Abstract
The observation of a reconstructed Fermi surface via quantum oscillations in holedoped cuprates opened a path towards identifying broken symmetry states in the pseudogap regime. However, such an identification has remained inconclusive due to the multifrequency quantum oscillation spectra and complications accounting for bilayer effects in most studies. We overcome these impediments with highresolution measurements on the structurally simpler cuprate HgBa_{2}CuO_{4+δ} (Hg1201), which features one CuO_{2} plane per primitive unit cell. We find only a single oscillatory component with no signatures of magnetic breakdown tunnelling to additional orbits. Therefore, the Fermi surface comprises a single quasitwodimensional pocket. Quantitative modelling of these results indicates that a biaxial charge density wave within each CuO_{2} plane is responsible for the reconstruction and rules out crisscrossed charge stripes between layers as a viable alternative in Hg1201. Lastly, we determine that the characteristic gap between reconstructed pockets is a significant fraction of the pseudogap energy.
Introduction
The identification of broken symmetry states, particularly in the pseudogap region, is essential for understanding the cuprate phase diagram. The surprising discovery of a small Fermi surface from quantum oscillations (QOs) in underdoped YBa_{2}Cu_{3}O_{6+x} (Y123) (ref. 1) motivated proposals for a crystal lattice symmetrybreaking order parameter^{2,3,4,5,6,7,8,9} that reconstructs either the large Fermi surface identified in overdoped cuprates^{10,11,12} or the Fermi arcs of the pseudogap state^{13,14,15}.
The spectrum of QOs is, in principle, a distinct probe of the Fermi surface morphology and is thus a signature of the broken symmetry state^{2,3,4,5,6,7,8,9}. The ubiquity of shortrange charge density wave (CDW) in underdoped cuprates^{16,17,18,19,20,21,22} make CDW a natural choice as the order responsible for Fermi surface reconstruction. However, despite the availability of exquisitely detailed QO studies in Y123 (refs 8, 23, 24, 25), its complicated multifrequency spectrum has prevented a consensus on the exact model for reconstruction^{8,9,24,25,26,27,28}. Part of the difficulty stems from the crystal structure of Y123, in particular the bilayer splitting of the elementary Fermi pockets due to the two CuO_{2} planes per unit cell. Different models are sensitive to the magnitude, symmetry and momentum dependence of the bilayer coupling^{8,26,27,28}, which are controversial. Furthermore, neither diffraction nor QO experiments in the cuprates have yet been able to address the crucial question as to whether the two orthogonal CDW vectors spatially coexist in the same CuO_{2} plane or whether stripes alternate in a crisscross manner on consecutive CuO_{2} planes^{29,30}.
Apart from the Ybased bilayer compounds^{1,31}, HgBa_{2}CuO_{4+δ} (Hg1201) is the only other holedoped cuprate for which QOs have been detected^{32} in the pseudogap regime. Importantly, in addition to featuring a very highT_{c} (≈97 K at optimal doping), Hg1201 has a tetragonal crystal symmetry consisting of only one CuO_{2} plane per primitive unit cell. This means that the analysis of the experimental data on this compound is free from complications associated with bilayer coupling and orthorombicity.
Here we show that highresolution measurements of up to ten cycles of the QOs in Hg1201 permit a resolution of the reconstructed electronic structure. Using pulsed magnetic fields extending to 90 T combined with contactless resistivity measurements, we find the QOs in Hg1201 to be remarkably simple featuring: a single oscillation frequency with a monotonic magnetic field dependence characteristic of a single Fermi surface pocket. We find quantitative agreement between the observed single QO frequency and that from a diamondshaped electron pocket resulting from biaxial CDW reconstruction^{18,33}. There are no signatures of the predicted additional small holelike pocket^{9} reported for Y123 (ref. 25). This could be due to the antinodal states, which constitute these hole pockets, being gapped out or strongly supressed by the pseudogap phenomena. We also determine a very small c axis transfer integral for Hg1201, which precludes a model based on an alternating crisscross pattern of uniaxial charge stripes on consecutive CuO_{2} planes^{30}. The absence of signatures of magnetic breakdown tunnelling to neighbouring sections of the Fermi surfaces (such as the putative small hole pockets^{9}) provides a lower bound estimate of ≈20 meV for the relevant gap. Importantly, this is a significant fraction of the antinodal pseudogap energy^{34}. Overall, our results point to biaxial CDW reconstruction acting on the short nodal Fermi arcs produced by the pseudogap phenomena.
Results
QO measurements in pulsed magnetic fields
The typical sample quality and magnetic field requirements for observing QOs is exponentially dependent on the condition ω_{c}τ≈1, where ω_{c}=eB/m* is the cyclotron frequency and 1/τ is the scattering rate. For the Hg1201 samples studied here, ω_{c}τ≈0.35 at B=45 T on average. Thus, compared with Y123, which has ω_{c}τ≈1.2 (ref. 35) at the same field, QO measurements in Hg1201 are much more challenging. To overcome this, we use a highsensitivity contactless resistivity method, wherein the sample forms part of a proximity detector oscillator circuit^{36}, and extremely large magnetic fields. Changes Δf in the proximity detector oscillator circuit frequency f in an applied magnetic field B are directly related to the changes in the complex penetration depth of the sample^{37} and hence the inplane resistivity. We focus on hole doping where a plateau in the T_{c} dome occurs, which is also the region where detailed QO measurements in Y123 have focused, as indicated in Fig. 1. Figure 2a shows Δf/f for an underdoped Hg1201 sample UD71 (T_{c}=71 K) in an applied magnetic field. The large increase in Δf/f at B≈35 T corresponds to the transition from the superconducting to the resistive state. A derivative of the data with respect to magnetic field clearly reveals QOs in the resistive state, without the need for background removal (see Fig. 2b).
Single QO frequency in Hg1201
Figure 2c shows QOs after removing the background as described in Methods. The dashed line is a good fit of the data (solid line) to the expected QO waveform for a single quasitwodimensional Fermi surface with no warping and no magnetic breakdown tunnelling (see equation (1) in Methods). Warping and magnetic breakdown introduces other frequency components manifest as a beat or nonmonotonic amplitude modulation, which are absent in our data. In Fig. 3a, we show highresolution data obtained by averaging multiple magnetic field shots for UD71 and an additional Hg1201 sample of slightly higher T_{c} (T_{c}=74 K, labelled UD74). Seven and ten full oscillations are resolved for UD71 and UD74, respectively. For both samples, the observed QOs are well captured by the single frequency fit (dotted lines in Fig. 3a), which yields oscillation frequencies of F=847(15) T and 893(15) T for UD71 and UD74, respectively.
To set limits on the amplitude of additional QO frequencies, we determine the residuals by subtracting the singlefrequency fits from the data in Fig. 3b. The residuals for both Hg1201 samples do not show evidence for additional oscillatory components. On further comparing the Fourier transform of the data with the Fourier transform of the fit (darkshaded regions in Fig. 3d,e), both can be seen to have the same line shape, thereby providing further evidence for the absence of additional QO frequencies. The Fourier transform of the residuals (lightshaded regions in Fig. 3d,e) are devoid of prominent peaks, consistent with it representing the noise floor of the experiment. From the noise floor, we can infer that in order for additional QO frequencies to go undetected, they must fall below ≈8% of the Fourier amplitude of the observed QO frequency. It is tempting to attribute the weak modulating background we observe in the derivative of the raw data in sample UD71 (Fig. 2b), to the additional low frequency oscillation previously reported for Y123 (ref. 25). However, as discussed in Supplementary Note 1 and Supplementary Figs 1 and 2, the lack of such a modulation in sample UD74 and the disappearance of the modulation in UD71 at slightly elevated temperatures (in contradiction with the light mass of the small pocket reported for Y123 (ref. 25)), leads us to conclude that this is a background effect unrelated to slow QOs.
Our finding of a single frequency in Hg1201 contrasts significantly with the multiple frequencies present in Y123 (refs 23, 39) evident from the data with the same number of oscillation periods as for Hg1201 UD74 (see Fig. 3). In contrast to the residuals obtained for Hg1201, the residual for Y123 (see Fig. 3b), again obtained on subtracting a fit to the dominant QO frequency (of F≈530 T), reveals a distinctive beat pattern resulting from the interference between the two remaining QO components whose amplitudes are ≈40 and 50% of the dominant frequency (FT of the residual shown Fig. 3c).
Limits on the Fermi surface warping and c axis hopping
In quasitwodimensional metals, the interplane hopping leads to warping of the cylindrical Fermi surface, yielding two oscillation frequencies originating from minimum and maximum extremal crosssections. Although our observation of a single QO frequency rules out very large warping, small warping can manifest in observable nodes in the magnetic fielddependent QO amplitude. This is represented by an additional amplitude factor, R_{w}, which is parametrized by the separation between the two frequencies 2ΔF_{c} (see Methods). To illustrate this point, in Fig. 4 we fit the data with several different fixed values of ΔF_{c} in R_{w}. The absence of nodes in the experimental data enables us to make an upper bound estimate of ΔF_{c}<16 T. Using m*≈2.7m_{e} (see Fig. 5) and , we obtain a c axis hopping of t_{⊥}<0.35 meV for Hg1201, revealing it to be at least 1,000 times smaller than the nearestneighbour hopping (t=460 meV^{40}) within the CuO_{2} planes. Our upper bound is also 25 times smaller than the bare value determined from local density approximation (LDA) calculations (t_{⊥}=10 meV^{40}), reflecting a large quasiparticle renormalization.
Our ability to set a firm upper bound estimate for t_{⊥} in Hg1201 contrasts with the situation in Y123, where estimates of the c axis warping are challenging to separate from the effects of bilayer coupling. Estimates for ΔF_{c} in Y123 range from ≈15 to 90 T depending on whether the observed beat pattern originates from the combined effects of bilayersplitting and magnetic breakdown tunnelling^{8} or Fermi surface warping^{23,24,25}.
Fermi surface reconstruction by biaxial CDW
The simple crystalline structure of Hg1201 makes it the ideal system for relating the kspace area of the observed Fermi surface pocket to prior photoemission^{41} and Xray scattering^{18} measurements. The former constrains the unreconstructed Fermi surface, whereas the later provides the magnitude of the reconstruction wave vector. Following Allais et al.^{9}, we require that the large unreconstructed holelike Fermi surface of area A_{UFS} accommodate 1+p carriers, where p is the hole doping defined relative to the halffilled band. We then proceed to translate the Fermi surface multiple times by the wavevectors (Q_{CDW},0) and (0,Q_{CDW}), and their combinations in Fig. 6a. Here we have assumed a biaxial reconstruction scheme. Further details of the calculation is described in the Methods section.
The biaxial CDW reconstruction (shown in Fig. 6) yields a diamondshaped electron pocket (depicted in red) flanked by smaller hole pockets (depicted in blue) accompanied by additional open Fermi surface sheets (shown in Supplementary Fig. 3). Although the parameters for Hg1201 are slightly different than for Y123, the topology of the reconstructed Fermi surface is essentially the same.
Using the Onsager relation , where A_{e} is the area of the electron pocket and A_{h} is the area of the hole pocket, the calculated QO frequencies are F_{e}=885 T and F_{h}=82 T for the electron and hole pocket, respectively. F_{e} is remarkably close to that observed for Hg1201: 847 and 893 T for UD71 and UD74, respectively. However, F_{h} is not observed in our experiment (see Supplementary Note 1 and Supplementary Figs 1 and 2).
If we instead consider a purely uniaxial reconstruction, that is, stripe CDW order, our calculation with the same parameters for Hg1201 yields an ovalshaped holelike pocket at the antinodal regions of the original Fermi surface with a frequency of 590 T and no futher closed pockets (see Supplementary Fig. 4). This is in much poorer agreement to the measured QO frequency and furthermore disagrees with Hall effect measurements, which imply the existence of a predominantly electronlike reconstructed Fermi surface^{38}.
In Fig. 6b we show the same calculation for Y123 with x=0.58 (p≈0.106). The agreement between the measured dominant frequency (534 T) and calculated electronpocket frequency F_{e}=380 T is less satisfactory. However, here (similar to Allais el al.^{9}) we have neglected some of the complications in Y123 such as the bilayer coupling and orthorhombic crystal structure. The photoemission data are also made more difficult to interpret in Y123, owing to the necessity of surface Kdoping to reach the desired hole doping of p≈11% (ref. 12). These uncertainties highlight the utility of studying the structurally simpler Hg1201.
Limits on magnetic breakdown tunnelling across band gaps
It was recently proposed by Allais et al.^{9} that a sufficiently large magnetic field enables magnetic breakdown tunnelling to occur between the electron and hole pockets shown in Fig. 6, providing a possible explanation for one or more of the observed cluster of three frequencies in Y123 (ref. 25). The probability of magnetic breakdown tunnelling increases with magnetic field, giving rise to new orbits and a reduction in the elementary QO amplitudes by . Here, and are the tunnelling and reflection amplitudes, whereas n_{p} and n_{q} are the number of breakdown tunnelling and Bragg reflection events encountered en route around the orbit, respectively^{42}. The magnetic breakdown probability is given by P=exp{−B_{0}/B}, where is the characteristic breakdown field, in which is the approximate Fermi energy and E_{g} is the band gap separating adjacent sections of Fermi surface.
Magnetic breakdown can manifest itself in two ways in our data. The first is by way of a reduction of the primary QO amplitude at higher magnetic fields. For the diamondshaped electron pocket in Fig. 6, the amplitude is reduced by to account for the four Bragg reflection points at the tips of the diamond. In a Dingle plot of the magnetic field dependence of the QO amplitude (Fig. 7), this should be discerned as deviations from a straight line for small 1/B. Accordingly, we fit the Dingle plot to where (solid lines in Fig. 7b), yielding B_{0}≈200 T and ≈250 T for UD74 and UD71, respectively. These large values for B_{0} are consistent with no observable effects of magnetic breakdown (that is, a straightline Dingle plot); therefore, we take the fitted B_{0} as lower bound values.
The second manifestation of magnetic breakdown is through the appearance of new QO frequencies corresponding to sums and differences of the areas of the Fermi surfaces involved in the tunnelling process. Magnetic breakdown between the electron and hole pocket^{9,25} in our reconstruction model results in additional QO frequencies of the form F_{eh,n}=F_{e}−nF_{h}, in which n is an integer. Based on our modelling, F_{h}≈80 T, meaning that the frequencies F_{e−h,n} are sufficiently distinct from F_{e} to be discernible in the raw and Fouriertransformed data in Fig. 3b,d,e. The noise floor of ≈8% of the dominant oscillation amplitude A_{e} provides an upper limit for the amplitude A_{e−h,1}, the leading magnetic breakdown frequency (n=1). Using the inequality
we obtain a second lower bound of B_{0}≈200 T. Here we have assumed a similar m* and scattering rate values for the various combination orbits, whereas the factor of two for A_{e−h} accounts for the two possible orbits involving one of the two holelike pockets in Fig. 6a.
We have shown above that both the Dingle plot and the absence of additional Fourier peaks above the noise floor provide mutually consistent large lower bound estimates for B_{0}. The most conservative of these (that is, B_{0} 200 T) enables a lower bound estimate of E_{g} 20 meV to be made for the band gap between the observed electron and presumed hole pockets in Fig. 6.
Discussion
Our observation of a simple monotonic waveform of a single QO frequency in Hg1201 and a single Fermi surface crosssectional area that is compatible with photoemission and Xray scattering measurements are essential for resolving issues relating to the nature of the CDW ordering. One of these concerns is whether the two chargeordering wavevectors (Q_{CDW},0) and (0,Q_{CDW}) coexist in the same CuO_{2} plane^{9,28,33} or whether stripes alternate in a crisscross manner on consecutive CuO_{2} planes^{29,30}. In the absence of a coupling between CuO_{2} planes, crisscross stripes lead to open Fermi surface sheets running in orthogonal directions on adjacent planes. The effect of the interplane coupling is to introduce a hybridization^{30}. Whereas a strong coupling in the range ∼10–100 meV occurs within the bilayers in Y123 and Y124 (ref. 43), no such coupling occurs in singlelayer Hg1201 and only a very weak coupling provided by the interlayer c axis hopping determined here to be t_{c}<0.4 meV can exist. In the context of crisscross stripe order, the effect of such a weak c axis hopping is to introduce a very small gap of order t_{c} in magnitude between the electron and hole pockets in Fig. 6, which would then have a very small characteristic magnetic breakdown field of B_{0}∼0.1 T (several orders of magnitude smaller than the lower bound constraint on B_{0} determined from our experimental results). The magnetic breakdown amplitude reduction factor R_{MB} for B_{0}=0.1 T would be so small that it would render the electron pocket not observable in experimentally relevant magnetic fields of 40 T. Our observations of a single electron pocket and small c axis hopping therefore rule out crisscross stripes as a viable route for creating observable Fermi surface pockets in Hg1201 at high magnetic fields.
Although the CDW correlations detected with Xray scattering are a natural candidate for the cause of Fermisurface reconstruction, an open question concerns whether the correlation length is sufficiently large to support QOs. A small correlation length of the order parameter can manifest as additional damping of the QO amplitude in the Dingle term (see Methods), thus suppressing the effective mean free path l^{44}. For Y123, the CDW correlation length at T_{c} and B=0 T is ξ_{CDW}≈65 Å^{16}. The effective mean free path, l≈200 Å^{35} obtained from QO measurements is of the same order of magnitude as ξ_{CDW}. For Hg1201, both ξ_{CDW} and l are similarly reduced compared with Y123: ξ_{CDW}≈20 Å^{18} at T∼T_{c} and l=85 Å (average of UD71 and UD74 and consistent with prior Hg1201 results^{32}). Thus, it appears that the QO effective mean free path might be correlated with the CDW domain size. Alternatively, both l and ξ_{CDW} could be similarly affected by disorder or impurities. The relatively small l for Hg1201 indicates that the CDW need not be long ranged, even at low temperatures and high magnetic fields, to yield the reconstructed Fermi surface observed here. Although ξ_{CDW} in Y123 increases at low temperatures and high magnetic fields, it remains rather small (≈100–400 Å)^{45,46}.
Another issue concerns the origin of the E_{g}20 meV gap separating the diamondshaped electron pocket from adjacent sections of Fermi surface in Hg1201. There are two possible CDW Fermi surface reconstruction scenarios that have been discussed in the literature. One of these involves the folding of the large Fermi surface^{9}, as shown in Fig. 6, which is expected to produce small hole pockets and open sheets in addition to the observed electron pocket. In such a scenario, E_{g} would then simply correspond to the CDW gap 2Δ_{CDW}. The alternative scenario is that the reconstructed Fermi surface occurs by connecting the tips of Fermi arcs produced by a preexisting or coexisting pseudogap state^{18,20}. In this scenario, we would expect the small hole pockets to be gaped out by the pseudogap causing E_{g}, then to correspond to the pseudogap energy. Two observations suggest the latter scenario to be more applicable to the underdoped cuprates. First, we find no evidence for QOs originating from the hole pocket, either by direct observation or by way of magnetic breakdown combination frequencies. Second, the Fermi arc, which refers to the region in momentum space over which the photoemission spectral weight is strongest, is seen to be very similar in length to the sides of the electron pocket in Fig. 6a,b (for both Hg1201 and Y123). The spectral weight drops off precipitously beyond the tips of the pocket. We note that although lowfrequency QOs in Y123 have been attributed to small hole pockets, this low frequency could also originate from Stark quantum interference effects associated with bilayer splitting^{8}. Alternatively, the pseudogap phenomena could also menifest as strong scattering at the antinodal regions, thus preventing an observation of such pockets in Hg1201. Recent hightemperature normalstate transport measurements in Hg1201 have also been interpreted in terms of Fermiliquidlike^{47,48} Fermi arcs^{49}.
Our findings in Hg1201 have direct implications for the interpretation of QO measurements made in other cuprate materials. If we assume a similar gap size between Hg1201 and Y123, the large E_{g} suggests that magnetic breakdown combination frequencies involving the electron and small hole pocket^{9,25} cannot be responsible for the complicated beat pattern associated with closely spaced frequencies in Y123 (ref. 8, 23) and Y124 (ref. 50). The splitting of the main frequency into two or more components must therefore be the consequence of the bilayer coupling in those systems^{8,28,50} or a stronger interlayer c axis hopping.
The biaxial reconstruction confirmed here for Hg1201 has also been proposed for Y123 (refs 8, 26, 28), which is supported by ultrasound measurements in high fields^{51}. However, Xray measurements show apparent localstripe CDW domains at high temperatures^{52}, which presumably become long ranged and possibly arranged in a crisscross pattern of stripes at low temperatures and high fields^{27,30}. Recent Xray measurements on Y123 show a new magnetic field induced threedimensional CDW centred at c axis wave vector L=1 r.l.u.^{45} only along the CuO chain directions^{46}, which breaks the mirror symmetry of the CuO_{2} bilayers. The role of bilayer coupling and CuO chains for this stripelike ordering tendency is still an open question and its relevance for Hg1201, which features neither, is unclear. The stripe picture is attractive because of natural analogies to singlelayered Labased cuprates^{53} and its implications for the role of nematicity (broken planar rotational symmetry) for the cuprate phase diagram^{54}. However, neutronscattering experiments have found that the typical signatures of spin stripes are absent in the magnetic excitations of Hg1201 (ref. 55). Despite the appearance of a new uniaxial threedimensional order in Y123, the CDW wavevector, with a smaller c axes correlation length, is still clearly observed in both planar directions in magnetic fields up to ∼17 T (ref. 46). For Hg1201, we have shown here that the CDW that causes the Fermisurface reconstruction is biaxial. It thus remains an open question as to whether electronic nematicity is generic to the cuprates, particularly in tetragonal Hg1201.
Methods
Samples
Hg1201 single crystals were grown using a selfflux method^{56}. As grown crystals have T_{c}≈80 K. Postgrowth heat treatment in N_{2} atmosphere at 400 °C and 450 °C was used to achieve T_{c}=74 K (hole concentration p=0.097) and T_{c}=71(2) K (p=0.09), respectively. T_{c} was determined from constantfield (DC) susceptibility measurements. The 95% level transition width of both samples is 2 K. The hole concentration p is determined based on the phenomenological Seebeck coefficient scale^{57}.
The YBCO crystal was flux grown and heat treated to obtain oxygen content x=0.58 with T_{c}=60 K and hole doping p=0.106 at the University of British Columbia, Canada^{58}.
Pulsed field measurements
High magnetic field measurements were performed at the PulsedField Facility at Los Alamos National Laboratory. The magnet system used consists of an inner and outer magnet. The outer magnet is first generator driven relatively slowly (∼3 s total width) between 0 and 37 T, followed by a faster (∼15 ms) capacitor bank driven pulse to 90 T.
Fitting QOs
We fit the field dependence to , where the first term is a polynomial representing the nonoscillatory background and A_{osc} is the oscillatory component. In the case of a single Fermi surface cylinder, the QOs are described by the Lifshitz–Kosevitch form^{42}
where F is the frequency of QOs, γ is the phase and A_{0} is a temperature and fieldindependent prefactor. Here, R_{T}, R_{D}, R_{S}, R_{MB} and R_{W} are the thermal, Dingle, spin, magnetic breakdown and warping damping factors, respectively^{8,24}. R_{T}=αT/[Bsinh(αT/B)] where accounts for the thermal broadening of the Fermi–Dirac distribution relative to the cyclotron energy and m^{*}=2.7 m_{e}, determined for one of our samples as shown in Fig. 5, is the quasiparticle effective mass (m_{e} being the free electron mass). Meanwhile, R_{D}=exp(−πl_{c}/l), where is the cyclotron radius and l is the mean free path. To lowest order, warping of a cylindrical Fermi surface leads to an amplitude reduction factor of the form R_{w}=J_{0}(2πΔF_{c}/B) in which J_{0} is a zerothorder Bessel function and is the difference in frequency between the minimum and maximum crosssections of the warped cylinder. As our experiments are performed at fixed angle (that is, B  c), we neglect R_{S} by setting it to unity. As discussed in the main text, our data shows no signatures of magnetic breakdown tunnelling or warping; thus, we also set R_{MB} and R_{W} to unity. Limits on these two terms are discussed in the Results section.
Calculation of reconstructed Fermi surface
The unreconstructed Fermi surface is calculated with the dispersion where the tightbinding parameters are ^{40} for Hg1201 and (0.35, −0.112, 0.007 and 0) eV for YBCO. μ is the chemical potential, and and where k_{x} and k_{y} are the planar wavevectors. We required that the tightbinding parameters produce a Fermi surface in agreement with the photoemission data and have carrier number 1+p where p=0.12 and 0.11 for the Hg1201 and Y123 samples, respectively, on which the photoemission data were taken. Hence, 1+p=2A_{UFS}/A_{UBZ}, where A_{UFS} and A_{UBZ} are the areas of the unreconstructed Fermi surface and Brillouin zone, respectively. Before calculating the reconstructed Fermi surface, only μ is adjusted to match the hole doping p=0.095 and p=0.106 on which the QO data were taken for Hg1201 and Y123, respectively.
Following ref. 33, the reconstructed Fermi surface is determined by diagonalizing a Hamiltonian considering translations of the biaxial CDW wavevector , where n_{x} and n_{y} are the number of translations in the planar directions. Strictly speaking, reconstruction by observed incommensurate CDW wavevectors requires an infinite number of terms in the Hamiltonian to obtain all the bands. However, as Δ<<t, the inclusion of highorder terms in the Hamiltonian gives rise to a hierarchy of higherorder gaps that are exponentially small and thus do not effect the primary closed orbits resulting from our calculation, which we restrict to nine terms. Supplementary Fig. 3 shows all the bands resulting from our reconstruction calculation.
We use Δ_{CDW}/t=0.1 for the ratio of the CDW order parameter magnitude to the inplane hopping^{9}. This implies Δ_{CDW}=46 meV, based on band structure determination of t (ref. 40), which is larger than the lower bound value determined from our analysis of magnetic breakdown tunnelling in the main text, but sufficiently small that it does not adversely affect the sizes of the pockets. Reducing the ratio to zero increases the area of the reconstructed pockets by only ≈3%.
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
Additional information
How to cite this article: Chan, M. K. et al. Single reconstructed Fermi surface pocket in an underdoped singlelayer cuprate superconductor. Nat. Commun. 7:12244 doi: 10.1038/ncomms12244 (2016).
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Acknowledgements
The work performed at Los Alamos National Laboratory, was supported by the US Department of Energy BES ‘Science at 100T’ grant number LANLF100. The National High Magnetic Field Laboratory  PFF facility is funded by the National Science Foundation Cooperative Agreement Number DMR1157490, the State of Florida and the U.S. Department of Energy. Work at the University of Minnesota was supported by the Department of Energy, Office of Basic Energy Sciences, under Award Number DESC0006858. N.B. acknowledges the support of FWF project P2798. We thank Ruixing Liang, W.N. Hardy and D.A. Bonn at UBC, Canada, for generously supplying the Y123 crystal measured as part of this work. We aknowledge fruitful discussion with S.E. Sebastian. We also thank the Pulsed Field Facility, Los Alamos National Lab engineering and technical staff for experimental assistance.
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Affiliations
Pulsed Field Facility, National High Magnetic Field Laboratory, Los Alamos National Laboratory, Mail Stop E536, Los Alamos, New Mexico 87545, USA
 M. K. Chan
 , N. Harrison
 , R. D. McDonald
 , B. J. Ramshaw
 & K. A. Modic
School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA
 M. K. Chan
 , N. Barišić
 & M. Greven
Technische Universität Wien, Wiedner Haupstrasse 810, 1040 Vienna, Austria
 N. Barišić
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Contributions
M.K.C., N.H., R.D.M., B.J.R., K.A.M. and N.B. performed contactless resistivity measurements in pulsed magnetic fields on Hg1201 and Y123. M.K.C. designed the experiment and analyzed the data. M.K.C. synthesized and prepared the Hg1201 samples. N.H. supervised the work at Los Alamos National Lab. M.G. supervised the work at the University of Minnesota. M.K.C. and N.H. wrote the manuscript with critical input from all authors.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to M. K. Chan or N. Harrison.
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