Quantum oscillations in a bilayer with broken mirror symmetry: a minimal model for YBa 2 Cu 3 O 6+δ
QQuantum oscillations in a bilayer with broken mirror symmetry: a minimal model forYBa Cu O δ Akash V. Maharaj , Yi Zhang , B.J. Ramshaw , and S. A. Kivelson Department of Physics, Stanford University, Stanford, California 94305, USA. and Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA. (Dated: September 11, 2018)Using an exact numerical solution and semiclassical analysis, we investigate quantum oscillations(QOs) in a model of a bilayer system with an anisotropic (elliptical) electron pocket in each plane.Key features of QO experiments in the high temperature superconducting cuprate YBCO can bereproduced by such a model, in particular the pattern of oscillation frequencies (which reflect “mag-netic breakdown” between the two pockets) and the polar and azimuthal angular dependence ofthe oscillation amplitudes. However, the requisite magnetic breakdown is possible only under theassumption that the horizontal mirror plane symmetry is spontaneously broken and that the bi-layer tunneling, t ⊥ , is substantially renormalized from its ‘bare’ value. Under the assumption that t ⊥ = ˜ Zt (0) ⊥ , where ˜ Z is a measure of the quasiparticle weight, this suggests that ˜ Z (cid:46) /
20. Detailedcomparisons with new YBa Cu O . QO data, taken over a very broad range of magnetic field,confirm specific predictions made by the breakdown scenario.
Quantum oscillations (QOs) are a spectacular conse-quence of the presence of a Fermi surface. Their obser-vation in the high T c cuprate superconductors com-bined with recent observations of charge density wavecorrelations , have led to a compelling view of thenon-superconducting “normal” state of the underdopedcuprates at high fields, H > H c , and low temperatures, T (cid:28) T c . In this regime, small electron-like Fermi pocketsarise from reconstruction of a larger hole-like Fermi sur-face presumably due to translation symmetry breaking inthe form of bidirectional charge-density-wave (CDW)order .However, to date, no theory of Fermi-surface recon-struction by a simple CDW can simultaneously accountfor the Fermi pockets and the relatively small magnitudeof the measured specific heat, which presumably re-flects the persistence a pseudo-gap that removes otherportions of the original (large) Fermi surface. Thus,rather than trying to infer the origin of the Fermi pock-ets, we explore a generic model of a single bilayer splitpocket to elucidate general features that can most easily account for the salient features of the QOs.Specifically, we focus on the bilayer cuprate YBCO, inwhich quantum oscillations have been studied in great-est detail. The frequency of the QOs and the negativevalues of various relevant transport coefficients establishthe existence of an electron pocket enclosing an area oforder 2% of the Brillouin zone. A typical spectrum ofQOs in underdoped YBCO is shown in Fig. 1. Whilethere is some suggestive evidence of more than one basicfrequency—which might suggest more than one pocketper plane —we instead adopt and further elucidatea suggestion of Harrison and Sebastian that the“three-peak” structure of the spectrum of oscillation fre-quencies reflects magnetic breakdown orbits associatedwith a single, bilayer-split pocket. In refining this sug-gestion, we show that, although many aspects of theQO experiments can be successfully accounted for in thisway, the requisite magnetic breakdown is forbidden in the ( T ) A m p li t u d e ( a . u . )
35 45 55Field ( T ) T o r q u e ( a . u . ) FIG. 1: Typical Fourier transform of QOs of the magnetictorque for underdoped YBa C O . ( T c = 60 K, p ≈ T ≈ . presence of a mirror symmetry that exchanges the planesof the bilayer; thus, a heretofore unnoticed implication isthat the high field phase must spontaneously break thissymmetry. Other striking features of the quantum os-cillations are the existence of prominent “spin zeros” and a strong C symmetric dependence of the oscillationamplitudes on the in-plane component of the magneticfield with no evidence of the enhancement at the “Ya-maji angle” expected from the simplest “neck and belly”structure of a quasi-2D Fermi surface . We show that all these experimental features are con-sistent with a simple model in which there is an ellipti-cal Fermi pocket in each of the planes of a bilayer, with a r X i v : . [ c ond - m a t . s t r- e l ] O c t their principal axes rotated by π/ component of whatever orderedstate exists in this range of T and B . We assume a (cid:126)k in-dependent coupling between the layers within a bilayer, t ⊥ , and we neglect all inter-bilayer coupling, t (cid:48)⊥ ≈
0. Aswe will discuss in Sec. IV, both these assumptions seemmore natural in the context of experiments and band-structure calculations of YBCO than those made by Se-bastian et al. in their pioneering treatment of this sameproblem. Specifically, Sebastian et al. assumed a strong (cid:126)k dependence associated with a presumed vanishing of t ⊥ in certain crystallographic directions, a significant rolefrom a non-zero t (cid:48)⊥ , and broken translation symmetry inthe c-direction ; these do not feature in our minimalmodel.Finally, we have uncovered a quantitative issue withpotential qualitative implications for magnetic break-down. The magnitude of t ⊥ sets the size of the gapbetween bilayer split Fermi surfaces thus controlling theimportance of magnetic breakdown orbits. Because ournumerical approach treats magnetic breakdown exactly(rather than using a Zenner tunneling approach), we areuniquely placed to examine this effect. We have foundthat in order for magnetic breakdown to play a signif-icant role in the relevant range of B , it is necessary toassume that t ⊥ is a factor of 20 or more smaller thanits “bare” value t (0) ⊥ , which can be estimated either fromband-structure calculations or from angle resolvedphotoemission (ARPES) studies of overdoped YBCO. .As was emphasized both in ARPES measurements andprevious theoretical studies , the ratio, ˜ Z ≡ t ⊥ /t (0) ⊥ ,is a measure of the degree of single particle interlayer co-herence, and so is related to the quasiparticle weight.This implies that the quasiparticles responsible for theQOs are very strongly renormalized, with ˜ Z (cid:46) . Logic and Organization of the Paper
In Sec. I, we define an explicit lattice model of non-interacting electrons with a band-structure engineered toproduce the desired small elliptical electron-like Fermipockets (shown in Fig. 2), and describe the numericalalgorithm we have used to obtain exact results for thismodel as a function of an applied magnetic field. To ori-ent ourselves, in Sec. II we sketch the semiclassical anal-ysis (including the effects of magnetic breakdown) whichwill allow us to associate the oscillation frequencies wewill encounter with the geometry of the Fermi surface.We then present results of the numerical analysis of themodel in Sec. III: In Fig. 3 we present the ideal QO spec-trum, while in Fig. 4 we exhibit the way in which higher - π - π π π - π - π π π k x k y (a) t ⊥ = 0 - π - π π π - π - π π π k x k y (b) t ⊥ = 0 . t a FIG. 2: The Fermi surface of the bilayer system in (a) theabsence ( t ⊥ = 0) and (b) the presence ( t ⊥ = 0 . t a ) of anisotropic interlayer tunneling t ⊥ . The parameters used are t b = t a / µ = − . t a . Note that we have zoomed into an area that is one quarter of the full (unreconstructed)Brillouin zone. harmonics are rapidly suppressed by a non-infinite quasi-particle lifetime. We then present spectra that resultwhen the range of magnetic fields analyzed is confined torealistically accessible values, discussing both qualitativeand quantitative trends as parameters are tuned (see Fig.5). We also study the polar and azimuthal angular de-pendence of the QOs (see Fig. 6 and 7), and develop ac-curate semiclassical arguments to interpret our numericalresults (see Figs. 7 and 8). Finally, in Sec. IV we discussthe implications of our results for the interpretation ofexperiments in the cuprates, including comparison withnewly presented QO data taken on YBa Cu O . , whichis used to test key features of the magnetic breakdownscenario discussed here. We also discuss the connectionwith other related theoretical work. I. THE MODEL
We study a tight-binding model of electrons hoppingon two coupled layers, each consisting of a square lat-tice with purely nearest-neighbor hopping elements. Inthe presence of an arbitrarily oriented magnetic field theHamiltonian of this model has the form H = (cid:88) (cid:104) (cid:126)r i ,(cid:126)r j (cid:105) ; σ (cid:88) λ − t (cid:126)r i − (cid:126)r j ; λ (cid:16) e i Φ ij c † (cid:126)r i ,λ,σ c (cid:126)r j ,λ,σ + H.c. (cid:17) + (cid:88) (cid:126)r i ; σ (cid:88) λ π ˜ gBσc † (cid:126)r i ,λ,σ c (cid:126)r i ,λ,σ (1) − (cid:88) (cid:126)r i ; σ t ⊥ (cid:16) e i Φ zi c † (cid:126)r i , ,σ c (cid:126)r i , ,σ + H.c. (cid:17) where c † (cid:126)r i ,λ,σ is an electron creation operator at position (cid:126)r i in layer λ = 1 , σ = ± /
2, and t (cid:126)r i − (cid:126)r j ; λ de-notes the appropriate hopping matrix element in layer λ ,while t ⊥ is the (isotropic) hopping between each layer inthe bilayer and ˜ g controls the strength of Zeeman split-ting. Here, Φ ij = (cid:82) (cid:126)r i (cid:126)r j A ( r ) d r is the phase obtained byan electron hopping from site (cid:126)r j to (cid:126)r i in units in which (cid:126) c/e = 1, while Φ zi is the phase obtained upon tunnel-ing from one layer to the next at position (cid:126)r i . To ob-tain perpendicularly oriented elliptical pockets we set t ˆ x ;1 = t ˆ y ;2 = t a , and t ˆ y ;1 = t ˆ x ;2 = t b . In the absenceof a magnetic field this Hamiltonian can be diagonalizedto give the spectrum E ± ( k ) where k = ( k x , k y ) is a twodimensional Bloch wavevector, with E ± ( k ) = ε + ( k ) ± (cid:113) ε − ( k ) + t ⊥ (2) ε ± ( k ) = − ( t a ± t b ) cos( k x ) − ( t b ± t a ) cos k y . (3)The Fermi surface with and without interlayer tunneling,with the choice of t b = t a / µ = − . t a is shown in Fig. 2.In the absence of t ⊥ , the addition of a magnetic fieldmaps Eq.1 to two copies of the Hofstadter problem. Uponcoupling the layers, and for fields at arbitrary polar ( θ )and selected azimuthal angles ( φ ), we can always picka gauge that preserves translation symmetry along thein-plane direction of the magnetic field, ˆ e . This allowsus to take the Fourier transform along ˆ e , and map Eq. 1to a modified Harper’s equation. For simplicity, we willconsider the case in which the magnetic field lies in the y − z plane, with the generalization to arbitrary orienta-tion deferred to Appendix A. With B = B (0 , sin θ, cos θ ),we can choose the gauge A = (0 , π Φ x, − π Φ x tan θ ) , (4)where Φ = B cos θ is the density of magnetic flux quantaper x − y lattice plaquette (in units in which the plaquettearea is 1).Upon Fourier transforming the Hamiltonian in the ˆ y direction we have H = (cid:80) k y ,σ ˆ H k y ,σ :ˆ H k y ,σ = (cid:88) x,λ (cid:110) t ˆ x,λ (cid:16) c † ( x +1 ,k y ); λ ; σ + c † ( x − ,k y ); λ ; σ (cid:17) + (cid:20) t ˆ y,λ cos (2 π Φ x − k y ) + 4 π ˜ g Φ σ cos θ (cid:21) c † ( x,k y ); λ ; σ (cid:27) c ( x,k y ); λ ; σ + (cid:88) x t ⊥ (cid:16) e − i π Φ a c tan θ c † x,k y ;2; σ c x,k y ;1; σ + H.c. (cid:17) (5)where a c is the ratio of inter-bilayer spacing to the in-plane lattice constant. Eq. 5 has three properties thatmake it particularly amenable to numerical analysis: (i)the two spins σ = ± / H are independent of k y in the thermo-dynamic limit , allowing us to suppress the k y summa-tion; 3) the resulting one-dimensional problem concern-ing ˆ H k y ,σ is a block tri-diagonal matrix, whose inverse(and by extension, the Green’s function) can be calcu-lated recursively as described in Appendix C, allowing efficient evaluations of its physical properties on systemsizes as large as L x ∼ sites along the ˆ x direction. Inthe remainder of the paper, we will be presenting calcula-tions of QOs in the density of states (DOS) ρ at chemicalpotential µ , defined as ρ ( µ ) = − πL x Tr (cid:16) Im[ ˆ G ] (cid:17) = − πL x (cid:88) x,λ Im[ G ( x,λ ) , ( x,λ ) ( µ )] (6)where G ( x,λ ) , ( x,λ ) ( µ ) represents the diagonal entry of theGreen’s functionˆ G ( µ ) = (cid:104) ( µ + iδ ) ˆ I − ˆ H k y ,σ (cid:105) − , (7)The small imaginary term iδ gives a finite lifetime to theelectrons and broadens the Landau levels. Choice of Parameters
For a range of values, the qualitative aspects of ourresults do not depend sensitively on the values of most ofthe parameters that enter the model (with the exceptionof the pattern of magnetic breakdown, which we shallsee is extremely sensitive to the value of t ⊥ ). However,to facilitate comparison with experiment, we chose pa-rameters so that the k -space area enclosed by the ellipti-cal Fermi pockets in the absence of interlayer tunnelingis S ≈ T = 1 . BZ , the mean cyclotron effectivemass m ∗ ∼ . m e , and the electron’s spin g factor is g = 2. (See Appendix D for further discussion.) In theabsence of any direct experimental information concern-ing the ellipticity of the Fermi pockets, we have arbitrar-ily adopted a moderate anisotropy, √ √ t b = t a / µ = − . t a , and ˜ g = 0 .
87. Since all our calculationsare carried out at T = 0, the overall scale of energiesis unimportant, but when referring to quantitative fea-tures of the electronic structure of YBCO, we will take t a = 400meV, in which case a characteristic inverselifetime is δ = 0 . t a ≈ (2 ps) − . We convert fluxquanta per unit cell, Φ, into units of the actual mag-netic field B , by using a unit cell area of YBCO to be ν unit cell = 3 . × . B is relatedto the flux per unit cell (in units of the flux quantum) by B = ( h/e ) × (Φ /ν unit cell ) ≈ Φ × t ⊥ , and the inverse lifetime δ aretreated as unknowns; exploring the changes in the QOspectrum which occur as they vary is one of the principlepurposes of this study. II. SEMICLASSICAL CONSIDERATIONS
Before undertaking the numerical solution of thismodel, it is useful to outline the results of a semiclassi- ( Tesla ) ρ ( μ ) (a) ρ vs. B / B ( T - ) ρ ( μ ) (b) ρ vs. 1 /B ✏✏ ↵ ↵ + + ( Tesla ) FT [ ρ ( μ ) ] + + ✏ T n = 4 T n = 1 ✏ T n = 4 T n = 1 ↵ T n = 2 T n = 4 (c)Fourier transform of ( b ) FIG. 3: QOs of the DOS for very small broadening δ =0 . t a (long lifetimes) and t ⊥ = 0 . t a , in the absence of aZeeman coupling (˜ g = 0). Panels ( a ) and ( b ) show the calcu-lated DOS ρ vs. B and 1 /B ; panel ( c ) is the Fourier transformof panel ( b ). Each peak indicates a characteristic frequency ofQOs and the corresponding semiclassical orbits are also illus-trated above. The number of equivalent semiclassical orbits, n , is indicated below each orbit, and we have explicitly shownthe two distinct classes of γ orbits. A relatively large rangeof magnetic field is used 4 T < B < T to capture all ofthe QO frequencies. The system size is L x = 2 . cal analysis to anticipate the basic structure of the QOsin the simplest situation in which B is perpendicular tothe planes. As we are considering weakly coupled bi-layers, we will always assume that t ⊥ (cid:28) t ≡ √ t a t b , sothe bilayer split Fermi surfaces have narrowly avoidedcrossings at four symmetry related points, as shown inFig. 2b. Electrons adhere strictly to semiclassical orbitsonly so long as (cid:126) ω c (cid:28) t ⊥ /t since magnetic breakdownat these four points becomes significant otherwise. (Here ω c ∼ φt is the cyclotron frequency.) Taking this magneticbreakdown into account, there are five distinct classesof semiclassical orbits, as shown in the middle panel ofFig. 3, each enclosing a k -space area which, when con-verted into an oscillation frequency, correspond to fiveoscillation frequencies separated by ∆ f ≈ T for themodel parameters we have defined. (These correspondto the frequencies labeled α , β , γ , δ , and (cid:15) in the spec-trum in the lower panel of the figure, whose calculationis discussed in the next section). ρ ( μ ) = 0 . t a = 0 . t a = 0 . t a ρ ( μ ) B cos ( θ )( T - ) ρ ( μ ) = 0 . t a = 0 . t a FIG. 4: The evolution of QOs in the DOS ρ ( µ ) for variousvalues of the inverse quasiparticle lifetime, δ . The interlayertunneling t ⊥ = 0 . t a and the rest of the parameters aredetailed at the end of Sec. I. The largest and smallest orbits represent the truestructure of the Fermi surface, so these two frequen-cies ( α and (cid:15) ) must dominate the QO spectrum when (cid:126) ω c (cid:28) t ⊥ /t . Conversely, in the limit (cid:126) ω c (cid:29) t ⊥ /t ,where to good approximation we can set t ⊥ = 0, thespectrum is dominated by the central frequency ( γ ), inwhich the electron orbits are confined to a single planeof the bilayer, and hence correspond to the ellipses inFig. 2a. More complex spectra, including those withthe three peak structure seen in experiment, occur onlywhen (cid:126) ω ∼ t ⊥ /t . This, we shall see, allows us to estimatethe magnitude of t ⊥ directly from experiment.We will return again to a semiclassical analysis, below,in order to understand still more subtle features of theQO spectrum which appear when the magnetic field istilted relative to the Cu-O plane. III. NUMERICAL RESULTS
In presenting our results, we will adopt two comple-mentary approaches. We first study an idealized theo-retical limit of infinitesimally small broadening ( δ → t ⊥ , and subsequently exam-ining the angular dependences. While we predominantlyhighlight the robust qualitative features of this model, wealso focus on the quantitative aspects of magnetic break- ( Tesla ) FT [ ρ ( μ ) ] t ⟂ = t a , δ = t a ( Tesla ) t ⟂ = t a , δ = t a ( Tesla ) t ⟂ = t a , δ = t a ( Tesla ) t ⟂ = t a , δ = t a ( Tesla ) t ⟂ = t a , δ = t a ( Tesla ) FT [ ρ ( μ ) ] t ⟂ = t a , δ = t a ( Tesla ) t ⟂ = t a , δ = t a ( Tesla ) t ⟂ = t a , δ = t a ( Tesla ) t ⟂ = t a , δ = t a ( Tesla ) t ⟂ = t a , δ = t a FIG. 5: The raw Fourier transform the density of states oscillations as the interlayer tunneling t ⊥ is increased (left to right),and the inverse lifetime δ is increased (top to bottom). Larger values of t ⊥ suppress the central frequencies and enhance thesatellite frequencies which correspond to orbits of the true bilayer split Fermi surface. Shorter quasiparticle lifetimes (larger δ )lead to decreased harmonic content. The field range used here is 20 T < B < T , with 2 data points. down, which are treated exactly in our numerical studies.In Fig. 3 we show the density of states as a functionof magnetic field strengths for a c -directed field. The toppanels show data where the broadening is infinitesimal at δ = 0 . t a and there is no Zeeman splitting. Each Lan-dau level is split due to the presence of two coupled layers,while the peaks in the density of states rise linearly with B as expected for free fermions. The lower panel of Fig. 3shows the Fourier transform of this data over a largerange of magnetic fields (4 T < B < T ). Here thehigh harmonic content of the oscillations is clearly seen,with comparable-in-magnitude first and second harmon-ics. For the first harmonics, there are five peaks clusteredaround a central frequency of f = 530 T , as expected fromsemiclassical considerations, while at higher frequenciesthere are all the expected harmonic combinations givingrise to a complicated spectrum. A. Dependence on interlayer tunneling and lifetime
We now study the model over an experimentally realis-tic range of magnetic fields with a finite Zeeman coupling,˜ g = 0 .
87. Fig. 4 and 5 show the evolution of the QOs asthe interlayer tunneling t ⊥ and Landau level broadening δ are varied, where we have reduced the range of mag-netic field to 20 T < B < T to conform roughly withthe range explored by current experiments in YBCO. Thefigures are constructed from 2 data points. As is clearfrom Fig. 4 the form of the oscillations is radically al-tered as the lifetime is decreased ( δ in Eq. 7 is increased),with the sharp Landau level structure of the spectrum be-coming broadened. This leads to oscillations with littleharmonic content, while the amplitude of the oscillatorysignal is also sharply suppressed.Fig. 5 shows the Fourier transform of ρ as both theinterlayer tunneling t ⊥ is increased (from left to right)and the inverse lifetime δ is increased (from top to bot-tom). Several qualitative features of the results are im- mediately apparent. (1) As the inverse lifetime δ is in-creased (and the oscillations of ρ become less singular),the peaks in the Fourier transform are also broadenedwhile the higher frequency peaks are preferentially sup-pressed in amplitude, leading to oscillations with littleharmonic content. This has a simple semiclassical inter-pretation: higher frequency peaks correspond to longersemiclassical orbits and so are suppressed in amplitudeby the decreasing quasiparticle lifetime. (2) The com-petition between different semiclassical orbits is sensi-tively controlled by the interlayer tunneling t ⊥ : as t ⊥ isincreased, the gaps between bonding and anti-bondingFermi surfaces increase, and the weight of QOs rapidlyshifts from the central frequency at 530 T (correspondingto the 3rd orbit in Fig. 3 which involves two magneticbreakdowns across the true Fermi surface of the bilayer)to the side frequencies at (530 ± T (corresponding tothe second and fourth semiclassical orbits in Fig. 3),and is eventually dominated by the outermost frequen-cies at (530 ± T (reflecting the ‘true’ bonding andanti-bonding Fermi surfaces of the bilayer).Indeed, a particularly appealing feature of our ap-proach is its exact treatment of magnetic breakdown.The immediate quantitative observation from Fig. 5 isthat maintaining the large (experimentally observed) ra-tio of the amplitude of the central 530 T frequency tothat of the satellite frequencies at 530 ± T requires verysmall values of the interlayer tunneling t ⊥ < . t a . Thisis at least an order of magnitude below the typical valuesof t ⊥ ∼ . t a assigned by band structure studies andARPES studies on overdoped YBCO, but agrees re-markably with ARPES measurements of the underdopedregime. We discuss the consequences of this observationin Sec. IV.
500 1000 15000.00000.00050.00100.00150.00200.00250.00300.0035 f ( Tesla ) FT [ ρ ( μ ) ] ° ° ° ° ° ° ° ° ° ●● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●
20 40 60 80 θ - - ( arbunits ) ●● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●
20 40 60 80 θ - - ( arbunits ) ●● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●
20 40 60 80 θ - - - ( arbunits ) (a) (b) 530T, Fit: A cos ⇥ . ⇡ ✓ ⇤ (c) 440T, Fit: A cos ⇥ . ⇡ ✓ ⇤ (d) 620T, Fit: A cos ⇥ . ⇡ ✓ ⇤ FIG. 6: (a) The Fourier transform of the DOS QOs for variouspolar angles θ of the magnetic field (cid:126)B (different angles havebeen arbitrarily offset). (b)-(d) Extracted peak heights in theFourier transform (we have used both amplitude and phaseinformation) versus the polar angle θ . The parameters are δ =0 . t a , t ⊥ = 0 . t a . The dashed lines are the theoreticalfits of the angular dependence due to spin splitting (Eq. 8)which is caused by Zeeman effect. B. Polar angle ( θ ) dependence of the QOs We now move on to cases where the magnetic field istilted away from the principal c -axis of this model andstudy the dependence of the QOs on the polar and az-imuthal angles, θ and φ ; we also comment briefly on thecorresponding dependence seen in YBCO. Key experi-mental features include the presence of spin zeroes near θ ≈ . ◦ and θ ≈ . ◦ , with the notable absence of aYamaji angle that is typical of simple s -wave warping ofa three dimensional Fermi surface. Spin zeros (as wellas the general θ dependence) arise due to Zeeman split-ting of spinful electrons. This coupling effectively shiftsthe chemical potential (and hence the area of each or-bit) oppositely for each species σ , by an amount that isproportional to the applied field δf = σγB . Such a B dependent shift of the Fermi surface area for each spinspecies becomes a shift of the bare (spinless) frequency f of oscillations, so that the amplitude of oscillationsfor the p ’th harmonic acquires a field independent (but θ dependent) amplitude: ρ (cid:18) B , θ (cid:19) ∝ (cid:88) σ = ± / cos (cid:18) πp ( f + σγB ) B cos θ (cid:19) = 2 cos (cid:16) πpγ cos θ (cid:17) cos (cid:18) πp f B (cid:19) . (8)A more careful analysis shows that this field indepen-dent amplitude takes the form A ( θ ) = cos (cid:16) πpg m ∗ m e cos θ (cid:17) where in practice the factor πpgm ∗ / m e is related to ourdefinition of ˜ g as discussed in Appendix D.Fig. 6(a) shows the polar angle θ dependence of theFourier transform of QOs for the model system in Eq.1. The azimuthal angle is fixed at φ = 45 ◦ throughoutthe calculation. As expected, no Yamaji-like resonance isseen because of the absence of a truly three-dimensionaldispersion. Fig. 6(b)-(d) shows the θ dependence of theQO amplitude A ( θ ) at the three main frequencies. Wesee characteristic spin-zeroes in the primary frequency f = 530 T near θ = 51 . ◦ and θ = 63 . ◦ . The dashedlines show fits of the amplitude to the form given in Eq.8 - remarkable agreement is found. We note that thepositions of the spin zeroes are different for the QOsat frequencies 440 T , 530 T and 620 T , despite the factthat the g -factor (our parameter ˜ g ) has been definedto be the same for all orbits. This robust feature ofour model can be attributed to the different effectivemass of the these three orbits which enters the formcos ( πpgm ∗ / m e cos θ ), and is explored further in Ap-pendix D. C. Azimuthal angle ( φ ) dependence of the QOs Another notable feature of QO experiments in YBCOis the dependence of the amplitude of the oscillations onthe azimuthal angle φ . The oscillation amplitudes exhibita four-fold anisotropy, which increases with increasingpolar angle θ . Here, we show that these features canbe reproduced in our model of a single bilayer, with thecaveat that strong anisotropy is only natural for selectedorbits that involve both layers of the bilayer ( β orbit at440 T , δ orbit at 620 T and γ orbit at 530 T ).In analyzing the behavior of QOs for different az-imuthal angles, much information can be gleaned fromsemiclassical intuitions. First, note that the semiclassi-cal orbits γ at the central 530 T frequency in Fig. 3 arepredominantly confined to a single layer of the bilayer.Such 2 d orbits are only affected by the field perpendic-ular to the layer, therefore no observable azimuthal de-pendence is expected. On the other hand, all other semi-classical orbits shown in Fig. 3 involve tunneling eventsfrom one layer to the next, upon which electrons may ob-tain a phase proportional to the horizontal magnetic field B sin θ . This means that there is weak four-fold depen-dence arising from γ orbits, wherever the signal is domi-nated by the 530 T frequency; conversely, a large four-fold B sin ✓ sin zk x k y kc Trajectory in Layer 1
Trajectory in Layer 2 (a)3D Cartoon plot of semiclassical orbit in the bilayer(b)Corresponding φ dependence of the QOs at 620 T and 440 T given in Eq. 9 FIG. 7: (a) A schematic diagram showing how the in-planecomponent of the flux enclosed by a given semiclassical or-bit ( δ orbit in Fig. 3) is determined. The red curve is thesemiclassical orbit, while grey ellipses are the Fermi surfaces.Note that the in-plane directions are in momentum space,while the vertical separation is in real space. The vertical re-gion enclosed (shaded gray) has a (real space) area of δk(cid:96) B c .(b) The φ dependence of QO amplitude A ( φ ) as in Eq. 9 forvarious values of the polar angle θ for δk = 0 . δk = 0 . θ and/or δk . modulation arises from the 530 ± T frequencies, and sois pronounced near to spin zero angles { θ , θ } of themain 530 T frequency.Within the semiclassical framework, we can obtain ananalytic expression for the amplitude as a function of az-imuthal angle φ by determining the amount of in-planedirected magnetic flux enclosed by a given breakdownorbit. Fig 7(a) shows the geometry of a particular break-down orbit for QOs at 620 T , where the total horizontalflux is the real space area corresponding to the shadedregion, multiplied by the field component B sin θ sin φ .Semiclassically, we find the real space area enclosed bythe orbit to be δk(cid:96) B × c , where (cid:96) B = h/eB cos θ is thesquare of the magnetic length, and δk is the distance be-tween the (avoided) crossings of the Fermi surfaces (seeFig. 7(a)). Thus, the in-plane flux enclosed by this orbitis Φ yz = δk (cid:126) eB c × B tan θ cos φ = cδk tan θ sin φ with our FIG. 8: The relative amplitude of the QOs at frequency 620 T for polar angles θ = 63 . ◦ and θ = 51 . ◦ , respectively. TheQO amplitude at φ = 0 is maximal and set as the unit 1for each of the data sets. The solid curve is the theoreticalexpectation value according to Eq. 9 and the expected C rotation symmetry is clearly present. The parameters used inour numerical calculations of the DOS QOs are t ⊥ = 0 . δ = 0 .
001 and 11
T < B < T .FIG. 9: The relative amplitude of the QOs at frequency 530 T for polar angles θ = 60 ◦ . The solid curve is the fit to theo-retical form given by Eq. 10 with M = − .
5. The parame-ters used in our numerical calculations of the DOS QOs are t ⊥ = 0 . δ = 0 .
001 and 11
T < B < T . choices of units. Similarly, there are three other possi-ble enclosed fluxes related by C rotations, and given byΦ − yz = − Φ yz , and Φ ± xz = ± cδk tan θ cos φ . The result-ing Φ j , j = ± xz, ± yz each give an additional constantinitial phase to the in-plane fluxes that determine theQOs of the corresponding reconstruction, which add upto give the overall amplitude: A ( φ ) ∝ (cid:88) j exp ( i Φ j ) ∝ xz + 2 cos Φ yz (9) ∝ cδk tan θ cos φ ) + 2 cos ( cδk tan θ sin φ )Examples of the azimuthal angular dependence givenby Eq. 9 are shown in Fig. 7(b), whose form guarantees C rotation symmetry. For smaller values of the polarangle θ (and thus a smaller overall factor cδk tan θ ), the φ angular dependence is suppressed. We note that themagnitude of the anisotropy depends sensitively on δk .An example of the QO amplitude variation for a larger δk = 0 . δk = 0 . T has the same result as Eq. 9,while at 530 T we need to consider both the γ and γ orbits: A (cid:48) ( φ ) ∝ M + 2 cos ( cδk tan θ (cos φ + sin φ ))+ 2 cos ( cδk tan θ (cos φ − sin φ )) (10)where M is a complex constant for the contribution from γ orbits and sensitively depends on the parameters in-cluding t ⊥ and B .Another immediate consequence of this expression isthat the maximum in QO amplitudes at the side fre-quencies at (530 ± T occurs when the field is alignedwith the principal axes of the ellipses. Given that exper-imentally, the maximum of the oscillation amplitudes isseen to occur for fields along the a and b crystallographicdirections, it is natural that the principal axes of suchelliptical pockets must lie along the a and b directions,i.e. such azimuthal dependence seemingly rules out pro-posals where the principal axes of the Fermi pockets areoriented at 45 ◦ to the a and b crystallographic directions.Returning to the model at hand, we calculated numeri-cally the density of states QOs with selected values of theazimuthal angle tan φ = 0 , , , , T and polar angles θ = 63 . ◦ and θ = 51 . ◦ are shown in Fig. 8 and is fully consistentwith the semiclassical expression derived in Eq. 9. Inparticular, the selected θ values are the spin zeros of thecentral frequency at f = 530 T of the QOs, where the ef-fect of the side frequencies at (530 ± T are enhanced.In addition, four-fold anisotropy is also seen for the QOamplitudes at frequency 530 T and polar angle θ = 60 ◦ asshown in Fig. 9, and fits well to Eq. 10 with parameter M = − . IV. IMPLICATIONS FOR THE CUPRATES
We have shown that a simple model of criss-crossedelliptical electron pockets can reasonably account for themost striking experimental observations of QOs in the bilayer cuprate YBCO. In particular, we have shownthat a three peak structure in the Fourier transformof QOs follows naturally from the ansatz of brokenmirror symmetry and weak bilayer splitting. Thechoices of tight-binding and Zeeman-splitting parametersthat best capture this physics have been analyzed semi-quantitatively. We have also demonstrated that majorfeatures of both the azimuthal and polar angular depen-dence of the QOs can be qualitatively reproduced by thissimplified model of a single bilayer.A central feature of our analysis involves the small ef-fective interlayer tunneling t ⊥ required to account forthe prominence of the central 530 T frequency relativeto those at 530 ± T . In certain situations, a singu-lar k dependence of the bare interlayer tunnel-ing, t (0) ⊥ ( k ) ≈ t (0) ⊥ (cos k x − cos k y ) , arises due to the lo-cal quantum chemistry. In this case the small value ofthe effective t ⊥ could reflect the location of the electronpockets along the “nodal” direction in the Brillouin zonewhere | k x | = | k y | , rather than any non-trivial many-bodyeffect. However, there are strong reasons to doubt thatthe bilayer tunneling in YBCO has such strong k depen-dence. On theoretical grounds, LDA studies havefound that the tunneling between the ‘dimpled’ planesof a YBCO bilayer remains substantial even along thenodal direction with t (0) ⊥ ( k n ) ≈ t (0) ⊥ ( k an ) ≈ where analmost isotropic bilayer splitting of ∆ ε k n = 2 t ⊥ ( k n ) =2 Zt (0) ⊥ ( k n ) ≈ ε k an ≈ Z ≈ .
5. This isin sharp contrast to underdoped samples, where despitethe theoretical (LDA) prediction of a doping indepen-dent t (0) ⊥ , the nodal bilayer splitting is difficult to re-solve. These experiments give an upper bound of thenodal quasiparticle weight in the underdoped regime of Z n < . Z n ≈ .
03. Such es-timates agree remarkably well with our estimate of theeffective value of t ⊥ necessary to account for the QO’sin underdoped YBCO. The constraint of the quasiparti-cle weight ˜ Z (cid:46) .
05, strongly suggests that the effectiveFermi liquid parameter t ⊥ = ˜ Zt (0) ⊥ is renormalized signif-icantly downwards. A. Comparison with previous proposals
There have been many proposals for the origin ofthe Fermi surface reconstruction in the cuprates. Givenrecent observations of (seemingly ubiquitous ) incom-mensurate CDW order, a prime candidate for the Fermisurface is one where nodally located electrons pocketsare produced by incommensurate CDWs which are atleast bi-axial, involving ordering at (cid:126)Q x = ( Q, , / (cid:126)Q y = (0 , Q, / . In this scenario, a diamond shaped, nodallylocated electron pocket is split by bilayer tunneling (withthe above mentioned (cos k x − cos k y ) form factor), withall three observed frequencies involving orbits where theelectron tunnels from one layer to the next. The nodallocation also serves to suppress simple isotropic ( s -wave)hoping in the c axis direction, leading to an absence of aYamaji resonance.The model discussed in the present paper, while simi-lar in spirit to that of Harrison and Sebastian, possessescrucial differences of symmetry and effective dimension-ality. Under the assumption that QO experiments probethe physics of a single bilayer, mirror symmetry betweenthe two layers of this bilayer must be broken in order forbreakdown orbits to be present in a purely c -axis directedmagnetic field – otherwise a conserved bilayer parity as-sociated with the split Fermi surfaces would prevent allmagnetic breakdown (see Appendix B). Indeed, there isevidence for such broken symmetry in the low field chargeorder. Once mirror symmetry is broken, a naturalconsequence is that the central 532 T frequency reflects asemiclassical orbit where electrons are confined to a sin-gle layer of the bilayer, and if so, is naturally the mostprominent in the regime of small interlayer tunneling. We have demonstrated that the experimental observa-tions can be generally accounted for in the context of aminimal model of a single bilayer. In contrast to previousproposals, this model requires no specific 3 d -structureof the Fermi surface, and makes no specific assump-tions about the nature of the order that reconstructs theFermi surface; given that recent high field X-ray scatter-ing experiments have given evidence of an unexpected,distinct high-field character of the CDW order, we viewthis lack of specificity as a virtue. B. Further tests from experiments in YBa Cu O . The magnetic breakdown scenario makes two specificpredictions for QO experiments in bilayer cuprates:1. Oscillations taken over a sufficiently large fieldrange should show five spectral features distributedsymmetrically about the main frequency, plus mul-tiple higher harmonics from combination orbits.2. The weight of the various frequency components ofthe quantum oscillations should be field-dependent,with orbits that require fewer breakdown eventsdominating at low fields.Fig. 10 shows torque magnetometry data taken onYBa Cu O . at 1.5 kelvin. Multiple spectral compo-nents, beyond the three main peaks identified in previousstudies but consistent with those presented in section III, → → ( T ) A m p li t u d e ( a . u . )
25 35 45 55Field ( T ) T o r q u e ( a . u . ) α β γ δ ϵ FIG. 10: Fourier transform of torque quantum oscillation inYBa Cu O . . Analysis of the full field range, from 18.5 to62.6 Tesla (red curve), reveals spectral features not present inFig. 1, but that correspond well with the frequencies shownin Fig. 3. Analysis of the oscillations between 18.5 and 26Tesla only (blue curve) show that spectral weight is shiftedaway from the main γ peak, and toward the sidelobes. Theblue curve has been multiplied by a factor of 10 and trun-cated at 700 T for clarity. Note that spectral features below ≈
150 T are removed as part of the background-subtractionprocedure, and thus this data does not address the possibil-ity of a 90 T frequency that has been reported in transportmeasurements . are clearly visible with this extended field range (18.5 to62.6 Tesla). Appendix E demonstrates that these peaks(particularly α and (cid:15) ) are not artifacts of the Fouriertransform, but are instead physical components of theoscillatory signal.Transforming the data over a limited low-field range,from 18.5 to 26 T (blue curve in Fig. 10), shows thatthe main 530 T peak is indeed no longer dominant.Semiclassically , the probability of tunneling throughany one of the four junctions between the bilayer splitFermi surfaces (Fig. 2) is P = e − B /B , where B is thecharacteristic breakdown field. The probability of avoid-ing breakdown (Bragg reflection) at a junction is (1 − P ).While this expression is not exact (unlike the breakdowntreatment in section III), particularly at fields large com-pared to B , it gives intuition as to why the spectralweight shifts at lower fields: the γ orbit shown in Fig.3 requires four breakdown events, while the α ( (cid:15) ) orbitrequires none and the β ( δ ) orbit requires two. Note thatthe field range used to obtain the blue curve in Fig. 10is insufficient to resolve the splitting of these peaks. Fi-nally, the dominance in amplitude of lower frequenciesover higher frequencies originates in the suppression oflarger orbits due to quasiparticle scattering .0 V. ACKNOWLEDGEMENTS
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0) and (0 , Q ) chargedensity wave order parameters occurring simultaneouslyin the same domain. Such a state could preserve C sym-metry, as in “checkerboard order,” or, as in the proposed“criss-cross stripe” phase of Ref. 40, could break this andother point-group symmetries. This is to be contrastedwith unidirectional or stripe order where a single wavevector CDW occurs per domain which necessarily impliesbreaking of C rotational symmetry. More complex states in which CDW order coexists withother orders, including an incommensurate dDW anda CDW in a FL* phase , have been proposed whichmay offer a way to reconcile these observations. While the Yamaji angle is not experimentally observedclose to its expected value θ = 59 ◦ , we note that if theunit cell were to be somehow doubled in the c -axis direc-tion, a Yamaji angle would be possible near to 38 ◦ , which isexactly where an enhancement is observed. Because thereis no compelling evidence for c -axis unit cell doubling, weignore such a possibility in this work. Instead, the ‘beat’near to 40 ◦ in our numerical analysis arises because of spin-zeroes of the satellite peaks. We note that recent X-ray diffraction experiments inpulsed magnetic fields up to 30 T have observed coherent(long ranged) charge density wave order which does not double the unit cell in the c direction While ARPES studies provide a direct measure of the elec-tronic spectral function, and are therefore sensitive to theexact quasiparticle residue Z , this is not the same param-eter which enters into the effective Fermi liquid parame-ter t ⊥ = ˜ Zt (0) ⊥ in QO experiments. Here, ˜ Z is a measureof interlayer coherence, which in the limit of degenerateinter-layer perturbation theory becomes exactly the quasi-particle residue. The γ orbits involve tunneling from one layer to the next,and do not contribute in the limit of t ⊥ = 0. In order to smoothen the Fourier transform, we have addedanother 2 zeroes to the ends of the data, and also em-ployed a Kaiser window function with width parameter α = 2 to eliminate ringing effects. We emphasize that thisprocedure introduces no additional harmonic content tothe data. We note that such broken mirror symmetries may be de-tected by probing higher rank tensorial response functionsin transport experiments.
Nevertheless, this a breakdown orbit of the ‘true’ bondingand anti-bonding Fermi surfaces of the bilayer QOs are suppressed by a factor of e − B D /B where the Din-gle reduction factor B D can be written in terms of onlythe momentum-space circumference of the Fermi surface C F and the mean-free path l free , via B D = (cid:126) C F el f ree . Appendix A: Form of the Hamiltonian for general angles φ For additional azimuthal angles as tan φ = 1 /M where M ∈ Z (or equivalently tan φ = M by symmetry): (cid:126)B = B (cid:16) ˆ z cos θ + ˆ x sin θM/ (cid:112) M + 1 + ˆ y sin θ/ (cid:112) M + 1 (cid:17) (A1)we can no longer keep the translation symmetry along the ˆ y direction for arbitrary B with the chosen Landau gauge,however, we can define the new magnetic unit cell with the new lattice vectors ˆ x (cid:48) = ˆ x, and ˆ y (cid:48) = M ˆ x + ˆ y along (cid:126)B inplane or equivalently x (cid:48) = x − M y , y (cid:48) = y . Once again we can choose a proper gauge so that the translation symmetryalong the ˆ y (cid:48) direction is preserved: A = (cid:16) , π Φ( x − M y ) , − π Φ a c tan θ ( x − M y ) / (cid:112) M + 1 tan θ (cid:17) = (cid:16) , π Φ x (cid:48) , − π Φ x (cid:48) a c tan θ/ (cid:112) M + 1 (cid:17) (A2)where Φ = B cos θ is the magnetic flux through the plaquette in the x − y plaquette, and Φ a c tan θ/ √ M + 1 is theflux through the x − z plaquette. The hopping matrix elements no longer depend on y (cid:48) , therefore we can Fouriertransform into the corresponding k (cid:48) y momentum basis. The resulting Hamiltonian (for each k (cid:48) y and spin σ ) becomes:ˆ H k (cid:48) y ,σ = (cid:88) x,λ t x,λ (cid:104) c † x +1 ,λ c x,λ + H.c. (cid:105) + 4 π ˜ g Φ σ cos θ c † x,λ c x,λ (A3)+ (cid:88) x,λ t y,λ (cid:104) c † x − M,λ c x,λ exp (cid:0) i π Φ x − k (cid:48) y (cid:1) + H.c. (cid:105) + (cid:88) x t ⊥ (cid:104) c † x, c x, exp (cid:16) i π Φ xa c tan θ/ (cid:112) M + 1 (cid:17) + H.c. (cid:105) where we have suppressed the k (cid:48) y and σ labels in the fermion operators. The Hamiltonian is still block tri-diagonaland its physical properties including DOS can be efficiently calculated using recursive Green’s function method. Appendix B: Mirror symmetry and the absence of breakdown frequencies
Here we discuss in further detail the absence of magnetic breakdown when a mirror symmetry relating the twoplanes of the bilayer is present. The essence of this symmetry argument is the following: in the presence of amagnetic field semiclassical dynamics correctly captures the motion of electrons, while magnetic breakdown is allowedas long as there exist matrix elements that take electrons from one orbit to the next. However, if there is a mirrorplane perpendicular to the magnetic field, the mirror parity of the states remains a good quantum number even inthe presence of a magnetic field. There are necessarily no matrix elements between states with different quantumnumbers, and so breakdown processes are forbidden by this symmetry. We emphasize that this argument is alsoapplicable in the limit of a single bilayer, i.e. when k z is not a good quantum number.This symmetry may be viewed at a more operational level by considering the Hamiltonian of a bilayer with identicaldispersions ε ( k ) in each layer. In the absence of a field, this takes the form H = (cid:88) k Ψ k ˆ H k Ψ k = (cid:88) k = k x ,k y ( c † k , c † k , ) (cid:18) ε ( k ) t ⊥ ( k ) t ⊥ ( k ) ε ( k ) (cid:19) (cid:18) c k , c k , (cid:19) (B1)where t ⊥ ( k ) is the (in general) momentum dependent tunneling between layers.Mirror symmetry relating the two layers of the bilayer is akin to the statement that the Hamiltonian commuteswith the x -Pauli matrix, ˆ τ x : (cid:104) ˆ H k , ˆ τ x (cid:105) = 0 , where ˆ τ x = (cid:18) (cid:19) (B2)It should be clear that this operation swaps the two planes of the bilayer, and so implements that mirror operationthat we are referring to. The addition of a magnetic field B is typically implemented via a Peierls substitution,resulting in a dramatic change to the structure of the Hamiltonian and eigenstates. In particular, working in Landau3 - - - - - - k x k y f = 2887 T f = 2704 Tf + = 3092 T ( Tesla ) FT [ ρ ( μ ) ] t ⟂ = δ = t a , Mirror Symmetric ( Tesla ) FT [ ρ ( μ ) ] t ⟂ = t a Cos [ ] , Mirror Symmetric - - - - - - k x k y (a)Mirror symmetric f = 2700 Tf + = 2894 Tf = 2521 T - - - - - - k x k y ( Tesla ) FT [ ρ ( μ ) ] t ⟂ = δ = t a , Broken Mirror Symmetry - - - - - - k x k y ( Tesla ) FT [ ρ ( μ ) ] t ⟂ = t a Cos [ ] , Broken Mirror Symmetry Eccentricity ⇥ (b)Broken mirror symmetry FIG. 11: Fourier transforms of QOs in the density of states for two models (a) with mirror symmetry and (b) without mirrorsymmetry. In (a) we consider identical Fermi surfaces with an interlayer tunneling of the form t ⊥ ( k ) = t ⊥ cos (2 k y ), while in(b) the mirror symmetry is weakly broken by considering orthogonal Fermi surfaces with a weak mass anisotropy ( t b = 0 . t a )and the interlayer tunneling is once more t ⊥ ( k ) = t ⊥ cos (2 k y ). The bilayer bonding and anti bonding Fermi surfaces arealmost identical in both cases, yet the QO frequencies are dramatically different: mirror symmetry forbids breakdown orbitsin (a). gauge we only preserve translation invariance in a single direction, so in general the eigenstates will be labeled by ageneralized Landau level index, n , and transverse momentum, k y . However, as long as the magnetic field does notbreak this mirror symmetry, i.e. B = B z ˆ z , it remains the case that eigenstates of ˆ H are also eigenstates of ˆ τ x , i.e.[ ˆ H, ˆ τ x ] = 0 (B3)ˆ H | n, k y , ±(cid:105) = E ( n,k y , ± ) | n, k y , ±(cid:105) (B4)ˆ τ x | n, k y , ±(cid:105) = ±| n, k y , ±(cid:105) (B5)Note that these are the exact eigenstates of the system, and they are necessarily orthogonal. Also notice that none ofthese statements depend on the form of the interlayer tunneling t ⊥ ( k ).The absence of magnetic breakdown is then most easily understood by considering the structure of the energyspectrum. Oscillations in any physical quantity arise because of periodicity in the structure of the energy spectrum asa function of 1 /B . The discrete two-fold mirror symmetry means that the Hamiltonian separates into two independentblocks, so that the energy spectrum for these + and − sectors can be solved independently. Because these sectorscan be treated as independent systems, as the magnetic field is varied, each sector produces a single fundamentalfrequency in quantum oscillations. This results in two (possibly degenerate) quantum oscillation frequencies, withneither magnetic breakdowns nor beat (sum or difference) frequencies.Fig. 11(a) and 11(b) provide confirmation of these symmetry arguments. In Fig. 11(a) we have considered identicaldispersions ε ( k ) = − t (cos k x + cos k y ) − µ with t = 1 and µ = − . t , and t ⊥ ( k ) = − . t cos (2 k y ). This form of theinterlayer tunneling is both technically simple to implement, and produces nodes in the bilayer splitting. As is clearfrom the Fourier transform, no magnetic breakdown is present, and only two fundamental frequencies are seen whenthe interlayer tunneling is present. In Fig. 11(b) we weakly break the symmetry by considering dispersions of theform ε ( k ) = − t a cos k x + t b cos k y ) − µ in one layer, and ε ( k ) = − t b cos k x + t a cos k y ) − µ in the next layer, with t b = 0 . t a . In the absence of interlayer tunneling, only one frequency is seen in QOs (these pockets have identicalareas), but a finite interlayer tunneling leads to multiple breakdown orbits.4 Appendix C: Recursive Green’s function method for the DOS of a tri-diagonal block Hamiltonian
As is shown in the main text, the Hamiltonian in k y and σ basis only involves finite-range coupling and is blocktri-diagonal ˆ H k y ,σ = . . . ... . . . ˆ h x − ,σ ˆ t ˆ t ˆ h x,σ ˆ t ˆ t ˆ h x +1 ,σ . . . ... . . . , (C1)ˆ h x,σ = (cid:18) t y, cos(2 π Φ x − k y ) + π ˜ gBσ cos θ t ⊥ exp( i π Φ a c x tan θ ) t ⊥ exp( − i π Φ a c x tan θ ) 2 t y, cos(2 π Φ x − k y ) + π ˜ gBσ cos θ (cid:19) , (C2)ˆ t = (cid:18) t x, t x, (cid:19) (C3)We are interested in the DOS ρ σ ( µ ) of spin σ electrons at chemical potential µ defined as ρ σ ( µ ) = − πL x Tr (cid:16) Im[ ˆ G σ ( µ )] (cid:17) (C4)ˆ G σ ( µ ) = (cid:104) ( µ + iδ ) I − ˆ H k y ,σ (cid:105) − (C5)where we have used the fact that the physical quantities are independent of k y in the thermodynamic limit to suppressthe summation over the k y index.To obtain the diagonal elements of the Green’s function ˆ G σ ( µ ), we note the inverse of the following block tri-diagonalmatrix may be calculated recursivelyˆ G − σ ( µ ) = ( µ + iδ ) I − ˆ H k y ,σ = a , a , a , a , a , a , a , a , . . . . . . . . . (C6)where a i,i = ( µ + iδ ) I − ˆ h x,σ and a i,i +1 = a i,i +1 = ˆ t . This is accomplished by the following recursive algorithm, whichconsists of two independent sweeps (and hence the computation is linear in the size L x ):For increasing i = 1 , , . . . , N − c Li = − a i +1 ,i ( d Li ) − , (C7)with d L = a , and d Li = a i,i + c Li − a i − ,i ; for decreasing i = N, N − , . . . , c Ri = − a i − ,i ( d Ri ) − , (C8)where d RN = a N,N and d Ri = a i,i + c Ri +1 a i +1 ,i , then the diagonal blocks of ˆ G σ ( µ ) = (cid:104) ( µ + iδ ) I − ˆ H k y ,σ (cid:105) − are givenby ˆ G i,i = ( − a i,i + d Li + d Ri ) − , i = 1 , , , . . . , N (C9) Appendix D: Effective masses of electron pockets and Zeeman splitting coefficient ˜ g
1. Value of ˜ g coefficient for Zeeman splitting in our tight-binding model The effective mass of a band structure is defined as m ∗ = (cid:126) π ∂S k ∂µ (D1)5 - - - - - - k x k y Uncoupled pockets, μ =- t a - - - - - - k x k y Coupled layers, t ⟂ = t a , μ =- t a - - - - - - μ A r ea ( T ) Pocket Area
Pocket Area: 531T Outer Pocket Area: 715TInner Pocket Area: 348T 531T ✏ ↵ ↵ ✏ µ/t a µ = . t a A k ( T e s l a ) T FIG. 12: Different slopes of S k versus µ suggest the effective masses are different for the different orbits. The areas of α , γ and (cid:15) orbits are obtained from exact calculations of the Fermi surface, while for β and δ orbits the areas are based on interpolationbetween the α , γ and (cid:15) orbits (shown as the thinner lines). The vertical line is the value of µ = − . t a chosen throughoutour calculations. where S k is the k -space area enclosed by the Fermi surface at chemical potential µ .The dispersion relation in our tight-binding model in one of the single layers is (equivalent to the 3rd orbit in Fig.3) (cid:15) k = − t a cos k x a − t b cos k y b (cid:104) − t a − t b + t a k x a + t b k y b (D2)near the bottom of the band, where a and b are the sizes of the unit cell. At chemical potential µ the Fermi surfaceis close to an ellipsis with k x = (cid:113) µt a a and k y = (cid:113) µt b b , thus the area enclosed by the Fermi surface S k = πk x k y = πµab √ t a t b (D3)The effective mass of the model near the band bottom is m ∗ = (cid:126) ab √ t a t b (D4)By definition, the Zeeman splitting is E Zeeman = ± g µ B B = ± gπ (cid:126) abm e Φcos θ = ± π √ t a t b gm ∗ m e Φcos θ (D5)where µ B = e (cid:126) / m e is the Bohr magneton and Φ is the dimensionless quantity of the number of magnetic fluxquantum Φ = h/e per x − y plaquette.Note that g = 2 for electron spin and m ∗ /m e (cid:104) . t a = 1 and t b = 1 / E Zeeman (cid:104) ± . × π Φ / cos θ (D6)In fact, the quadratic approximation in (cid:15) k in Eq. D2 underestimates the effective mass m ∗ due to the higherorder terms we have neglected. A more careful treatment and comparison between the numerical and theoretical θ dependence suggests the best choice is E Zeeman (cid:104) ± . × π Φ / cos θ (D7)suggesting ˜ g = 0 .
87 in connection with Eq. 5.
2. Effective mass for different semiclassical orbits
While ˜ g = 0 .
87 determines the effective mass of the electron pocket in a single layer and the central peak in the QOpower spectrum, it is conceivable that the effective mass of the other viable semiclassical cyclotron orbits associatedwith the side peaks be different, as their enclosed areas are necessarily modified - Fig. 12 shows the enclosed areas ofthese orbits as the chemical potential is varied, and the effective mass extracted from the corresponding slope accordingto Eq. D1 is fully consistent with that obtained from the fit to QO amplitude versus θ angle of the magnetic field (cid:126)B in Fig. 6.6 FIG. 13: The inset shows two simulated data sets: one is apodized with a boxcar function (black), and the other uses equationE1 with α = 1 . Appendix E: Fourier Transform Analysis
Fast-Fourier transforms of finite data sets are known to introduce frequency ‘artifacts’ into power-spectrum plots.These artifacts originate in the choice of how the data is truncated. For example, a ‘boxcar’ function—whereby thesignal is simply truncated at the start and end—introduces high-frequency components due to the sharp cutoffs atthe data boundaries. Modern signal processing solves this through ‘apodization’, whereby the data is brought tozero in some way at the boundary. The choice of apodization function depends on what features in the data are ofinterest.The data in Fig. 10 were processed using a Kaiser window, designed to resolve closely-spaced frequencies whilesuppressing side-lobes (at the expense of absolute amplitude determination, which was not important for this analysis).The weighting function w for N data points is defined as w ( n ) = I (cid:32) πα (cid:114) − (cid:16) nN − − (cid:17) (cid:33) I ( πα ) , (E1)where I is the zeroth modified Bessel function of the first kind and α controls the roll-off of the weighting function(chosen to be 1.7 for this work). Fig. 13 shows the effect of such a windowing function on a signal and its Fouriertransform.Simulated QO data is shown in the inset of Fig. 14. The data contains only the three central frequencies: 440, 530,and 620 T. Specifically, the function is τ = e − /B (cid:18) cos (cid:18) π B − π (cid:19) + cos (cid:18) π B − π (cid:19) + cos (cid:18) π B − π (cid:19)(cid:19) . (E2)Note the lack of side-lobes near 350 and 710 T: this demonstrates that the α and (cid:15) peaks in Fig. 10 are not artifactsof the data analysis.7 FIG. 14: The inset shows simulated data from equation E2 before the window is applied. The Fourier transform uses the Kaiserwindow with α = 1 ..