cid:88) ν = t ν cos( Φ ν x − k ν ) + m (cid:88) j = V j cos( Q j x − θ j ) a † x ,(cid:126) k a x ,(cid:126) k Thus, for given (cid:126) k , the generic problem is equivalent to aproblem in one dimension. Moreover, it is immediately ap-parent from Eq. 6 that there is a formal equivalence betweenhigher dimensional problems with a magnetic field and lowerdimensional problems with more components of the CDW;e.g. with all V j = Φ per plaquette and the cor-responding one-dimensional problem (governed by Harper’sor the “almost Mathieu” equation ) of a particle in a sinu-soidal potential. ? Duality relations:
The problem of present interest is thatof the two-dimensional crystal subjected to a uniform flux Φ and a unidirectional CDW potential V cos[ Qx − θ ]. We nowsee that this is equivalent both to a one-dimensional systemwith the doubly periodic potential, U ( (cid:126) r ) = V cos[ Qx − θ ] + t y cos[ Φ x − k y ], and the anisotropic three-dimensional crystalwith hopping matrix elements t x , t y and t z = V / x , ˆ y , and ˆ z directions subjected to a uniform mag-netic field with flux Φ through each xy plaquette and flux Q through each xz plaquette (see Fig. 2). We will use the one-dimensional representation of the problem as the basis of ournumerical study. However, we gain physical insight into thesolution and approximate analytic understanding by viewingit as a translationally invariant (crystalline) three-dimensionalproblem in a tilted magnetic field. We also note that higherharmonic components of the CDW potential with wave vec-tors nQ , n ∈ Z , correspond to the same net magnetic field butfurther neighbor hoppings in the ˆ z direction thus more generic k z dispersions.There is still the issue of the sum over (cid:126) k . However, for theone-dimensional problem with a multi-component ICDW po-tential for which Q i / Q j is irrational for all i (cid:44) j , it is easyto see and straightforward to prove that in the thermodynamiclimit , the spectrum is independent of the the phases θ j , that isto say that there is a sliding symmetry for each component ofthe CDW. Translated to the magnetic flux problem, since Φ ν isgenerically an irrational multiple of Φ , this implies that (ex-cept in fine tuned circumstances), the spectrum of Eq. 6 willbe independent of (cid:126) k . Thus, for all thermodynamic quantities,the sum over (cid:126) k simply produces a degeneracy factor of N ⊥ foreach eigenstate. Notation and Units:
We will henceforth suppress the index (cid:126) k on all quantities, ( e.g. a x ,(cid:126) k will be represented as a x ). We FIG. 2. Upper panel: A one-dimensional model with two incommen-surately periodic potentials, one with amplitude 2 and wave vector Φ (blue curve) and the other with amplitude V and wave vector Q (redcurve). Middle panel: The equivalent two-dimensional square latticemodel with an ICDW (red curve) and a magnetic flux (blue arrow) Φ = π B per plaquette. Lower panel: The e ff ective cubic latticemodel in three dimensions. Here the hopping amplitudes are 1 alongthe ˆ x and ˆ y directions (solid lines), and V / z direction(dashed lines), and an additional magnetic flux of Q is now presentthrough the xz plaquettes (the red arrow). will specialize to the case in which the two-dimensional bandstructure is isotropic, t x = t y = t , and will use units of energysuch that t =
1. Recall that we have adopted units of lengthsuch that the lattice constant is 1, and units of magnetic field B = Φ / π such that Φ = π . IV. RESULTS OF NUMERICAL EXPERIMENTS
In our numerical studies, we compute two physical quanti-ties which are related to the Green’s function for the doublyincommensurate one-dimensional problem with open bound-ary conditions, Eq. 6 with d = n =
1. The Green’sfunction for each k y is defined as the matrix inverse: G k y ( x , x (cid:48) ) = (cid:16) µ + i δ − H k y (cid:17) − x , x (cid:48) (7)where µ is the chemical potential and i δ is a very small imagi-nary part to round o ff the singularities, which for large systemscan always be chosen so that, to any desired accuracy, the re-sults are δ independent. We calculate both the DOS ρ at theFermi level ρ k y ( µ ) = − π L (cid:88) x Im G k y ( x , x ) (8) FIG. 3. (Upper panel) the DOS ρ and (lower panel) the localizationlength λ versus 1 / B calculated from various values of k y as well by astracing / averaging over all k y (taking L y = L = without a CDW ( V = S k = . S BZ . and the localization length λ that defines the exponential de-cay of the Green’s function G k y (1 , L ) ∼ exp( − L /λ k y ) betweentwo ends of the system. Further details on our numericalmethods are presented in the Appendix A. A. Results for a commensurate CDW
We first benchmark our methods on the same model with noCDW ( V =
0) or a commensurate CDW ( Q = π , V = . µ = − .
2, corresponding toslight hole doping away from half filling to avoid complica-tions that arise from (fine-tuned) Fermi surface nesting on abipartite lattice when µ =
0; our conclusions are readily gen-eralizable to other chemical potentials and shapes of the Fermisurface.To study the QOs for V =
0, we extract the localizationlength λ and DOS ρ over a range of magnetic field on a L = system. As shown in Fig. 3, both the DOS ρ (up-per panel) and the localization length λ (lower panel) exhibita clear single-period oscillation. By linearly fitting the loca-tions of the peaks we obtain the period ∆ (1 / B ) = . S k = π ∆ (1 / B ) = . S BZ (9)where S BZ = (2 π ) is the area of the entire Brillouin zone ofthe square lattice. For comparison, given the zero-field disper-sion relation of Eq. 3 with the assumed value of the chemicalpotential, the Fermi surface enclosed area in the absence ofany CDW is S k = . S BZ . To highlight the high accuracyof our results, we have also marked the theoretically expectedlocations of the maxima from Eq. 2 at the top of each figure.We have checked the expected k y independence of the re-sults by repeating the calculation for di ff erent choices of k y aswell as by averaging over k y as shown in Fig. 3. Henceforthwe suppress the k y label for both ρ and λ .As another benchmark, we have studied QOs in the pres-ence of a commensurate CDW: V = .
16 and Q = π for L = × ; the results for the DOS and the localization lengthare shown in the upper and middle panels of Fig. 4, respec-tively. A Fourier transform (Fig. 4 insets) reveals peaks andhigher harmonics ? that correspond to a fundamental period of ∆ (1 / B ) = . S k = . S BZ (cid:48) , where S BZ (cid:48) = S BZ / S k = . S BZ (cid:48) .Strictly speaking, at any finite magnetic field, the QO pe-riods are magnetic-field dependent due to corrections to thesemiclassical results associated with magnetic breakdown.This behavior is illustrated in the lower panel of Fig. 4,which is the same model but with larger values of B (1 / B ∈ [120 , ∆ (1 / B ) = .
245 are clearlyseen, which corresponds to the Fermi surface area in the ab-sence of the CDW potential ( V = ∆ (1 / B ) = . B. Results for an ICDW
We now consider systems with ICDW, and discuss the con-sequences within various regimes of magnetic field strengthand CDW amplitude.
1. Reconstructed Fermi-surface in the perturbative regime
Many studies of QOs in an ICDW use perturbative ar-guments to evaluate the reconstructed Fermi surfaces. As in-dicated before, this approach is partially justified in the limitof weak CDW potentials V (cid:28) t , where the result can be vi-sualized dynamically as electrons precessing in semiclassicalorbits around a closed Fermi surface, and only occasionally FIG. 4. The DOS ρ (upper panel) and the localization length (middlepanel) versus the inverse magnetic field 1 / B for a system of length L = × with parameters µ = − . V = .
16 and Q = π . In-sets: the Fourier transform of the QOs exhibits a clear peak at period ∆ (1 / B ) = .
464 corresponding to the fundamental period. Lowerpanel: the QOs behavior of the DOS and the localization length overa range of larger magnetic fields B . FIG. 5. The localization length λ versus the inverse magnetic field1 / B for a system of length L = . × , µ = − . Q = . V = .
16 over various ranges of the magneticfield. The last figure is the Fourier transform of the QOs for the range1 / B ∈ [280 , ff erent choices of k y and θ are shown in di ff erent colorsand their results clearly collapse onto the same curve.FIG. 6. The DOS ρ versus the inverse magnetic field 1 / B for a systemof length L = × , µ = − . Q = . V = .
16 over various ranges of the magnetic field. The last figure is theFourier transform of the QOs for the range 1 / B ∈ [280 , ff erent choicesof k y and θ are shown in di ff erent colors; the results clearly collapseonto the same curve. picking up momentum ± Q to jump from one section of theFermi surface to another.To have a secure basis for our discussion, we begin by ap-plying the same numerical technique described above to theincommensurate case. As far as we are aware of, this is thefirst numerical solution of the fully incommensurate prob- FIG. 7. Red crosses: The calculated QOs characteristic periodsversus the ICDW wave vector Q with V = . µ = − . L = × . The black solid curve is the theoretical values derivedfrom the area of the first-order reconstructed Fermi surfaces. lem. To begin with, we consider a CDW with wave vector Q = = π (1 /π ) and potential V = .
16 with µ = − .
2; theresults are shown in Fig. 5 (localization length λ ) and Fig. 6(DOS ρ ). For better convergence and accuracy we work withsystems of size L = . × for the localization length and L = × for the DOS calculations. We find QOs with periodcorresponding to the original (unreconstructed) Fermi surface ∆ (1 / B ) = .
245 which are dominant when the field is not toosmall, while another with ∆ (1 / B ) = .
271 becomes increas-ingly prominent at lower fields; each of these fundamentals isaccompanied by a complicated spectrum of higher harmonics.In lowest order perturbation theory with a momentum transferof ± Q , the area enclosed by the reconstructed Fermi pocket(illustrated schematically as the light yellow patch in Fig. 1)is S k = . S BZ of the original Brillouin zone, which cor-responds to a period of ∆ (1 / B ) = .
12 as we have marked inFig. 5 and 6 together with corresponding higher harmonics.As expected, the above results are independent of the spe-cific choices of k y and θ .To further test the perturbative approach, we repeated theabove calculations with a series of ICDWs with wave vec-tors in the range Q ∈ (2 π/ , π/
3) and with fixed values of V and µ . The main oscillation periods obtained from the peaklocations in the Fourier transformed spectra are indicated bythe crosses in Fig. 7. For comparison, we also include thetheoretical QO period corresponding to the pocket sizes pre-dicted from the lowest order perturbation. The consistency isremarkable.
2. Breakdown of perturbation theory as B → We have anticipated that the finite-order perturbative resultsinevitably break down for small enough magnetic fields. Sinceany gaps generated in n th order perturbation theory are of or-der W ( V / W ) n , the characteristic magnetic field scale above FIG. 8. The Fourier transformations of the QOs of the DOS ρ withparameters V = . Q = . µ = − . V = . Q = . µ = − . L = × . which these gaps are eliminated by magnetic breakdown isexpected to scale with V according to B n ∼ V n . This leadsto the cascade of field ranges that has already been exhibitedin the commensurate case above. The di ff erence is that, whilein the case of commensurability q no new gaps are generatedbeyond q th order perturbation theory, in the incommensuratecase no termination of the cascade is expected. For numeri-cal experiments, it is simpler to study this working in a fixedrange of magnetic fields and increasing the magnitude of theCDW potential V , rather than by going to still lower fields.We thus calculate the QOs of the DOS for somewhat largermagnitude of the CDW potential. The results for two illustra-tive examples: Q = . V = . Q = . V = . µ = − . L = × in these calculations. Di ff er-ent ranges of the magnetic field are shown in di ff erent colorsto emphasize the cross-over between di ff erent regimes.For Q = .
3, distinct points on the Fermi surface are con-nected by Q and 2 Q , so we expect the QOs to be determinedprincipally by the area of the unperturbed Fermi surface forhigh fields, B > B ∼ ( V / W ) , the first order perturbatively FIG. 9. The localization length λ in the presence of a strong ICDW V = . µ = − . Q = . Q = . Q = . λ with V = . µ = − . Q / B =
200 for the same range ofmagnetic field B (green) with L = . × . reconstructed Fermi surface for B > B > B ∼ ( V / W ) , andthe second order reconstructed Fermi surface for B > B > B ∼ ( V / W ) . We do not report the data for the highest fields, B > B but in the upper panel of Fig. 8 one can observea crossover from the dominant oscillations having period of ∆ (1 / B ) = .
97 in the higher field range to ∆ (1 / B ) = . V , respectively. On the other hand, for Q = . Q does not span the Fermisurface, so no new gaps on the Fermi surface are opened insecond order in perturbation theory. However, scattering withmomentum Q and 3 Q (modulo 2 π ) opens distinct gaps on theFermi surface, and indeed the expected crossover from a dom-inant oscillation period of ∆ (1 / B ) = .
07 at relatively highfields to ∆ (1 / B ) = .
32 at somewhat lower fields is observed,in good agreement with the expected periods from the pertur-bative construction.We conclude from the numerical results for V smaller than W and not too small magnetic fields that perturbation theoryprovides a grossly satisfactory description of the structure ofthe QOs, and that it also provides a vivid account of the man-ner in which the perturbative results to any given order breakdown for low enough magnetic fields.
3. Absence of all QOs for large V
For large V > W , clearly perturbation theory is expected tohave no regime of validity. On the other hand, the behaviorcan be well understood in a dual picture using the mapping ofthe problem onto the properties of a three-dimensional crystalwith no CDW but in the presence of a tilted e ff ective magneticfield, (cid:126) B e ff = (cid:2) B ˆ z + Q ˆ y / π (cid:3) , as we will discuss in Sec. V. Inparticular, we will show that for V larger than a characteristic ~B eff ~B eff FIG. 10. Zero-field Fermi surfaces of the three-dimensional e ff ec-tive theory for (left) a strong ICDW with V = . µ = − . V = . µ = − .
2. The cross sectionsperpendicular to (cid:126) B e ff are shown as the blue lines. magnitude V c (defined as the point at which a Lifshitz transi-tion occurs in the band-structure of the dual three-dimensionalcrystal), there are no well defined QOs at all!To illustrate the basic phenomenology, we show in Fig. 9the localization length as a function of B in the presence of areasonably strong CDW potential with V = . µ = − . Q = .
2, 1 .
6, and 2 .
0. Also shown in the figure is the cor-responding curve in the unphysical case in which the CDWordering vector varies in proportion to the magnetic field as Q = α B , with the constant α =
200 chosen so that Q is oforder 1 in the range of fields represented in the figure. Mani-festly, the three curves with fixed Q show no periodic oscilla-tions at all, and indeed relatively weak magnetic field depen-dence of any sort. However, when we vary Q in proportion to B we do see perfectly periodic QOs, albeit with a rather un-usual harmonic content. These features are easily understoodin the dual three-dimensional picture, which we now discuss. V. THE DUAL THREE-DIMENSIONAL VIEW OFQUANTUM OSCILLATIONS
The advantage of the dual three-dimensional representationof the problem is that it allows us to treat the CDW potential V non-perturbatively, by incorporating it into the band-structure.Then provided the magnitude of (cid:126) B e ff = ( B ˆ z + Q ˆ y / π ) is suf-ficiently small (in a sense we will define below) we can againtreat the electron dynamics semiclassically, although now ona three-dimensional Fermi surface. A. Numerical tests of duality
Although the duality is exact in the thermodynamic limit,we begin by performing numerical tests to establish the ac-curacy of the approach for the system sizes used in this study.For this purpose, we use the same numerical approach to solvethe one-dimensional problem with a doubly incommensuratepotential, and compare the results to expectations based onthe three-dimensional e ff ective theory. This is made sim-pler if we consider the unphysical situation in which both B and Q are varied simultaneously, while keeping the ra-tio B / Q fixed, so that the direction of the e ff ective magnetic FIG. 11. Fourier transformations of the QOs of the DOS ρ with astrong ICDW V = . µ = − . Q / B =
50 over a range of1 / B ∈ [500 , ∆ (1 / B ) = .
175 aswell as its higher harmonics. The system size is L = . × . field (cid:126) B e ff = ( B ˆ z + Q ˆ y / π ) remains fixed. Semiclassically,QO frequencies are simply given by the maximum and mini-mum cross-section areas of the three-dimensional Fermi sur-face perpendicular to (cid:126) B e ff , and the numerical results in Fig. 9confirm QOs are present, even for large V . Strong CDW ( V = . µ = − . ffi ciently large that the Fermi surface in thethree-dimensional e ff ective theory is closed. Fig. 10 (leftpanel) provides an illustration of the three-dimensional Fermisurface structure, which is topologically equivalent to a sphereand closed in the ˆ z direction. We numerically obtain QOs ofthe DOS for Q / B =
50 over a range of 1 / B ∈ [500 , L = . × , and the resulting Fouriertransforms is shown in Fig. 11. A clear single peak is ob-served at the expected position as predicted from the Fermisurface maximum cross-section area S BZ / S k = .
174 per-pendicular to (cid:126) B e ff . Weak CDW ( V = . µ = − . (cid:126) B e ff . In particular, we calculate the QOs of the DOS overa range of 1 / B ∈ [1000 , L = . × , and the resulting Fourier transforms for thecases of Q / B = Q / B =
20 are shown in Fig. 12. A cleardouble peak structure is observed in both calculations. Again,the peak positions are consistent with the theory deductionfrom maximal and minimal Fermi surface cross section areas S BZ / S MAXk = .
399 and S BZ / S MINk = .
089 for the Q / B = S BZ / S MAXk = . S BZ / S MINk = . Q / B =
20 case, respectively.
FIG. 12. Fourier transformations of the QOs of the DOS ρ for a weakCDW V = .
16 and µ = − . / B ∈ [1000 , Q / B = .
089 and 2 .
399 as well as their higher harmonics;Lower panel: Q / B =
20, the peaks are at 2 . .
239 as wellas their higher harmonics. The inset is an enlargement over the rangeof the double peaks. The system size is L = . × . B. Results
In the physical case, Q is generally expected to be approx-imately independent of magnetic field. (Although note thatthere are cases, such as the famous field-induced SDW statesin (TMTSF) ClO , in which Q is strongly B dependent. ) Inthis case, both the magnitude and direction of (cid:126) B e ff varies as afunction of B . In terms of the three-dimensional semiclassicaltrajectories, this means that the orientation of the orbits, andhence extremal cross-section areas S ⊥ k ( B ), vary as a functionof the applied magnetic field B , in addition to its usual de-pendence on the band structure parameters t , V , and µ . Sincewe are typically most interested in the small B limit, the mostimportant cases are those in which B e ff is only tilted slightlyaway from the ˆ y directions, and hence the semiclassical orbitslie close to the x − z plane. Maxima in the semiclassical DOS FIG. 13. The value of the orbital phase factor S k / π B e ff over a rangeof magnetic field B : 1 / B ∈ [0 , V = . µ = − . Q = . Q = . Q = . S k is the maximal area of cross-sections perpen-dicular to (cid:126) B e ff for each given B . Results for fixed Q / B =
200 arealso shown for comparison (cyan). Peaks in the DOS occur when-ever these curves cross half-integer values, shown in the figures asthe horizontal dashed lines. are therefore found when1 (cid:112) B + ( Q / π ) = (cid:32) n + (cid:33) π S ⊥ k ( B ) (10)From this equation it is clear that QOs which are per-fectly periodic in 1 / B are no longer possible for generic three-dimensional Fermi surfaces (i.e. generic cross-section areas S ⊥ k ( B )). There is also a qualitative di ff erence in the semiclassi-cal trajectories for V < V c , for which the Fermi surface formsa warped cylinder which is open in the ˆ z direction as shown inthe right panel of Fig. 10, vs. V > V c , in which case the Fermisurface is closed in the ˆ z direction, as shown in the left panel ofthe figure. Here V c ( µ ) is the critical value of V at which a Lif-shitz transition occurs at which the three-dimensional Fermisurface changes its topology.
1. Absence of QOs for V > V c So long as V > V c , there is a well defined maximal cross-sectional area of Fermi surface contours perpendicular to ˆ y .Moreover, for small B , the maximal cross-sectional area per-pendicular to ˆ B e ff , S ⊥ k (cid:46) S ⊥ k , MAX , is close to and boundedby the maximum area for cross sections perpendicular to ˆ y .Therefore, even if B is changed by an arbitrarily large factor,so long as B (cid:28) Q , neither the magnitude of (cid:126) B e ff , nor thevalue of S ⊥ k changes substantially, and hence little variation isexpected of the electronic structure. For example, we show inFig. 13 the value S k / π B e ff over a range of 1 / B with param-eters V = . µ = − . Q = .
2, 1 .
6, and 2 .
0. Thesecurves do not intersect the half integers and are constrained tothe quantum limit with only few Landau levels occupied for0reasonably small B , thus explaining the absence of QOs re-ported in Fig. 9. (The “ghost” of QOs in the Q = . S k / π B e ff passes close to ahalf-integer value, where due to its vicinity to resonance thephysical quantities are sensitive to small variation of the or-bital phase factor). This can be contrasted with the case witha fixed Q / B = S k is invariant and S k / π B e ff islinear in B e ff , producing periodic QOs as a function of 1 / B .Interestingly, for the same V = . µ = − . Q = π/
2, the model can be consid-ered in the conventional framework of Sec. II B, and is a metalwith one piece of closed Fermi surface. As expected, numer-ical calculations show clear and perfectly periodic QOs. Yet,we have shown that for a nearby incommensurate wave vec-tor Q = .
2. Breakdown of semiclassical approximation for V < V c For small values of V / W , the perturbative picture is useful.Nevertheless several new e ff ects become apparent, even in thislimit, when viewed from the dual perspective.This is the regime where the three-dimensional Fermi sur-face is a warped cylinder as shown in the right panel of Fig.10. The projections of the extremal three-dimensional Fermisurface cross sections S ⊥ k ( B ) into the x - y plane, which we de-note as S (cid:107) k ( B ), is related to S ⊥ k ( B ) by S (cid:107) k ( B ) = (cid:18) BB e ff (cid:19) S ⊥ k ( B ) , (11)and some typical projections are shown in Fig. 14. The“rippled” nature of the projected Fermi surface cross sec-tions is a consequence of “neck and belly” variations alongthe ˆ z direction induced by the k z dispersion in the e ff ec-tive three-dimensional Fermi surface; despite its complex-ity, the total area S (cid:107) k ( B ) is close to that of the original two-dimensional Fermi surface in the absence of the CDW (redcurves in Fig. 14), since the contributions from numerous“neck” and “belly” parts become averaged out. We there-fore see that despite the absence of truly periodic oscillationsin 1 / B , there exist approximately periodic oscillations since S ⊥ k ( B ) / (cid:112) B + ( Q / π ) = S (cid:107) k ( B ) / B ∼ const / B .Nevertheless, since there are typically two or more extremalorbits of the three-dimensional Fermi surface, there are con-tributions from multiple S (cid:107) k ( B ) and corresponding “fine struc-tures” in the QOs. As an example, we show in Fig. 15 theQOs of both the localization length and the DOS with param-eters V = . µ = − . Q = . / B ∈ [80 , S (cid:107) k ( B ) / B .As we move to smaller values of magnetic field B , (cid:126) B e ff be-comes closer to ˆ y and its perpendicular planes closer to the FIG. 14. The Fermi surfaces projected to the xy plane for parameters V = . µ = − . Q / B =
80 (upper panel) and Q / B = Q / B and V large Q / B and V Dual 3D model large Q / B and V small Q / B and V TABLE I. The duality between the magnetic breakdown parameterregimes of the original two-dimensional system and its dual three-dimensional version. xz plane, thus the S ⊥ k ( B ) cross sections span more Brillouinzones in the ˆ z direction. Inevitably, the extremal S (cid:107) k ( B ) be-comes closer and the fine structure gets suppressed. If we re-duce B further, however, the projected Fermi surface becomesvery complex and singular, as shown in the lower panel of Fig.14. This is where the three-dimensional representation, whilestill exact, ceases to be useful due to a catastrophic magneticbreakdown of the semiclassical approximation.Interestingly, there is a ‘duality’ between the magneticbreakdown scenarios in the original two-dimensional systemand its dual three-dimensional model: the regime of the mag-netic breakdown (QO period approximately determined by the1 FIG. 15. The QOs of the localization length λ (upper panel) andDOS ρ (lower panel) as calculated numerically and compared withtheoretical simulations over a range of relatively large magnetic field1 / B ∈ [80 , V = . µ = − . Q = . L = . × Fermi surface without ICDW) in the original two-dimensionalsystem corresponds in the dual three-dimensional model to theregime without magnetic breakdown; on the other hand, thetwo-dimensional reconstructed Fermi surface (without mag-netic breakdown) is relevant when magnetic breakdown be-comes important in the dual three-dimensional system. Thequalitative results on the parameter regimes for the magneticbreakdowns in the dual scenarios are summarized in Table I.We discuss more details in Appendix B.
3. V c is a crossover While a sharp Lifshitz transition occurs in the three-dimensional band structure, this does not imply that there isa sharp transition in the QO spectrum at V = V c . It is im-portant to remember that for fixed Q , | (cid:126) B e ff | ≥ Q , so magneticbreakdown e ff ects are significant wherever small gaps occurin the three-dimensional band structure. Of course, this sit- uation necessarily arises as V approaches V c . Consequently,in the exact solution, V c marks a crossover in behavior – acrossover that is increasingly sharp as Q and B get smaller. VI. GENERALIZATIONS OF THE DUALITY RELATIONSA. Localization transition in an incommensurate potential
The problem of a one-dimensional chain in the presenceof an ICDW is a classic problem that has been widelystudied . The corresponding eigenvalue equation is knownas the “almost Mathieu equation.” The dual mapping of thisproblem to a two-dimensional crystal in the presence of amagnetic field is equivalent to the Hofstadter problem with in-commensurate flux density, B = Q / π , and with hopping ma-trix elements 1 and V / x and ˆ y directions, respec-tively. (For V =
2, the one-dimensional equation is known asHarper’s equation.)This equation has been the subject of numerous studies fo-cusing on the supposed “metal-insulator” transition which isthought to occur as a function of V ; from delocalized stateswhich occur for small V to purely localized states which oc-cur for large V . In particular, it has been claimed that for V <
2, all states are delocalized while for V > V < Q is truly irrational. We discuss this aspect of theproblem in detail in a forthcoming paper, but here we sketchthe basic point.In a magnetic field, the electron orbits in the two-dimensional momentum space are on the Fermi surface,which is given by the dispersion relation µ = (cid:15) k = k x + V cos k y . Since the velocity of an electron at the Fermi energyis everywhere perpendicular to the Fermi surface, the semi-classical motion is localized in the ˆ x direction (which is thespatial direction in the original one-dimensional problem) un-less the Fermi surface is open along the ˆ y direction. Thus,in the semiclassical approximation, for fixed µ , states at theFermi level are delocalized provided V ∈ ( − + | µ | , − | µ | ),and otherwise are e ff ectively localized. To test the validity ofthis semiclassical argument, we have numerically computedthe Green’s function G (1 , L ) with the same parameters usedto generate the two-dimensional Fermi surfaces in the upperpanel of Fig. 16, assuming an incommensurate wave vector Q = /
31; the results are shown in the lower panel of Fig. 16.Clearly, the semiclassical results are fully consistent with nu-merical calculations, in contradiction to claims that all statesare delocalized for V < µ (cid:44) Q plays the role of B e ff in this discussion. We have worked witha relatively small value of Q where the semiclassical resultsare expected to be more or less reliable, but for larger valuesof Q , magnetic breakdown should be exceedingly important,especially when V is smaller but close to the transition V c .Therefore, for fixed µ (cid:44)
0, the critical value of V is shifted2 FIG. 16. The Fermi surfaces (upper panel) and the Green’s function − log( G (1 , L )) versus L (lower panel) with parameters V = . µ = − . V = . µ = − . x direction) and V = µ = − . y direction), respectively. An exponentialsuppression is observed for the former two cases while the latter caseindicates de-localization. from the semiclassical value derived above. This e ff ect hasalso been verified in our numerical calculations. B. Bidirectional ICDW in a magnetic field
It is straightforward to generalize similar higher-dimensional dual models and lower-dimensional numericalmethods to a bidirectional ICDW. For a d -dimensionalhyper-cubic lattice in the presence of a uniform magnetic field and H = (cid:80) (cid:126) k H (cid:126) k with H = − (cid:88) x , y ,(cid:126) k t x a † x + , y ,(cid:126) k a x , y ,(cid:126) k + t x a † x − , y ,(cid:126) k a x , y ,(cid:126) k (12) + t y e i Φ y x ,(cid:126) k a † x , y + ,(cid:126) k a x , y ,(cid:126) k + t y e − i Φ y x a † x , y − ,(cid:126) k a x , y ,(cid:126) k + a † x , y ,(cid:126) k a x , y ,(cid:126) k × d (cid:88) ν = t ν cos( Φ ν x − k ν ) + V x cos ( Q x x + k z ) + V y cos (cid:16) Q y y + k w (cid:17) where (cid:126) k is the Bloch vector for the d − x and ˆ y directions. We have also denoted the initialphase of the CDW along the ˆ x and ˆ y directions as k z and k w ,respectively.Similar to the duality argument in Sec. III, by summingover both the k z and k w , the above system is equivalent to a d + ff ective model with two additional hop-pings V x / V y / z and ˆ w directions, and twoadditional magnetic fluxes Q x through each xz plaquette, and Q y through each yw plaquette, respectively. On the other hand,for e ffi cient numerical calculations we can suppress the (cid:126) k in-dices and consider the resulting two-dimensional system. Weleave more details to future work. VII. CONCLUSIONS
For crystals - and this includes the case of a commensu-rate CDW – the existence of a sharp Fermi surface has theprofound consequence that there exist exactly periodic oscil-lations in physical quantities as a function of 1 / B in the limit T → B →
0, and in the absence of disorder. In this pre-cise sense, the breakdown of Bloch’s theorem in the presenceof an ICDW of arbitrary magnitude V eliminates the sharpdefinition of the Fermi surface and the exactly periodic char-acter of the associated QOs. For small V , approximately pe-riodic oscillations survive over parametrically broad rangesof B , but inevitably as B is reduced toward 0, the period ofthe oscillations shifts as smaller and smaller gaps opened inincreasingly high order in perturbation theory become rele-vant. A crossover in the properties occurs at V = V c ∼ W ,beyond which all periodic oscillations, and indeed essentiallyall magnetic field dependence of the electronic structure, iseliminated.In real materials, disorder or finite temperature T limits therange of applicability of our results. At very low fields, wheneither T or the Dingle temperature T ∗ exceeds the cyclotronenergy ω c , all QOs are lost in a way that is independent of thenature of the CDW order.We would like to thank Boris Spivak, Andre Broido, DannyBulmash, Abhimanyu Banerjee, Yingfei Gu and Xiao-liangQi for insightful discussions. This work is supported by theStanford Institute for Theoretical Physics (YZ), the NationalScience Foundation through the grant No. DMR 1265593(SAK) and DOE O ffi ce of Basic Energy Sciences under con-tract No. DEAC02-76SF00515(AM).3 Appendix A: Green’s function and recursive relations
Inspired by the recursive Green’s function method, herewe propose an e ffi cient method to calculate some targetedGreen’s functions and their related physical measurable. First,it is straightforward to derive: H ky − µ − i δ = h h h h . . .. . . . . . where h x = (cid:16) π Φ x + k y (cid:17) + V cos ( Qx + φ ) − µ − i δ . Here δ denotes a very small imaginary part to round o ff the singu-larities and account for a finite line width for the energy lev-els and will be useful later in the Green’s function. For largeenough systems the detailed value of δ does not make essen-tial changes to our results and conclusions. In particular, theGreen’s function for each k y is given by the matrix inverse: G k y = (cid:16) µ + i δ − H k y (cid:17) − For simpler notation, let’s define: D ij ≡ det h i h i + h i + . . .. . . . . . h j which has the following recursive relations: D ij = h j D ij − − D ij − D ij = h i D i + j − D i + j or equivalently: D ij / D ij − = h j − D ij − / D ij − D ij / D i + j = h i − D i + j / D i + j with the initial conditions: D = h , D = h h − D LL = h L and D L − L = h L − h L − The localization length:
The Green’s function between two ends of the system alongthe ˆ x direction is exponentially suppressed as the systemlength L increases: | G (1 , L ) | ∼ exp ( − L /λ ). λ is the localiza-tion length - an important property that characterizes transportbehaviors of the system. To calculate G (1 , L ) we need to traceover k y : G (1 , L ) = (cid:88) y G ( { , y } ; { L , y } ) = (cid:88) k y G k y (1 , L ) yet we have shown in the main text that in the presence of asmall magnetic field Φ , the localization length derived from G (1 , L ) is equivalent to that from each individual G k y (1 , L ).Then, by standard matrix inversion procedure, it is not hardto obtain: G k y (1 , L ) = ( − L / D L which can be e ffi ciently derived using the recursive relations.We can perform a log-linear fit with respect to L for the local-ization length λ .Further, the above expression is consistent if we use G (1 , x ) ∼ exp ( − x /λ ) to extract λ with a fixed system size L : G k y (1 , x ) = ( − x D x − L / D L . The density of states(DOS):
The DOS at chemical potential µ is another important phys-ical quantity that we mainly focus on: ρ ( µ ) = − π LL y (cid:88) x , k y Im G k y ( x , x ) = − π L (cid:88) x Im G k y ( x , x )Again, we have shown in the main text that the DOS aver-aged over k y is equivalent to that for each specific k y . We alsonote that: − G k y ( x , x ) = D x − D x + L / D L = D x − D x + L / (cid:16) D x − D x + L h x − D x − D x + L − D x − D x + L + (cid:17) (cid:104) (cid:16) h x − D x − / D x − − D x + L / D x + L (cid:17) − (A1)where in the last step we have neglected the term of 2 since2 / D L is exponentially small. − G k y ( x , x ) may be e ffi cientlyextracted with the recursive relations. Both λ and ρ shouldoscillate as a function of 1 / B with periods given by the cross-section area of the Fermi surface. Appendix B: Duality in magnetic breakdown scenarios
As we have shown in the main text, for V (cid:28) W and B (cid:28) Q , the dual three-dimensional Fermi surface cross sec-tion’s projected area onto the xy plane, S (cid:107) k , is almost invari-ant and asymptotically close to the original Fermi surface,therefore we expect QOs with a period ∆ (1 / B ) in the regimewhere the electron semiclassical orbit restores the originalFermi surface. The magnetic breakdown in the original two-dimensional system are given by the dual three-dimensionalmodel with no magnetic breakdown.On the other hand, in the limit B →
0, the magnetic break-down in the original two-dimensional system gets washed outand the QOs are dominated by the CDW reconstructed Fermisurface. However, in this limit, the Fermi surface in the dualthree-dimensional model also gets increasingly complex andsingular (see Fig. 14 lower panel), and magnetic breakdownbecomes essential. The electrons can tunnel between the rip-ples given (cid:96)δ k > (cid:96) is the magnetic length (that isalmost constant since B e ff ∼ Q / π ) and δ k is the distance be-tween two neighboring ripples, thus denser (larger Q / B ) and4 FIG. 17. Upper panel: The localization length λ and the DOS ρ (in-sets) with parameters V = . µ = − . Q / B = / B ∈ [1000 , Q ∈ [1 , B gets smaller. Lower panel: ifwe reduce B even further: 1 / B ∈ [2800 , L = . × . stronger (larger V ) ripple structures are favorable to the mag-netic breakdown, which is exactly opposite to the regimes ofthe original two-dimensional system, see Table I.To manifest the impact of magnetic breakdown in the dualthree-dimensional model, we present in Fig. 17 the numeri-cal results on both the localization length and the DOS witha fixed ratio Q / B = / B ∈ [1000 , ff erentfrom those in Fig. 12. This can be explained as the follow-ing: the magnetic breakdown introduces tunnelings betweenthe ripples and e ff ectively increases the area enclosed by theelectron trajectory S (cid:107) k ( B ). As we lower the magnetic field B e ff ∝ B , the distance the electrons can tunnel through is re-duced, thus S (cid:107) k ( B ) is a decreasing function of 1 / B . When thishappens, the orbital phase factor S (cid:107) k ( B ) / B is slower than 1 / B ,the periodicity in 1 / B is lost and the interval between the os-cillations becomes and larger, see Fig. 17 upper panel. On theother hand, when the magnetic field B e ff ∝ B is reduced fur-ther so that the magnetic breakdown becomes unimportant inthe dual three-dimensional model, the periodicity should berestored. This is exactly what we observe in Fig. 17 lowerpanel. S. Chakravarty, Science, , 735 (2008); S. Chakravarty, H-Y.Kee Proc. Natl. Acad. Sci. USA , 8835 (2008) N. Doiron-Leyraud et al. , Nature , 565-568 (2007). S. E. Sebastian et al., Phys. Rev. Lett. , 196403 (2012). S. E. Sebastian et al., Nature , 61-64 (2014). Andrea Allais, Debanjan Chowdhury, and Subir Sachdev, NatureCommunications 5, 5771 (2014). M. H. Cohen and L. M. Falikov, Phys. Rev. Lett. , 231(1961). E. I. Blount, Phys. Rev. Lett. , 114(1960). Jean-Michel Carter, Daniel Podolsky, and Hae-Young Kee, Phys.Rev. B , 064519(2010). R. G. Chambers, Proc. Phys. Soc. London , 701(1966). S. Aubry and G. Andr´e, Ann. Israel Phys. Soc. 3, 133 (1980). J. B. Sokolo ff , Physics Reports No. 4, 189(1985). A. J. Millis and M. R. Norman, Phys. Rev. B , 220503 (2007). H. Yao, D-H. Lee, and S. A. Kivelson, Phys. Rev. B , 012507(2011). Jonghyoun Eun, Zhiqiang Wang, and Sudip Chakravarty, Proc.Natl. Acad. Sci. USA (33), 13198 (2012). P. M. Chaikin, J. Phys. I France , 1875 (1996).16