The Mixed State of a π -Striped Superconductor
aa r X i v : . [ c ond - m a t . s up r- c on ] N ov The Mixed State of a π -Striped Superconductor M. Zelli, ∗ Catherine Kallin, and A. John Berlinsky
Department of Physics and Astronomy, McMaster University, Hamilton, Canada (Dated: November 3, 2018)A model of an anti-phase modulated d-wave superconductor has been proposed to describe thedecoupling between Cu-O planes in 1 / − x Ba x CuO . Unlike a uniform d-wave supercon-ductor, this model exhibits an extended Fermi surface. Within Bogoliubov-de Gennes theory, westudy the mixed state of this model and compare it to the case of a uniform d-wave superconductor.We find a periodic structure of the low-energy density of states, with a period that is proportionalto B , corresponding to Landau levels that are a coherent mixture of particles and holes. Theseresults are also discussed in the context of experiments which observe quantum oscillations in thecuprates, and are compared to those for models in which the Fermi surface is reconstructed due totranslational symmetry breaking in the non-superconducting state and to a model of a Fermi-arcmetal. I. INTRODUCTION
High temperature superconductivity in the under-doped cuprates condenses from a so-called pseudogapphase whose properties are distinctly different from thoseof a conventional metal.
Below a relatively high tem-perature, T ∗ , a gap which may or may not be connectedto superconductivity starts to develop in the excitationspectrum and affects the temperature dependence of alltransport processes. At lower temperatures, above butcloser to T c , superconducting fluctuations in the form ofa disordered vortex-antivortex liquid grow up until long-range d-wave superconducting order appears at T c . The nature of the pseudogap phase has been the sub-ject of much study and debate. One can characterize thebehavior in this phase as “non-Fermi-liquid-like”, whichtypically means that the sharp fermionic excitations ofa Fermi liquid are broadened even close to the Fermisurface (FS), although the situation in the pseudogap issomewhat more complicated. ARPES observes “Fermiarcs” – sections of 2D FS which appear to terminate atgaps and which become shorter, possibly tending towardnodal points, as T is lowered. No sharp quasiparticlesare observed near the anti-nodal points at ( ± π,
0) and(0 , ± π ).The origin of this non-Fermi-liquid-like behavior ishotly disputed. Some studies connect it to resonating va-lence bonds or preformed pairs, while others associateit with exotic forms of fluctuating or static spatial or-der such as charge or spin density waves or singlet ortriplet D-density waves. This unsatisfactory state of understanding was com-pounded in 2007 by the observation of quantum oscilla-tions, first in the Hall resistivity and shortly afterwardin Shubnikov-de Haas and in de Haas-van Alphen mea-surements, below T c at fields that are comparable to, al-though typically smaller than, H c . Since such fieldsare still weak relative to Fermi energy scales, one mightexpect that, once superconductivity is destroyed by alarge magnetic field, the underlying resistive state wouldbe the same pseudogap state as exists above T c , that is,a non-Fermi liquid. However quantum oscillations are normally associated with sharp, closed Fermi surfaces ofa Fermi liquid. Furthermore, it is not straightforwardto connect the FS areas determined by quantum oscilla-tions with the Fermi arc observed in ARPES. Howeverit has been noted that commensurate static translationalsymmetry breaking, due to charge or spin density waves,could reorganize the large hole FS of the undistorted lat-tice into a number of smaller hole and electron pocketsand that the small electron pockets could account for thequantum oscillations, while sections of the larger holepockets coincide with the Fermi arc.
The explana-tion for the arcs is then that the spectral weight dueto the periodic perturbation of the charge-density wave(CDW) or spin-density wave (SDW) is large on the arcsthat are observed by ARPES and small on the remainderof the FS hole pockets. Such a result is consistent witha simple picture of zone folding due to a weak periodicsuperlattice potential.In this paper, we consider a variation of this picture inwhich the periodic superlattice arises from a modulationof the d-wave superconducting gap function.
Sucha modulation has been invoked to explain an interestingphenomenon called the 1 / Most lanthanumcuprates exhibit singular behavior in the doping depen-dence of various low-temperature properties around 1 / Furthermore,the stacking arrangement assumed for this model resultsin a zero Josephson coupling between nearest cuprate lay-ers which explains the apparent dynamical decoupling ofcuprate layers observed in transport measurements of 1 / − x Ba x CuO . In this paper, we study sucha modulated d-wave gap in the presence of large mag-netic fields using lattice Bogoliubov-de Gennes (BdG)theory to determine if quantum oscillations, associatedwith Landau level formation, occur.The remainder of this paper is organized as follows. InSec. 2, we describe the model for a π -striped supercon-ductor. This section shows the density of states (DOS),the spectral functions and the FS in zero field for differ-ent amplitude gaps for this model. In Sec. 3, we establishthe generation of Landau levels in the DOS by a magneticfield and the effect of doping and of the gap amplitude onthe Landau level spectra. In Sec. 4, the specific heat iscalculated to make some connections to the experiments.Finally, in Sec. 5, we discuss the results and comparethem to other models and to relevant experiments. II. THE MODEL AND METHOD
The two-dimensional tight-binding model for a π -striped superconductor is described by the followingmean-field Hamiltonian H = H + X x,y ∆ { cos( q x x )[ c † x,y ↑ c † x +1 ,y ↓ − c † x,y ↓ c † x +1 ,y ↑ ](1) − cos( q x ( x − / c † x,y ↑ c † x,y +1 ↓ − c † x,y ↓ c † x,y +1 ↑ ] + H.C. } where c † x,yσ creates an electron with spin σ on site ( x, y ).By setting q x = π/
4, the Hamiltonian describes a systemwith a d-wave-type order parameter that has a sinusoidalmodulation with 8-site periodicity in the x direction. H ,the kinetic part of the Hamiltonian, has the dispersion ǫ = − t (cos( k x ) + cos( k y )) − t cos( k x ) cos( k y ) − µ in k space where t and t are the first and second nearestneighbor hopping terms. Due to the periodic modulationof the order parameter, the superconducting condensateoccurs at a non-zero q , and a particle with wave vector k is coupled to ones with wave vectors − k ± q x . We shallsee that this property of a striped superconductor hascrucial effects on its low-energy properties.There are two possible stable configurations for theorder parameter of the π -striped superconducting model.One configuration is the site-centered configuration inwhich the node of the modulation lies on a site. Theother one is the bond-centered configuration in which thenode lies on a bond. The calculations in this work aredone for the latter configuration which is shown in Fig.1. However, the qualitative behavior of the system in thepresence of a magnetic field is similar for the site-centeredcase.We solve the model by diagonalizing the BdGHamiltonian. The local density of states is defined as D ( i, E ) = 2 Nl X n =1 [ | u n ( i ) | δ ( E − E n ) + | v n ( i ) | δ ( E + E n )](2)where l is the number of sites in one magnetic unit cell, N is the number of magnetic unit cells and ( u n ( i ), v n ( i ))is the position eigenstate of the n -th positive-energystate. Due to translational symmetry, the local densityof the system is the same for all unit cells. The spectral FIG. 1: Position dependence of the pairing gap for the bond-centered configuration using color coding on bonds. The cir-cles in the middle of the plaquettes specify the positions ofvortices for a l = 8 magnetic field unit cell whose boundary isshown by the dashed line. In the singular gauge, the vorticesat white (dark) circles are only seen by particles (holes). Thelower part of the figure shows the varying gap amplitude as afunction of x .FIG. 2: DOS of a π -striped superconductor for various valuesof the pairing gap amplitude. The second nearest neighborhopping for this DOS calculation is set to zero and the chem-ical potential µ is adjusted to yield 1 / weight function in the extended Brillouin zone (BZ) isdefined as A ( k, E ) = 2 N l Nl X n =1 [ | u n ( k ) | δ ( E − E n ) + | v n ( k ) | δ ( E + E n )](3) FIG. 3: The spectral weight (left) and FS (right) for fourvalues of the pairing gap ∆ a) 0 .
05, b) 0 .
1, c) 0 . . where u n ( k ) ( v n ( k )) is the Fourier transform of u n ( i )( v n ( i )). In the reduced BZ scheme, one sums A ( k, E )over the eight coupled k in the extended BZ that can befolded back to one point in the reduced BZ. The DOS asa function of energy can be obtained from the positionaverage of the local density of states or the wave vectoraverage of the spectral weight function.The DOS of a homogeneous d-wave superconductorvanishes linearly at low-energy. In contrast, a π -stripedsuperconductor has a non-zero DOS at zero energy. Thelow-energy dependence of the DOS for various values ofthe pairing gap amplitude, ∆, is shown in Fig. 2. Forsimplicity, the second nearest neighbor hopping in the ki-netic part of the Hamiltonian has been set to zero and allthe energy quantities are written in units of the nearestneighbor hopping t which is set to 1. The chemical po-tential, µ , is adjusted to yield 1 / q x as shown in Fig. 3(a). Consequently the DOS at zeroenergy does not change significantly with respect to theunperturbed case. For intermediate values of ∆, in theapproximate range (0 . . ∆ . . .
1. TheFermi velocity associated with these loops is small andconsequently they contribute considerably to the DOSat zero energy. This is why there is a peak in the DOSfor ∆ = 0 .
1. For larger values of ∆, in the approximaterange (0 . . ∆ . . . III. RESULTS IN A MAGNETIC FIELD
The above results are all for zero magnetic field. Amagnetic field is incorporated into the model using theFranz-Tesanovic singular gauge transformation.
Inthis approach, the magnetic unit cell has linear size l ,where l is measured in units of the lattice constant. Eachunit cell has two vortices; one is seen only by particlesand the other seen only by holes. We position the vor-tices at the nodes of the order parameter, as shown inFig. 1 for the case of l = 8. We take l to be an integermultiple of 8 which is the period of the order parameter.The magnetic field and l are related by B = φ /l where φ is the flux quantum. For example, taking the latticeconstant a = 3 . A , l = 32 corresponds to B = 28 T.Hereafter, we express the magnetic field in terms of l .In this section, we investigate how the DOS structure FIG. 4: Low-energy DOS of a π -striped superconductor with∆ = 0 . t and µ = − .
226 in the presence of magnetic fieldsof l = 24 (top) and l = 32 (bottom). of the model changes as a function of the pairing am-plitude ∆ in the presence of a magnetic field. First, weconsider small values of ∆ where one can expect to un-derstand the effect of the interaction based on a simpleperturbative picture. For ∆ = 0 .
01, the spectral weightin the absence of a magnetic field exhibits only small gapsat four points of the unperturbed FS. It is similar to Fig.3(a) except that the gaps are smaller.The low-energy DOS structures for ∆ = 0 .
01 and twomagnetic fields of l = 32 and l = 24 are shown in Fig. 4.The most striking feature of this figure is the appearanceof Landau levels that are equally spaced in energy withthe spacing proportional to B . Furthermore, the presenceof a small perturbative interaction, ∆, causes the low-energy Landau levels to be slightly broadened and alsopartially reflected to the other side of the Fermi energydue to particle-hole scattering. In fact, each Landau levelfor ∆ = 0 is split into two peaks with the second peakhaving much smaller weight for small ∆, as seen in Fig.4. The sum of the number of states in these two peaksequals the degeneracy of a Landau level.From a semiclassical point of view, particles can keepundergoing Larmor precession by tunnelling through thegaps since the gaps are small for ∆ = 0 .
01. This is theso-called magnetic breakdown phenomenon. A particlecan also be Andreev scattered as a hole into a state of − k ± q . This process explains the reflected part of eachLandau level with smaller weight in Fig. 4. This pic-ture is motivated by the work of Pippard, who studiedthe cyclotron motion of nearly free electrons in the pres-ence of a weak periodic potential that induces gaps inthe Fermi surface. For this case, when the periodic po- tential is weak, electrons can tunnel through the gaps,following the unperturbed FS trajectory, or they may beBragg scattered onto a different cyclotron orbit leadingto broadening. The main difference between Pippard’smodel and the π -striped superconducting model is thatthe superconducting periodic potential scatters electronswith wave vector k into holes with wave vector − k ± q andvice versa. Thus electrons either tunnel through gaps in-duced by the periodic potential or scatter into hole states.Note that magnetic breakdown occurs even if the magni-tude of the gap in the FS is larger than ~ ω c . For intermediate values of ∆ (0 . . ∆ . . FIG. 5: DOS for ∆ = 0 .
25 and magnetic field of l = 32 shownas a function of positive and negative energies separately. Theband structure spans energies from − − µ to 4 − µ . However,the DOS is only shown in the − < E < .
25 and magnetic fields of l = 40 (top) and l = 32 (bottom). For larger values of ∆ (0 . . ∆ . . .
25 and l = 32 is shown in Fig. 6 forpositive and negative energies up to E = 1 separately.Remarkably, we again observe periodic behavior of thelow-energy DOS as a function of E with a spacing thatvaries linearly with B. This is illustrated in Fig. 6 for∆ = 0 .
25 and two values of the magnetic field, l = 40and l = 32. Note the splitting of each Landau level intoa strong and weak peak seen for small ∆, Fig. 4, doesnot occur in this larger ∆ range, where the original largeFS is not accessible to the quasi-particles.In Fig. 6, the DOS has a minimum, or possibly a verysmall gap, at E = 0 for l = 40. However, for l = 32it appears that the two Landau levels closest to E = 0are joined together and the DOS at E = 0 has a nonzerovalue. In general, we find that, for l = 8 m where m isan integer, if m is even, the DOS at E = 0 is nonzeroand if m is odd, the DOS is zero at E = 0. This is acommensurability effect that results in oscillation of the DOS at E = 0 as a function of l or 1 / √ B and is discussedfurther in appendix A. −4 −3 Landau l e v e l s pa c i ng FIG. 7: Low energy Landau level spacing as a function of1 /l for ∆ = 0 .
25 and µ = − .
3. The spacing is defined as E ( N ) /N where E ( N ) is the minimum in the DOS betweenthe N -th and N + 1-th Landau levels and is shown for N = 2(triangle) and N = 10 (circle). The line is a linear fit to thedata that goes through the origin. We have calculated the spacing of the low energy Lan-dau levels for a wide range of fields for ∆ = 0 .
25 and µ = − . E ( N ) /N where E ( N ) is the mini-mum in the DOS between the N -th and N + 1-th Lan-dau levels and is essentially independent of N provided E ( N ) . . /l ∝ B for N = 2 and N = 10. The slopeof the Landau level spacing versus B is inversely propor-tional to the DOS at E = 0. By comparison, we find thatthe slope is about half as large and the DOS at E = 0about twice as large for ∆ = 0.Furthermore, the number of states in each peak isnearly the same as that of a Landau level. In general,in the presence of a magnetic field, the n -th peak on theleft of E = 0 can have a degeneracy slightly different from a Landau level degeneracy. However, the n -th peakon the right compensates so that the number of states ofthe two peaks together is always twice that of a Landaulevel. This shows that the Landau levels are a coherentmixture of particles and holes together and the particle-hole scattering is playing a role in the formation of theLandau levels. The reason for the small difference of thenumber of states in each peak from the exact degener-acy of a Landau level is that the low-energy DOS in theabsence of a magnetic field is asymmetric around E = 0except at half filling, as shown in Fig. 8. −1.5 −1 −0.5 0 0.5 1 1.500.20.40.60.811.21.4 EDOS FIG. 8: The DOS structure in the absence of a magnetic fieldfor ∆ = 0 .
25 and two dopings. Note the asymmetry at low E for 1 / It is worth contrasting the behavior of the π -stripedsuperconductor, Fig. 6, to the DOS structure of a homo-geneous d-wave superconductor. For the latter at half-filling, Landau levels are formed in the low-energy DOS,but the Landau levels are not equally spaced and thespacing scales as √ B around E = 0. This is a conse-quence of the nodal behavior at the Fermi energy. There-fore, quantum oscillations periodic in 1 /B are not ex-pected for a d-wave superconductor. For the remainderof this paper we refer to each peak of the type shownin Fig. 6 (that is, equally spaced with a spacing propor-tional to B ) as a Landau level. The fact that there is onlyone set of Landau levels and the number of states in eachpeak is equal to that of a Landau level indicates that allparts of the FS participate in the formation of low-energyLandau levels. As noted above, the case of large valuesof ∆, Fig. 6, is different from the case of very small val-ues of ∆, Fig. 4, in which the sum of the degeneracy ofthe two peaks equals the degeneracy of only one Landaulevel.Although Landau levels are observed in the low-energyDOS in the large ∆ regime (0 . . ∆ . . & . . . l = 48.The DOS for ∆ = 0 . E = 0 which can be attributed to the commen-surability effect. This is further discussed in appendix FIG. 9: Low-energy DOS for l = 48 and ∆ = 0 . . . E = 0 appears. A.As ∆ is increased, large gaps appear within the Fermiarcs. This is seen in the spectral function, shown in Fig.3(d) for ∆ = 0 . involves tunnelling of particles (holes) across thegaps within the Fermi arcs, one expects the magneticbreakdown phenomenon not to occur if the gaps are toolarge. This may explain why the Landau levels are sup-pressed for very large ∆. FIG. 10: Density of electrons versus − µ for the magnetic fieldof l = 16 and two cases of ∆ = 0 .
25 and ∆ = 0. Unlike ∆ = 0,the density does not exhibit a stepped behavior for ∆ = 0 . Landau-type quantum oscillatory behavior has previ-ously been discussed in the context of a particular modelof a ‘Fermi-arc metal’. In that model, parts of the FS ofa metal are artificially gapped out by restricting super-conducting pairing to the antinodal regions of momen-tum space in order to get a FS that consists of Fermiarcs. The π -striped model, which is based on a specificmicroscopic mechanism and has no such restriction, dif-fers from the Fermi-arc metal of Ref. 33 in that theone-electron spectral function has non-zero (but possiblyvery small) weight along continuous lines in k -space.At a more basic level, the behavior of a π -striped su-perconductor is strikingly different from that of a metal,in spite of the fact that both exhibit a Fermi surface. In ametal, the particle density n versus µ exhibits a stepped behavior in the presence of a constant magnetic field.In contrast, the particle density in a π -striped supercon-ductor changes smoothly as a function of µ as shownin Fig. 10 for a large magnetic field of l = 16. Fur-thermore, we find that the low-energy DOS behavior ofthe π -striped superconductor is rather insensitive to thechange in µ . In other words, no oscillatory behavior ofthe DOS at E = 0 is observed as µ is varied except forfinite size effects. This is in contrast to the result forthe simple Fermi-arc metal model. However, quantumoscillations are induced by changing the magnetic field,not the chemical potential, and consequently could stillbe observed for a π -striped superconductor. IV. SPECIFIC HEAT
In this section, we provide specific heat calculationsin the absence and presence of a magnetic field to makeconnections to experiments on the cuprates. In advance,we note that the √ B dependence of the Sommerfeld co-efficient, γ , in the cuprates is not present in the π -stripedsuperconducting model as there is a finite DOS at E = 0.However, it will be shown that some features of the spe-cific heat in the cuprates are consistent with this model.One can calculate the specific heat by using the relation-ship c = T ∂S∂T . For a system of quasiparticles, the entropyis given by S = − k B X pα [ f p ln f p + (1 − f p ) ln(1 − f p )] (4)where α is the spin state and f p is the Fermi-Dirac dis-tribution function given by f p = 11 + exp( ǫ p /k B T ) (5)and ǫ p is the energy of the quasiparticle associated withthe state p . Converting the sum over states in Eq. 4to an integral over energy brings in the DOS. It shouldbe noted that, in this study, the DOS is not calculatedself-consistently as there is no microscopic Hamiltoniandefined. Furthermore, it is assumed that the magnitudeof the pairing interaction is fairly constant at low temper-atures so that the quasiparticle spectrum is unchanged astemperature increases.The specific heat at zero field for a π -striped supercon-ductor as a function of temperature is shown in Fig. 11for ∆ = 0 .
25 at 1 / γ = 0 . k B t − per site and is di-rectly proportional to the DOS at E = 0, which is 0 . t − per site. The slope is about half of that of ∆ = 0 at 1 / .
25 and µ = − .
3. As expected, all the curves con-verge to that of zero field as the temperature increases.However, at very low temperatures, the specific heat be-havior for different fields is significantly affected by thecommensurability effect. This is seen in the nearly zeroslope of the curves for odd m (recall l = 8 m ) as T → FIG. 11: Specific heat in the presence of various fields asa function of temperature for ∆ = 0 .
25 and µ = − .
3. Thebehavior of the curves at very low temperatures is significantlyaffected by the commensurability effect. The heavy line showsthe specific heat in zero field for ∆ = 0 and µ = − . / .
25 in zero field (noted by l = ∞ ). Low temperature electronic specific heat measure-ments of a cuprate indicate a relatively large DOS at E = 0 which can not be explained by the presence ofdisorder in a d-wave superconductor. Taking the latticeconstant of a typical cuprate to be a = 3 . A , the specificheat effective mass becomes m ∗ /m = 0 . eV /t . A ratherwide range of values has been used for t . Within asimplified nearest-neighbor hopping only model as usedhere, we obtain m ∗ /m = 1 .
36 for t ≈ . eV . Thisvalue corresponds to γ ≈ .
98 mJ · K − · mol − whichis consistent with the specific heat measurements forcuprates in the absence of a magnetic field, γ ≈ . · K − · mol − . Riggs et al. have studied the low temperature spe-cific heat as a function of magnetic field up to very highfields (50 T) and observed quantum oscillations. Thisallowed them both to measure the magnitude of the spe-cific heat in what is presumably the normal state, andalso to determine the cyclotron effective mass associatedwith the quantum oscillations. Then if one assumes thenormal state has broken translation symmetry, model-ing the arrangement of electron and hole pockets in theBrillouin zone and using the measured cyclotron effec-tive mass, one can estimate what the specific heat shouldbe. The result is much larger than the specific heat thatthey observed.
The same problem was also noted ina theoretical study based on FS reconstruction wherethe calculated specific heat was larger than the measuredvalue.In our calculations for a π -striped superconductor, itwas found that, even though there exist several FS pock-ets in the BZ, a single set of Landau levels is observedabove and below the Fermi energy. The relation between the slope of Fig. 7, defining the cyclotron effective massobtained from the spacing of Landau levels, and the DOSat E = 0 is the same as for the ∆ = 0 case. This impliesthat, as for the ∆ = 0 case, the cyclotron effective mass, m c , is equal to the specific heat effective mass, m ∗ , forlarge values of ∆. Consequently, the quantum oscillationsin the specific heat and the magnitude of the specific heatwhich is observed in Ref. 12 could be consistent with thebehaviour of a π -striped superconductor state inducedby large magnetic fields, rather than a striped metallicstate with no pairing gap as is often assumed. However,as noted earlier, the ideal π -striped model (with no uni-form d -wave component) is not expected to give a √ B background, which also appears to be a feature of theexperiments. In addition, since we cannot study smallchanges in B we make no prediction about the spacingof the observed quantum oscillations. V. DISCUSSION AND CONCLUSION
In this work we have studied the requirements for hav-ing quantum oscillations in a model of a π -striped su-perconductor. For a large range of values of the pairinginteraction, the FS corresponds to closed loops while theone-particle spectral function exhibits Fermi arcs in k-space. Our main finding is that Landau levels are seen inthe low-energy DOS of the π -striped superconductor in alarge range of the magnetic field, which indicates the pos-sibility of quantum oscillations. We find that low-energyLandau level formation persists even though particle andhole levels are mixed by the pairing interaction. Othertheoretical studies of quantum oscillations in the cupratesare typically based on FS reconstruction of a metallicstate and involve multiple pockets and frequencies. Furthermore, the pockets associated with those studiesare located where the ARPES experiment shows a largepseudogap. By contrast, the π -striped superconductorexhibits a unique low-energy Landau level set that is onlydue to the Fermi arc part of the spectral weight function.Since our numerical studies are restricted to satisfy-ing l = 8 m , we cannot change the magnitude of themagnetic field continuously or in small steps. In addi-tion, the Landau levels are located symmetrically around E = 0 for the discrete values of the magnetic field thatwe can study. As a result it is not possible, from thesecalculations, to find the FS area associated with quan-tum oscillations that would be observed by conventionalexperimental methods. However, we can make conjec-tures about FS areas that might be observed, based onour analysis. We expect that any semiclassical trajectorydescribing the formation of Landau Levels should havethe following characteristics: 1) The trajectory shoulduse all parts of the FS. 2) Andreev scattering needs tooccur at least at one point during the Larmor precessionbecause Landau levels are a coherent mixture of parti-cles and holes. 3) Magnetic breakdown is likely involvedin Landau level formation because once the gaps withinthe Fermi arcs become too large, the Landau levels dis-appear.It is also useful to compare the behaviour of the π -striped superconductor to the Fermi-arc metal, in whicha new mechanism for quantum oscillations is proposedthat is not based on FS reconstruction. For that model,it was shown, based on a semiclassical approach, that thefrequency of quantum oscillations is proportional to theFermi arc length. In the π -striped superconductor, thefact that a quasiparticle with wave vector k is coupledto ones with wave vectors − k − q x and − k + q x pro-vides a different scattering mechanism which changes thesemiclassical motion of quasiparticles. Consequently, thesemiclassical trajectories of the two studies are expectedto be different. Although Landau level formation in theFermi-arc metal resembles what we have seen in the π -striped model, the DOS at E = 0 shows an oscillatorybehavior as a function of µ for the Fermi-arc metal whichis inconsistent with our study.We note that the observation of quantum oscillationscorresponding to small Fermi surface pockets supportsthe scenario of translational symmetry breaking andFermi surface reconstruction, whether due to charge orspin density waves or to modulation of the d-wave gap,as discussed in this paper. Indeed recent experimentalresults support the connection between stripe formationand quantum oscillations. Our calculations show thata modulated d-wave superconductor can support Lan-dau levels and quantum oscillations but we are unable tomake detailed comparisons to quantum oscillation exper-iments because of the restriction to commensurate vor-tex lattices. One might expect there to be observabledifferences between quantum oscillations in the presenceof charge or spin stripes and superconducting stripes,due to the Andreev reflection and particle-hole mixinginvolved in the formation of Landau levels in the lattercase. Therefore, it would be of interest to study modu-lated superconductivity within a framework which allowsthe magnetic field to be varied continuously to more di-rectly connect to the quantum oscillation experiments onthe cuprates. Possible approaches would be to use ran-dom vortex lattices, as was done by Chen and Lee, or to develop a semiclassical approximation that allowsmagnetic unit cells of arbitrary aspect ratios. Acknowledgments
The authors acknowledge useful discussions with Mar-cel Franz and Steven Kivelson. This work was supportedby the Natural Sciences and Engineering Research Coun-cil of Canada and the Canadian Institute for AdvancedResearch.
Appendix A: Commensurability effects
In this appendix, we discuss the commensurability ef-fect mentioned in Sec. III. We found that the DOS at E = 0 exhibits a periodic behavior as a function of l or1 / √ B for large values of ∆. If m is even, the two vorticesin a magnetic unit cell are in perfectly equivalent posi-tions with respect to the spatially modulated gap and theDOS at E = 0 is nonzero. In contrast, for odd m , thegap on the right of one vortex is positive but on the rightof the other vortex is negative as can be seen in Fig. 1.The DOS at E = 0 is zero for odd m .The same kind of commensurability effect is also seenin a uniform d-wave superconductor. There, due tostrong internodal scattering, the DOS at zero energy ex-hibits a periodic behavior as a function of k d l where k d is the k -space half distance between the nearest nodes ofthe d-wave superconductor. Specifically, depending onwhether n is odd or even in k d l = πn , the DOS around E = 0 shows a linear or gapped behavior. For a π -stripedsuperconductor, the relevant k -space half distance is π/ δl = 16 for the DOS at zeroenergy as a function of l . So the commensurability effectseen in a π -striped superconductor is most likely due tointerference effects. This suggests that the nonzero DOSat E = 0 for even m is due to constructive interferenceof particle and hole waves, while the gapped behavior forodd m is due to destructive interference. FIG. 12: Local density of electrons due to the low-energystates within 0.001t of E = 0 for l = 48 and ∆ = 0 . For very large ∆, a sharp peak develops near E = 0for even m only, as shown in Fig. 9. It appears that theorigin of the peak can be traced back to the non-zero DOSat E = 0 for smaller ∆, and consequently is related to thecommensurability effect. The fact that the low-energyLandau levels disappear when the peak at E = 0 is sharpsuggests that the commensurability effect is competingwith the Landau level formation. Fig. 12 shows the realspace representation of the the states under the sharppeak at E = 0 where a pattern of stripes of low andhigh particle density is clearly visible. On average, theratio of the density of the higher density stripes to thelower density is 2.35 for ∆ = 0 .
4, which is larger than thevalue 0.2 in the absence of a magnetic field (see appendixB). It is not clear whether these effects are due to theexistence of a strong commensurability effect or due tothe presence of large gaps within the Fermi arcs.
Appendix B: Periodic Andreev state
FIG. 13: High (red stripe) and low (blue stripe) density struc-ture of the low-energy particles relative to the modulated d-wave gap. High (low) density is indicated by a red (blue)stripe.
In this appendix, a type of Andreev state that is seenfor the low-energy particles (holes) in the absence of amagnetic field and persists in the presence of a magneticfield is discussed. We have already examined how thespectral weight of the low-energy states of a π -stripedsuperconductor changes as the pairing amplitude ∆ isvaried. We can also look at the real space representationof these states. Our main finding is that, for large enough values of ∆ where the shape of Fermi arc is assumed inthe spectral function (see Fig. 3), the real space rep-resentation of the low-energy states exhibits a periodicstripe structure with the periodicity of four lattice sites.The stripe structure corresponds to higher and lower den-sity of low-energy electrons and holes. Each stripe hasa width of two lattice constants and the higher densitystripes are located exactly where the order parameteris minimum as shown in Fig. 13. The density ratio ofthe higher density stripes to the lower density ones in-creases as ∆ increases. The ratio is approximately 1.5for ∆ = 0 . . ∗ Electronic address: [email protected] D. S. Marshall, D. S. Dessau, A. G. Loeser, C.-H. Park,A. Y. Matsuura, J. N. Eckstein, I. Bozovic, P. Fournier,A. Kapitulnik W. E. Spicer, and Z.-X. Shen.
Phys. Rev.Lett. , 76:4841, 1996. Tom Timusk and Bryan Statt.
Rep. Prog. Phys. , 62:61,1999. Z. A. Xu, N. P. Ong, Y. Wang, T. Kakeshita, andS. Uchida.
Nature , 406:486, 2000. H. Ding, M. R. Norman, T. Yokoya, T. Takeuchi, M. Ran-deria, J. C. Campuzano, T. Takahashi, T. Mochiku, andK. Kadowaki.
Phys. Rev. Lett. , 78:2628, 1997. M. R. Norman, T. Takeuchi, T. Takahashi, T. Mochiku,K. Kadowaki, P. Guptasarma, and D. G. Hinks.
Nature ,392:157, 1998. P. W. Anderson.
Science , 235:1196, 1987. Stuart Brown and George Gruner.
Scientific American ,270:50, 1994. S. V. Borisenko, A. A. Kordyuk, A. N. Yaresko, V. B.Zabolotnyy, D. S. Inosov, R. Schuster, B. B ˆA¨uchner,R. Weber, R. Follath, L. Patthey, and H. Berger.
Phys.Rev. Lett. , 100:196402, 2008. Sudip Chakravarty, R. B. Laughlin, Dirk K. Morr, andChetan Nayak.
Phys. Rev. B , 63:094503, 2001. N. Doiron-Leyraud, Cyril Proust, David LeBoeuf, JulienLevallois Jean-Baptiste Bonnemaison, Ruixing Liang,D. A. Bonn, W. N. Hardy, and Louis Taillefer.
Nature ,447:565, 2007. C. Jaudet, D. Vignolles, A. Audouard, J. Levallois,D. LeBoeuf, M. Nardone N. Doiron-Leyraud, B. Vignolle,A. Zitouni, R. Liang, D. A. Bonn, W. N. Hardy, L. Taille-fer, and C. Proust.
Phys. Rev. Lett. , 100:187005, 2008. Scott C. Riggs, O. Vafek, J. B. Kemper, J. B. Betts,A. Migliori, F. F. Balakirev, W. N. Hardy, Ruixing Liang,D. A. Bonn, and G. S. Boebinger.
Nature Physics , 7:332,2011. A. F. Bangura, J. D. Fletcher, A. Carrington, J. Levallois,M. Nardone, B. Vignolle, P. J. Heard, N. Doiron-Leyraud,D. LeBoeuf, L. Taillefer, S. Adachi, C. Proust, and N. E.Hussey.
Phys. Rev. Lett. , 100:047004, 2008. E. A. Yelland, J. Singleton, C. H. Mielke, N. Harrison,F. F. Balakirev, B. Dabrowski, and J. R. Cooper.
Phys.Rev. Lett. , 100:047003, 2008. Alain Audouard, C. Jaudet, D. Vignolles, R. Liang, D. A.Bonn, W. N. Hardy, L. Taillefer, and C. Proust.
Phys.Rev. Lett. , 103:157003, 2009. Suchitra E. Sebastian, N. Harrison, E. Palm, T. P. Murphy,C. H. Mielke, Ruixing Liang, D. A. Bonn, W. N. Hardy, and G. G. Lonzarich. Nature , 454:200, 2008. John Singleton, Clarina de la Cruz, R. D. McDonald, Shil-iang Li, Moaz Altarawneh, Paul Goddard, Isabel Franke,Dwight Rickel, C. H. Mielke, Xin Yao, and Pengcheng Dai.
Phys. Rev. Lett. , 104:086403, 2010. P. M. C. Rourke, A. F. Bangura, C. Proust, J. Levallois,N. Doiron-Leyraud, D. LeBoeuf, L. Taillefer, S. Adachi,M. L. Sutherland, and N. E. Hussey.
Phys. Rev. B ,82:020514, 2010. L. Onsager.
Phil. Mag. , 43:1006, 1952. Andrew J. Millis and M. R. Norman.
Phys. Rev. B ,76:220503, 2007. N. Harrison.
Phys. Rev. Lett , 102:206405, 2009. E. Berg, E. Fradkin, E. A. Kim, S. A. Kivelson,V. Oganesyan, J. M. Tranquada, and S. C. Zhang.
Phys.Rev. Lett. , 99:127003, 2007. E. Berg, Eduardo Fradkin, and Steven A. Kivelson.
Phys.Rev. B , 79:064515, 2009. Shirit Baruch and Dror Orgad.
Phys. Rev. B , 77:174502,2008. E. Berg, E. Fradkin, E. A. Kim, S. A. Kivelson,V. Oganesyan, J. M. Tranquada, and S. C. Zhang.
Rev.Mod. Phys. , 75:1201, 2003. Q. Li, M. H¨ucker, G. D. Gu, A. M. Tsvelik, and J. M. Tranquada.
Phys. Rev. Lett. , 99:067001, 2007. P. G. de Gennes.
Superconductivity of metals and alloys .Westview press, 1989. M. Franz and Z. Teˇsanovi´c.
Phys. Rev. Lett. , 84:554, 2000. O. Vafek, A. Melikyan, M. Franz, and Z. Teˇsanovi´c.
Phys.Rev. B , 63:134509, 2001. D. Shoenberg.
Magnetic Oscillations in Metals . CambridgeUniversity Press, 1984. A. B. Pippard.
Proc. Roy. Soc. A , 270:1, 1962. Philip W. Anderson. 1998. cond-mat/9812063. T. Pereg-Barnea, H. Weber, G. Refael, and M. Franz.
Na-ture Physics , 6:44, 2009. Hong Yao, Dung-Hai Lee, and Steven A. Kivelson. 2011.arXiv:1103.2115v1. Partha Goswami, Manju Rani, and Avinashi Kapoor.2010. arXiv:1007:5041v1. Zlatko Tesanovic.
Nature Physics , 7:283, 2011. S. Chakravarty and H. Y. Kee.
Proc. Natl. Acad. Sci.U.S.A. , 105:8835, 2008. F. Laliberte et al. Kuang-Ting Chen and Patrick A. Lee.
Phys. Rev. B ,79:180510, 2009. Ashot Melikyan and Oskar Vafek.