## Under the Dome

The superconducting transition temperature *T _{c}* of copper oxides has a dome-shaped dependence on chemical doping. Whether there is a quantum critical point (QCP) beneath the dome, and whether it is related to the enigmatic pseudogap, has been heavily debated. Two papers address this question in two different families of Bi-based cuprates. In (Bi,Pb)

_{2}(Sr,La)

_{2}CuO

_{6+δ},

**He**(p. 608) found that the Fermi surface (FS) undergoes a topological change as doping is increased, which points to the existence of a QCP at a doping close to the maximum in

*et al.**T*, seemingly uncorrelated with the pseudogap.

_{c}**Fujita**(p. 612) studied a range of dopings in Bi

*et al.*_{2}Sr

_{2}CaCu

_{2}O

_{8+δ}to find an FS reconstruction simultaneous with the disappearance of both rotational and translational symmetry breaking, the latter of which has been associated with the pseudogap. These findings point to a concealed QCP.

## Abstract

The existence of electronic symmetry breaking in the underdoped cuprates and its disappearance with increased hole density *p* are now widely reported. However, the relation between this transition and the momentum-space electronic structure underpinning the superconductivity has not yet been established. Here, we visualize the (intra–unit-cell) and (density-wave) broken-symmetry states, simultaneously with the coherent topology, for Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} samples spanning the phase diagram 0.06 ≤ *p* ≤ 0.23. We show that the electronic symmetry-breaking tendencies weaken with increasing *p* and disappear close to a critical doping *p*_{c} = 0.19. Concomitantly, the coherent topology undergoes an abrupt transition, from arcs to closed contours, at the same *p*_{c}. These data reveal that the topology transformation in cuprates is linked intimately with the disappearance of the electronic symmetry breaking at a concealed critical point.

The highest known superconducting critical temperature *T*_{c} (*1*–*3*) occurs atop the *T*_{c}(*p*) (where *p* is hole density) “dome” of hole-doped cuprates (Fig. 1A). In addition to the superconductivity, electronic broken-symmetry states (*4*) have also been reported at low *p* in many such compounds. Wave vector (intra–unit-cell) symmetry breaking, typically of 90°-rotational (C_{4}) symmetry, is reported in YBa_{2}Cu_{3}O_{6+δ}, Bi_{2}Sr_{2}CaCu_{2}O_{8+δ}, and HgBa_{2}CuO_{4+}* _{x}* (

*5*–

*14*). Finite wave vector (density-wave) modulations breaking translational symmetry, long detected in underdoped La

_{2–}

_{x}_{–}

*Nd*

_{y}*Sr*

_{y}*CuO*

_{x}_{4}and La

_{2–}

*Ba*

_{x}*CuO*

_{x}_{4}(

*15*,

*16*), are now also reported in underdoped YBa

_{2}Cu

_{3}O

_{7–δ}, Bi

_{2}Sr

_{2}CuO

_{6+δ}, and Bi

_{2}Sr

_{2}CaCu

_{2}O

_{8+δ}(

*17*–

*25*). Summarizing all such reports in Fig. 1A reveals some stimulating observations. First, although the and states are detected by widely disparate techniques and are distinct in terms of symmetry, they seem to follow approximately the same phase-diagram trajectory (shaded band Fig. 1A), as if facets of a single phenomenon (

*26*). The second implication is that a critical point (perhaps a quantum critical point) associated with these broken-symmetry states may be concealed beneath the

*T*

_{c}(

*p*) dome. Numerous earlier studies reported sudden alterations in many electronic and magnetic characteristics near

*p*= 0.19 (

*2*,

*3*,

*27*), but whether these phenomena are caused by electronic symmetry changes (

*28*) at a critical point was unknown.

In momentum space , the hole-doped cuprates also exhibit an unexplained transition in electronic structure with increasing hole density. Open contours, or “Fermi arcs” (*29*–*32*), are reported at low *p* in all compounds studied, whereas at high *p* closed holelike pockets surrounding (where *a*_{0} is the unit cell dimension) are observed (*33*, *34*). One possibility is that such a transition could occur due to the disappearance of an electronic ordered state, with the resulting modifications to the Brillouin zone geometry altering the topology of the electronic bands (*28*).

Our strategy, therefore, is a simultaneous examination of both the electronic structure and the broken-symmetry states, over a sufficiently wide range of *p* to include any concealed critical point. The objective is to search for a relation between the broken-symmetry states (*5*–*25*) and the Fermi surface (FS) topology (*29*–*34*). Fourier transform analysis of spectroscopic imaging scanning tunneling microscopy (SI-STM) is distinctive in that it allows access simultaneously to the broken-symmetry states (*26*, *35*) and to the electronic structure by using quasi-particle scattering interference (QPI) (*35*). The SI-STM tip-sample differential tunneling conductance (*e*, electron charge; *V*, tip-sample bias voltage) at location and energy *E* relates to density of electronic states as (*I*_{S} and *V*_{S} are arbitrary parameters) (*35*). In cuprates, it is necessary to study the ratio of the differential conductances because it suppresses the severe systematic errors due to the unknown denominator by division, thereby allowing and symmetries to be measured correctly (*35*).

The structure of coherent Bogoliubov quasi-particles is then detectable, because scattering between the eight density-of-states maxima at (circles in inset Fig. 1B) produces interference patterns with wave vectors in , the Fourier transform of . At low *p*, one finds that the predicted set of seven inequivalent Bogoliubov QPI wave vectors (inset Fig. 1B) exist only below an energy Δ_{0} (Fig. 1B), indicating that it is the limiting binding energy of a Cooper pair (*35*). At |*E*| > Δ_{0}, the dispersive disappear (Fig. 1B and movie S1) and are replaced by broken-symmetry states consisting of (i) spatial modulations with energetically quasi-static wave vectors and breaking translational symmetry (*24*, *25*, *35*, *36*) and (ii) (intra–unit-cell) breaking of C_{4} symmetry detectable either directly in (*11*, *36*) or at the Bragg wave vectors (*11*, *26*, *35*). But the complete doping dependence of these broken-symmetry signatures was unknown.

To determine the topology of coherent states across the phase diagram of Bi_{2}Sr_{2}CaCu_{2}O_{8+δ}, we use a recently developed approach (*37*) that requires measurement of only a single QPI wave vector, (blue arrow in Fig. 1B, inset). Brillouin zone geometry (Fig. 2A) means that is simply related to the FS states as
(1)Here is the superconducting energy gap whose energy-minimum follows the trajectory of . An image of the locations of for all 0 < |*E*| < Δ_{0} (pale blue region in Fig. 1B) then yields the FS location of coherent states that contribute to Cooper pairing. An efficient way to locate these states is to sum all the images to the energy Δ_{0}
(2)and then to plot the contour of within these images (see supplementary text section I and figs. S1 and S2) (*38*). The power of this procedure is demonstrated in the determination of the FS in Fig. 2B (supplementary text section II) (*38*). Applying this approach to determine the doping dependence of topology, we find very different results at low and high *p*. Figure 2C shows at *p* = 0.14, whereas Fig. 2D shows at *p* = 0.23. The most prominent difference between the two is that the contour of only spans four arcs in Fig. 2C, whereas it completes four closed curves surrounding in Fig. 2D. He *et al*. report similar phenomena in Bi_{2}Sr_{2}CuO_{6+δ} (*37*). In Fig. 2E, we show the complete doping dependence of measured over the full range of *p* (supplementary text section II) (*38*). A notable transition in topology is observed within the narrow range *p* ≈ 0.19 ± 0.01, wherein the arc of coherent Bogoliubov states typical of low *p* suddenly switches to the complete closed contour surrounding (supplementary text section II, fig. S3, and movie S2) (*38*).

Next we study the broken-symmetry states by examining measured simultaneously with the data in Fig. 2, but now for Δ_{0} < |*E*| < Δ_{1} (pink regions in Fig. 1B), where Δ_{1} is the maximum detectable gap [pseudogap at low *p* and maximum superconducting gap at high *p* (*35*)]. These images exhibit several distinct broken spatial symmetries whose evolution with *p* we explore. Figure 3A shows for *p* = 0.08, whereas Fig. 3B shows for *p* = 0.23, with their Fourier transforms shown in Fig. 3, C and D, respectively. The former exhibits the widely reported (*24*–*26*, *35*, *36*) quasi-static wave vectors and of states with local symmetry breaking along with the Bragg peaks (red circle), whereas in the latter, the quasi-static wave vectors and have disappeared. The broken C_{4}-symmetry states can be detected by using the lattice-phase–resolved nematic order parameter (*11*)
(3)The and are the Bragg vectors after the necessary transformation to nearly perfect lattice periodicity in so that real and imaginary components of the Bragg amplitudes, and , are well defined (*11*, *35*). The measured for *p* = 0.06 and for *p* = 0.23 are shown in Fig. 3, E and F, respectively (supplementary text section IV) (*38*). Here we see that the extensive order in observed at low *p* (*11*) has disappeared at high *p*, leaving nanoscale domains of opposite nematicity (*26*) probably nucleated by disorder. The doping dependence of , the intensity of the modulations in , is shown in Fig. 3G (supplementary text section III and figs. S4 and S5) (*38*), whereas the dependence of the spatially averaged magnitude of the C_{4} breaking is shown in Fig. 3H (supplementary text section IV and figs. S7 and S8) (*38*). These plots reveal that the more extended broken symmetry and the shorter-range ordering tendencies in modulations (*11*, *26*, *35*) disappear near a critical doping *p*_{c} ≈ 0.19.

Figure 4A is a schematic summary of our findings, from Bogoliubov QPI techniques (*32*, *35*, *37*), on the dependence of electronic structure with increasing *p*. Fig. 4B shows that the wave vectors of states at which Bogoliubov QPI disappears (circles) evolve along the lines with increasing *p*. Concomitantly, the quasi-static wave vectors and of broken-symmetry states also evolve on the same trajectory (squares). Thus, the and wave vectors of incommensurate (density-wave) modulations evolve with doping, as shown in Fig. 4C (*35*). Figure 4D shows the area of between the arc and the line (left inset) increasing proportional to hole density *p* (*32*, *35*); at *p* = 0.19, there is a transition to a diminishing area of electron count as 1-*p* for the closed-contour FS topology. Finally, we show in Fig. 4, E to G, that the critical point *p*_{c} ≈ 0.19 is associated microscopically with a transition to conventional *d*-wave Bogoliubov QPI on a complete FS (simulated in Fig. 4E and measured at *p* > *p*_{c} in 4G) from a highly distinct form of scattering (Fig. 4F) of unknown cause (*39*).

To recapitulate: With increasing hole density, the modulations (density waves) weaken and disappear at *p*_{c} ≈ 0.19 (Fig. 1A and Fig. 3, A to D and G). Concurrently, the broken-symmetry (intra–unit-cell nematic) states become progressively more disordered (*13*) and reach a zero average value at approximately the same *p*_{c} (Fig. 1A and Fig. 3, E to F and H). Simultaneously, the topology of coherent Bogoliubov quasi-particles (or the FS supporting their superconducting gap) undergoes an abrupt transition from arcs to closed contours (Figs. 2 and 4 and movie S2) (*38*). This key transformation of cuprate electronic structure is therefore linked directly with disappearance of electronic symmetry breaking. However, this phenomenology also exhibits many peculiar components unexpected within a simple FS reconstruction scenario. First, the coevolution and contiguous disappearance at *p*_{c} of the signatures of two distinct broken symmetries (Figs. 1A and 3) reinforce the deductions that they are microscopically closely related (*11*, *26*, *35*, *36*, *40*). Second, because the modulations exhibit wave vectors generated by scattering regions (hot spots) moving along the lines (Fig. 4, A to C) (*32*, *35*), FS nesting provides an inadequate explanation for the cuprate density waves. Third, the abrupt topology change at *p*_{c} (Fig. 2E and Fig. 4, A, F, and G) exhibits characteristics more reminiscent of an antinodal coherence recovery transition (*41*) than of a conventional band reorganization. Fourth, because the disappearance of the pseudogap is associated axiomatically with the reappearance of coherent antinodal states, and because the latter is precisely what occurs at *p*_{c} (Figs. 2E and 4A), the pseudogap (*1*–*3*) and the electronic symmetry breaking (*5*–*25*) must be intimately linked (Fig. 1A). Finally, as neither long-range order nor any associated quantum critical point can exist with quenched disorder (*40*), a nematic critical point, at which the electronic symmetry breaking between the two oxygen sites within the CuO_{2} unit cell (*11*, *26*, *35*, *36*) disappears, seems most consistent with our observations.

## Supplementary Materials

www.sciencemag.org/content/344/6184/612/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S9

Movies S1 and S2

## References and Notes

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**Acknowledgments:**We are particularly grateful to S. Billinge, J. E. Hoffman, S. A. Kivelson, D.-H. Lee, and A. P. Mackenzie for key scientific advice. We thank K. Efetov, E. Fradkin, P. D. Johnson, J. W. Orenstein, C. Pepin, S. Sachdev, and K. M. Shen for helpful discussions and communications. Experimental studies were supported by the Center for Emergent Superconductivity, an Energy Frontier Research Center, headquartered at Brookhaven National Laboratory (BNL) and funded by the U.S. Department of Energy under grant DE-2009-BNL-PM015, as well as by a Grant-in-Aid for Scientific Research from the Ministry of Science and Education (Japan) and the Global Centers of Excellence Program for Japan Society for the Promotion of Science. C.K.K. acknowledges support from the FlucTeam program at BNL under contract DE-AC02-98CH10886. J.L. acknowledges support from the Institute for Basic Science, Korea. I.A.F. acknowledges support from Fundação para a Ciência e a Tecnologia, Portugal, under fellowship number SFRH/BD/60952/2009. S.M. acknowledges support from NSF grant DMR-1120296 to the Cornell Center for Materials Research. Theoretical studies at Cornell University were supported by NSF grant DMR-1120296 to Cornell Center for Materials Research and by NSF grant DMR-0955822. The original data are archived by Davis Group, BNL, and Cornell University.