Superconductivity in the Iron Age Zlatko Tesanovic, Johns Hopkins University Zbt@Pha.Jhu.Edu

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Superconductivity in the Iron Age Zlatko Tesanovic, Johns Hopkins University [email protected] http://www.pha.jhu.edu/~zbt

Saturday, March 23, 13 Superconductivity in the Iron Age Zlatko Tesanovic, Johns Hopkins University [email protected] http://www.pha.jhu.edu/~zbt

Science Blockbuster of 2009

Saturday, March 23, 13 Superconductivity in the Iron Age Zlatko Tesanovic, Johns Hopkins University [email protected] http://www.pha.jhu.edu/~zbt

Science Blockbuster of 2009

Saturday, March 23, 13 Superconductivity in the Iron Age Zlatko Tesanovic, Johns Hopkins University [email protected] http://www.pha.jhu.edu/~zbt

#6- Iron-based Superconductors, which rivaled swine-flu for citations among scholars…

Science Blockbuster of 2009

Saturday, March 23, 13 Superconductivity in the Iron Age Zlatko Tesanovic, Johns Hopkins University [email protected] http://www.pha.jhu.edu/~zbt

o $ 1,000,000,000 question: How to make a 100 K iron-based

superconductor #6-? Iron-based Superconductors, which rivaled swine-flu for citations among scholars…

Science Blockbuster of 2009

Saturday, March 23, 13 Superconductivity in the Iron Age Zlatko Tesanovic, Johns Hopkins University [email protected] http://www.pha.jhu.edu/~zbt

o o $ 1,000,000,000 question: o$ 100 question: What is the Howtheory to make of iron-pnictidesa 100 K iron-based ?

superconductor #6-? Iron-based Superconductors, which rivaled swine-flu for citations among scholars…

Science Blockbuster of 2009

Saturday, March 23, 13 PHYSICAL REVIEW B 80, 024512 ͑2009͒

Valley density-wave and multiband superconductivity in iron-based pnictide superconductors

Vladimir Cvetkovic and Zlatko Tesanovic Institute for Quantum Matter and Department of Physics & Astronomy, The Johns Hopkins University, Baltimore, Maryland 21218, USA ͑Received 5 September 2008; revised manuscript received 22 June 2009; published 20 July 2009͒ The key feature of the Fe-based superconductors is their quasi-two-dimensional multiband Fermi surface. By relating the problem to a negative U Hubbard model and its superconducting ground state, we show that the defining instability of such a Fermi surface is the valley density-wave ͑VDW͒,acombined spin/charge density- wave at the wave vector connecting the electron and hole valleys. As the valley parameters change by doping or pressure, the fictitious superconductor experiences “Zeeman splitting,” eventually going into a nonuniform “Fulde-Ferrell-Larkin-Ovchinikov” ͑FFLO͒ state, an itinerant and often incommensurate VDW of the real world, characterized by the metallic conductivity from the ungapped remnants of the Fermi surface. When Zeeman splitting exceeds the “Chandrasekhar-Clogston” limit, the “FFLO” state disappears and the VDW is destabilized. Near this point, the VDW fluctuations and interband pair repulsion are essential ingredients of high-Tc superconductivity in Fe pnictides. RAPID COMMUNICATIONS

DOI: 10.1103/PhysRevB.80.024512PHYSICAL REVIEWPACS B number83, 020505(R)͑s͒: 74.20. (2011)Ϫz, 71.45.Lr, 74.70.Dd, 75.30.Fv

TheoryI. INTRODUCTION of the valley-density wave and͑xc͒, the hidden “superconducting” order in state iron is pnictides completely destroyed and so is the VDW in a true material. However, for ␦kF above but Recently, the superconductivity below 7 KJian in LaOFeP Kang and Zlatkonear ␦k Tec,sanoviˇ we considerc´ strong “superconducting” fluctuations ͑Ref. 1͒ led to the discovery of high Tc ϳ26 K in its doped and find that these VDW fluctuations can induce real super- Institute for Quantum Matter and Department2 of Physics & Astronomy, The Johns Hopkins University, Baltimore, Maryland 21218, USA sibling LaO1−xFxFeAs͑xϾ0.1͒. Even higher Tc’s were conductivity in Fe pnictides ͑see Fig. 1͒. In principle, one found by replacing(Received La 20 with November other rare 2010; earths, revised up to manuscript the cur- receivedcould 22 avoid December the mapping 2010; published to the negative 31 JanuaryU Hubbard 2011) model 3 rent record ofInT thec =55 limit K. ofThese perfect are nesting, the first the noncuprate physics of ironsu- pnictidesand argue is governed that the by VDW the density instability wave in formation pnictides at occurs the for 9 perconductorszone-edge͑SCs͒ exhibiting vector M. At such high high energies,Tc’s and various their spin- dis- (SDW),the same charge-, reasons and orbital/pocket- as the SDW instability (PDW) density found waves, in, say, Cr. covery has touched off a storm of activity.4 We find, however, that our “fictitious superconductivity” de- and their linear combinations, all appear equally likely, unified within the unitary order parameter of U(4) U(4) In this paper, we introduce a notable element into the scription is more appropriate to pnictides not only× due to its theoreticalsymmetry. debate by Nesting considering imperfections a unified and model low-energy of spin interactionsillustrative reduce purposes this symmetry but also because to that of it real allows materials. us to extend density-wave,Nevertheless, orbital density-wave, the generic ground structural state deformation, preserves a distinctthe analogy signature to of the its “FFLO” highly symmetric state and origins: multiband A SDW “SC,” i.e., and superconductivityalong one axis in of Fe the pnictides. iron lattice The is model predicted is simple to coexist with a perpendicular PDW, accompanied by weak charge but it containscurrents. the necessary This “hidden” physical order features. induces The the structural essential transition in our theory, naturally insures Ts ! TN ,andleads ingredientsto are orbital electron ferromagnetism and hole pockets and other͑valleys observable͒ of consequences. the

Saturday,quasi-two-dimensional March 23, 13 ͑2D͒ multiply connected Fermi surface ͑FS͒.5DOI:–7 To10.1103/PhysRevB.83.020505 extract the basic physics we consider spin- PACS number(s): 74.70.Xa, 75.30.Fv, 75.25.Dk less electrons first and only a single electron and a single hole band with identical band parameters. We then show that Thethis model discovery can be related of high-temperature to a 2D negative U superconductivityHubbard model, vector Q,withallofitsdifferentreincarnations—various 1,2 3 (HTS)the in ground iron pnictides state of whichhas sparked is known intense exactly—it research. isLike a spin-, charge-, and orbital/pocket-density waves (SDW, CDW, thesuperconductor. cuprates, the8 pnictidesIn real FeAs are materials, layered systems this fictitious and super- exhibit PDW, respectively), as well as their mutually orthogonal linear antiferromagnetismconductivity translates (AF) into at zero a fully doping gapped (x valley0), density- followed combinations—unified within a unitary U(4) U(4) order bywave HTS͑ beyondVDW͒, a some unified finite statex. representing3,4 Magnetic a combination order= in parent of parameter.22 At yet lower energies, however, as× the U(4) compoundsspin, charge, consists and orbitalof an AF density-waves spin chain along͑SDW/CDW/ODW the wave vector͒ U(4) symmetry-breaking interactions and the deviations from× (π,0)at orthe (0commensurate,π)intheunfoldedwave vectorBrilliouinM connecting zone (UBZ) the and two an val- FM perfect nesting come into play, the symmetry is reduced down leys. Next, we introduce two different fictitious5 “chemical spin chain alonge the perpendicularh direction. The dynamical to that of real materials. Nevertheless, provided there is a originpotentials,” of this AF␮ ! state␮ for is the hotly electron debated: and Within the hole the valleys, itinerant as significant segregation of scales in the effective Hamiltonian measured from the bottom and the top of the bands, electron model, the magnetic transition is ascribed to the of iron pnictides between the high-energy U(4) U(4)- respectively—this describes the effect of doping the parent FIG. 1. ͑Color online͒ Phase diagram of Fe pnictides, depicting× SDWiron-pnictide instability, compounds enhanced by and the corresponds near-nesting to among the external electron symmetric and the low-energy symmetry-breaking terms, the and hole pockets of the Fermi surface (FS).6–9 To ensure theground evolution state of our and fictitious its excitations superconductor bear a from distinct the fully signature gapped of their Zeeman splitting in our fictitious attractive Hubbard model. VDW insulator to the “FFLO superconductor”—a partially gapped “striped” spine order,h only one electron pocket is involved in highly symmetric origin. As ␦␮=␮ −␮ increases, so does this Zeeman splitting, and metallic VDW—to the real SC under the influence of the Zeeman SDW,eventually and the our spin-wave fictitious anisotropy superconducting arises state from approaches the electron to Our␦␮ picture is based on the itinerant model and re- 10,11 splitting ͑doping or pressure͒. The red dot on the vertical axis pockets’and exceeds finite ellipticity. the “Chandrasekhar-Clogston”In contrast, within limit, the giving localized way symbolizeslies on the the parent hierarchy compounds of and energy the regime scales below that it might separate be the 12,13 Heisenberg-typeto a nonuniform modelFulde-Ferrell-Larkin-Ovchinikovvarious frustrated couplings͑FFLOJ͒1a, physically“flavor”-conserving inaccessible. Insets: from FS the of ͑ “flavor”-changinga͒ the normal state in interactions the J1b,groundJ2 between state at neighboring an incommensurate spins conspire͑IC͒ wave to produce vector q the, folded ͑⌫ M͒ BZ ͑Ref. 18͒, ͑b͒ the VDW metal ͑computed with e h of quasiparticles↔ on the FS, composed of two hole (h1,h2) where ͉q͉ is set by ␦kF =kF −kF, and thus by doping 14x.,15 This the interband interaction set to unity͒—this is the C version of ͑c͒ observed magnetic order and the magnon anisotropy. and two electron (e1,e2)pockets(orvalleys)(Fig.4 1). This “FFLO state” is nothing but an IC VDW at the wave vector the continuum FFLO state Ref. 21 . The remaining states are fully In addition, the tetrahedral-to-orthorhombic structural hierarchy is further͑ assisted͒ by the differences in area and M+q. Finally, as ␦k ͑x͒ exceeds certain critical value ␦k gapped. transformation is observed,F accompanied by the AFc shape of different pockets being much smaller than their transition.16,17 The AF-ordered moment is linearly 1098-0121/2009/80͑2͒/024512͑8͒ 024512-1 common overall features;©2009 hence The American the U(4) Physicale U(4) Societyh symmetry. proportional to orthorhombicity on change in x, and Such hierarchy, quantified in Ref. [ 22],× does not reflect a 18 both transitions disappear for x>xc. Magnetoelastic deep underlying principle; rather, it is an accident of the coupling was suggested as being responsible for the close 19 particular semimetallic character of pnictides and a screened relation between two transitions. In this approach, the Coulomb repulsion.23 But be that as it may, the hierarchy structural transition is driven by magnetic interactions.20 is well obeyed in all parent compounds and we use it as However, in the 1111 compounds, the structural transition an organizing framework to derive the following results: temperature T is consistently above the AF one, T ,atany s N (i) The ground state of parent pnictides is the combination x.5 Furthermore, the in-plane resistivity anisotropy develops of a SDW along the wave vector (π,0) or (0,π)intheUBZ well above TN in the presence of uniaxial pressure, and hints at the appearance of a form of order near T .21 One possible and a spin-singlet density wave (DW) along the perpendicular s direction; (ii) the spin-singlet DW is predominantly a PDW, explanation for Ts >TN is that magnetic fluctuations are much stronger than those associated with structural order. with a tiny admixture of a CDW, and is imaginary, i.e., it In this Rapid Communication, we advance another physical represents a modulated pattern of weak currents on interiron picture to account for this evident close relation between bonds. This PDW is difficult to detect and is dubbed the “hidden” order; (iii) the imaginary PDW at Q (π,0) [or the structural and magnetic transitions: the two are just = different faucets of one and the same type of ordering of (0,π)] induces real CDW at 2Q (0,0), differing from the = much higher, U(4) U(4) symmetry. This high symmetry CDW similarly generated by the SDW. The resulting broken × characterizes the dynamics of pnictides within the high- orbital symmetry between ex and ey pockets (Fig. 1)drivesthe energy regime, extending from the energies of order of the observed tetragonal-to-orthorhombic transition; and (iv) the bandwidth D down to those set by Ts TN .Thisregimeis predicted electronic structure of the ground state has numerous governed by “perfect” nesting and the ensuing∼ tendency toward observable consequences, some of which we explore. Our formation of a valley-density wave (VDW) at the nesting results are generic for the 1111 and 122 materials, and—with

Eremin, Knolle

Key assumption I:

Hirschfeld, Kuroki, Bernevig, Thomale, Chubukov, Eremin,

Key assumption II:

Saturday, March 23, 13 U(4)£U(4) Theory of Valley-Density Wave (VDW)

V. Cvetkovic and ZT, PRB 80, 024512 (2009); J. Kang and ZT, arXiv:1011.2499

U(4)£U(4) symmetry unified spin and pocket/orbital flavors

Saturday, March 23, 13 Hierarchy of RG Energy Scales U, W >> G1, G2 V. Cvetkovic and ZT, PRB 80, 024512 (2009); U(4)£U(4) Theory of Valley-Density Wave (VDW) J. Kang and ZT, arXiv:1011.2499

TS (K) TN (K) mord (B) D. K. Pratt, et al., LaFeAsO 155arxiv/0903.2833 137 0.36 CeFeAsO 155 140 0.83

PrFeAsO 153 127 0.48

NdFeAsO 150 141 0.9

CaFeAsF 134 114 0.49

SrFeAsF 175 120

CaFe2As2 173 173 0.8

SrFe2As2 220 220 0.94-1.0

BaFe2As2 140 140 0.9

VDW in Fe-pnictides (. . .) is a (nearly) U(4)£U(4) symmetric combination: SDW/CDW/ODW

Saturday, March 23, 13 “Near” U(4)£U(4) Symmetry and Experiments J. Kang and ZT, arXiv:1011.2499

Orbital “AF” Can this modulated current pattern be observed by neutrons? ¹SR?

Saturday, March 23, 13 week ending PRL 108, 107002 (2012) PHYSICAL REVIEW LETTERS 9 MARCH 2012

Friedel-Like Oscillations from Interstitial Iron in Superconducting Fe1 yTe0:62Se0:38 þ V. Thampy,1 J. Kang,1 J. A. Rodriguez-Rivera,2,3 W. Bao,4 A. T. Savici,5 J. Hu,6 T. J. Liu,6 B. Qian,6 D. Fobes,6 Z. Q. Mao,6 C. B. Fu,7,8,9 W. C. Chen,3,9 Q. Ye,5 R. W. Erwin,2 T. R. Gentile,9 Z. Tesanovic,1 and C. Broholm1,2 1Institute for Quantum Matter and Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USA 2NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA 3Department of Materials Science and Engineering, University of Maryland, College Park, Maryland 20740, USA 4Department of Physics, Renmin University of China, Beijing 100872, China 5NSSD, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA 6Department of Physics, Tulane University, New Orleans, Louisiana 70118, USA 7Department of Physics, Indiana University, Bloomington, Indiana 47408, USA 8Department of Physics, Shanghai Jiaotong University, Shanghai, 200240, China 9National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA (Received 23 September 2011; published 7 March 2012) Using polarized and unpolarized neutron scattering, we show that interstitial Fe in superconducting 1 Fe1 yTe1 xSex induces a magnetic Friedel-like oscillation that diffracts at Q 2 0 and involves >50 þ À ? ¼ð Þ neighboring Fe sites. The interstitial >2B moment is surrounded by compensating ferromagnetic four- 1 1 spin clusters that may seed double stripe ordering in Fe1 yTe. A semimetallic ﬁve-band model with 2 2 þ ð Þ Fermi surface nesting and fourfold symmetric superexchange between interstitial Fe and two in-plane nearest neighbors largely accounts for the observed diffraction.

DOI: 10.1103/PhysRevLett.108.107002 PACS numbers: 74.70.Xa, 74.25.Ha

While superconducting Fe1 yTe1 xSex shares band the difference data probe magnetic correlations that be- structure, Fermi surface [1], andþ a spinÀ resonance [2] with come static below 25 K. Polarized neutrons were used to Fe pnictide superconductors [3–5], the parent magnetic establish the magnetic origin and polarization of the scat- structures are surprisingly different. Figure 1(a) depicts the tering. Spin-polarized 3He gas held in glass cells within a Saturday, March 23, 13 distinct magnetic unit cells with single striped order for 122 vertical solenoid concentric with the sample rotation axis 1 1 was used to select the vertical component of neutron spin arsenides [qm 2 ; 2 ][6] versus double stripes for ¼ð1 Þ before and after detected scattering events [14]. The 5 meV Fe1 yTe [qm 2 ; 0 ][7,8]. In this Letter, we show that þ ¼ð Þ 1 flipping ratio was typically 56 and 8.4 for Bragg scattering short range ordered glassy magnetism at 2 ; 0 in super- ð Þ from Al2O3 and Fe1 yTe0:62Se0:38, respectively. The cor- conducting Fe1 yTe1 xSex (x 0:38) arises from magnetic þ þ À ¼ Friedel-like oscillations surrounding interstitial Fe forming responding sample depolarization factor of 0.825 was what we call a magnetic polaron. A critical role of interstitial T-independent between 4 and 30 K. A channel mixing iron to stabilize the lamellar structure [9], enhance magne- correction, obtained from the measured flipping ratio, 3 tism [10], and reduce the superconducting volume fraction and transmission correction for time-dependent He polarization ( 60–90 h)—averaging 60 (42) for the non- [11] was previously noted. Our results provide a quantitative microscopic view of the pivotal magnetic polaron. spin-flip (spin-flip) channel—was applied to T-difference We used three coaligned Fe1 yTe0:62Se0:38 single crystals with total mass 20 g and yþ 0:01 2 determined by energy-dispersive x-ray analysis.¼ Grownð byÞ a flux method [11], the samples are tetragonal (space group P4=nmm) with low temperature (T) lattice parameters a 3:791 A ¼ and c 6:023 A . Magnetization and specific heat mea- ¼ surements yielded Tc 14:0 2 K and a superconducting volume fraction of 92.9(7)%¼ ð Þ and 83(1)%, respectively [Fig. 5(b)]. Neutron scattering was performed by using the Multi Axis Crystal Spectrometer at National Institute of Standards and Technology Center for Neutron Research [12]. Twenty detection channels permitted mapping of elastic scattering throughout a reciprocal lattice plane FIG. 1 (color online). (a) Fe-plane magnetic order in the 122 [13]. High T measurements (T 25 K) provided back- and 11 parent compounds. (b) Half unit cell of Fe1 yTe1 xSex ¼ þ À ground to cancel the dominant elastic nuclear scattering, so showing the location of interstitial Fe in orange (FeI).

0031-9007=12=108(10)=107002(5) 107002-1 Ó 2012 American Physical Society REPORTS

hole pocket, then nodes should show up in the temperature limit (16). The nodes are intrinsic to trum without residual density of states at the superconducting gap function (5). The presence the superconducting gap function of the stoichi- Fermi level is only possible in an accidental case of nodes in the Fe-based superconductors is still ometric FeSe. The scattering-induced extrinsic in which scattering strength exactly matches a REPORTS very controversial (9–12). Here, we report the origin of the V-shaped spectrum in FeSe is quite specific value (17). onances in dependenceobservation on H.WhenH ofdecreases, nodal superconductivityresponding to all possible structural in iron configurationsunlikely.16. If M. the A. Ordal scatteringet al., Appl. Opt. strength22, 1099 (1983). is too weak, Examination of the electronic structure in resonances I and II show opposite spectral be- are stored. Spectral features can then be associated 17. See supporting material on Science Online. selenide (FeSe) by use of a low-temperature the gap is18. not C. L. G. closed; Alzar, M. A. ifG. Martinez, it is too P. Nussenzveig, strong, there is a the Brillouin zone (BZ) reveals the origin of havior, and the transmittance intensities of the with certain distortions. Novel methods of data scanning tunneling microscope (STM). We find finite residualAm. J. Phys. density70, 37 (2002). of states at the Fermi lev- the nodes as well as the symmetry of the order two resonance branches cross one another. In mining and inference would facilitate this task. 19. M. Fleischhauer, A. Imamoglu, J. P. Marangos, particular, resonancethat I evolves the from symmetry a broad pro- ofThis the concept order can parameter be further extended is byel. using In this extrinsicRev. Mod. Phys. 77 scenario,, 633 (2005). the V-shaped spec- parameter. In the unfolded BZ of FeSe (Fig. 1D), file to a narrower and suppressed resonance, polarization sensitivity as well as tomography-like 20. H. Xu, Y. Lu, Y. Lee, B. S. Ham, Opt. Express 18, 17736 twofold instead of fourfold. (2010). whereas resonance II grows in strength and be- spectroscopy from different directions. The real- 21. C. D. Mao, W. Q. Sun, N. C. Seeman, Nature 386, 137 comes more and moreFeSe pronounced. is the This simplest is due ization Fe-based of 3D superconductorplasmon rulers using nanoparticles REPORTS Fig. 1. STM(1997). characteriza- to the fact that when the middle rod is suc- and biochemical linkers is challenging, but 3D 22. E. Winfree, F. R. Liu, L. A. Wenzler, N. C. Seeman, Nature with an ambient-pressure transition temperature 394, 539 (1998). cessively shifted toward the bottom shorter rod nanoparticle assemblies16. M. A. with Ordal desiredet al., Appl. symmetries Opt.tion22, of 1099 the (1983). as-grown FeSe onances in dependence on H.WhenH decreases, responding to all possible structural configurations 23. J. P. Zheng et al., Nature 461, 74 (2009). pair and thereforeof couplesTc ~8Kthatcanincreaseto37Katapressure more strongly to it, and configurations17. have See supporting been successfully material on Science dem- Online. resonances I and II show opposite spectral be- are stored. Spectral features can then be associated films. (A)Topographicim-24. S. J. Tan, M. J. Campolongo, D. Luo, W. Cheng, Nature resonance II is enhanced.of 8.9 Simultaneously, GPa (1, 2). res- However,onstrated the very recentlyuncertainty18. C. L. G. (24 Alzar,, 25 M., in27 A.– G.30 Martinez,). These P. Nussenzveig,Nanotechnol., 10.1038/nnano.2011.49 (2011). havior, and the transmittance intensities of the with certain distortions. Novel methods of data Am. J. Phys. 70, 37 (2002).age (2.5 V, 0.1 nA, 200 by two resonance branches cross one another.onance I In is reducedmining because and inference the middle would rod facilitate is exciting this task. experimental achievements will pave the 225. A. J. Mastroianni, S. A. Claridge, A. P. Alivisatos, J. Am. shifted gradually awaythe stoichiometry from the top longer of rod Fe(Se,Te)road toward samples the realization19. M. Fleischhauer,(1– of3)has 3D A.plasmon Imamoglu,200 rulers J. P. nm Marangos,)ofaFeSefilmChem. Soc. 131, 8455 (2009). particular, resonance I evolves from a broad pro- This concept can be further extended by using Rev. Mod. Phys. 77, 633 (2005). pair and therefore couples less strongly to it. It in biological and soft-matter systems. 26. J. Sharma et al., Science 323, 112 (2009). file to a narrower and suppressed resonance, polarizationmade sensitivity it challenging as well as tomography-like to understand20. H. the Xu, Y. super- Lu, Y. Lee, B. S.(~30 Ham, Opt. unit Express27. cells P.18 Cigler,, 17736 thick). A. K. R. TheLytton-Jean, D. G. Anderson, M. G. Finn, is noteworthy that the spectral positions of the (2010). whereas resonance II grows in strength and be- spectroscopy from different directions. The real- S. Y. Park, Nat. Mater. 9, 918 (2010). two quadrupolarconducting resonances can and also provide normal statesReferences in the and21. materials. Notes C. D. Mao, W. Q. To Sun, N.step C. Seeman, heightNature 386 is, 5.5 137 Å. (In- comes more and more pronounced. This is due ization of 3D plasmon rulers using nanoparticles 28. M. R. Jones et al., Nat. Mater. 9, 913 (2010). information on theavoid spatial this structure complexity, change. we1. C. grew Sönnichsen, the B. stoichiomet-M.(1997). Reinhard, J. Liphardt, set) The29. crystal J. W. Zheng structure.et al., Nano Lett. 6, 1502 (2006). to the fact that when the middle rod is suc- and biochemical linkers is challenging,A. but P. Alivisatos, 3D 22.Nat. E. Biotechnol. Winfree, F.23 R., Liu, 741 L. (2005). A. Wenzler, N. C. Seeman, Nature Figure 4C shows the calculated spectral posi- 30. D. R. Han et al., Science 332, 342 (2011). cessively shifted toward the bottom shorter rod nanoparticleric FeSe assemblies single-crystalline with desired symmetries2. films G. L. Liu et on al., theNat.394 Nanotechnol. SiC(0001), 539 (1998).1, 47 (2006).(B)Atomic-resolutionSTMAcknowledgments: We thank T. Pfau and M. Dressel for useful tions of the two quadrupolar resonances in de- pair and therefore couples more strongly to it, and configurations have been successfully3. Y. W. dem- Jun et al.,23.Proc. J. Natl. P. Zheng Acad.et Sci. al., U.S.A.Nature106461, 17735, 74 (2009). discussions and comments and S. Hein for his material pendence of H. Whensubstrate the middle with rod is molecular shifted (2009). beam epitaxy24. S. J. Tan, (MBE) M. J. Campolongo,topography D. Luo, W. Cheng, (10Nature mV, 0.1 resonance II is enhanced. Simultaneously, res- onstrated very recently (24, 25, 27–30). These visualizations.2 N.L. and A.P.A. were supported by the NIH downward (see the gray region), the position of 4. L. Stryer, Annu. Rev.Nanotechnol. Biochem. 47,, 10.1038/nnano.2011.49 819 (1978).nA, 5 by (2011). 5Plasmon nm )ofFeSe Rulers Project, grant NIH NOT-OD-09-056. M.H., onance I is reduced because the middle rod is excitingin ultra-high experimental vacuum achievements (UHV) will5. pave J. A. Fan( the13et)andperformed al.,25.Science A. J.328 Mastroianni,, 1135 (2010). S. A. Claridge, A. P. Alivisatos, J. Am. resonance I stays nearly the same, whereas T.W., and H.G. were financially supported by the Deutsche 6. M. Hentschel et al., Chem.Nano Lett. Soc.10131, 2721, 8455 (2010). (2009).film. The bright spots cor- shifted gradually away from the topresonance longer rod II shiftsroadthe to toward lower STM the energies. realization experiment In contrast, of 3D plasmon on the rulers films in the same Forschungsgemeinschaft (DFG) (grants SPP1391 and pair and therefore couples less strongly to it. It in biological and soft-matter systems. 7. J. B. Lassiter et26. al., Nano J. Sharma Lett. et10 al, 3184., Science (2010).323, 112 (2009). FOR557), by The German Ministry of Science (grants 13N9048 when the middle rod is shifted upward (see the 8. C. Sönnichsen, A.27. P. P.Alivisatos, Cigler, A.Nano K. R. Lett. Lytton-Jean,5, 301respond (2005). D. G. Anderson, to the M. G. Finn, Se atoms is noteworthy that the spectral positions of the UHV system. The MBE growth of the FeSe films and 13N10146), and by Landesstiftung B. W.T.W. were also white region), resonance II does not show a prom- 9. S. Zhang, D. A. Genov,S. Y. Y. Park, Wang,Nat. M. Mater.Liu, X. Zhang,9, 918 (2010). supported by the DFG (grant GI 269/11-1) and by Deutsch- References and Notes in the top layer. a and b two quadrupolar resonances can alsoinent provide position change,is characterized whereas resonance by I shifts a typicalPhys. layer-by-layer Rev. Lett. 28.101, M. 047401 R. Jones mode, (2008).et al., Nat. Mater. 9, 913 (2010).Französische Hochschule–Université franco-allemande. on March 22, 2013 information on the spatial structure change. 1. C. Sönnichsen, B. M. Reinhard, J. Liphardt,10. N. Liu et al., Nat.29. Mater. J. W.8 Zheng, 758et (2009). al., Nano Lett.correspond6, 1502 (2006). to either of to significantly lowerasA. P. demonstrated energies. Alivisatos, Nat. Figure Biotechnol. 4 clearly in fig.23, 741 S1.11. (2005). N. The Liu et STM al., Nano topographic Lett. 10, 1103 (2010). Figure 4C shows the calculated spectral posi- 30. D. R. Han et al., Science 332, 342 (2011).Supporting Online Material demonstrates how2. G. the L. Liu fullet spectral al., Nat. Nanotechnol. behavior of1, 47 (2006).12. N. Papasimakis,Acknowledgments: V. A. Fedotov, N. I.We Zheludev, thank T.Fe-Fe Pfau and bond M. Dressel directions. for useful The images (Fig. 1, A and B, and fig. S1) revealed www.sciencemag.org/cgi/content/full/332/6036/1407/DC1 tions of the two quadrupolar resonances in de- 3. Y. W. Jun et al., Proc. Natl. Acad. Sci. U.S.A. 106S., L. 17735 Prosvirnin, Phys.discussions Rev. Lett. and101 comments, 253903 and(2008). S. Hein for his material the 3D plasmonic structure is correlated to the same conventionMaterials and Methods is used pendence of H. When the middle rod is shifted (2009). 13. P. Tassin, L. Zhang, T. Koschny, E. N. Economou, structural changesatomically in space and flat will andallow defect-free for Se-terminatedvisualizations. (001) N.L. and A.P.A. were supportedFigs. by S1 the to NIHS5 4. L. Stryer, Annu. Rev. Biochem. 47, 819 (1978).C. M. Soukoulis, Phys.Plasmon Rev. Lett. Rulers102 Project,, 053901 grant (2009). NIH NOT-OD-09-056. M.H., downward (see the gray region), the positionevaluation of of the magnitudes as well as the di- for a and b axes through- on March 22, 2013 resonance I stays nearly the same, whereas 5.surfaces J. A. Fan et al., withScience 328 large, 1135 terraces. (2010).14. N. Liu Theet al., seleniumNat. Mater.T.W.,7 and, 31 H.G. atom (2008). were financially supported3 by November the Deutsche 2010; accepted 12 May 2011 rections of structural6. M. changes. Hentschel Differentialet al., Nano Lett. spectra10, 2721 (2010).15. G. Granet, J. P. Plumey, J. Opt. A 4, S145 (2002).out. (C)Temperaturede-10.1126/science.1199958 resonance II shifts to lower energies. In contrast, spacing of the (1 × 1)–Se lattice (Fig.Forschungsgemeinschaft 1B) in the (DFG) (grants SPP1391 and that evaluate 3D7. conformational J. B. Lassiter et al., changesNano Lett. from10, 3184 (2010). FOR557), by The German Ministrypendence of Science (grants of 13N9048 differential when the middle rod is shifted upwardthe (see initial the configuration8. C. Sönnichsen, to successive A. P. Alivisatos, configu-Nano Lett. 5, 301 (2005). and 13N10146), and by Landesstiftung B. W.T.W. were also 9.topmost S. Zhang, D. A. layer Genov, Y. was Wang, 3.8 M. Liu, Å, X. Zhang, which is in good agree- white region), resonance II does not showrations a prom- offer even more distinct spectral features supported by the DFG (grantconductance GI 269/11-1) and by Deutsch- spectra (set- www.sciencemag.org inent position change, whereas resonance I shifts mentPhys. Rev. with Lett. 101 a, previous 047401 (2008). report (1). The synchrotronFranzösische Hochschule–Université franco-allemande. on March 22, 2013 and facilitate the10. identification N. Liu et al., Nat. of Mater. specific8, 758 mo- (2009). Direct Observationpoint, 10 of mV, Nodes 0.1 nA). to significantly lower energies. Figuretions. 4 clearly Figures S411. andx-ray N. Liu S5et (17 poweral.,)showhowthiscanNano Lett. diffraction10, 1103 (2010). exhibited a structural tran- Supporting Online Material(D)Schematicoftheun- demonstrates how the full spectral behavioraid the spectral of 12. analysis N. Papasimakis, of the 3D V. A. ruler Fedotov, system N. I. Zheludev, and Twofoldwww.sciencemag.org/cgi/content/full/332/6036/1407/DC1 Symmetry in sitionS. L. Prosvirnin, fromPhys. tetragonal Rev. Lett. 101, 253903 to orthorhombic (2008). symmetry the 3D plasmonic structure is correlatedupon to minute the structural tilting and twisting. Materials and Methods folded Brillouin zone and 13. P. Tassin, L. Zhang, T. Koschny, E. N. Economou, structural changes in space and will allow for at 90 K for FeSe (14). In the low-temperatureFigs. S1 to S5 or- To demonstrateC. the M. fundamental Soukoulis, Phys. concepts Rev. Lett. 102 of, 053901 (2009). the Fermi surface (green evaluation of the magnitudes as well as the di- FeSe Superconductor 3D plasmon rulers,14.thorhombic N. we Liu usedet al., inNat. our Mater. phase,study7, a 31 com- (2008). the Fe-Fe lattice’3sconstantdif- November 2010; accepted 12ellipses). May 2011 The nodal lines rections of structural changes. Differentialplex spectra sequence of15. nanolithography G. Granet, J. P. Plumey, steps.J. Opt. The A re-4, S145Can-Li (2002). Song,1,210.1126/science.1199958Yi-Lin Wang,2 Peng Cheng,1 Ye-Ping Jiang,1,2 Wei Li,1 Tong Zhang,1,2 Zhi Li,2 that evaluate 3D conformational changessulting from optical responseference of the between 3D plasmon the ruler twoKe close-packed He,2 Lili Wang,2 Jin-Feng directions Jia,1 Hsiang-Hsuanfor coskxcos Hung,ky and3 Congjun (coskx Wu,+3 Xucun Ma,2* Downloaded from 1 1,2 the initial configuration to successivehas configu- been correlated with the particle plasmon www.sciencemag.org was 0.012 Å at 20 K. ThisXi Chen, difference* Qi-Kun is Xue too small cosky ) gap functions are indicated by black and red dashed lines, respectively. The sizes of all pockets rations offer even more distinct spectralresonances features of the individual nanostructure assem- www.sciencemag.org We investigated the electron-pairing mechanism in an iron-based superconductor, iron selenide and facilitate the identification of specificbly. The mo- sameDirect conceptsto be can resolved be applied Observation with to single STM, so Fig. 1B of appears Nodes as a are exaggerated for clarity. The black arrow indicates the direction of nesting. (FeSe), using scanning tunneling microscopy and spectroscopy. Tunneling conductance spectra of tions. Figures S4 and S5 (17)showhowthiscanmetallic nanocrystals joined together by oligo- square lattice. stoichiometric FeSe crystalline films in their superconducting state revealed evidence for a gap aid the spectral analysis of the 3D rulernucleotides system or peptides (21–24), giving rise to a andThe Twofold scanning tunneling Symmetryfunction spectroscopy with nodal lines. in Electron (STS) pairing with twofold symmetry was demonstrated by direct upon minute structural tilting and twisting.new generation of plasmon rulers with unprec- imaging of quasiparticle excitations in the vicinity of magnetic vortex cores, Fe adatoms, and Se edented ability to monitorprobes the the sequence quasiparticle of events density of states and mea- To demonstrate the fundamental concepts of vacancies. The twofold pairing symmetry was further supported by the observation of striped that occur duringFeSe a wide variety Superconductor of macromo- 3D plasmon rulers, we used in our study a com- sures the superconductingelectronic gap nanostructures at the Fermi in the en- slightly Se-doped samples. The anisotropy can be explained in plex sequence of nanolithography steps.lecular The transformations re- Can-Li Song, in three1,2 dimensions.Yi-Lin Wang, Me-2 Peng Cheng,1 Ye-Ping Jiang,1,2 Wei Li,1 Tong Zhang,1,2 Zhi Li,2

terms of the orbital-dependent reconstruction of electronic structure inFeSe.Downloaded from sulting optical response of the 3D plasmontallic nanoparticles ruler Keergy He, of2 differentLili (E Wang,F)( lengths152 Jin-Feng). In or sizes Fig. Jia, 1C,1 Hsiang-Hsuan we show Hung, the tunneling3 Congjun Wu,3 Xucun Ma,2* has been correlated with the particlecould plasmon be attachedXi Chen,spectra at different1* Qi-Kun on positions the Xue sample1,2 of the in Fig. 1Aespite at various intense experimental temper- investiga- “Josephson tunneling” (6, 7)andangle-resolved resonances of the individual nanostructureDNA assem- or protein (25, 26), and each metallic ele- tion, the pairing symmetry in the recently photoemission spectroscopy (8)experiments.If Downloaded from We investigated the electron-pairing mechanism in an iron-based superconductor, iron selenide bly. The same concepts can be appliedment to single may move individuallyatures. The or collectively spatial homogeneity in discovered of the iron STS (Fe) spec-–based supercon- the sign change occurs on a single electron or (FeSe), using scanning tunneling microscopyD and spectroscopy. Tunneling conductance spectra of metallic nanocrystals joined togetherthree by oligo- dimensions.tra Dark-field (fig. S2)microspectroscopy further demonstratesductors remains the elusive high (1 quality–3). Phonon-mediated stoichiometric FeSe crystalline films in their superconducting state revealed evidence for a gap nucleotides or peptides (21–24), givingof the rise scattering to a or the extinction spectrum would pairing in conventional superconductors is typi- 1State Key Laboratory for Low-Dimensional Quantum Physics, functionof the with MBE nodal lines. samples. Electron At pairing a temperature with twofold symmetry below T wasc, demonstrated by direct new generation of plasmon rulers withoffer unprec- a useful tool to identify the 3D arrangement cally isotropic, leading to s-wave symmetry. Un- Department of Physics, Tsinghua University, Beijing 100084, imaging of quasiparticle excitations in the vicinity of magnetic vortex cores, Fe adatoms,the People and’sRepublicofChina. Se 2Institute of Physics, Chinese edented ability to monitor the sequenceof of the events different constituentsthe spectra in real exhibit time, as it two is conventional conductance pairing peaks mechanisms, and a such as spin vacancies. The twofold pairing symmetry was further supported by the observation ofAcademy striped of Sciences, Beijing 100190, the People’sRepublicof that occur during a wide variety ofunambiguously macromo- correlated to very distinct and rich fluctuations, may give rise to an order parameter 3 electronicgap centered nanostructures at the in the Fermi slightly energy. Se-doped The samples. maximum The anisotropy can be explainedChina. Department in of Physics, University of California, San lecular transformations in three dimensions.spectral Me- features. As in the case of nuclear mag- with its sign change over the Fermi surfaces and Diego, La Jolla, CA 92093–0319, USA. terms of the orbital-dependent reconstruction of electronic structure in FeSe. tallic nanoparticles of different lengthsnetic or resonance, sizes 3Dof plasmon the superconducting rulers could use a gapa pairing∆0 symmetry=2.2meVishalf such as sT (4, 5). The sT *To whom correspondence should be addressed. E-mail: could be attached at different positionslookup of thedatabase whereof theespite the energy optical intense spectra between experimental cor- thescenario investiga- two conductance is supported“Josephson by peaks. tunneling the phase-sensitive” (6, 7)[email protected] (X.M.); [email protected] (X.C.) DNA or protein (25, 26), and each metallic ele- Thetion, most the pairing striking symmetry feature in the of recently the spectraphotoemission at 0.4 spectroscopy K, (8)experiments.If ment may move individually or collectively in discovered iron (Fe)–based supercon- the sign change occurs on a single electron or 1410 D 17 JUNE 2011 VOL 332 SCIENCE www.sciencemag.org three dimensions. Dark-field microspectroscopy ductorsanalogous remains elusive to the (1– cuprate3). Phonon-mediated high-Tc superconductors of the scattering or the extinction spectrum would pairing(15 in), conventional is the V-shaped superconductorsdI/dV isand typi- the1State linear Key Laboratory depen- for Low-Dimensional Quantum Physics, offer a useful tool to identify the 3D arrangement cally isotropic, leading to s-wave symmetry. Un- Department of Physics, Tsinghua University, Beijing 100084, 2 of the different constituents in real time, as it is conventionaldence of pairing the quasiparticle mechanisms, such density as spin ofthe states People’sRepublicofChina. on en- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, the People’sRepublicof unambiguously correlated to very distinct and rich fluctuations,ergy near may giveEF.Thisfeatureexplicitlyrevealsthe rise to an order parameter China. 3Department of Physics, University of California, San spectral features. As in the case of nuclear mag- withexistence its sign change of lineover the nodes Fermi in surfaces the superconducting and Diego, La Jolla, CA gap 92093–0319, USA. netic resonance, 3D plasmon rulers could use a a pairing symmetry such as sT (4, 5). The sT *To whom correspondence should be addressed. E-mail: lookup database where the optical spectra cor- scenariofunction. is supported At elevated by the phase-sensitive temperatures,[email protected] the V-shaped (X.M.); [email protected] (X.C.) spectra in Fig. 1C smear out as the superconduct- 1410 17 JUNE 2011ing VOL gap 332 disappearsSCIENCE abovewww.sciencemag.orgTc. We suggest that the nodal superconductivity

Saturday,exists March 23, only 13 in FeSe with a composition close to stoichiometry. By introducing Te into the compound, the ternary Fe(Se,Te) becomes a nodeless Fig. 2. The vortex core states. (A)STS(setpoint,10mV,0.1nA)onthecenterofavortexcore.(B)Zero-bias sT-wave superconductor, which is characterized conductance map (40 × 40 nm2;setpoint,10mV,0.1nA)forasinglevortexat0.4Kand1Tmagneticfield. by a fully gapped tunneling spectrum in the low- (C and D)Tunnelingconductancecurvesmeasuredat equally spaced (2 nm) distances along a and b axes.

www.sciencemag.org SCIENCE VOL 332 17 JUNE 2011 1411 Progress on the physics of the underdoped cuprates

Zlatko Tesanovic Memorial Symposium

Johns Hopkins University March 22, 2013

Subir Sachdev

HARVARD Saturday, March 23, 13 Max Metlitski Erez Berg

HARVARD Saturday, March 23, 13 Outline 1. Update on cuprate experiments

2. Antiferromagnetism in metals: d-wave superconductivity

3. Low energy theory, emergent pseudospin symmetry, and bond order

4. Unrestricted Hartree-Fock-BCS

5. Quantum Monte Carlo without the sign problem

Saturday, March 23, 13 Outline 1. Update on cuprate experiments

2. Antiferromagnetism in metals: d-wave superconductivity

3. Low energy theory, emergent pseudospin symmetry, and bond order

4. Unrestricted Hartree-Fock-BCS

5. Quantum Monte Carlo without the sign problem

Saturday, March 23, 13 Strange Metal

Saturday, March 23, 13 Vol 447 | 31 May 2007 | doi:10.1038/nature05872 LETTERS

Quantum oscillations and the Fermi surface in an underdoped high-Tc superconductor Nicolas Doiron-Leyraud1, Cyril Proust2, David LeBoeuf1, Julien Levallois2, Jean-Baptiste Bonnemaison1, LETTERS Ruixing Liang3,4, D. A. Bonn3,4, W. N.NATURE Hardy3,4| Vol& Louis 447 Taillefer| 31 May1,4 2007 Nature 447, 565 (2007)

Despite twenty years of research, the phase diagram of high- frequency and mass in the vortex state as in the field-induced normal 1,2 of inverse field, by subtracting the monotonic background (shown for atransition-temperature4 superconductors remains enigmatic .A state above the upper critical field Hc2(0) (for example, ref. 7). They are all temperatures in Supplementary Fig. 2). This shows that the oscilla- central issue is the origin1 / (530 of the T) differences in the physical prop- caused by the passage of quantized Landau levels across the Fermi level erties of these copper oxides doped to opposite sides of the super- as the applied magnetic field is varied, and as such they are considered tions are periodic in 1/B, as is expected of oscillations that arise conducting2 region. In the overdoped regime, the material behaves the most robust and direct signature of a coherent Fermi surface. The from Landau quantization. A Fourier transform yields the power as a reasonably conventional metal, with a large Fermi surface3,4. inset of Fig. 2 shows the 2 K isotherm and a smooth background curve. The underdoped regime, however, is highly anomalous and We extract the oscillatory component, plotted in Fig. 3a as a function

spectrum, displayed in Fig. 3b, which consists of a single frequency, appears) to have no coherent Fermi surface, but only disconnected 0 5,6 F 5 (530 6 20) T. In Fig. 3c, we plot the amplitude of the oscillations ‘FermiΩ arcs’ . The fundamental question, then, is whether underdoped copper oxides have a Fermi surface, and if so, whether a 250 (m as a function of temperature, from which we deduce a carrier mass it isxy topologically different from that seen in the overdoped

R –2 T*

regime.∆ Here we report the observation of quantum oscillations m* 5 (1.9 6 0.1)m0, where m0 is the bare electron mass. Within error 200 – T in the electrical resistance of the oxygen-ordered copper oxide N bars, both F and m* are the same in sample B, for which the current J is YBa2Cu3O6.5, establishing the existence of a well-defined Fermi parallel to the b axis (see Supplementary Fig. 1). Oscillations of the surface–4 in the ground state of underdoped copper oxides, once 150 same frequency are also observed in R (in both samples), albeit with a superconductivity is suppressed by a magnetic field. The low oscil- xx lation frequency reveals a Fermi surface made of small pockets, in 100 smaller amplitude. We note that while at 7.5 K the oscillations are still contrast–6 to the large cylinder characteristic of the overdoped (K) Temperature

regime. Two possible interpretations are discussed: either a small AF insulator perceptible, they are absent at 11 K, as expected from thermally 50 T pocket0.015 is part of the band structure 0.020 specific to YBa2Cu3O6.5 0.025or Superconductor c damped quantum oscillations (see Supplementary Fig. 5). small pockets arise from a topological change at a critical point –1 in the phase diagram. Our understanding1 / B (T ) of high-transition- 0 While quantum oscillations in YBa2Cu3O61y (YBCO) have been 0.0 0.1 0.2 0.3 temperature (high-Tc) superconductors will depend critically on 8–10 Hole doping, p the subject of a number of earlier studies , the data reported so far b 1.5which of these two interpretations provesc –7 to be correct. Saturday,The March electrical 23, 13 resistance of two samples of ortho-II ordered do not exhibit clear oscillations as a function of 1/B and, as such, have YBa Cu O was measured in a magnetic field of up to 62 T applied b p = 0.10 c p = 0.25 11 2 3 6.5 not been accepted as convincing evidence for a Fermi surface . normal to the CuO2 planes (B c). (Sample characteristics and details 1.0 jj ) –8 Furthermore, we note that all previous work was done on oriented of the measurements are given in the MethodsT section.) With a Tc of

57.5 K, these samples have a hole doping / per planar copper atom of (a.u.) powder samples as opposed to the high-quality single crystals used in p 5 0.10, that is, they are well into theA underdoped region of the A 0.5 the present study. phase diagram (see Fig. 1a). Angle-resolvedln ( –9 photoemission spectroscopy (ARPES) data for underdoped Na2 2 xCaxCu2O2Cl2 (Na- Quantum oscillations are a direct measure of the Fermi surface CCOC) at precisely the same doping (reproduced in Fig. 1b from 2 0.0ref. 6) shows most of the spectral intensity to be concentrated in a area via the Onsager relation: F 5 (W0/2p )Ak, where W0 5 (2.07 3 Figure 1 | Phase diagram of high-temperature superconductors. small0123 region near the nodal position (p/2, p135/2), suggesting24 a Fermi 215 2 a, Schematic doping dependence of the antiferromagnetic (TN) and 10 )Tm is the flux quantum, and Ak is the cross-sectional area of surface brokenF up(kT into) disconnected arcs, while ARPES studiesT (K) on superconducting (Tc) transition temperatures and the pseudogap crossover overdoped Tl2Ba2CuO61d (Tl-2201) at p 5 0.25 reveal a large, con- temperature T* in YBCO. The vertical lines at p 5 0.1 and p 5 0.25 mark the the Fermi surface normal to the applied field. A frequency of 530 T Figure 3 | Quantum oscillations in YBCO. a, Oscillatory part of the Hall implies a Fermi surface pocket that encloses a k-space area (in the a–b tinuous cylinder (reproduced in Fig. 1c from ref. 4). positions of copper oxide materials discussed in the text: ortho-II ordered resistance, obtainedThe Hall resistance by subtractingRxy as a function the monotonic of magnetic field background is displayed (shownYBa2Cu3O6.5 inand Na-CCOC, located well into the underdoped region, and 22 Tl-2201, well into the overdoped region, respectively. b, c, Distribution of plane) of Ak 5 5.1 nm , that is, 1.9% of the Brillouin zone (of area the inset ofin Fig. Fig. 2 for for sampleT 5 A,2 and K), in as Supplementary a function Fig. of 1 forinverse sample magneticB, where field, 1/B. 2 oscillations are clearly seen above the resistive superconducting trans- ARPES spectral intensity in one quadrant of the Brillouin zone, measured b 5 c 5 4p /ab). This is only 3% of the area of the Fermi surface cylinder The backgroundition. Note at each that a vortex temperature liquid phase is is given believed in to Supplementaryextend well above ( Fig.), on 2. Na-CCOC at p 0.1, and ( ), on Tl-2201 at p 0.25 (reproduced 21 from ref. 6 and ref. 4, with permissions from K. M. Shen and A. Damascelli, measured in Tl-2201 (see Fig. 1c), whose radius is kF < 7 nm . In the b, Power spectrumthe irreversibility (Fourier field, transform) beyond our highest of the field oscillatory of 62 T, which part may for therespectively).T 5 2K These respectively reveal a truncated Fermi surface made of remainder, we examine two scenarios to explain the dramatic differ- isotherm, revealingexplain why aR singlexy is negative frequency at these low at temperatures,F 5 (530 6 as20) opposed T, which to ‘Fermi arcs’ at p 5 0.10, and a large, roughly cylindrical and continuous positive at temperatures above Tc. Nevertheless,22 quantum oscillations Fermi surface at p 5 0.25. The red ellipse in b encloses an area Ak that ence between the small Fermi surface revealed by the low frequency of correspondsare to known a k-space to exhibit area theAk very5 5.1 same nm diagnostic, from characteristics the Onsager of relationcorresponds to the frequency F of quantum oscillations measured in YBCO. 2 F 5 (W0/2p 1 )Ak . Note that the uncertainty of 4% on F is not given2 by the quantum oscillations reported here for YBa2Cu3O6.5 and the large De´partement de physique and RQMP, Universite´ de Sherbrooke, Sherbrooke, Canada J1K 2R1. Laboratoire National des Champs Magne´tiques Pulse´s(LNCMP),UMRCNRS-UPS- width of theINSA peak 5147, (a Toulouse consequence 31400, France. 3Department of the small of Physics number and Astronomy, of University oscillations), of British Columbia, but by Vancouver, Canada V6T 1Z4. 4Canadian Institute for Advanced Research, cylindrical surface observed in overdoped Tl-2201. The first scenario Toronto, Canada M5G 1Z8. the accuracy with which the position of successive maxima in a can be assumes that the particular band structure of YBa2Cu3O6.5 is differ- 565 determined. c, Temperature dependence of the oscillation© 2007 Natur amplitudee Publishing GroAup, ent and supports a small Fermi surface sheet. In the second, the plotted as ln(A/T) versus T. The fit is to the standard Lifshitz–Kosevich electronic structure of overdoped copper oxides undergoes a trans- formula, whereby A/T 5 [sinh(am*T/B)]21, which yields a cyclotron mass formation as the doping p is reduced below a value pc associated with m* 5 (1.9 6 0.1)m0, where m0 is the free electron mass. a critical point. Band structure calculations for stoichiometric YBCO (y 5 1.0), doping14,15 appear to be in broad agreement with this electronic which is slightly overdoped (with p 5 0.2), show a Fermi surface structure. However, recent band structure calculations16 performed consisting of four sheets12,13, as reproduced in Fig. 4a: two large specifically for YBa2Cu3O6.5, which take into account the unit cell cylinders derived from the CuO2 bi-layer, one open surface coming doubling caused by the ortho-II order, give a Fermi surface where the from the CuO chains, and a small cylinder associated with both chain small cylinder is absent, as shown in Fig. 4b. This leaves no obvious and plane states. The latter sheet, for example, could account for the candidate Fermi surface sheet for the small orbit reported here. low frequency reported here. ARPES studies on YBCO near optimal The fact that the same oscillations are observed for currents along a and b suggests that they are not associated with open orbits in the 0.06 chain-derived Fermi surface sheet. In YBCO, the CuO chains along 1.5 K B 2 K the b axis are an additional channel of conduction, responsible for an 2.5 K anisotropy in the zero-field resistivity r(T) of the normal state (above J II a 3 K 3.5 K 4.2 K 0.04 a b ) Ω

( 0.06 xy R – ) Ω

( Y Y 0.02 0.05 xy R –

0.04 45 50 55 60 65 B (T) 0.00 3035 40 45 50 55 60 65 Γ XXΓ B (T) Figure 4 | Fermi surface of YBCO from band structure calculations. Figure 2 | Hall resistance of YBa2Cu3O6.5. Rxy as a function of magnetic a, Fermi surface of YBa2Cu3O7 in the kz 5 0 plane (from ref. 13, with field B, for sample A, at different temperatures between 1.5 and 4.2 K. The permission from O. K. Andersen), showing the four bands discussed in the field is applied normal to the CuO2 planes (B ||c) and the current is along the main text. b, Fermi surface of ortho-II ordered YBa2Cu3O6.5 in the kz 5 0 a axis of the orthorhombic crystal structure (J ||a). The inset shows a zoom plane (from ref. 16, with permission from T. M. Rice). In both a and b the on the data at T 5 2 K, with a fitted monotonic background (dashed line). grey shading indicates one quadrant of the first Brillouin zone. 566 © 2007 Nature Publishing Group FigureTwofold2: Experimental twisted quantum Fermi surface oscillations from for different staggered angles order compared in with simulationsAPS March for a two fundamental-warped cylinder model. ( ) Measured oscillations in the contactless an underdoped high Tc superconductorA meeting 2013 conductivity plotted versus 1/B cos θ (the projection of the field along the cˆ-axis) at θ =0, 1.3, 1 2 2 2,3 B2.00004 11.3, 12,Suchitra 16.3, 18, E. Sebastian, 21.3, 26.3,∗ N. 31.3, Harrison, 36.3, 38,F. F. 41.3, Balakirev, 45.2, 46.3,M. M. 48, Altarawneh, 49, 49.4, 50.1, 50.6, 51.4, 51.5, 52, 52.3,Ruixing 52.5, Liang, 52.9,4 53.1,,5 D. A. 54.4, Bonn, 54.9,4,5 W. 55.5, N. Hardy, 56, 56.2,4,5 G. 56.95, G. Lonzarich, 57.2, 57.4,1 58.15, 58.2, 59.4, 59.6, 60.6, 61.2, 61.4, 61.7, 62.5, 62.6, 62.7, 63.2, 63.4, 63.7, 64.1, 64.5, 65.5, 66, 66.3, 68.1, Saturday, March 23, 13 69.4 and1Cavendish 70.6◦, Laboratory, all at φ Cambridge45◦. The University, inset shows JJ Thomson a schematic Avenue, Cambridge of the angles CB3 OHE,θ and U.K,φ with respect ≈ to the crystalline2National axes. High (B) Magnetic Simulated Field oscillations Laboratory, LANL, at the Los same Alamos, angles NM 8754, and fields as (A) for two Fermi surface cylinders3Department exhibiting of Physics, Mu’tah a fundamental University, Mu’tah, warping Karak, [21, 61710, 22, 23, Jordan, 24], for parameters listed in [31]4Department previously of Physics fit andto the Astronomy, restricted University experimental of British Columbia, range withinVancouver the V6T dotted 1Z4, Canada, line accessed in earlier experiments.5Canadian The Institute striking for amplitude Advanced Research, enhancement Toronto M5G expected 1Z8, Canada, in the vicinity of the Yamaji angle in B is absent in the experimental data in A.(C) A schematic showing the degeneracy in the cyclotron orbit cross-sectional area yielding an amplitude enhancement at the Yamaji angle (green). ( ) Schematic of experimentally measured quantum oscillations in β-(ET)2IBr2 We presentD quantum oscillation measurements in underdoped YBa2Cu3O6+x from [18], for which θYama ji 18◦. Here, the same fundamental sinusoidal warping occurs over a broad range in magnetic≈− fields tilted with respect to the planes and for two slightly different Fermi surface cross-sections, giving rise to an additional beat. rotated in-plane, which reveal that the fundamental warping of the Fermi sur-

face expected for the tetragonal symmetry of the YBa2Cu3O6+x Brillouin zone

is suppressed, and is instead replaced by a small amplitude warping of an unexpected two-fold twisted symmetry. The twisted5 Fermi surface geometry

enables the unique identiﬁcation of a staggered form of order that tranforms the symmetry of the Brillouin zone, and importantly, locates the Fermi surface pockets at the nodal region of the original Brillouin zone. The suppression of the fundamental warping further provides a potential explanation for the ob-

served anisotropy in optical conductivity that characterises the pseudogap.

1 LETTER doi:10.1038/nature10345

Magnetic-field-induced charge-stripe order in the high-temperature superconductor YBa2Cu3Oy

Tao Wu1, Hadrien Mayaffre1, Steffen Kra¨mer1, Mladen Horvatic´1, Claude Berthier1, W. N. Hardy2,3, Ruixing Liang2,3, D. A. Bonn2,3 & Marc-Henri Julien1

11–13 Electronic charges introduced in copper-oxide (CuO2) planes reconstruction . Thus, whatever the precise profile of the static generate high-transition-temperature (Tc) superconductivity but, charge modulation is, the reconstruction must be related to the trans- under special circumstances, they can also order into filaments lational symmetry breaking by the charge ordered state. called stripes1. Whether an underlying tendency towards charge The absence of any splitting or broadening of Cu2E lines implies a order is present in all copper oxides and whether this has any one-dimensional character of the modulation within the planes and relationship with superconductivity are, however, two highly con- imposes strong constraints on the charge pattern. Actually, only two troversial issues2,3. To uncover underlying electronic order, mag- types of modulation are compatible with a Cu2F splitting (Fig. 2). The netic fields strong enough to destabilize superconductivity can be first is a commensurate short-range (2a or 4a period) modulation used. Such experiments, including quantum oscillations4–6 in running along the (chain) b axis. However, this hypothesis is highly YBa2Cu3Oy (an extremely clean copper oxide in which charge unlikely: to the best of our knowledge, no such modulation has ever order has not until now been observed) have suggested that super- been observed in the CuO2 planes of any copper oxide; it would there- conductivity competes with spin, rather than charge, order7–9. Here fore have to be triggered by a charge modulation pre-existing in the we report nuclear magnetic resonance measurements showing that filled chains. A charge-density wave is unlikely because the finite-size high magnetic fields actually induce charge order, without spin chains are at best poorly conducting in the temperature and doping 11,14 order, in the CuO2 planes of YBa2Cu3Oy. The observed static, uni- range discussed here . Any inhomogeneous charge distribution directional, modulation of the charge density breaks translational such as Friedel oscillations around chain defects would broaden rather symmetry, thus explaining quantum oscillation results, and we than split the lines. Furthermore, we can conclude that charge order argue that it is most probably the same 4a-periodic modulation occurs only for high fields perpendicular to the planes because the as in stripe-ordered copper oxides1. That it develops only when NMR lines neither split at 15 T nor split in a field of 28.5 T parallel superconductivity fades away and near the same 1/8 hole doping to the CuO2 planes (along either a or b), two situations in which as in La22xBaxCuO4 (ref. 1) suggests that charge order, although superconductivity remains robust (Fig. 1). This clear competition visibly pinned by CuO chains in YBa2Cu3Oy, is an intrinsic pro- between charge order and superconductivity is also a strong indication pensity of the superconducting planes of high-Tc copper oxides. that the charge ordering instability arises from the planes. The ortho II structure of YBa2Cu3O6.54 (p 5 0.108, where p is the The only other patterncompatiblewith NMR data is an alternation of hole concentration per planar Cu) leads to two distinct planar Cu more and less charged Cu2F rows defining a modulation with a period NMR sites: Cu2F are those Cu atoms located below oxygen-filled of four lattice spacings along the a axis (Fig. 2). Strikingly, this corre- 10 chains, and Cu2E are those below oxygen-empty chains . The main sponds to the (site-centred) charge stripes found in La22xBaxCuO4 at discovery of our work is that, on cooling in a field H0 of 28.5 T along the doping levels near p 5 x 5 0.125 (ref. 1). Being a proven electronic c axis (that is, in the conditions for which quantum oscillations are instability of the planes, which is detrimental to superconductivity2, resolved; see Supplementary Materials), the Cu2F lines undergo a stripe order not only provides a simple explanation of the NMR splitting profound change, whereas the Cu2E lines do not (Fig. 1). To first order, but also rationalizes the striking effect of the field. Stripe order is also this change can be described as a splitting of Cu2F into two sites having fully consistent with the remarkable similarity of transport data in both different hyperfine shifts K 5 Æhzæ/H0 (where Æhzæ is the hyperfine YBa2Cu3Oy and in stripe-ordered copper oxides (particularly the 11–13 field due to electronic spins) and quadrupole frequencies nQ (related to dome-shaped dependence of T0 around p 5 0.12) . However, stripes the electric field gradient). Additional effects might be present (Fig. 1), must be parallel from plane to plane in YBa2Cu3Oy, whereas they are but they are minor in comparison with the observed splitting. Changes perpendicular in, for example, La22xBaxCuO4. We speculate that this in field-dependent and temperature-dependent orbital occupancy (for explains why the charge transport along the c axis in YBa2Cu3Oy example dx2{y2 versus dz2{r2 ) without on-site change in electronic becomes coherent in high fields below T0 (ref. 15). If so, stripe fluctua- density are implausible, and any change in out-of-plane charge density tions must be involved in the incoherence along c above T0. or lattice would affect Cu2E sites as well. Thus, the change in nQ can Once we know the doping dependence of nQ (ref. 16), the difference only arise from a differentiation in the charge density between Cu2F DnQ 5 320 6 50 kHz for p 5 0.108 implies a charge density variation sites (or at the oxygen sites bridging them). A change in the asymmetry as small as Dp 5 0.03 6 0.01 hole between Cu2Fa and Cu2Fb. A parameter and/or in the direction of the principal axis of the electric canonical stripe description (Dp 5 0.5 hole) is therefore inadequate 25 field gradient could also be associated with this charge differentiation, at the NMR timescale of ,10 s, at which most (below T0) or all but these are relatively small effects. (above T0) of the charge differentiation is averaged out by fluctuations 5 21 The charge differentiation occurs below Tcharge 5 50 6 10 K for faster than 10 s . This should not be a surprise: the metallic nature of p 5 0.108 (Fig. 1 and Supplementary Figs 9 and 10) and 67 6 5 K for the compoundLETTER at all fields is incompatible with full charge order, even doi:10.1038/nature10345 p 5 0.12 (Supplementary Figs 7 and 8). Within error bars, for each of if this order is restricted to the direction perpendicular to the stripes17. the samples Tcharge coincides with T0, the temperature at which the Actually, there is compelling evidence of stripe fluctuations down to Magnetic-field-induced charge-stripe18 order in the Hall constant RH becomes negative, an indication of the Fermi surface very low temperatures in stripe-ordered copper oxides , and indirect high-temperature superconductor YBa2Cu3Oy 1Laboratoire National des Champs Magne´tiques Intenses, UPR 3228, CNRS-UJF-UPS-INSA, 38042 Grenoble, France. 2Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada. 3Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada.Tao Wu1, Hadrien Mayaffre1, Steffen Kra¨mer1, Mladen Horvatic´1, Claude Berthier1, W. N. Hardy2,3, Ruixing Liang2,3, D. A. Bonn2,3 & Marc-Henri Julien1

8SEPTEMBER2011|VOL477|NATURE|191 11–13 ©2011 Macmillan Publishers Limited.Electronic All rights charges reserved introduced in copper-oxide (CuO2LETTER) planes RESEARCHreconstruction . Thus, whatever the precise profile of the static generate high-transition-temperature (Tc) superconductivity but, charge modulation is, the reconstruction must be related to the trans- under special circumstances, they can also order into filaments lational symmetry breaking by the charge ordered state. called stripes1. Whether an underlying tendency towards charge The absence of any splitting or broadening of Cu2E lines implies a a 100 b order is present in all copper oxides and whether this has any one-dimensional character of the modulation within the planes and relationship120 with superconductivity are, however, two highly con- imposes strong constraints on the charge pattern. Actually, only two 28.5 T troversial issues2,3. To uncover underlying electronic order, mag- types of modulation are compatible with a Cu2F splitting (Fig. 2). The ) 33.5 T netic fields strong enough to destabilize superconductivity can be

–1 first is a commensurate short-range (2a or 4a period) modulation 10–1 4–6 15 T used. Such experiments, including quantum oscillations in running along the (chain) b axis. However, this hypothesis is highly (ms 1 15 T YBa2Cu3Oy (an extremely clean copper oxide in which charge unlikely: to the best of our knowledge, no such modulation has ever T

1/ order has not until now been observed) have suggested that super- been observed in the CuO2 planes of any copper oxide; it would there- conductivity80 competes with spin, rather than charge, order7–9. Here fore have to be triggered by a charge modulation pre-existing in the 10–2 we report nuclear magnetic resonance measurements showingSuperconducting that filled chains. A charge-density wave is unlikely because the finite-size high magnetic fields actually induce charge order, without spin chains are at best poorly conducting in the temperature and doping 11,14 order, in the (K) CuO2 planes of YBa2Cu3Oy. The observed static, uni- range discussed here . Any inhomogeneous charge distribution T c d directional, modulation of the charge density breaks translational such as Friedel oscillations around chain defects would broaden rather 0.06 symmetry, thus explaining quantum oscillation results, and we than split the lines. Furthermore, we can conclude that charge order 28.5 T argue that it40 is most probably the same 4a-periodic modulation occurs only for high fields perpendicular to the planes because the ) 33.5 T 1

–1 as in stripe-ordered copper oxides . That it develops only when NMR lines neither split at 15 T nor split in a field of 28.5 T parallel s 0.04 superconductivity fades away and near the sameField-induced 1/8 hole doping to the CuO2 planes (along either a or b), two situations in which ( 2

charge order T as in La22xBaxCuO4 (ref. 1) suggests that charge order, although superconductivity remains robust (Fig. 1). This clear competition

1/ visibly pinned bySpin CuO chains in YBa2Cu3Oy, is an intrinsic pro- between charge order and superconductivity is also a strong indication 0.02 pensity of the superconductingorder planes of high-Tc copper oxides. that the charge ordering instability arises from the planes. 15 T 15 T The ortho II0 structure of YBa2Cu3O6.54 (p 5 0.108, where p is the The only other patterncompatiblewith NMR data is an alternation of hole concentration0.04 per planar0.08 Cu) leads to two0.12 distinct planar Cu0.16 more and less charged Cu2F rows defining a modulation with a period 0 NMR sites: Cu2F are those Cu atoms locatedp (hole/Cu) below oxygen-filled of four lattice spacings along the a axis (Fig. 2). Strikingly, this corre- 10 e 2.0 f chains, and Cu2E are those below oxygen-empty chains . The main sponds to the (site-centred) charge stripes found in La22xBaxCuO4 at Figure 4 | Phase diagram of underdoped YBa2Cu3Oy. The charge ordering discovery of our work is that, on cooling in a field H0 of 28.5 T along the doping levels near p 5 x 5 0.125 (ref. 1). Being a proven electronic temperature T (defined as the onset of the Cu2F line splitting; blue open 2 1.5 Saturday,c axis March (that 23, is, 13 in thecharge conditions for which quantum oscillations are instability of the planes, which is detrimental to superconductivity ,

resolved;circles) see Supplementary coincides with T Materials),0 (brown plus the signs), Cu2F the lines temperature undergo a at whichstripe orderthe Hall not only provides a simple explanation of the NMR splitting 1.0 profoundconstant change,RH whereaschanges the its Cu2E sign. linesT0 is do considered not (Fig. 1). as To the first onset order, of thebut Fermi also surface rationalizes the striking effect of the field. Stripe order is also 0 50 100 0 50 100 11–13 T (K) T (K) this changereconstruction can be described. The as a continuous splitting of Cu2F line represents into two sites the having superconductingfully consistent with the remarkable similarity of transport data in both differenttransition hyperfine temperature shifts KT5c. TheÆhzæ/ dashedH0 (where lineÆh indicateszæ is the hyperfine the speculativeYBa2 natureCu3Oy ofand in stripe-ordered copper oxides (particularly the 11–13 field duethe to extrapolation electronic spins) of the and field-induced quadrupole frequencies charge order.nQ (related The magnetic to dome-shaped transition dependence of T0 around p 5 0.12) . However, stripes g 8 27 the electrictemperatures field gradient). (Tspin) Additional are from muon-spin-rotation effects might be present (mSR) (Fig. data 1), (greenmust stars) be parallel. T0 from plane to plane in YBa2Cu3Oy, whereas they are but theyand areT minorspin vanish in comparison close to the with same the critical observed concentration splitting. Changesp 5 0.08.perpendicular A scenario of in, for example, La22xBaxCuO4. We speculate that this in field-dependentfield-induced and spin temperature-dependent order has been predicted orbital for p occupancy. 0.08 (ref. (for 8) byexplains analogy why with the charge transport along the c axis in YBa Cu O 4 2 3 y exampleLa1.855dx2{Sry20.145versusCuOd4z,2 for{r2 ) which without the on-site non-magnetic change ground in electronic state switchesbecomes to coherent in high fields below T0 (ref. 15). If so, stripe fluctua- densityantiferromagnetic are implausible, and order any in change fields in greater out-of-plane than a fewcharge teslas density (ref. 7 andtions references must be involved in the incoherence along c above T0. or latticetherein). would Our affect work, Cu2E however, sites as shows well. Thus, that spin the change order does in n notQ can occur upOnce to , we30 know T. the doping dependence of nQ (ref. 16), the difference Intensity (arb. units) 0 0 20 40 60 80 100 only ariseIn contrast, from a differentiation the field-induced in the charge charge order density reported between here Cu2F raises theDn questionQ 5 320 6 of50 kHz for p 5 0.108 implies a charge density variation T (K) sites (orwhether at the oxygen a similar sites field-dependent bridging them). charge A change order in the actually asymmetry underliesas the small field as Dp 5 0.03 6 0.01 hole between Cu2Fa and Cu2Fb. A parameter and/or in the direction of the principal axis of the electric canonical stripe description (Dp 5 0.5 hole) is therefore inadequate dependence of the spin order in La22xSrxCuO4 and YBa2Cu3O6.45. Error bars 25 Figure 3 | Slow spin fluctuations instead of spin order. a, b, Temperaturefield gradient could also be associated with this charge differentiation, at the NMR timescale of ,10 s, at which most (below T0) or all 63 represent the uncertainty in defining the onset of the NMR line splitting (Fig. 1f dependence of the planar Cu spin-lattice relaxation rate 1/T1 for p 5 0.108but these are relatively small effects. (above T0) of the charge differentiation is averaged out by fluctuations and Supplementary Figs 8–10). 5 21 (a) and p 5 0.12 (b). The absence of any peak/enhancement on cooling rules The charge differentiation occurs below Tcharge 5 50 6 10 K for faster than 10 s . This should not be a surprise: the metallic nature of out the occurrence of a magnetic transition. c, d, Increase in the 63Cu spin–spinp 5 0.108 (Fig. 1 and Supplementary Figs 9 and 10) and 67 6 5 K for the compound at all fields is incompatible with full charge order, even 17 relaxation rate 1/T2 on cooling below ,Tcharge, obtained from a fit of the spin-p 5 0.12fluctuations (Supplementary strongly Figs 7 enhances and 8). Within the errorspin–lattice bars, for (1/ eachT1 of) andif this spin–spin order is restricted to the direction perpendicular to the stripes . a 139 echo decay to a stretched form s(t) exp(2(t/T2) ), for p 5 0.108 (c) andthe samples(1/T2)Tcharge relaxationcoincides rates with betweenT0, the temperatureTcharge and atTspin whichfor the LaActually, nuclei. there For is compelling evidence of stripe fluctuations down to / 63 18 p 5 0.12 (d). e, f, Stretching exponent a for p 5 0.108 (e) and p 5 0.12 (f). TheHall constantthe moreRH stronglybecomes negative, hyperfine-coupled an indication ofCu, the the Fermi relaxation surface ratesvery low become temperatures in stripe-ordered copper oxides , and indirect 5 deviation from a 2 on cooling arises mostly from an intrinsic combination1Laboratoire of so National large des that Champs the Magne Cu´tiques signal Intenses, is gradually UPR 3228, CNRS-UJF-UPS-INSA, ‘wiped out’ 38042 on coolingGrenoble, France. below2Department of Physics and Astronomy, University of British Columbia, Vancouver, 3 63 Gaussian and exponential decays, combined with some spatial distributionBritish of ColumbiaTcharge V6T 1Z1,(refs Canada. 18,Canadian 23, 24). Institute In for contrast, Advanced Research, the Toronto,Cu(2) Ontario M5G signal 1Z8, Canada. here in T values (Supplementary Information). The grey areas define the crossover 2 YBa2Cu3Oy does not experience any intensity loss and 1/T1 does not 8SEPTEMBER2011|VOL477|NATURE|191 temperature Tslow below which slow spin fluctuations cause 1/T2 to increase show any peak or enhancement©2011 as aMacmillan function Publishers of temperature Limited. All (Fig. rights 3). reserved and to become field dependent; note that the change of shape of the spin-echo Moreover, the anisotropy of the linewidth (Supplementary decay occurs at a slightly higher (,115 K) temperature than T . T is slow slow Information) indicates that the spins, although staggered, align mostly slightly lower than Tcharge, which is consistent with the slow fluctuations being a consequence of charge-stripe order. The increase of a at the lowest along the field (that is, c axis) direction, and the typical width of the 2 1/2 temperatures probably signifies that the condition cÆhz æ tc = 1, where tc is central lines at base temperature sets an upper magnitude for the static 23 the correlation time, is no longer fulfilled, so that the associated decay is no spin polarization as small as gÆSzæ # 2 3 10 mB for both samples in longer a pure exponential. We note that the upturn of 1/T2 is already present at fields of ,30 T. These consistent observations rule out the presence of 15 T, whereas no line splitting is detected at this field. The field therefore affects magnetic order, in agreement with an earlier suggestion based on the the spin fluctuations quantitatively but not qualitatively. g, Plot of NMR signal presence of free-electron-like Zeeman splitting6. intensity (corrected for a temperature factor 1/T and for the T2 decay) against In stripe-ordered copper oxides, the strong increase of 1/T on 5 5 2 temperature. Open circles, p 0.108 (28.5 T); filled circles, p 0.12 (33.5 T). cooling below T is accompanied by a crossover of the time decay The absence of any intensity loss at low temperatures also rules out the presence charge of magnetic order with any significant moment. Error bars represent the added of the spin-echo from the high-temperature Gaussian form exp(2K(t/T )2) to an exponential form exp(2t/T )18,23. A similar uncertainties in signal analysis, experimental conditions and T2 measurements. 2G 2E All measurements are with H ||c. crossover occurs here, albeit in a less extreme manner because of the absence of magnetic order: 1/T2 sharply increases below Tcharge and the evidence (explaining the rotational symmetry breaking) over a broad decay actually becomes a combination of exponential and Gaussian temperature range in YBa2Cu3Oy (refs 14, 19–22). Therefore, instead decays (Fig. 3). In Supplementary Information we provide evidence of being a defining property of the ordered state, the small amplitude of that the typical values of the 1/T2E below Tcharge imply that antiferro- the charge differentiation is more likely to be a consequence of stripe magnetic (or ‘spin-density-wave’) fluctuations are slow enough to 17 212 order (the smectic phase of an electronic liquid crystal ) remaining appear frozen on the timescale of a cyclotron orbit 1/vc < 10 s. partly fluctuating (that is, nematic). In principle, such slow fluctuations could reconstruct the Fermi sur- 25,26 In stripe copper oxides, charge order at T 5 Tcharge is always accom- face, provided that spins are correlated over large enough distances panied by spin order at Tspin , Tcharge. Slowing down of the spin (see also ref. 9). It is unclear whether this condition is fulfilled here. The

Thermodynamic phase diagram of static charge order in underdoped YBa2Cu3Oy David LeBoeuf1*, S. Krämer2, W. N. Hardy3,4, Ruixing Liang3,4, D. A. Bonn3,4 and Cyril Proust1,4*

The interplay between superconductivity and any other charge order is most likely uniaxial4. In zero field, long-range competing order is an essential part of the long-standing charge fluctuations in YBCO were recently observed with resonant debate on the origin of high-temperature superconductivity soft X-ray scattering (RSXS) up to 150 K and 160 K for p in cuprate materials1,2. Akin to the situation in the heavy 0.11 (ref. 5) and p 0.133 (ref. 6), respectively, whereas hard= fermions, organic superconductors and pnictides, it has been X-ray scattering experiments= suggest that a charge order develops proposed that the pairing mechanism in the cuprates comes below 135 K for p 0.12 (ref. 7). All measurements identify from fluctuations of a nearby quantum phase transition3. charge fluctuations= at two wave vectors corresponding to an Recent evidence for charge modulation4 and its associated incommensurate periodicity of approximately 3.2 lattice units. 5–7 fluctuations in the pseudogap phase of YBa2Cu3Oy makes The identification of a thermodynamic phase transition is thus charge order a likely candidate for a competing order. However, important to determine where long-range charge order exists in the a thermodynamic signature of the charge-ordering phase phase diagram and particularly whether static order occurs only in transition is still lacking. Moreover, whether the charge high magnetic fields. modulation is uniaxial or biaxial remains controversial. Here Here we report sound velocity measurements, a thermodynamic we address both issues by measuring sound velocities in probe, in magnetic fields large enough to suppress superconduc- YBa2Cu3O6.55 in high magnetic fields. We provide the first tivity. The sound velocity is defined as vs pcij /⇢, where ⇢ is the thermodynamic signature of the competing charge-order phase density of the material, c @2F/@" @" =(ref. 15), F is the free ij = i j transition in YBa2Cu3Oy and construct a field–temperature energy and "i is the strain along direction i (in the contracted Voigt phase diagram. The comparison of different acoustic modes notation). Changes in the elastic constants cij are expected whenever indicates that the charge modulation is biaxial, which differs a strain-dependent phase transition occurs. Owing to their high from a uniaxial stripe charge order. sensitivity, sound velocity measurements are a powerful probe for In most La-based cuprate superconductors, static order of both detecting such phase transitions, in particular charge ordering in spin and charge (so-called stripe order) has been unambiguously strongly correlated electron systems16. identified by spectroscopic and thermodynamic probes1,2. At We have measured several elastic constants (see Supplementary low temperature, magnetic fields weaken superconductivity and Table S1 for the description of the elastic modes) in high at the same time reinforce the magnitude of such orders8–10. magnetic fields in an underdoped YBCO sample with T 60.7K 6.55 c = As the superconducting transition temperature (Tc) in the La- corresponding to a hole doping p 0.108 (ref. 17). Figure 1a,b based materials is substantially lower than in other cuprate shows the field dependence of the= relative variation of the materials, it has been argued that stripe order is detrimental to sound velocity 1vs/vs corresponding to the c11 mode, at different high-temperature superconductivity. In underdoped YBa Cu O temperatures. At T 4.2 K, the softening of the elastic constant at 2 3 y = (YBCO), there is now compelling evidence of competing order the vortex lattice melting field Bm 20 T corresponds to the first- ⇡ 18,19 even though Tc 94 K at optimal doping. The discovery of order melting transition from a vortex lattice to a vortex liquid quantum oscillations= 11 combined with the negative Hall12 and (see Supplementary Information for more details). At T 29.5 K, 13 = Seebeck coefficients at low temperature has demonstrated this anomaly shifts to lower field (Bm 5 T) and because the pinning that the Fermi surface of underdoped YBCO undergoes a potential becomes less effective, the⇡ magnitude of the change of reconstruction at low temperature and consists of at least one c at the melting transition becomes smaller20. At T 29.5 K and 11 LETTERS= electron pocket. A comparative study of thermoelectric transport above Bm, a sudden increase of the elastic constant can clearly be PUBLISHED ONLINE: 23 DECEMBER 2012 | DOI: 10.1038/NPHYS2502 in underdoped YBCO and in La1.8 xEu0.2SrxCuO4 (a cuprate in resolved at Bco 18 T, which corresponds to a thermodynamic = which stripe order is well established) has been interpreted as signature of a phase transition. Whereas Bco is almost temperature charge stripe order causing reconstruction of the Fermi surface at independent at low temperature, it increases rapidly between 35 and 14 low temperature . High-field nuclear magnetic resonance (NMR) 50 K (see red arrows in Fig. 1b). For T 50 K, no change of c11 can measurements have revealed that the translational symmetry be resolved up to the highest field. Owing to the difference in the Thermodynamicof the CuO2 planes in YBCO phase is broken by the diagram emergence temperature of dependence static of Bm and chargeBco, the phase transition at Bco of a modulation of the charge density at low temperature4. cannot originate from vortices. Figure 2 shows the phase diagram in In addition, NMR measurements show that the modulation which both Bm and Bco deduced from sound velocity measurements orderis in observed underdoped above a threshold magnetic field YBa andLETTERS suggest2Cu that 3Oare plottedy as a function of temperature. The identification of this PUBLISHED ONLINE: 23 DECEMBER 2012 | DOI: 10.1038/NPHYS2502 David LeBoeuf1*, S. Krämer2, W. N. Hardy3,4, Ruixing Liang3,4, D. A. Bonn3,4 and Cyril Proust1,4* Thermodynamic1Laboratoire phase National diagram des Champs Magnétiques of static Intenses, charge UPR 3228, (CNRS-INSA-UJF-UPS), Toulouse 31400, France, 2Laboratoire National des Champs Magnétiques Intenses, UPR 3228, (CNRS-INSA-UJF-UPS), Grenoble 38042, France, 3Department of Physics and Astronomy, University of British order in underdopedColumbia, Vancouver YBa V6TCu 1Z1, Canada,O 4Canadian Institute for Advanced Research, Toronto M5G 1Z8, Canada. *e-mail: [email protected]; The interplay between superconductivity2 3 y and any other 4 LETTERS [email protected] PHYSICS. DOI: 10.1038/NPHYS2502 charge order is most likely uniaxial . In zero field, long-range David LeBoeufcompeting1*, S. Krämer order2, W. is N. Hardy an essential3,4, Ruixing Liang part3,4, ofD. A. the Bonn long-standing3,4 and Cyril Proust1,4*charge fluctuations in YBCO were recently observed with resonant debate on the origin of high-temperature superconductivity a NATURE PHYSICS VOL 9 FEBRUARYTco 2013 www.nature.com/naturephysicssoft X-ray scattering (RSXS) up to 150 K and 160 K for p 79 4.2 K 1,2 | | | = The interplayin cuprate between superconductivity materials and. any Akin other to the situation in the4 heavy 0.11 (ref. 5) and p 0.133 (ref. 6), respectively, whereas hard 9.9 K charge order is most likely uniaxial . In zero field, long-range c11 © 2013 Macmillan Publishers Limited. All rights= reserved 2 competingfermions, order is an essential organic part of superconductors the long-standing charge and fluctuations pnictides, in YBCO were it recently has observed been with resonant 14.9 K debate on the origin of high-temperature superconductivity X-ray scattering experiments suggest that a charge order develops 30 Static soft X-ray scattering (RSXS) up to 150 K and 160 K for p 19.6 K in cuprateproposed materials1,2. Akin that to the the situation pairing in the mechanism heavy 0.11 (ref. 5 in) and thep 0 cuprates.133 (ref. 6), respectively, comes whereas hard= . charge order = below 135 K for p 0 12 (ref. 7). All measurements identify 24.9 K fermions, organic superconductors and pnictides, it has been X-ray scatteringTNMR experiments suggest that a charge3 order develops = proposed thatfrom the pairing fluctuations mechanism in the of cuprates a nearby comes below quantum 135 Kco for p phase0.12 (ref. transition7). All measurements. charge identify fluctuations at two wave vectors corresponding to an from fluctuationsRecent of a evidence nearby quantum for phase charge transition modulation3. charge fluctuations4 and= at two its wave associated vectors correspondingincommensurate to an periodicity of approximately 3.2 lattice units. 0 Recent evidence for charge modulation4 and its associated incommensurate periodicity of approximately 3.2 lattice units. ) 5–7 5–7 ¬4 fluctuationsfluctuationsin the pseudogap phasein theof YBa pseudogap2Cu3Oy makes The phase identification of YBa of a thermodynamic2Cu3Oy makes phase transitionThe is thus identification of a thermodynamic phase transition is thus charge order a likely candidate for a competing order. However, importantB to determine where long-range charge order exists in the

(10 co s

charge order (T) a likely candidate for a competing order. However, v a thermodynamic signature of the charge-ordering phase phase diagram and particularly whether static order occursimportant only in to determine where long-range charge order exists in the B / 15 s v transitiona is still thermodynamic lacking. Moreover, whether signature the charge ofhigh the magnetic charge-ordering fields. phase ∆ phase diagram and particularly whether static order occurs only in modulation is uniaxial or biaxial remains controversial. Here Here we report sound velocity measurements, a thermodynamic ¬2 Vortex liquid we addresstransition both issues by is measuring still sound lacking. velocities Moreover, in probe, in magnetic whether fields large the enough charge to suppress superconduc-high magnetic fields. B YBa2Cu3O6.55 in high magnetic fields. We provide the first tivity. The sound velocity is defined as vs pcij /⇢, where ⇢ is the CO thermodynamicmodulation signature of the is competing uniaxial charge-order or biaxial phase density remains of the material, controversial.c @2F/@" @" =(ref. Here15), F is the freeHere we report sound velocity measurements, a thermodynamic YBCO ij = i j transition in YBa2Cu3Oy and constructVortex a solid field–temperature energy and "i is the strain along direction i (in the contracted Voigt Bm we address both issues by measuringB p = sound 0.108 velocities in probe, in magnetic fields large enough to suppress superconduc- phase diagram. The comparison of different acoustic modes mnotation). Changes in the elastic constants cij are expected whenever indicates that the charge modulation is biaxial, which differs ¬4 YBa2Cu3O60.55 in high magnetica fields. strain-dependent We phase provide transition the occurs. first Owing to theirtivity. high The sound velocity is defined as vs pcij /⇢, where ⇢ is the from a uniaxialthermodynamic stripe charge order.0 signature 20 of the competingsensitivity, 40 sound charge-order velocity 60 measurements phase are a powerful probedensity for of the material, c @2F/@" @" =(ref. 15), F is the free In most La-based cuprate superconductors, static orderTemperature of both (K)detecting such phase transitions, in particular charge ordering in ij i j spin and charge (so-called stripe order) has been unambiguously strongly correlated electron systems16. = 0 10 20 30 transition in YBa2Cu3Oy and construct a field–temperature energy and " is the strain along direction i (in the contracted Voigt identified by spectroscopic and thermodynamic probes1,2. At We have measured several elastic constants (see Supplementary i Figure 2 Thermodynamic phase diagram. Magnetic field–temperature B (T) low temperature,phase magnetic diagram. fields weaken The superconductivity comparison and Table of different S1 for the description acoustic of the modes elastic modes)notation). in high Changes in the elastic constants cij are expected whenever | 8–10 b 0 at the same timephase reinforce diagram the of magnitude underdoped of such YBCO orders (p 0..108)magnetic obtained fields from in an the underdoped YBCO6.55 sample with Tc 60.7K As the superconductingindicates transition that temperature the charge (T ) in modulation the= La- corresponding is biaxial, to a hole doping whichp 0. differs108 (ref. 17). Figurea= strain-dependent1a,b phase transition occurs. Owing to their high c 29.5 K anomalies seen in the elastic constantc c11 (Fig. 1). Black squares indicate the 11 based materials is substantially lower than in other cuprate shows the field dependence of the= relative variation of the 34.8 K fromtransition a uniaxial from a vortex stripe lattice charge to a vortex order. liquid at Bm, which cannot be sensitivity, sound velocity measurements are a powerful probe for materials, it has been argued that stripe order is detrimental to sound velocity 1vs/vs corresponding to the c11 mode, at different Saturday,40.1 March K 23, 13 high-temperatureInresolved superconductivity. most above La-based 40 InK. Red underdoped circlescuprate correspond YBa2Cu superconductors,3Oy totemperatures. the phase transition At T static4.2 K, the order softening of of both the elastic constantdetecting at such phase transitions, in particular charge ordering in = 42.2(YBCO), K there istowards now compelling static charge evidence order of at competingBco, as observed order the in c vortex11. The lattice error melting bars on field theBm 20 T corresponds to the first- 16 spin and charge (so-called stripe order) has been unambiguously⇡ strongly18,19 correlated electron systems . 43.5even K though Tcfield94 scale K atB optimal(B ) doping. correspond The to discovery the width of oforder the transition melting transition in the from a vortex lattice to a vortex liquid ¬2 quantum oscillations= 11 combinedm co with the negative Hall12 and (see Supplementary Information for more1,2 details). At T 29.5 K, 44.8 K 13 identifiedderivative (raw by data) spectroscopic of c (B). The charge-order and thermodynamic transition is almost probes . At = We have measured several elastic constants (see Supplementary Seebeck coefficients at low temperature11 has demonstrated this anomaly shifts to lower field (Bm 5 T) and because the pinning BCO 50 K ⇡

) that the Fermilowtemperature surface temperature, of underdoped independent magnetic YBCO up to undergoes40 K.fields Above a potential weaken 40 K the becomes ﬁeld superconductivity scale lessB effective,co at the magnitude and of the changeTable of S1 for the description of the elastic modes) in high 20 ¬4 ⇡ reconstruction atwhich low temperature charge order and sets consists in rises. of Inat the least Supplementary one c11 at the Information,melting transition we becomes smaller 8.– At10T 29.5 K and electron pocket.at the A comparative same study time of thermoelectric reinforce transport the magnitudeabove B , a sudden of increase such of the orders elastic constant. can= clearlymagnetic be fields in an underdoped YBCO6.55 sample with Tc 60.7K

(10 m s argue that the overall behaviour of the charge-order phase boundary in this v in underdoped YBCO and in La1.8 xEu0.2SrxCuO4 (a cuprate in resolved at Bco 18 T, which corresponds to a thermodynamic = / . s As the superconducting transition temperature= (Tc) in the La- corresponding to a hole doping p 0 108 (ref. 17). Figure 1a,b v ¬4 which stripe orderB–T isdiagram well established) is consistent has been with interpreted a theoretical as modelsignature of superconductivity of a phase transition. in Whereas Bco is almost temperature ∆ = charge stripebased ordercompetition causing materials reconstruction with a density-wave is of the substantially Fermi state surface21. The at green lowerindependent diamond than at low is thetemperature, in other it increases cuprate rapidly betweenshows 35 and the field dependence of the relative variation of the 14 low temperature . High-field nuclearNMR magnetic resonance (NMR) 50 K (see red arrows in Fig. 1b). For T 50 K, no change of c11 can temperature Tco 50 10 K at which NMR experiments detect the onset 1 / measurementsmaterials, have revealed it that has the= been translational± argued symmetry thatbe stripe resolved up order to the highest is detrimental field. Owing to the to differencesound in the velocity vs vs corresponding to the c11 mode, at different of the CuO planesof a charge in YBCO modulation is broken at a by field theB emergence28.5 T in YBCOtemperature at doping dependencep 0. of11 B and B , the phase transition at B high-temperature2 superconductivity.= In underdoped= m YBaco 2Cu3Oy temperatures.co At T 4.2 K, the softening of the elastic constant at of a modulation(ref. of4 the). Within charge the density error at bars, low this temperature onset temperature4. cannot originate agrees with from vortices.our Figure 2 shows the phase diagram in = In addition,(YBCO), NMRfindings. measurements Dashed there lines show is arenow that guides the compelling modulation to the eye. which evidence both Bm and ofBco deduced competing from sound order velocity measurementsthe vortex lattice melting field Bm 20 T corresponds to the first- ¬6 is observed above a threshold magnetic field and suggest that are plotted as a function of temperature. The identification of this ⇡ 18,19 even though Tc 94 K at optimal doping. The discovery of order melting transition from a vortex lattice to a vortex liquid quantumwith a density-wave oscillations= order11 21combined(see discussion with in the the Supplementary negative Hall12 and (see Supplementary Information for more details). At T 29.5 K, 0 10 20 30 Information).13 For T below 40 K or so, static charge order sets = 1Laboratoire NationalSeebeck des Champscoefficients Magnétiques Intenses, UPR at 3228, low (CNRS-INSA-UJF-UPS), temperature Toulouse has 31400, France, demonstrated2Laboratoire National desthis Champs anomaly shifts to lower field (Bm 5 T) and because the pinning B (T) in only above a threshold field of 18 T, akin to the situation in Magnétiques Intenses, UPR 3228, (CNRS-INSA-UJF-UPS), Grenoble 38042, France, 3Department of Physics and Astronomy, University of British ⇡ Columbia, VancouverthatLa V6T2 thex 1Z1,Srx Canada,CuO Fermi4 4(Canadianx surface0. Institute145) in for Advanced which of underdoped aResearch, magnetic Toronto field M5G YBCO 1Z8, is necessary Canada. *e-mail: [email protected] a potential; becomes less effective, the magnitude of the change of = 20 Figure 1 Field dependence of the sound velocity in [email protected] destabilize. superconductivity at low temperature and to drive and the consists system of to ata least one c11 at the melting transition becomes smaller . At T 29.5 K and | 9 YBa2Cu3Oy. a,b, Field dependence of the longitudinal mode c11 magnetically ordered state . Close to the onset temperature of = NATURE PHYSICSelectronVOL 9 FEBRUARY pocket. 2013 www.nature.com/naturephysics A comparative study of thermoelectric transport above79 Bm, a sudden increase of the elastic constant can clearly be (propagation q and polarization u of the sound wave along a axis) in | static| charge order,| Tco, the threshold field Bco sharply increases underdoped YBCO (p 0.108) at different temperatures from T 4.2Ktoin underdopedand the phase boundary YBCO© 2013 and tends Macmillan in to Publishers La become1. 8Limited.xEu All vertical. 0rights.2Sr reservedxCuO This4 is(a in cuprate in resolved at Bco 18 T, which corresponds to a thermodynamic = = = T 24.9K(a), and from T 29.5KtoT 50 K (b). The curves are shiftedwhichagreement stripe with order the theoretical is well established) phase of competing has been order with interpreted as signature of a phase transition. Whereas B is almost temperature = = = co for clarity. The measurements were performed in static magnetic field up tochargesuperconductivity stripe order that causing predicts reconstruction that superconducting of fluctuations the Fermi surface at independent at low temperature, it increases rapidly between 35 and 28 T. Black arrows indicate the field Bm corresponding to the vortex lattice lowhave temperature no significant14. High-field effect on charge nuclear order magnetic in this resonance part of (NMR) 50 K (see red arrows in Fig. 1b). For T 50 K, no change of c can melting. At low temperature, the loss of the vortex lattice compression the phase diagram. 11 modulus can be estimated and is in agreement with previous studies (see measurementsWe now turn have to the revealed analysis of that the symmetry the translational of the charge symmetry be resolved up to the highest field. Owing to the difference in the Supplementary Information). For T 40 K, B cannot be resolved. Red > m ofmodulation. the CuO2 Inplanes the framework in YBCO of the is Landau broken theory by of phase the emergence temperature dependence of Bm and Bco, the phase transition at Bco arrows indicate the field Bco where the charge-order phase transition oftransitions, a modulation an anomaly of inthe the charge elastic constant density occurs at at low a phase temperature4. cannot originate from vortices. Figure 2 shows the phase diagram in occurs. This transition is not related to vortex physics because it is also transition only if a coupling in the free energy F g Qm "n (where c = mn seen in acoustic modes c44 and c55 (Fig. 3 and Supplementary Fig. S3), Inm addition,and n are integers NMR and measurementsgmn is a coupling constant)show that between the the modulation which both Bm and Bco deduced from sound velocity measurements which are insensitive to the flux line lattice because those modes involve is observedorder parameter aboveQ and a threshold the strain " is magnetic symmetry allowed, field and that is, suggest that are plotted as a function of temperature. The identification of this atomic motions parallel to the vortex flux lines (u H c). only if Qm and "n transform according to the same irreducible k k representation22. In Fig. 3 we compare the field dependence at phase stabilized by the magnetic field above B is straightforward. T 20 K of four different modes c , c , c and c that display co = 11 44 55 66 High-field NMR measurements in YBCO at similar doping have an anomaly at Bco. To explain the presence of such coupling for shown that charge order develops above a threshold field Bco > 15 T1 all these modes, we rely on group theory arguments. YBCO is 2 RMN Laboratoire National des Champs Magnétiques Intenses, UPR 3228, (CNRS-INSA-UJF-UPS), Toulouse 31400, France, Laboratoire National des Champs and below Tco 50 10 K (ref. 4). Given the similar field and an orthorhombic system (point group D2h), and given the even 3 temperature scales,= it is± natural to attribute the anomaly seen inMagnétiquescharacter of Intenses, the strains UPR we have 3228, only (CNRS-INSA-UJF-UPS), to consider the character table Grenoble 38042, France, Department of Physics and Astronomy, University of British 4 the elastic constant at Bco to the thermodynamic transition towardsColumbia,of point Vancouver group D2 shown V6T in 1Z1,Table Canada,1. Canadian Institute for Advanced Research, Toronto M5G 1Z8, Canada. *e-mail: [email protected]; the static charge order. [email protected] represent the different. symmetric charge modulations that The phase diagram in Fig. 2 shares common features with the transform according to each irreducible representation of the point theoretical phase diagram of superconductivity in competition group D2 and to determine to which acoustic mode they couple, we NATURE PHYSICS VOL 9 FEBRUARY 2013 www.nature.com/naturephysics 79 | | | 80 NATURE PHYSICS VOL 9 FEBRUARY 2013 www.nature.com/naturephysics | | | © 2013 Macmillan Publishers Limited. All rights reserved © 2013 Macmillan Publishers Limited. All rights reserved R EPORTS model that proposed that the circulating super- (x, y). We use this technique of combined Å (or L Ϸ 7.8 Ϯ 1.3a0). This is substantially currents weaken the superconducting order pa- electronic background subtraction and energy greater than the measured (21) core radius. It rameter and allow the local appearance of a integration to enhance the signal-to-noise ra- is also evident in Figs. 1B and 2A that the coexisting spin density wave (SDW) and HTSC tio of the vortex-induced states. In Bi-2212, LDOS oscillations have stronger spectral phase (23)surroundingthecore.Inamore these states are broadly distributed in energy weight in one Cu-O direction than in the Ϯ12 recent model, which is an extension of (5)and around Ϯ7 meV (21), so S Ϯ1 (x, y, B) effec- (22), the effective mass associated with spin tively maps the additional spectral strength fluctuations results in an AF localization length under their peaks. 12 that might be substantially greater than the core Figure 1B is an image of S1 (x, y, 5) radius (30). An associated appearance of charge measured in the FOV of Fig. 1A. The loca- density wave order was also predicted (31) tions of seven vortices are evident as the whose effects on the HTSC quasi-particles darker regions of dimension ϳ100 Å. Each should be detectable in the regions surrounding vortex displays a spatial structure in the inte- the vortex core (23). grated LDOS consisting of a checkerboard To test these ideas, we apply our recently pattern oriented along Cu-O bonds. We have developed techniques of low-energy quasi-par- observed spatial structure with the same pe- ticle imaging at HTSC vortices (21). We choose riodicity and orientation, in the vortex-in- to study Bi-2212, because YBCO and LSCO duced LDOS on multiple samples and at have proven nonideal for spectroscopic studies fields ranging from 2 to 7 T. In all 35 vortices because their cleaved surfaces often exhibit studied in detail, this spatial and energetic nonsuperconducting spectra. Our “as-grown” structure exists, but the checkerboard is more Bi-2212 crystals are generated by the floating clearly resolved by the positive-bias peak. zone method, are slightly overdoped with Tc ϭ We show the power spectrum from the 89 K, and contain 0.5% of Ni impurity atoms. two-dimensional Fourier transform of 12 12 12 2 They are cleaved (at the BiO plane) in cryogen- S1 (x, y,5),PS[S1 (x, y,5)]ϭ{FT͓S1 ͑x, y,5)]}, ic ultrahigh vacuum below 30 K and immedi- in Fig. 2A and a labeled schematic of these ately inserted into the STM head. Figure 1A results in Fig. 2B. In these k-space images, on March 1, 2013 shows a topographic image of the 560 Å square the atomic periodicity is detected at the points area where all the STM measurements reported labeled by A, which by definition are at here were carried out. The atomic resolution (0,Ϯ1) and (Ϯ1,0). The harmonics of the and the supermodulation (with wavelength supermodulation are identified by the sym-

ϳ26 Å oriented at 45° to the Cu-O bond direc- bols B1 and B2. These features (A, B1, and tions) are evident throughout. B2) are observed in the Fourier transforms of To study effects of the magnetic field B on all LDOS maps, independent of magnetic the superconducting electronic structure, we field, and they remain as a small background

12 www.sciencemag.org first acquire zero-field maps of the differential signal in PS[S1 (x, y, 5)] because the zero- tunneling conductance (G ϭ dI/dV)measured field and high-field LDOS images can only at all locations (x, y)inthefieldofview(FOV) be matched to within 1 Å before subtraction. 12 of Fig. 1A. Because LDOS(E ϭ eV) ϰ G(V), Most importantly, PS[S1 (x, y, 5)] reveals where V is the sample bias voltage, this results new peaks at the four k-space points, which Fig. 1. Topographic and spectroscopic images of in a two-dimensional map of the local density correspond to the spatial structure of the vor- the same area of a Bi-2212 surface. (A)Atopo- of states LDOS(E, x, y, B ϭ 0). We acquire tex-induced quasi-particle states. We label graphic image of the 560 Å field of view (FOV ) in these LDOS maps at energies ranging from –12 their locations C. No similar peaks in the which the vortex studies were carried out. The meV to ϩ12 meV in 1-meV increments. The B spectral weight exist at these points in the supermodulation can be seen clearly along with Downloaded from some effects of electronic inhomogeneity. The field is then ramped to its target value, and, after two-dimensional Fourier transform of these Cu–O–Cu bonds are oriented at 45° to the su- any drift has stabilized, we remeasure the topo- zero-field LDOS maps. permodulation. Atomic resolution is evident graph with the same resolution. The FOV To quantify these results, we fit a Lorent- throughout, and the inset shows a 140 Å square R EPORTS 12 FOV at ϫ2magnificationtomakethiseasierto where the high-field LDOS measurements are zian to PS[S1 (x, y, 5)] at each of the four model that proposed that the circulatingto be made super- is then(x, matched y). We to use that this in Fig.technique 1A ofpoints combined labeledÅ C (or in Fig.L Ϸ 2B.7.8 Ϯ We1.3 finda ). that This they is substantiallysee. The mean Bi-Bi distance apparent here is 0 a ϭ 3.83 Å and is identical to the mean Cu-Cu currents weaken the superconductingwithin1Å( order pa-ϳ0.25aelectronic)bycomparingcharacter- background subtractionoccur and energy at k-spacegreater radius than 0.062 the measured ÅϪ1 with (21 width) core radius.0 It 0 distance in the CuO plane ϳ5Åbelow.(B)Amap rameter and allow the local appearance of a integration to enhance the signal-to-noise ra- is also evidentϪ1 in Figs. 1B and 2A that the istic topographic/spectroscopic features. Final- ␴ϭ0.011 Ϯ 0.002 Å . Figure 2C shows of S12(x, y, 5) showing the additional LDOS in- coexisting spin density wave (SDW) and HTSC tio of the vortex-induced states. In Bi-2212, LDOS12 oscillations have stronger spectral1 ly, we acquire the high-field LDOS maps, the value of PS[S1 (x, y, 5)] measured along duced by the seven vortices. Each vortex is ap- phase (23)surroundingthecore.InamoreLDOS(E, x, y, B), atthese the same states series are broadly of energies distributedthe in dashed energy lineweight in Fig. in 2B. one The Cu-O central direction peak thanparent in the as a checkerboard at 45° to the page recent model, which is an extension of (5)and around Ϯ7 meV (21), so S Ϯ12(x, y, B) effec- as the zero-field case. Ϯ1 associated with long-wavelength structure, orientation. Not all are identical, most likely be- (22), the effective mass associated with spin tively maps the additional spectral strength To focus preferentially on B field effects, the peak associated with the atoms, and the cause of the effects of electronic inhomogeneity. fluctuations results in an AF localization length under their peaks. RTheEPORTS units of S12(x, y, 5) are picoamps because it we define a type of two-dimensional map: peak12 due to the vortex-induced quasi-particle 1 that might be substantially greater than the core Figure 1B is an image of S (x, y, 5) represents ⌺dI/dV⅐⌬V.Inthisenergyrange,the states1 are all evident. The vortex-induced strong antiferromagnetic spin fluctuations are radius (30). An associated appearance of charge measuredE2 in the FOV of Fig. 1A. The loca- maximum integrated LDOSincluded at a (22–26 vortex). Common is ϳ3 elements of their density wave order was also predicted (31) tions of seven vortices are evidentstates as identified the A by Four this means Unit occur Cell at (Ϯ1/4, PeriodicpA, as compared with thepredictions zero field include integrated the following: (i) the prox- E2 S E (x, y, B) ϭ ͓LDOS͑E, x, y, B͒ imity of a phase transition into a magnetic or- whose effects on the HTSC quasi-particles1 darker͸ regions of dimension ϳ1000) and Å. Each (0, Ϯ1/4) to within the accuracy of the LDOS of ϳ1 pA. The latter is subtracted from should be detectable in the regions surrounding vortexE1 displays a spatial structuremeasurement. in thePattern inte- Equivalently, of Quasi-Particle the checkerboard the former States to give a maximumdered state contrast can be revealed of ϳ2 when the supercon- pA. We also note that theductivity integrated is weakened differen- by the influence of a the vortex core (23). Ϫ LDOSgrated͑E, x, LDOSy,0͔͒dE consisting(1) of a checkerboardpattern evidentSurrounding in the LDOS has Vortex spatial peri- Cores in vortex (22–26), (ii) the resulting magnetic order, To test these ideas, we apply our recently pattern oriented along Cu-O bonds.odicity We have 4a oriented along the Cu-O bonds. tial conductance betweeneither 0 and spinϪ (22200, 23, meV25)ororbital( is 24, 26), will 0 200 pA because all measurements reported in developed techniques of low-energywhich quasi-par- represents theobserved integral spatial of all structure additional with theFurthermore, same pe- the widthBi ␴Srof theCaCu LorentzianO coexist with superconductivity in some region ticle imaging at HTSC vortices (21). We choose riodicity and orientation, in the vortex-in- 2 2 2 8؉␦this paper were obtainednear at the a junction core, and (iii) resis- this localized magnetic spectral density induced by the B field be- yields a spatial correlation1 length1,2 for these 1tance of 1 gigaohm1 setorder at a will bias generate voltage associated of spatial modula- to study Bi-2212, because YBCO and LSCO duced LDOS on multiple samples andJ. at E. Hoffman, E. W. Hudson, * K. M. Lang, V. Madhavan, 3 3 1,2 tions in the quasi-particle density of states (23– tween the energies E1 and E2 at each location LDOS oscillationsH. of Eisaki,L ϭ (1/† S.␲␴ Uchida,) Ϸ 30J.Ϯ C.5 Davis–200‡ mV. have proven nonideal for spectroscopic studies fields ranging from 2 to 7 T. In all 35 vortices 26). Given the relevance of such predictions to because their cleaved surfaces often exhibit studied in detail, this spatial and energetic the identification of alternative ordered states, nonsuperconducting spectra. Our “as-grown” structure exists, but thewww.sciencemag.org checkerboard isScanning more tunneling SCIENCE microscopy VOL 295 is used 18to image JANUARY the additional 2002 quasi-particle determination of the magnetic467 and electronic Bi-2212 crystals are generated by the floating clearly resolved by the positive-bias peak.states generated by quantized vortices in the high critical temperature superstructure of the HTSC vortex is an urgent conductor Bi Sr CaCu O . They exhibit a copper-oxygen bond–oriented priority. zone method, are slightly overdoped with T ϭ We show the power spectrum from the 2 2 2 8ϩ␦ c “checkerboard” pattern, with four unit cell (4a0) periodicity and a ϳ30 ang- Information on the magnetic structure of 89 K, and contain 0.5% of Ni impurity atoms. two-dimensional Fourier transformstrom of decay length. These electronic modulations may be related to the mag- HTSC vortices has recently become available 12 12 12 netic2 field–induced, 8a periodic, spin density modulations with decay length from neutron scattering and nuclear magnetic They are cleaved (at the BiO plane) in cryogen- S1 (x, y,5),PS[S1 (x, y,5)]ϭ{FT͓S1 ͑x, y,5)]}, 0 of ϳ70 angstroms recently discovered in La Sr CuO . The proposed ex- resonance (NMR) studies. Near optimum dop- ic ultrahigh vacuum below 30 K and immedi- in Fig. 2A and a labeled schematic of these 1.84 0.16 4 planation is a spin density wave localized surrounding each vortex core. General ing, some cuprates show strong inelastic neutron on March 1, 2013 ately inserted into the STM head. Figure 1A results in Fig. 2B. In these k-space images,theoretical principles predict that, in the cuprates, a localized spin modulation scattering (INS) peaks at the four k-space points shows a topographic image of the 560 Å square the atomic periodicity is detected at theof points wavelength ␭ should be associated with a corresponding electronic modu- (1/2 Ϯ␦,1/2)and(1/2,1/2Ϯ␦), where ␦ϳ1/8 area where all the STM measurements reported labeled by A, which by definition arelation at of wavelength ␭/2, in good agreement with our observations. and k-space distances are measured in units of here were carried out. The atomic resolution (0,Ϯ1) and (Ϯ1,0). The harmonics of the 2␲/a0.Thisdemonstratestheexistence,inreal on March 1, 2013 and the supermodulation (with wavelength supermodulation are identified byTheory the indicates sym- that the electronic structure of nodes, the local density of electronic states space, of fluctuating magnetization density with the cuprates is susceptible to transitions into a (LDOS) inside the core is strongly peaked at the spatial periodicity of 8a0 oriented along the ϳ26 Å oriented at 45° to the Cu-O bond direc- bols B1 and B2. These features (A,variety B1 of, ordered and states (1–10). Experimentally, Fermi level. This peak, which would appear in Cu-O bond directions, in the superconducting tions) are evident throughout. B2) are observed in the Fourier transformsantiferromagnetism of (AF) and high-temperature tunneling studies as a zero bias conductance phase. The first evidence for field-induced mag- To study effects of the magnetic field B on all LDOS maps, independent ofsuperconductivity magnetic (HTSC) occupy well-known peak (ZBCP), should display a four-fold sym- netic order in the cuprates came from INS ex- regions of the phase diagram, but, outside these metric “star shape” oriented toward the gap periments on La Sr CuO by Lake et al. the superconducting electronic structure, we field, and they remain as a small background 1.84 0.16 4 regions, several unidentified ordered states exist. nodes and decaying as a power law with (27). When La1.84Sr0.16CuO4 is cooled into the 12 www.sciencemag.org first acquire zero-field maps of the differential signal in PS[S1 (x, y, 5)] becauseFor the example, zero- at low hole densities and above the distance. superconducting state, the scattering intensity at tunneling conductance (G ϭ dI/dV)measured field and high-field LDOS imagessuperconducting can only transition temperature, the un- Scanning tunneling microscopy (STM) stud- these characteristic k-space locations disappears at all locations (x, y)inthefieldofview(FOV) be matched to within 1 Å before subtraction.identified “pseudogap” state exhibits gapped ies of HTSC vortices have revealed a very dif- at energies below ϳ7meV,openingupa“spin www.sciencemag.org of Fig. 1A. Because LDOS(E ϭ eV) ϰ G(V), Most importantly, PS[S12(x, y, electronic5)] reveals excitations (11). Other unidentified ferent electronic structure from that predicted by gap.” Application of a 7.5 T magnetic field 1 ordered states, both insulating (12)andconduct- the pure d-wave BCS models. Vortices in below 10 K causes the scattering intensity to where V is the sample bias voltage, this results new peaks at theSaturday, four March k-space 23, 13 points, which Fig. 1. Topographic and spectroscopic images of ing (13), exist in magnetic fields sufficient to YBa2Cu3O7 (YBCO) lack ZBCPs but exhibit reappear with strength almost equal to that in the in a two-dimensional map of the local density correspond to the spatial structurequench of the superconductivity. vor- the same Categorization area of a of Bi-2212 the additional surface. quasi-particle (A)Atopo- states at Ϯ5.5 meV normal state. These field-induced spin fluctua- of states LDOS(E, x, y, B ϭ 0). We acquire tex-induced quasi-particle states.cuprate We electronic label orderedgraphic states image and of clarifica- the 560 Å( field19), whereas of view those (FOV in ) in Bi2Sr2CaCu2O8ϩ␦ (Bi- tions have a spatial periodicity of 8a0 and wave these LDOS maps at energies ranging from –12 their locations C. No similar peakstion of intheir the relationshipwhich to HTSC the vortex are among studies the were2212) also carried lack ZBCPs out. The (20). More recently, the vector pointing along the Cu-O bond direction. key challenges insupermodulation condensed matter physics can be seenadditional clearly quasi-particle along with states at Bi-2212Downloaded from vorti- Their magnetic coherence length LM is at least meV to ϩ12 meV in 1-meV increments. The B spectral weight exist at these points in the Downloaded from today. some effects of electronicces inhomogeneity. were discovered at The energies near Ϯ7meV 20a ,althoughthevortex-corediameterisonly field is then ramped to its target value, and, after two-dimensional Fourier transform of these 0 Because the suppressionCu–O–Cu of superconductiv- bonds are oriented(21). atThus, 45° a common to the su-phenomenology for low- ϳ5a0.Morerecently,studiesbyKhaykovichet any drift has stabilized, we remeasure the topo- zero-field LDOS maps. ity inside a vortexpermodulation. core can allow one Atomic of the resolutionenergy quasi-particles is evident associated with vortices is al. (28)onarelatedmaterial,La2CuO4ϩy,found graph with the same resolution. The FOV To quantify these results, we fitalternative a Lorent- ordered statesthroughout, (1–10)toappearthere, and the inset showsbecoming a 140 apparent. Å square Its features include (i) the field-induced enhancement of elastic neutron where the high-field LDOS measurements are zian to PS[S12(x, y, 5)] at eachthe of electronicthe four structureFOV of at HTSCϫ2magnificationtomakethiseasierto vortices has absence of ZBCPs, (ii) a radius for the actual scattering (ENS) intensity at these same incom- 1 attracted wide attention. Initially, theoretical ef- vortex core (where the coherence peaks are mensurate k-space locations, but with L Ͼ to be made is then matched to that in Fig. 1A points labeled C in Fig. 2B. We find that they see. The mean Bi-Bi distance apparent here is M forts focused on thea quantizedϭ 3.83 vortex Å and in a is Bard- identicalabsent) to the of ϳ mean10 Å Cu-Cu (21), (iii) low-energy quasi- 100a0.Thus,field-inducedstaticAForderwith within1Å( 0.25a )bycomparingcharacter- occur at k-space radius 0.062 ÅϪ1 with width 0 ϳ 0 een-Cooper-Schrieffer (BCS) superconductor particle states at Ϯ5.5 meV (YBCO) and Ϯ7 8a0 periodicity exists in this material. Finally, Ϫ1 distance in the CuO plane ϳ5Åbelow.(B)Amap istic topographic/spectroscopic features. Final- ␴ϭ0.011 Ϯ 0.002 Å . Figurewith 2C d showsx2-y2 symmetry (14–1812 ). These models meV (Bi-2212), (iv) a radius of up to ϳ75 Å NMR studies by Mitrovic´ et al. (29)explored of S1 (x, y, 5) showing the additional LDOS in- ly, we acquire the high-field LDOS maps, the value of PS[S12(x, y, 5)] measuredincluded along predictionsduced that, because by the of seven the gap vortices.within Each which vortex these isstates ap- are detected (21), and the spatial distribution of magnetic fluctuations 1 (v) the absence of a four-fold symmetric star- near the core. NMR is used because 1/T ,the LDOS(E, x, y, B), at the same series of energies the dashed line in Fig. 2B. The central peak parent as a checkerboard at 45° to the page 1 shaped LDOS. inverse spin-lattice relaxation time, is a measure as the zero-field case. associated with long-wavelength1Department structure, of Physics,orientation. University of California, Not all Berke- are identical, most likely be- 2 Because d-wave BCS models do not explain of spin fluctuations, and the Larmor frequency To focus preferentially on B field effects, the peak associated with the atoms,ley, CA and 94720–7300, the cause USA. Materials of the Sciences effects Divi- of electronicthis phenomenology, inhomogeneity. new theoretical approach- of the probe nucleus is a measure of their loca- sion, Lawrence BerkeleyThe National units Laboratory. of S12(x, Berke- y, 5) are picoamps because it we define a type of two-dimensional map: peak due to the vortex-induced quasi-particleley, CA 94720, USA. 3Department of Superconductiv-1 es have been developed. Zhang (5)andArovas tions relative to the vortex center. In YBCO at ity, University of Tokyo,represents Tokyo 113-8656,⌺dI/dV Japan.⅐⌬V.Inthisenergyrange,theet al. (22)firstfocusedattentiononmagnetic B 13 T, the 1/T of 17Orisesrapidlyasthe states are all evident. The vortex-induced ϭ 1 E2 maximum integrated LDOSphenomena at a vortex associated is ϳ with3 HTSC vortices with core is approached and then diminishes inside states identified by this means occur*Present at (Ϯ address:1/4, Department of Physics, Massachu- E setts Institute of Technology,pA, as Cambridge, compared MA 02139– with theproposals zero field that integrated a magnetic field induces antifer- the core. These experiments are all consistent S 2(x, y, B) ϭ ͓LDOS͑E, x, y, B͒ E1 ͸ 0) and (0, Ϯ1/4) to within the accuracy4307, USA. of the LDOS of ϳ1 pA. The latterromagnetic is subtracted order localized from by the core. More with vortex-induced spin fluctuations occurring E1 †Present address: Departmentthe former of Applied to give Physics, a maximumgenerally, contrast new theories of ϳ describe2 vortex-induced outside the core. measurement. Equivalently, the checkerboardStanford University, Stanford, CA 94305–4060, USA. pA. We also note that theelectronic integrated and magnetic differen- phenomena when the Theoretical attention was first focused on the Ϫ LDOS͑E, x, y,0͔͒dE (1) pattern evident in the LDOS has spatial‡To whom peri- correspondence should be addressed. E- odicity 4a oriented along the Cu-Omail: [email protected] bonds. tial conductance betweenanticipated 0 and Ϫ200 effects meV of strong is correlations and regions outside the core by a phenomenological 0 200 pA because all measurements reported in which represents the integral of all additional Furthermore, the width ␴ of the Lorentzian 466 this paper were18 obtained JANUARY at 2002 a junction VOL 295 resis- SCIENCE www.sciencemag.org spectral density induced by the B field be- yields a spatial correlation length for these tance of 1 gigaohm set at a bias voltage of tween the energies E1 and E2 at each location LDOS oscillations of L ϭ (1/␲␴) Ϸ 30 Ϯ 5 –200 mV. www.sciencemag.org SCIENCE VOL 295 18 JANUARY 2002 467 RESEARCH ARTICLE

where z is the tip’s surface-normal coordinate, V is the relative sample-tip bias, and N r→, E is the sample’s local-density-of-states (LDOS)ð Þ at An Intrinsic Bond-Centered Electronic lateral locations r→ and energy E. Unmeasurable effects due to the tunneling matrix elements, the tunnel-barrier height, and z variations from elec- Glass with Unidirectional Domains tronic heterogeneity are contained in f r→, z (see supporting online text 1). For a simpleð metallicÞ in Underdoped Cuprates system where f r→, z is a featureless constant, Eq. 1 shows thatð spatialÞ mapping of the dif- 1 1 1,2 1 3 4 5 Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien, T. Hanaguri, M. Azuma, ferential tunneling conductance dI=dV r→, V 5 6 2,4 2,7 1,8 M. Takano, H. Eisaki, H. Takagi, S. Uchida, J. C. Davis * yields N r→, E eV . However, for the stronglyð Þ correlatedð electronic¼ Þ states in a lightly hole- Removing electrons from the CuO2 plane of cuprates alters the electronic correlations sufficiently doped cuprate, the situation is much more to produce high-temperature superconductivity. Associated with these changes are spectral-weight complex. In theory (4), the correlations cause transfers from the high-energy states of the insulator to low energies. In theory, these should be the ratio Z(V) of the average density-of-states detectable as an imbalance between the tunneling rate for electron injection and extraction—a for empty states N E eV to that of filled tunneling asymmetry. We introduce atomic-resolution tunneling-asymmetry imaging, finding states N E −eV ð to¼þ becomeÞ asymmetric by ð ¼ Þ virtually identical phenomena in two lightly hole-doped cuprates: Ca1.88Na0.12CuO2Cl2 and an amount Bi2Sr2Dy0.2Ca0.8Cu2O8+d. Intense spatial variations in tunneling asymmetry occur primarily at the planar oxygen sites; their spatial arrangement forms a Cu-O-Cu bond-centered electronic pattern N E eV 2n Z V ≡ ð ¼þ Þ ≈ 2 without long-range order but with 4a0-wide unidirectional electronic domains dispersed ð Þ N E eV 1 n ð Þ − þ throughout (a0: the Cu-O-Cu distance). The emerging picture is then of a partial hole localization ð ¼ Þ within an intrinsic electronic glass evolving, at higher hole densities, into complete delocalization Spectral-weight sum rules (5) also indicate that and highest-temperature superconductivity. the ratio R r→ of the energy-integrated N r→, E for empty statesð Þ E > 0 to that of filled statesðE <0Þ on March 1, 2013 etallicity of the cuprate CuO2 planes just above the chemical potential m (Fig. 1B). But is related to n by derives (1) from both oxygen 2p and precisely how these spectral-weight transfers W → copper 3d orbitals (Fig. 1A). Coulomb result in cuprate high-temperature supercon- c N r, E dE M → Z 0 ð Þ R r ≡ interactions lift the degeneracy of the relevant ductivity remains controversial. ð Þ 0 N r→, E dE d-orbital, producing lower and upper d-states sepa- Recently, it has been proposed that these Z −∞ ð Þ rated by the Mott-Hubbard energy U (Fig. 1B). doping-induced correlation changes might be 2n r→ nt ð Þ O 3 The lower d-states and oxygen p-state become observable directly as an asymmetry of electron ¼ 1 − n r→ þ U ð Þ hybridized, yielding a correlated insulator with tunneling currents with bias voltage (4, 5)— ð Þ www.sciencemag.org charge-transfer gap D (Fig. 1B). The “hole-doping” electron extraction at negative sample bias being Here t is in-plane hopping rate and Wc satisfies process, which generates highest-temperature su- strongly favored over electron injection at posi- “all low-energy scales” < Wc < U. perconductivity, then removes electrons from the tive sample bias. Such effects should be de- As a test of such ideas, we show in Fig. 1C CuO2 plane, creating new hole-like electronic tectable with a scanning tunneling microscope the predicted evolution of the tunneling states with predominantly oxygen 2p character (STM). The STM tip-sample tunneling current asymmetry (TA) with n from (4), and in Fig. (2). This is a radically different process than hole- is given by 1D we show the measured evolution of spatially doping a conventional semiconductor because, averaged TA in a sequence of lightly hole-doped eV when an electron is removed from a correlated → → → Ca2-xNaxCuO2Cl2 samples with different x.We I r, z, V f r, z N r, E dE 1 Downloaded from insulator, the states with which it was correlated ð Þ¼ ð Þ 0 ð Þ ð Þ see that the average TA is indeed large at low x Z are also altered fundamentally. Numerical modeling of this process (3)indicatesthatwhen Fig. 1. (A)Relevantelectronic n holes per unit cell are introduced, the correlation orbitals of the CuO2 plane: Cu 3d changes generate spectral-weight transfers from orbitals are shown in orange and both filled and empty high-energy bands— oxygen 2p orbitals are shown resulting in the creation of ~2n new empty states in blue. A single plaquette of four Cu atoms is shown within RESEARCH ARTICLE the dashed square box, and a 1Laboratory of Atomic and Solid State Physics, Department single Cu-O-Cu unit is within of Physics, Cornell University, Ithaca, NY 14853, USA. the dashed oval. (B) Schematic 2Department of Advanced Materials Science, University of direct test of such ideas has3 not been possibleenergy levels in thewith CuO2 problemsplane in lightly doped cuprates. For rotational symmetry breaking in the tunneling Tokyo, Kashiwa, Chiba 277-8651, Japan. Département de and the effects of hole doping becausePhysique, neither Université the de Sherbrooke, real-space Sherbrooke,RESEARCH electronic QC J1K struc-ARTICLEexample, a standard dI/dV image, although well patterns (35). 2R1, Canada. 4Magnetic Materials Laboratory, RIKEN, upon it. (C) The expected tun- 5 tureWako, of the Saitama ECG 351-0198, state, Japan. norInstitute that for of Chemical an individualneling asymmetrydefined, between elec- is not a direct imagewhere ofz is the the tip LDOS’s surface-normal (see coordinate, V The new proposals (4, 5) for tunneling Research, Kyoto University, Uji, Kyoto 601-0011, Japan. tron extraction (negative bias) is the relative sample-tip bias, and N r→, E is “cluster,6 ” could be determined directly as no supporting online text 1). Moreover, there are ð asymmetryÞ measurements provide a notable National Institute of Advanced Industrial Science and and injection (positive bias) from the sample’s local-density-of-states (LDOS) at 7 → suitableTechnology, imaging Tsukuba, techniques Ibaraki 305-8568,An Japan. existed. IntrinsicDepart- (4)wherelowvaluesof Bond-CenteredtheoreticalZ occur at concerns Electronic that, inlateral Ca locationsNar andCuO energyClE. Unmeasurable, solution to problems with standard dI/dV ment of Physics, University of Tokyo, Bunkyo-ku, Tokyo effects due2-x to the tunnelingx matrix2 2 elements, the 8 D Design113-0033, Japan.of TACondensed studies Matter in Physics Ca andNa Materi- CuOlow holeCl densitiesthen.( topmost)Mea- CuO plane maytunnel-barrier be in height, an “ andextraor-z variations from elec-imaging because Eqs. 2 and 3 have a crucial Glass1.88 with0.12sured Unidirectional2 doping2 dependence of Domains2 tronic heterogeneity are contained in f r→, z (see als Science Department, Brookhaven National Laboratory, ð Þ andUpton, Bi Sr NY 11973,Dy USA.Ca Cu O . STM-basedaverage im- tunnelingdinary asymmetry” state (34) or that interferencesupporting online text between 1). For a simple metallicpractical advantage. If we define the ratios 2 2 0.2 0.8 2 8+ → d system where f r, z is a featureless constant, in Underdoped Cuprates → → *To whom correspondence should be addressed. E-mail: in Ca2-xNaxCuO2Cl2.a.u.,arbi- Eq. 1 shows thatð spatialÞ mapping of the dif- aging might appear an appropriate1 tool1 to1,2 ad- two1 tunneling3 4 trajectories5 through the 3pz-Cl Z r, V and R r, V in terms of the tunneling [email protected] Y. Kohsaka, C. Taylor, K.trary Fujita, units.A. Schmidt, C. Lupien, T. Hanaguri, M. Azuma, ferential tunneling conductance dI=dV r→, V 5 6 2,4 2,7 1,8 + ð Þ ð Þ dress such issues. But dI/dM.V Takano,imagingH. Eisaki, isH. Takagi, fraughtS. Uchida,orbitalsJ. C. Davis adjacent* to a dopantyields NaN r→ion, E eV may. However, cause for the stronglyð currentÞ correlatedð electronic¼ Þ states in a lightly hole- 1380 Removing electrons9 MARCH from the 2007 CuO2 plane VOL of cuprates 315 altersSCIENCE the electronicwww.sciencemag.org correlations sufficiently doped cuprate, the situation is much more to produce high-temperature superconductivity. Associated with these changes are spectral-weight complex. In theory (4), the correlations cause transfers from the high-energy states of the insulator to low energies. In theory, these should be the ratio Z(V) of the average density-of-states dI → detectable as an imbalance between the tunneling rate for electron injection and extraction—a for empty states N E eV to that of filled → dV r, z, V tunneling asymmetry. We introduce atomic-resolution tunneling-asymmetry imaging, finding states N E −eV ð to¼þ becomeÞ asymmetric by Z r, V ð þ Þ 4a ð ¼ Þ ≡ dI → virtually identical phenomena in two lightly hole-doped cuprates: Ca1.88Na0.12CuO2Cl2 and an amount ð Þ r, z, −V ð Þ Bi2Sr2Dy0.2Ca0.8Cu2O8+d. Intense spatial variations in tunneling asymmetry occur primarily at the dV planar oxygen sites; their spatial arrangement forms a Cu-O-Cu bond-centered electronic pattern N E eV 2n ð Þ Z V ≡ ð ¼þ Þ ≈ 2 without long-range order but with 4a0-wide unidirectional electronic domains dispersed ð Þ N E eV 1 n ð Þ − þ throughout (a0: the Cu-O-Cu distance). The emerging picture is then of a partial hole localization ð ¼ Þ within an intrinsic electronic glass evolving, at higher hole densities, into complete delocalization Spectral-weight sum rules (5) also indicate that → and highest-temperature superconductivity. the ratio R r→ of the energy-integrated N r→, E

on March 1, 2013 I r, z, V for empty statesð Þ E > 0 to that of filled statesðE <0Þ → R r , V ≡ ð → þ Þ 4b etallicity of the cuprate CuO2 planes just above the chemical potential m (Fig. 1B). But is related to n by derives (1) from both oxygen 2p and precisely how these spectral-weight transfers ð Þ I r, z, −V ð Þ W → copper 3d orbitals (Fig. 1A). Coulomb result in cuprate high-temperature supercon- c N r, E dE ð Þ M → Z 0 ð Þ R r ≡ interactions lift the degeneracy of the relevant ductivity remains controversial. ð Þ 0 N r→, E dE d-orbital, producing lower and upper d-states sepa- Recently, it has been proposed that these Z −∞ ð Þ rated by the Mott-Hubbard energy U (Fig. 1B). doping-induced correlation changes might be 2n r→ nt ð Þ O 3 The lower d-states and oxygen p-state become observable directly as an asymmetry of electron ¼ 1 − n r→ þ U weð Þ see immediately from Eq. 1 that the un- hybridized, yielding a correlated insulator with tunneling currents with bias voltage (4, 5)— ð Þ → www.sciencemag.org charge-transfer gap D (Fig. 1B). The “hole-doping” electron extraction at negative sample bias being Here t is in-plane hopping rate and Wc satisfiesknown effects in f r, z are all canceled out process, which generates highest-temperature su- strongly favored over electron injection at posi- “all low-energy scales” < Wc < U. ð Þ → perconductivity, then removes electrons from the tive sample bias. Such effects should be de- As a test of such ideas, we show in Fig.by 1C the division process. Thus, Z r, V and CuO2 plane, creating new hole-like electronic tectable with a scanning tunneling microscope the predicted evolution of the tunneling → ð Þ Saturday, March 23, 13 states with predominantly oxygen 2p character (STM). The STM tip-sample tunneling current asymmetry (TA) with n from (4), and in Fig.R r, V not only contain important physical (2). This is a radically different process than hole- is given by 1D we show the measured evolution of spatiallyð Þ → doping a conventional semiconductor because, averaged TA in a sequence of lightly hole-dopedinformation (4, 5) but, unlike N r, E , are also eV when an electron is removed from a correlated → → → Ca2-xNaxCuO2Cl2 samples with different x.We ð Þ I r, z, V f r, z N r, E dE 1 Downloaded from insulator, the states with which it was correlated ð Þ¼ ð Þ 0 ð Þ ð Þ see that the average TA is indeed large at lowexpressiblex in terms of measurable quantities Z are also altered fundamentally. Numerical only. We have confirmed that the unknown modeling of this process (3)indicatesthatwhen Fig. 1. (A)Relevantelectronic → on March 1, 2013 n holes per unit cell are introduced, the correlation orbitals of the CuO2 plane: Cu 3d factors f r, z are indeed canceled out in Eq. 4 changes generate spectral-weight transfers from orbitals are shown in orange and ð Þ both filled and empty high-energy bands— oxygen 2p orbitals are shown (see supporting online text and figures 2). resulting in the creation of ~2n new empty states in blue. A single plaquette of four Cu atoms is shown within the dashed square box, and a To address the material-specific theoret- 1Laboratory of Atomic and Solid State Physics, Department single Cu-O-Cu unit is within of Physics, Cornell University, Ithaca, NY 14853, USA. B ical concerns (34, 35), we have designed a 2 the dashed oval. ( ) Schematic Department of Advanced Materials Science, University of energy levels in the CuO plane Tokyo, Kashiwa, Chiba 277-8651, Japan. 3Département de 2 sequence of identicalTA-imagingexper- Physique, Université de Sherbrooke, Sherbrooke, QC J1K and the effects of hole doping 2R1, Canada. 4Magnetic Materials Laboratory, RIKEN, upon it. (C) The expected tun- iments in two radically different cuprates: Wako, Saitama 351-0198, Japan. 5Institute for Chemical neling asymmetry between elec- Research, Kyoto University, Uji, Kyoto 601-0011, Japan. tron extraction (negative bias) 6 National Institute of Advanced Industrial Science and and injection (positive bias) from strongly underdoped Ca1.88Na0.12CuO2Cl2 Fig. 4. (A and D) R maps of Na-CCOC and Dy-Bi2212,7 respectively (taken at 150 mV from areas in Technology, Tsukuba, Ibaraki 305-8568, Japan. Depart- (4)wherelowvaluesofZ occur at ment of Physics, University of Tokyo, Bunkyo-ku, Tokyo (Na-CCOC; critical temperature Tc ~21K) the blue boxes of Fig. 3,113-0033, C and Japan. D).8Condensed The Matter fields Physics and of Materi- viewlow are hole densities (A) 5.0n.(D)Mea- nm by 5.3 nm and (B) 5.0 nm by als Science Department, Brookhaven National Laboratory, sured doping dependence of and Bi Sr Dy Ca Cu O (Dy-Bi2212; T ~ 5.0 nm. The blue boxesUpton, in NY (A) 11973, and USA. (D) indicate areasaverage of tunneling Fig. asymmetry 4, B and C, and Fig. 4, E and F, 2 2 0.2 0.8 2 8+d c www.sciencemag.org *To whom correspondence should be addressed. E-mail: in Ca2-xNaxCuO2Cl2.a.u.,arbi- 45 K). As indicated schematically in Fig. 2, B respectively. (B and E) [email protected] R map withintrary units. equivalent domains from Na-CCOC and Dy- Bi2212, respectively (blue boxes of Fig. 4, A and D). The locations of the Cu atoms are shown as and C, they have completely different crystal- black crosses. (C and 1380F) Constant-current topographic9 MARCH 2007 images VOL 315 simultaneouslySCIENCE www.sciencemag.org taken with Fig. 4, B lographic structure, chemical constituents, and and E, respectively. Imaging conditions are (C) 50 pA at 600 mV and (F) 50 pA at 150 mV. The dopant species and sites in the termination markers show atomic locations, used also in Fig. 4, B and E. The fields of view of these images are layers lying between the CuO2 plane and the shown in Fig. 3, A and B, as orange boxes. STM tip. Na-CCOC has a single CuO2 layer

Fig. 5. (A)Locationsrelativeto Downloaded from the O and Cu orbitals in the CuO2 plane where each dI/dV spectrum at the surfaces of Fig. 4, C and F, and shown in Fig. 5B, is measured. Spectra are measured along equivalent lines labeled 1, 2, 3, and 4 in both domains of Fig. 4, B and E, and Fig. 5A. (B)Differentialtunnelingcon- ductance spectra taken along parallel lines through equivalent domains in Na-CCOC and Dy-Bi2212. All spectra were taken under identical junction conditions (200 pA, 200 mV). Numbers (1 to 4) correspond to trajectories where these sequences of spectra were taken. Locations of the trajectories, relative to the domains, are shown between Fig. 4B (C) and 4E (F) by arrows.

1382 9 MARCH 2007 VOL 315 SCIENCE www.sciencemag.org LETTERS PUBLISHED ONLINE: 14 OCTOBER 2012 | DOI: 10.1038/NPHYS2456

Direct observation of competition between superconductivity and charge density wave order in YBa2Cu3O6.67 J. Chang1,2*, E. Blackburn3, A. T. Holmes3, N. B. Christensen4, J. Larsen4,5, J. Mesot1,2, Ruixing Liang6,7, D. A. Bonn6,7, W. N. Hardy6,7, A. Watenphul8, M. v. Zimmermann8, E. M. Forgan3 and S. M. Hayden9

8,12 Superconductivity often emerges in the proximity of, or in YBa2Cu3O6.67 (with ortho-VIII oxygen ordering , Tc 67 K competition with, symmetry-breaking ground states such as and p 0.12, where p is the hole concentration per= planar antiferromagnetism or charge density waves1–5 (CDW). A Cu). We= find that a CDW forms in the normal state below number of materials in the cuprate family, which includes the T 135 K. The charge modulation has two fundamental CDW ⇡ high transition-temperature (high-Tc) superconductors, show wave vectors qCDW q1 (1,0,0.5) and q2 (0,2,0.5), where 5–7 = = = spin and charge density wave order . Thus a fundamental 1 0.3045(2) and 2 0.3146(7), with no significant field- or question is to what extent do these ordered states exist temperature-dependence⇡ ⇡ of these values. The CDW gives rise for compositions close to optimal for superconductivity. to satellites of the parent crystal Bragg peaks at positions such Here we use high-energy X-ray diffraction to show that as Q (2 1,0,0.5). Although the satellite intensities have a a CDW develops at zero ﬁeld in the normal state of strong= temperature± and magnetic field dependence, the CDW is superconducting YBa Cu O (T 67 K). This sample has not field-induced and is unaffected by field in the normal state. 2 3 6.67 c = a hole doping of 0.12 per copper and a well-ordered oxygen Below Tc it competes with superconductivity, and a decrease of 8 chain superstructure . Below Tc, the application of a magnetic the CDW amplitude in zero field becomes an increase when ﬁeld suppresses superconductivity and enhances the CDW. superconductivity is suppressed by field. A very recent paper14 Hence, the CDW and superconductivity in this typical high-Tc reports complementary resonant soft X-ray scattering experiments material are competing orders with similar energy scales, performed on (Y,Nd)Ba2Cu3O6 x as a function of doping and in + and the high-Tc superconductivity forms from a pre-existing the absence of a magnetic field. The results are broadly in agreement CDW environment. Our results provide a mechanism for the with our zero field data. 9 formation of small Fermi surface pockets , which explain the Figure 1a,g shows scans through the (2 1,0,0.5) and (0,2 negative Hall and Seebeck effects10,11 and the ‘T plateau’12 in ,0.5) positions at T 2 K. Related peaks were observed at c 2 = this material when underdoped. (2 1,0,0.5) and (4 1,0,0.5) (see Supplementary Fig. S3). The Charge density waves in solids are periodic modulations of con- incommensurate+ peaks are not detected above 150 K (Fig. 1c). From duction electron density. They are often present in low-dimensional the peak width we estimate that the modulation has an in-plane systems such as NbSe (ref. 4). Certain cuprate materials such as correlation length ⇠ 95 5 Å (at 2 K and 17 T—see Methods). 2 a ⇡ ± La2 x yNdySrxCuO4 (Nd-LSCO) and La2 xBaxCuO4 (LBCO) also The existence of four similar in-plane modulations ( 1,0) and show charge modulations that suppress superconductivity near x (0, ) indicates that the modulation is associated with± the (nearly = ± 2 1/8 (refs 6,7). In some cases, these are believed to be unidirectional square) CuO2 planes rather than the CuO chains. The present 2,3 in the CuO2 plane, and have been dubbed ‘stripes’ . There is now a experiment cannot distinguish between 1 q and 2 q structures, mounting body of indirect evidence that charge and/or spin density that is, we cannot tell directly whether modulations along the a and waves (static modulations) may be present at high magnetic fields b directions co-exist in space or occur in different domains of the in samples with high Tc: quantum oscillation experiments on un- crystal. However, Bragg peaks from the two CDW components have derdoped YBa2Cu3Oy (YBCO) have revealed the existence of at least similar intensities and widths (Fig. 1b,g) despite the orthorhombic one small Fermi surface pocket9,10, which may be created by a charge crystal structure, which breaks the symmetry between them. This 11 modulation . More recently, nuclear magnetic resonance (NMR) suggests that q1 and q2 are coupled, leading to the co-existence of studies have shown a magnetic-field-induced splitting of the Cu2F multiple wave vectors, as seen in other CDW systems such as NbSe2 lines of YBCO (ref. 13). An important issue is the extent to which the (ref. 4). The scan along the c ⇤ direction in Fig. 1d has broad peaks 2,3 tendency towards charge order exists in high-Tc superconductors . close to l 0.5 reciprocal lattice units (r.l.u.), indicating that the Here we report a hard (100 keV) X-ray diffraction study, in CDW is weakly=± correlated along the c direction, with a correlation magnetic fields up to 17 T, of a detwinned single crystal of length ⇠c of approximately 0.6 lattice units.

1Institut de la Matière Complexe, Ecole Polytechnique Fedérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland, 2Paul Scherrer Institut, Swiss Light Source, CH-5232 Villigen PSI, Switzerland, 3School of Physics and Astronomy, University of Birmingham, Birmingham, B15 2TT, UK, 4Department of Physics, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark, 5Laboratory for Neutron Scattering, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland, 6Department of Physics and Astronomy, University of British Columbia, Vancouver, Canada, 7Canadian Institute for Advanced LETTERSResearch, Toronto, Canada, 8Hamburger Synchrotronstrahlungslabor (HASYLAB) at Deutsches Elektronen-Synchrotron (DESY), 22603 Hamburg, PUBLISHED ONLINE: 14 OCTOBER 2012 | DOI: 10.1038/NPHYS2456Germany, 9H. H. Wills Physics Laboratory, University of Bristol, Bristol, BS8 1TL, UK. *e-mail: johan.chang@epﬂ.ch.

NATURE PHYSICS VOL 8 DECEMBER 2012 www.nature.com/naturephysics 871 Direct observation of competition between | | | superconductivity and charge density wave order in YBa2Cu3O6.67

1,2 3 3 4 4,5 1,2 NATURE PHYSICSJ. ChangDOI:*, E. 10.1038/NPHYS2456 Blackburn , A. T. Holmes , N. B. ChristensenNATURE, J. Larsen PHYSICS, J. Mesot , DOI: 10.1038/NPHYS2456 LETTERS Ruixing Liang6,7, D. A. Bonn6,7, W. N. Hardy6,7, A. Watenphul8, M. v. Zimmermann8, E. M. Forgan3 LETTERS and S. M. Hayden9 ab a 200 b 200300 160 Superconductivity often emerges in the proximity of, or in YBa Cu O (with ortho-VIII oxygen ordering8,12, T 67 K T∗ YBCO YBCO p = 0.12 2 3 6.67 T c = competition with, symmetry-breakingT C ground states such as and p 0.12, where p is the hole concentrationN per planar antiferromagnetism or charge density waves1–5 (CDW). A Cu). We= find that a CDW forms250 in the normal state below TCDW 2 K number of materials in the cuprate family, which includes the TCDW 135 K. The charge modulation has two fundamental 150 ⇡ 150 120 TH high transition-temperature (high-Tc) superconductors,Q show = (1.695,wave 0, vectors 0.5) qCDW q1 (1,0,0.5) and q2 (0,2,0.5), where 5–7 17 T = = ) = ) spin and charge density wave order . Thus a fundamental 1 0.3045(2) and 2 0.3146(7),200 with no significant field- or ¬1 ¬1 question is to what extent do these ordered states exist temperature-dependence⇡ ⇡ of these values. The CDW gives rise 15 T TH for compositions close to optimal for superconductivity.Q = (0, 3.691,to satellites 0.5) of (x4) the parent crystal Bragg peaks at positions such 66 K TCDW 100Here we use high-energy X-ray diffraction to show that as Q (2 1,0,0.5). Although the150 satellite intensities have a 80 = ± 100 a CDW develops at zero ﬁeld in the normal state of strong temperature and magnetic field dependence, the CDW is Tcusp superconducting7.5 T YBa Cu O (T 67 K). This sample has not field-induced and is unaffected by field in the normal state. 2 3 6.67 c = a hole doping of 0.12 per copper and a well-ordered oxygen Below Tc it competes with superconductivity, and a decrease of 8 Temperature (K) 100 Temperature (K) chain superstructure . Below Tc, the application of a magnetic the CDW amplitude in zero field becomes an increase when 50 TSDW 14 NMR Intensity (cnts s 50 40 T Intensity (cnts s ﬁeld suppresses superconductivity and enhances the CDW. superconductivityT is suppressed by field. A very recent paper T VL Hence,0 the T CDW and superconductivity in this typical high-T CDW c reports complementary resonant soft X-ray50 scattering experiments Vortex liquid material are competing orders with similar energy scales, performed on (Y,Nd)Ba2Cu3O6 x as a function of doping and in + Vortex and the high-Tc superconductivity forms from a pre-existing the absence of a magnetic field. The results are broadlyAF in agreement CDW0 environment. Our results provide a mechanism for the with our zero field data. SC solid 9 00 150 K 0 formation of small Fermi surface pockets , which explain the Figure 1a,g shows scans through the0.00 (2 1,0,0.5) and 0.05 (0,2 0.10 0.15 0.20 10,11 12 NATURE PHYSICS DOI: 10.1038/NPHYS2456 0 10 20 30 40 50 LETTERSnegative Hall and Seebeck effects and the ‘Tc plateau’ in 2,0.5) positions at T 2 K. Related peaks were observed at this0 material when underdoped. 50 100(2 ,0,0.5) 150 and (4 =,0,0.5) (see Supplementary0 Fig. 5 S3). The Doping, 10 p (holes/Cu) 15 µ 0H (T) + 1 1 Charge density waves in solids are periodicT modulations(K) of con- incommensurate peaks are not detected above 150 K (Fig. 1c). From H (T) aduction450 electron density. They are often present in low-dimensionalbc450 the peakFigure width we 4 estimatePhase that diagram the modulation of YBa hasCu450 anO in-plane. a, Doping dependence of the antiferromagnetic ordering temperature T , the incommensurate spin-density systems such as NbSe (ref. 4). Certain cuprate materials such as correlation length ⇠ 95 5 Å (at 2 K and 17 T—see2 3 Methods).7 x N 17 T ||2c Cut 117 T ||c a|⇡ ± Cut 1 17 T ||c Cut 1 30 c La2 x 400yNdySrxCuO4 (Nd-LSCO) and La2 xBaxCuO4 (LBCO) also400 The existencewave of order four similarTSDWd in-plane(green modulationstriangles;400 ref. ( 21),1,0) the and superconducting temperature Tc and the pseudogap temperature T⇤ as determined from the Nernst effect 0 T 0 T ± 0 T ) show charge modulations that suppress superconductivity near x (0, 2) indicates that the modulation) is associated with the29 (nearly 30 ) ) (black squares) and neutron diffraction) (purple squares). Notice that the Nernst effect indicates a broken rotational symmetry inside the pseudogap

¬3 350 350= ± ¬3 350 ¬1 1/8 (refs 6,7). In some cases, these are believed to be unidirectional¬1 square) CuO2 planes rather than the CuO chains.¬1 The present 29 12 2,3 region, whereas a translational12 symmetry preserving magnetic order is found by neutron scattering . Below temperature scale TH (black circles), a larger 10 in the300 CuO2 plane, and have been dubbed ‘stripes’ . There is now300 a experiment cannot distinguish between 10 1 q and 2 300q structures, × mounting body of indirect evidence that charge and/or spin density that is, we cannot tell directly whether× modulations along the a 26and ( and negative Hall coefﬁcient( was observed and interpreted in terms of a Fermi surface reconstruction. Our X-ray diffraction experiments show that in

(cnts s (cnts s (cnts s I waves250 (static modulations) may be present at high magnetic fieldsI 250 b directions co-exist in space or occur in differentI domains250 of the 11 YBCO p 0.12 incommensurate11 CDW order spontaneously breaks the crystal translational symmetry at a temperature TCDW that is twice as large as Tc. in samples with high Tc: quantum oscillation experiments on un- crystal. However, Bragg peaks from the two CDW components have 200 200 = NMR200 13 derdoped YBa2Cu3Oy (YBCO) have revealed the existence of at least similar intensitiesTCDW is and also widths much (Fig. larger 1b,g) despite than T the orthorhombic(red squares), the temperature scale below which NMR observes field-induced charge order . b, Field dependence of 9,10 2 K one10 small150 Fermi surface pocket , which may be created by a charge150 crystal structure, which breaks the symmetry between150 them. This 11T = 2 K T = 66T KCDW (filled red circles)10 and Tcusp (open squares),T = 150 K the temperature below which the CDW is suppressed by superconductivity, compared with TH (open δ modulation . More recently, nuclear magnetic resonance (NMR) suggests that q and q are coupled, leading to the co-existence of 1 2 δ 26 studies have1.64 shown 1.66 a magnetic-field-induced 1.68 1.70 1.72 splitting 1.74 of the Cu2F multiple1.64 1.66 waveblack vectors, 1.68 circle)1.70 as andseen 1.72 inTVL other(filled 1.74 CDW blue systems circles), such1.64 as NbSethe 1.66 temperature2 1.68 1.70 where 1.72 the 1.74 vortex liquid state forms . Error bars on TSDW, TH, TNMR, and T⇤ are explained in lines9 of YBCO (ref. 13).h in An ( importanth, 0, 0.5) (r.l.u.) issue is the extent to which the (ref. 4). Therefsh in scan ( 21,26,30,33.h, 0, along 0.5) the (r.l.u.)c ⇤ Thedirection error9 in barsFig. 1d on hasT broadand peakshT in (h,reflect 0, 0.5) (r.l.u.) the uncertainty in determining the onset and suppression temperature of CDW order from Fig. 2. 2,3 CDW cusp tendency towards charge order exists in high-Tc superconductors . close to l 0.5 reciprocal lattice units (r.l.u.), indicating that the Saturday,Here March we report 23, 13 a hard (100 keV) X-ray diffraction study, in CDW is weakly=± correlated along the c direction,66 withK a correlation d ef350 31 magnetic8 200 fields up to 17 T, of a detwinned single crystal of length ⇠c of approximately 0.6 lattice units.8

) detwinned and the Cu–O chains were ordered with the ortho-VIII structure by these various orders are ‘intertwined’50 K . In this context, we can

¬1 2 K Cut 2 Cut 3 standard procedures12. σ view our presentσ results as indicating300 0 T that the electron system 150 The diffracted intensities from the CDW, shown in Fig. 1, are composed of 1Institut7 de la Matière Complexe, Ecole Polytechnique Fedérale de Lausanne (EPFL),l CH-1015 Lausanne, Switzerland, 2Paul Scherrer7 Institut, Swiss Light has a tendency towards two) ground states: a charge density T cusp Source, CH-5232 Villigen PSI, Switzerland, 3School of Physics and Astronomy, University of Birmingham, Birmingham, B15 2TT, UK, 4Department¬1 250 of an incommensurate lattice modulation peak on a smoothly varying background. 100 5 wave, which breaks translational symmetry and involves electron– Width (r.l.u.) of (2- , 0, 0.5) Width (r.l.u.) of (2-

Width (r.l.u.) of (2- , 0, 0.5) Width (r.l.u.) of (2- The background along (h, 0, 0.5) mainly originates from the tails of the ortho-VIII Physics, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark, Laboratory for NeutronCut 3 Scattering, Paul Scherrer Institut, CH-5232 6 (0T) (cnts s 6 7 6 I hole correlations, versus superconductivity, which breaks gauge Villigen PSI, Switzerland, Department of Physics and Astronomy, University of BritishCut Columbia, 1 Vancouver, Canada,CutCanadian 4 Institute for200 Advanced peaks (see Supplementary Information). It varies strongly from one Brillouin 0 50 100 150 0 (cnts s 5 10 15 Research,50 Toronto, Canada, 8Hamburger Synchrotronstrahlungslabor (HASYLAB)0.5 at Deutschessymmetry Elektronen-Synchrotron and involves (DESY), electron–electron 22603I Hamburg, correlations. We note that zone to another; for example, the background around (2.7, 0, 0.5) is an order of 9 Germany, H. H. Wills Physics Laboratory, UniversityT (K) of Bristol, Bristol, BS8 1TL, UK. *e-mail: johan.chang@epﬂ.ch. 150 H (T) magnitude larger than around (1.7, 0, 0.5). The background has essentially no

(17 T)- the q-vectors of the CDW lie close to the separation of pieces of I 0 k field dependence (Fig. 1a–c) so subtracting the zero-field from high-field data is NATURE PHYSICS VOL 8 DECEMBER 2012 www.nature.com/naturephysics Fermi surface that have maximum superconducting871 gap at optimal | | | Cut 2 100 a simple way to eliminate the background. This reveals the field-enhanced signal Figure 2 Competition between¬1.0 charge–density-wave ¬0.5 0.0 0.5 order1.0 and superconductivity. a, Temperature0.3 dependence–0.05 of the peak 0.00 intensity0.05 at (1.695, 0, 0.5) doping and have the same sign of the order parameter. inside the superconducting state (Fig. 1a–d). | h 0.3 (0, 0, 0) k in (1.694, k, 0.5) (r.l.u.) (circles) and (0, 3.691, 0.5) (squares)l in for (1.696, different 0, 1) (r.l.u.) applied magnetic fields. The square data points have been multiplied by a factor of four. In the normalAs there is a weak temperature dependence in the background (Fig. 1a–c), it is not possible to eliminate it by subtracting a high-temperature curve. Therefore, state, there is a smooth onset of the CDW order. In the absence of an appliedMethods magnetic field there is a decrease in the peak intensity below Tc. This trend g 200 h 200 i 200 to obtain the temperature dependences shown in Fig. 2, we fitted the data to Our experiments used 100 keV hard X-ray synchrotron radiation from the can be reversed by the application65 K of a magnetic field.Cut 4b, Magnetic field dependence of the Cut lattice 4 modulation65 peak K intensity at (1.695,0,0.5)Cut 4 fora different Gaussian function G(Q) and modelled the background by a second-order DORIS-III storage ring at DESY, Hamburg,0 Germany. T We installed a recently polynomial B(Q) c c Q c Q2. The constants c , c and c have a small ) temperatures. At T 2 K,) the peak0 T intensity grows approximately) linearly with magnetic field up to the highest applied field. c,d, Gaussian linewidth of the 0 1 2 0 1 2 ¬1 ¬1 ¬1 developed 17 T horizontal cryomagnet designed for beamline use on the triple-axis but significant temperature= + dependence.+ The low counting statistics resulted in = 150 150 150 x0.625 (1.695, 0, 0.5) CDW modulation plotted versus temperature and field respectively.diffractometer The at raw beamline linewidth, BW5. The including sample was a contribution mounted by gluing from it over the a instrumental hole Gaussians fitting equally well as other possible lineshapes such as Lorentzians. in a temperature-controlled aluminium plate within the cryomagnet vacuum and (cnts s resolution, is field-independent (cnts s in the normal state (T > T ). In contrast, (cnts s the CDW order becomes more coherent below T , once a magnetic field isThe applied. signal-to-background ratio is best for the (2 1, 0, 0.5) peak due I I c I c 65 K was thermally shielded by thin Al and aluminized mylar foils glued to this plate. The the weaker structural ortho-VIII peak (see Supplementary Fig. S2). From the This effect ceases once the amplitude starts to be suppressed owing100 to0 competition T sample temperature with superconductivity. could be controlled100 over The the vertical range 2–300 dashed K. The lines incoming in a,c illustrate the 100 Gaussian fits to the (2 1, 0, 0.5) satellite peak at 2 K and 17 T we can estimate and outgoing beams passed through 1 mm thick aluminium⇠ cryostat vacuum connection between these two1.64 features 1.66 of 1.68 the 1.70 data 1.72 that 1.74 define the Tcusp2.26temperatures. 2.28 2.30 2.32 All 2.34 other 2.36 lines are guides3.64 to 3.66 the 3.68 eye. Error3.70 3.72 bars 3.74 indicate standardthe correlation length ⇠ along the three crystal axis directions. We define ⇠ 1/, windows, which gave a maximum of 10 input and output angles relative to 2 2 0.5 = ⇠ ± where (meas R ) is the measured Gaussian standard deviation corrected for deviations of the fit parameters describedk in (0, k in, 0.5) Methods. (r.l.u.) kthe in (0, field k, direction,0.5) (r.l.u.) which was parallel to the sample ck inaxis (0, within k, 0.5) <(r.l.u.)1. Between = 1 the instrument resolution R and expressed in Å . Along the a axis direction, we the beam access windows and the sample plate, there were further aluminium 3 2 1 find 6.4 10 r.l.u. 1.1 10 Å , and hence ⇠a 95 5 Å. Deconvolving Figure 1 Incommensurate charge–density-wave order. Diffracted intensityfoil in thermal reciprocal radiation space shieldsQ (h at,k liquid,l) h nitrogena⇤ kb⇤ temperature.lc⇤ where a A⇤20 2 mm2⇡/a square, b⇤ 2⇡/b = ⇥ ⌘ ⇥ = ± = = + + = = the poor instrumental resolution along the b axis direction for the (2 1, 0, 0.5) The intensitiesand c of the|2⇡/c incommensurate, with lattice parameters Bragga 3.81 peaks Å, b 3. are87 Å sensitive (Supplementaryaperturemay Fig. collimated S1), bec 11 related. the72 Å. incoming Four to different beam, phonon so scans that in it anomalies reciprocalpassed mainly space, through, projected which the part into suggest that in ⇤ = = = = peak yields a similar correlation length ⇠b ⇠a. to atomic displacementsthe first Brillouin parallelzone, are shown to schematicallythe total inscatteringe. a–c, Scans vectoralong (h,of0, the0.YBCO5) sample for temperatures over near the holep and in1 the magnetic/8 aluminium there fields are plate, (applied anomalies greatly along reducing the in crystal background the underlying charge ⇠ scattering by the plate. Further⇡ slits before the analyser and the detector removed Q. The comparativelyc-direction) as small indicated. contribution An incommensurate to latticeQ modulation,along the peakedc ⇤ at (2susceptibility1, 0, 0), where for1 0q.3045(2),(0,0. emerges3). as the temperature is lowered Received 18 June 2012; accepted 18 September 2012; 6 scattering by the cryostat windows= and nitrogen shields. The scattering plane below 135 K. The intensity of the satellite in b is of the order 2 10 weaker than the (2, 0, 0) reflection.⇡ This becomes field-dependent below the published online 14 October 2012 direction from l 0.5r.l.u. means that our signal for⇥ a (h,(a⇤–c ⇤) wasCuprate horizontal. superconductors The cryomagnet was mounted show on stronga rotation stage spin with correlations, a and zero-field= superconducting transition temperature Tc 67 K. The full-widthgoniometer half-maximum giving instrumentaltilt about the resolution field axis. is The shown sample by horizontal was initially lines mounted in b,f.By 0, 0.5) peak is dominated by displacements parallel= to the a the interplay between spin3 and charge correlations may be at the deconvolving the resolution from the Gaussian fits to the data taken at 17with T and its 2a K,axis an nearlyh-width horizontal. of a 6 The.4 10goniometer r.l.u. corresponding allowed the exact to a correlation alignment of References = ⇥ direction. (Therelength will⇠a 1/ alsoa of 95 be5 displacementsÅ was found (see Methods). paralleld, The to field-induced the c this signal axisheart usingI(17 of T) the the (2I(0 0 high- T) 0) at BraggT Tc peak2phenomenon. K is and modulated could also along be Theused (1.695, for spin low-resolution0, l) correlations and peaks at 1. are Mathur, largely N. D. et al. Magnetically mediated superconductivity in heavy fermion = ± = direction but weapproximately are essentiallyl 0.5. insensitivef, Scan along (1.695, to themk, 0.5). in The our poor present resolutionscans alongdynamic, in the thek-directionb⇤ direction. with did Magnetic not energies allow fields accurate up were to determination applied several with hundred the of sample the width heated meV. along YBa Cucompounds.O Nature 394, 39–43 (1998). =± 2 3 6 x (1.695, k, 0.5), but we estimate a value of 0.01 r.l.u., comparable to that alongabove (hT,c 0,; it 0.5), was indicatingthen field-cooled similar to coherence base temperature. lengths along Whena fields- and wereb-axis applied, directions. 2. Kivelson,+ S. A. et al. How to detect fluctuating stripes in the high-temperature scattering geometry). Our data indicate that the incommensurateminorand changes La2 in thex(Ba position,Sr)x andCuO angle4 of theshow sample incommensurate holder were observed; these magneticsuperconductors. order, Rev. Mod. Phys. 75, 1201–1241 (2003). g–i, Scans along (0, k, 0.5). Incommensurate peaks are found in several Brillouin zones, for example, at positions+Q (0,2 2,0.5) and (0,4 2,0.5), peaks are much stronger if they are satellites of strong Braggwerewhich corrected can by use be of horizontal enhanced and vertical by= suppressing motion± stages under superconductivity the cryostat 3. Vojta, with M. an Lattice symmetry breaking in cuprate superconductors: Stripes, where 2 0.3146(7), see also Supplementary Fig. S3. The vertical dashed line in g indicates 1 whereas the line in a indicates 2. The lattice modulation peaks of the form (⌧= (2n,0,0)) at positions such as ⌧ q .rotationapplied stage, and magnetic by realigning field on the21–24 (2 0; 0) this Bragg has peak. some During analogies temperature with thenematics, CDW and superconductivity. Adv. Phys. 58, 699–820 (2009). was fitted to= a Gaussian function (solid lines in a–d,f–i) on a background± 1 scans, (dashed realignment lines) modelled on the by(2 0 a 0) second-order Bragg peak was polynomial. performed Error automatically bars are determined at every 4. Moncton, D. E., Axe, J. D. & DiSalvo, F. J. Neutron scattering study of the This indicatesby that counting the statistics. satellites are caused by a modulationtemperatureorder point observed to ensure here. that all The measurements magnetic were centred. order After is static results on the charge–density1 meV wave transitions in 2H–TaSe2 and 2H–NbSe2. Phys. Rev. B 16, of the parent crystal structure. The fact that the scattering ishad beenfrequency obtained with scale the a ofaxis neutron horizontal, the diffraction sample was remounted and has with been detected⇠801–819 in(1977). In zero field, the intensity of the CDW Bragg peak (Fig. 2) growsthe b axisours horizontal13 indicate for further that the measurements. CDW is not The accompanied YBa2Cu3O6.67 bysample magnetic had or- 5. Demler, E., Sachdev, S. & Zhang, Y. Spin-ordering quantum transitions of peaked at l 0.5r.l.u. means that neighbouring bilayers are lightly doped YBa Cu O 3 for p 0.082 (ref. 21), and moderately > dimensions a b c 3.1 1.72 0.63 mm6 xand mass 18 mg. The superconducting superconductors in a magnetic field. Phys. Rev. Lett. 87, 067202 (2001). =±on cooling to Tc, below which it is partially suppressed. For T Tc, der, and⇥ this⇥ = is confirmed⇥ ⇥ by soft+ X-ray measurements,. which would modulated in antiphase. The two simplest structures (Fig. 3a,b)transitiondoped temperature La2 xTSrc xCuO67 K (width:4 for 10%–90%p 140.141 1 K) (ref. was derived 24). from The YBa6.2Cu Tranquada,3O6.67 J. M. et al. Evidence for stripe correlations of spins and holes in a magnetic field applied along the c direction has no effect. Belowa zero-field-cooledalso be sensitive magnetization to= fluctuating curve at order 0.1 mT. . The= Charge single density crystal was modulations 99% copper oxide superconductors. Nature 375, 561–563 (1995). compatible withTc ourit causes data an (see increase Supplementary of the intensity Information) of the CDW signal involve (Figs 1a (pin solids1/8) will sample always studied involve both here a is modulation expected of to the have electronic a relatively large the neighbouringand 2). CuO At T planes2 K, the in intensity the bilayer grows with being applied displaced magnetic in field spincharge⇡ gap, andh a! periodic20 meVdisplacement (ref. 25), of the in atomic its magnetic positions15. We excitations at 2 = NATURE PHYSICS VOL 8 DECEMBER 2012 www.nature.com/naturephysics 875 the same (bilayer-centred)(Fig. 2b) and shows or opposite no signs of (chain-centred) saturation up to 17 T. directions, The magnetic loware moretemperature,| sensitive¯ ⇡| to making the atomic it| displacements unlikely that than it to orders the charge magnetically. field also makes the CDW more long-range ordered (Fig. 2c). In modulation because ions with large numbers of electrons (as in 13,14 resulting in thezero maximum magnetic amplitude field, the q-width of the varies modulation little with being temperature. on AsYBCO) discussed dominate earlier, the scattering this is (see confirmed Supplementary by other Information). measurements , the CuO planes or CuO chains respectively. In their 2 q form, so the CDW13 does not seem to be accompanied by spin order. 2 However, below Tc in a field, the CDW order not only becomes NMR data suggest that CDW order only appears below these structuresstronger, would but lead also to becomes the in-plane more coherent, ‘checkerboard’ down to a temperature pattern Moreover,T 67 K and thereH > 9 is T, no whereas obvious with X-rays relationship we observe between CDW orderqCDW and the T below which the intensity starts to decrease (Figs 2 and 4). in⇡ zero field up to 135 K. This apparent discrepancy may arise shown in Fig. 3c.cusp Scanning tunnelling microscopy studies of other wave vector of the incipient spin fluctuations qSF (0.1,0) of underdoped cupratesAll of this16 and is clear of evidence field-induced for competition CDW correlations between CDW in and similarlyfrom differing doped timescales samples of25. various probes (see Supplementary⇡ 17 superconducting orders. Information for further discussion). X-ray diffraction experiments vortex cores alsoNon-resonant support theX-ray tendency diffraction towards is sensitive checkerboard to modulations of areIt usually is interesting interpreted to as note measuring that T theCDW staticcorresponds order of a given approximately 18 formation , althoughcharge density disorder and magnetic can cause moments. small In our stripe case, domains the expected withstructure,TH (Fig. but, 4), if the performed temperature with wide at which energy Hall acceptance, effect measurements are to mimic checkerboardmagnetic cross-section order19. is several Our observation orders of magnitude of a smaller CDW than suggestalso sensitive that Fermi to short-lived surface structures. reconstruction Thus, it is begins possible26. that A CDW that our observed signal, which must therefore be due to charge scatter- the observed CDW is quasi-static and only frozen on the NMR ing. NMR measurements on a sample of the same composition as timescale ( 3 ns) at high fields and lower temperatures. NATURE PHYSICS VOL 8 DECEMBER 2012 www.nature.com/naturephysics ⇡ 873 | | | 872 NATURE PHYSICS VOL 8 DECEMBER 2012 www.nature.com/naturephysics | | |

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c c ( % ) % ( T ∆ / T / B36.00002 2013 meeting March APS energy with respect to the site charge density N(1 2 − Low T phase diagram bi )= ni and the complex bond pairing amplitude " #αβ 2 NQ = c† c† /(b b )(whereb is the hole density of a doped antiferromagnet ij "J iα jβ# i j i at site i and denotes the Sp(2N)-invariant antisym- metric tensor),J while maintaining certain local and global constraints. There have been a number of related earlier p 4 mean-ﬁeld studies17,buttheyhaveall(withtheexcep-

tion of Ref. 11) restricted attention to the case where bi and Q are spatially uniform (note that Q has the | ij| | ij| q same symmetry signature as the bond charge density, and is therefore a measure of its value). However such solu- A1 ) tions are usually unstable, and at best metastable, at low J ’/

J doping; here we have attempted to find the true global minima of the saddle-point equations, while allowing for (or arbitrary spatial dependence: such a procedure leads to N Bond considerable physical insight, and also leads to solutions broken d-wave X order in accord with recent experimental observations. First, at δ =0alongA1 we find the fully dimerized, insulating spin-Peierls (or 2 1bondcharge-densitywave) Spin broken solution18 in which Q is× non-zero only on the bonds | ij| order A2 shown in Fig. 1. Moving to small non-zero δ along A1, our numerical search always yielded lowest energy states 0 Doping δ with broken, consisting of bond-centered charge-density wavesC19 with a p 1unitcell,asshowninFig.1.We M. VojtaFIG. and 1. Schematic,S. Sachdev, Physical proposed, Review ground Letters state 83 phase, 3916 diagram (1999) always found p to be× an even integer, reflecting the dimer- of S. Sachdevas a function and N. ofRead, the Int. doping J. Mod.δ forPhys. physically B 5, 219 reasonable(1991) H ization tendency of the δ =0solution.Withineachp 1 values of t, J and V .Theverticalaxisrepresentsaparam- × Saturday, March 23, 13 unit cell, we find that the holes are concentrated on a eter which measures the strength of quantum spin fluctuations—it increases linearly with N but can also be tuned con- q 1region,withatotallinearholedensityofρ#.Akey property× is that q and ρ remain finite, while p , tinuously by J !/J.Themagnetic symmetry is broken # M →∞ in the hatched region, while symmetry is broken (with ac- as δ 0. Indeed, the values of q and ρ# are deter- C → companying charge-density modulation) in the shaded region; mined primarily by t, J,andthenearest-neighborvalue there are numerous additional phase transitions at which the of Vij = Vnn,andareinsensitivetoδ and longer range detailed nature of the or symmetry breaking changes - parts of Vij.Forδ 0, we found that q =2wasoptimum M C → these are not shown. For δ =0, symmetry is broken only for a wide range of parameter values, while larger values M below the critical point X,while symmetry is broken only of q (q 4) appear for smaller values of Vnn;specifically C ≥ above X.Thesuperconducting symmetry is broken for all we had q =2,ρ# =0.42 at t/J =1.25, Vnn/t =0.6, and S δ > 0atlargeN;forsmallerN,the can be restored at q =4,ρ# =0.8att/J =1.25, Vnn/t =0.5. The limit small δ by additional breaking alongS the vertical axis for V 0leadstoq which reflects the tendency to C nn the states in the inset–this is not shown. The superconduc- phase→ separation in→∞ the “bare” t J model. The evolu- tivity is pure d-wave only in the large δ region were and tion of p with δ is shown in Fig.− 2. Note that there is a C are not broken. The arrow A1 represents the path along M large plateau at p =4arounddopingδ =1/8, and, for which quantitative results are obtained in this paper, while some parameter regimes, this is the last state before A2 is the experimental path. The nature of the symme- C C is restored at large δ;indeedp =4isthesmallestvalue try breaking along path A1 is also sketched: the thick and of p for which our mean-field theory has solutions with dashed lines indicate varying values of Qij (proportional to 1,2 | | bi not spatially uniform. Experimentally ,apinningof the bond charge density) on the links, while the circles rep- 2 the charge order at a wavevector K =1/4isobserved, resent bi (proportional to the site hole density). The charge and we consider it significant that this value emerges nat- densities on the links and sites not shown take values con- urally from our theory. sistent with the symmetries of the figures shown. We expect Our large-N theory only found states in which the that the nature of the symmetry breaking will not change C ordering wavevector K was quantized at the rational significantly as we move from A1 to A2,andacrossthephase boundary where is broken: this suggests the appearance plateaus in Fig. 2. However, for smaller N we expect that of collinearly polarizedM spin-density waves, which break both irrational, incommensurate, values of K will appear, and and ,andwhichundergoan‘anti-phase’shiftacrossthe interpolate smoothly between the plateau regions. C M hole-rich stripes16. In our large-N theory, each q-width stripe above is aone-dimensionalsuperconductor,whiletheintervening (q p)-width regions are insulating. However, fluctua- tion− corrections will couple with superconducting regions,

2 Outline 1. Update on cuprate experiments

2. Antiferromagnetism in metals: d-wave superconductivity

3. Low energy theory, emergent pseudospin symmetry, and bond order

4. Unrestricted Hartree-Fock-BCS

5. Quantum Monte Carlo without the sign problem

Saturday, March 23, 13 Outline 1. Update on cuprate experiments

2. Antiferromagnetism in metals: d-wave superconductivity

3. Low energy theory, emergent pseudospin symmetry, and bond order

4. Unrestricted Hartree-Fock-BCS

5. Quantum Monte Carlo without the sign problem

Saturday, March 23, 13 Fermi surface+antiferromagnetism

Metal with “large” Fermi surface

The electron spin polarization obeys

iK r S⌃(r, ) = ⌃⇥ (r, )e · ⇥ where K is the ordering wavevector.

Saturday, March 23, 13 Fermi surface+antiferromagnetism

Metal with “large” Fermi surface

Saturday, March 23, 13 Fermi surface+antiferromagnetism

Fermi surfaces translated by K =(, ).

Saturday, March 23, 13 Fermi surface+antiferromagnetism

“Hot” spots

Saturday, March 23, 13 Fermi surface+antiferromagnetism

Electron and hole pockets in antiferromagnetic phase with antiferromagnetic order parameter ~' =0 h i6

Saturday, March 23, 13 Fermi surface+antiferromagnetism Increasing SDW order

~' =0 ~' =0 h i6 h i Metal with electron Metal with “large” and hole pockets Fermi surface r

S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997). Saturday, March 23, 13 Fermi surface+antiferromagnetism Increasing SDW order

Rest of the talk

~' =0 ~' =0 h i6 h i Metal with electron Metal with “large” and hole pockets Fermi surface r

S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997). Saturday, March 23, 13 Pairing by SDW ﬂuctuation exchange

We now allow the SDW ﬁeld ⌦⌅ to be dynamical, coupling to electrons as H = ⌦⌅ c† ⌦⇥ c . sdw q · k, ⇥ k+K+q,⇥ k,q,,⇥ Exchange of a ⌦⌅ quantum leads to the eective interaction 1 Hee = V⇥,⇤⌅(q)c† ck+q,⇥cp† ,⇤ cp q,⌅, 2 k, q p,⇤,⌅ k,,⇥ where the pairing interaction is ⇤ V (q)=⌦⇥ ⌦⇥ 0 , ⇥,⇤⌅ ⇥ · ⇤⌅ 2 +(q K)2 2 with ⇤0 the SDW susceptibility and the SDW correlation length.

Saturday, March 23, 13 BCS Gap equation

In BCS theory, this interaction leads to the ‘gap equation’ for the pairing gap k ck c k . ⇤⌅ ⇥ ⇤⇧

3⇥0 p k = 2 2 +(p k K) 2 2 p 2 ⇤ + ⇤ ⇥ p p ⌅ Non-zero solutions of this equation require that and have opposite signs when p k K. k p ⇥

Saturday, March 23, 13 Pairing “glue” from antiferromagnetic ﬂuctuations

V. J. Emery, J. Phys. (Paris) Colloq. 44, C3-977 (1983) D.J. Scalapino, E. Loh, and J.E. Hirsch, Phys. Rev. B 34, 8190 (1986) K. Miyake, S. Schmitt-Rink, and C. M. Varma, Phys. Rev. B 34, 6554 (1986) S. Raghu, S.A. Kivelson, and D.J. Scalapino, Phys. Rev. B 81, 224505 (2010) Saturday, March 23, 13 ck† ↵c† k = "↵S(cos kx cos ky) D E

S S

Unconventional pairing at and near hot spots

Saturday, March 23, 13 Outline 1. The “modern era” of cuprate experiments

2. Antiferromagnetism in metals: d-wave superconductivity

3. Low energy theory, emergent pseudospin symmetry, and bond order

4. Unrestricted Hartree-Fock-BCS

5. Quantum Monte Carlo without the sign problem

Saturday, March 23, 13 Outline 1. The “modern era” of cuprate experiments

2. Antiferromagnetism in metals: d-wave superconductivity

3. Low energy theory, emergent pseudospin symmetry, and bond order

4. Unrestricted Hartree-Fock-BCS

5. Quantum Monte Carlo without the sign problem

Saturday, March 23, 13 Fermi surface+antiferromagnetism

“Hot” spots

Saturday, March 23, 13 Low energy theory for critical point near hot spots

Saturday, March 23, 13 Theory has fermions 1,2 (with Fermi velocities v1,2) and boson order parameter ~' , interacting with coupling

v1 v2

1 fermions Ar. Abanov and 2 fermions A. V. Chubukov, Phys. Rev. Lett. 84, 5608 occupied occupied (2000).

Saturday, March 23, 13 Theory has fermions 1,2 (with Fermi velocities v1,2) and boson order parameter ~' , interacting with coupling

v1 v2

1 fermions M. A. Metlitski and 2 fermions S. Sachdev, Phys. Rev. B 85, 075127 occupied occupied (2010)

Saturday, March 23, 13 Theory has fermions 1,2 (with Fermi velocities v1,2) and boson order parameter ~' , interacting with coupling

v1 v2

This low-energy theory is invariant underky particle-hole transformation. Particles and

holes both have spin S=1/2, and have only kx spin-spin interactions. Theory has an emergent SU(2)4 symmetry.

1 fermions M. A. Metlitski and 2 fermions S. Sachdev, Phys. Rev. B 85, 075127 occupied occupied (2010)

Saturday, March 23, 13 ck† ↵c† k = "↵S(cos kx cos ky) D E

S S

Unconventional pairing at and near hot spots

Saturday, March 23, 13 ck† Q/2,↵ck+Q/2,↵ = Q(cos kx cos ky) D E After Q is ‘2kF ’ pseudospin wavevector rotation Q

M. A. Metlitski and Q S. Sachdev, Phys. Rev. B 85, 075127 (2010)

Unconventional particle-hole pairing at and near hot spots

Saturday, March 23, 13 ck† Q/2,↵ck+Q/2,↵ = Q(cos kx cos ky) D E After Q is ‘2kF ’ pseudospin wavevector rotation Q

M. A. Metlitski and Q S. Sachdev, Phys. Rev. B 85, 075127 (2010)

Q corresponds to incommensurate d-wave bond order or a quadrupole density wave

Unconventional particle-hole pairing at and near hot spots

Saturday, March 23, 13 Incommensurate d-wave bond order

Q Q M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, 075127 (2010)

ck† Q/2,↵ck+Q/2,↵ = Q(cos kx cos ky) Saturday, March 23, 13 D E Incommensurate d-wave bond order

Q Q M. A. Metlitski and Q S. Sachdev, Phys. Rev. B 85, 075127 (2010)

ck† Q/2,↵ck+Q/2,↵ = Q(cos kx cos ky) Saturday, March 23, 13 D E Incommensurate d-wave bond order

+1 “Bond density” measures amplitude for electrons to be in spin-singlet valence bond.

M. A. Metlitski and S. Sachdev, -1 Phys. Rev. B 85, 075127 (2010)

iQ (r+s)/2 ik (r s) cr†↵cs↵ = e · e · ck† Q/2,↵ck+Q/2,↵ Q k ⌦ ↵ X X D E where Q extends over Q =( Q , Q )withQ =2⇡/(7.3) and ± 0 ± 0 0

ck† Q/2,↵ck+Q/2,↵ = Q(cos kx cos ky) D E Note cr†↵cs↵ is non-zero only when r, s are nearest neighbors. Saturday, March 23, 13 ⌦ ↵ Incommensurate d-wave bond order

High T pseudogap: Fluctuating composite order parameter of nearly degenerate d-wave pairing and incommensurate d-wave bond order

K. B. Efetov, H. Meier, and C. Pepin, arXiv:1210.3276

ck† Q/2,↵ck+Q/2,↵ = Q(cos kx cos ky) Saturday, March 23, 13 D E Observed low T ordering

Saturday, March 23, 13 Outline 1. The “modern era” of cuprate experiments

2. Antiferromagnetism in metals: d-wave superconductivity

3. Low energy theory, emergent pseudospin symmetry, and bond order

4. Unrestricted Hartree-Fock-BCS

5. Quantum Monte Carlo without the sign problem

Saturday, March 23, 13 Outline 1. The “modern era” of cuprate experiments

2. Antiferromagnetism in metals: d-wave superconductivity

3. Low energy theory, emergent pseudospin symmetry, and bond order

4. Unrestricted Hartree-Fock-BCS

5. Quantum Monte Carlo without the sign problem

Saturday, March 23, 13 Charge ordering in metals with antiferromagnetic spin correlations Charge ordering in metals with antiferromagnetic spin correlations Rolando La Placa and Subir Sachdev Rolando La Placa and Subir Sachdev Department of Physics, Harvard University, Cambridge MA 02138 Department of Physics, Harvard University, Cambridge MA 02138 (Dated: February 28, 2013) (Dated: February 28, 2013) Metals with antiferromagnetic spin correlations have an instability to the superconductivity of spin-singlet Metals with antiferromagnetic spin correlations have an instabilityCooper to the pairs superconductivity with d symmetry of spin-singlet (for the Fermi surface of the cuprates). Metlitski et al. (Phys. Rev. B 82, 075128 Cooper pairs with d symmetry (for the Fermi surface of the cuprates).(2010)) Metlitski notedet al. that(Phys. in two Rev. dimensions, B 82, 075128 in the low energy continuum theory, this superconductivity is degenerate (2010)) noted that in two dimensions, in the low energy continuumCharge theory,with ordering a this charge superconductivity in density metals wave with ordering is antiferromagnetic degenerate which has a local spind symmetry correlations under rotations about the lattice points. We with a charge density wave ordering which has a local d symmetry underpresent rotations an unrestricted about the Hartree-Focklattice points. computation We on a simple lattice model, and find that the d symmetry is Rolando La Placa and Subir Sachdev present an unrestricted Hartree-Fock computation on a simple latticedominant model, and for find a range that of the smalld symmetry ordering iswavevectors, including those observed in recent experiments. We note implicationsDepartment for of Physics,the phase Harvard diagram University, of the cuprates. Cambridge MA 02138 dominant for a range of small ordering wavevectors, including those observed in recent experiments.(Dated: February We 28, note 2013) implications for the phase diagram of the cuprates. Metals with antiferromagneticPACS numbers: spin correlations have an instability to the superconductivity of spin-singlet Cooper pairs with d symmetry (for the Fermi surface of the cuprates). Metlitski et al. (Phys. Rev. B 82, 075128 PACS numbers: (2010)) noted that in two dimensions, in the low energy continuum theory, this superconductivity is degenerate withIntroduction a charge density. A wave remarkable ordering which series has a local of experimentsd symmetry under [1–12] rotations aboutfrom the the lattice time-reversal-symmetry points. We breaking “d-density wave” present an unrestricted Hartree-Fock computation on a simple lattice model, and find that the d symmetry is Introduction. A remarkable series of experiments [1–12] havedominantfrom shed for the a new range time-reversal-symmetry light of small on ordering the enigmatic wavevectors, breaking underdoped including “d those-density region observed wave” of in recentstate experiments. [22], and We the note confusion in the nomenclature is unfortu- have shed new light on the enigmatic underdoped region of theimplicationsstate cuprate [22], for high the and phase temperature the diagram confusion of the superconductors. cuprates. in the nomenclature These is exper- unfortu- nate. As will become clear from our presentation below, it iments consistently detect a bi-directional density wave with is useful to formulate the various charge ordering configura- the cuprate high temperature superconductors. These exper- PACSnate. numbers: As will become clear from our presentation below, it iments consistently detect a bi-directional density wave with a periodis useful of about to formulate 3-4 lattice the various spacings charge at low ordering hole densities. configurations in terms of their symmetries of rotation about the Cu a period of about 3-4 lattice spacings at low hole densities. Moreover,tions in this terms density of their wave symmetries order is co-incident of rotation with about regions the Cu lattice points. In our approach, the “d-density wave” state of Introductionof. the A remarkable phase diagram series of where experiments quantum [1–12] oscillationsfrom the [13] time-reversal-symmetry were Ref. [22] breaking actually “d has-density a p wave”symmetry about the lattice points Moreover, this density wave order is co-incident withhave regions shed new lightlattice on the points. enigmatic In our underdoped approach, region the “ ofd-densitystate [22], wave” and state the confusion of in the nomenclature is unfortu-x,y of the phase diagram where quantum oscillationsthe [13] cuprate were highobserved temperatureRef. [22] in YBa actually superconductors.2Cu3 hasOy, a stronglypx,y Thesesymmetry supporting exper- aboutnate. the the Ashypothesis lattice will become points clear[17], from and our so we presentation will identify below, it byit the px,y label. observed in YBa2Cu3Oy, strongly supporting the hypothesisiments consistently[7,[17], 14–16] detect and a that so bi-directional we the will density identify density wave itwave by plays withthe apx central,y islabel. useful role to in formulate the the variousHere we charge will ordering present configura- an unrestricted Hartree-Fock-BCS on [7, 14–16] that the density wave plays a central rolea period in the of aboutformation 3-4Here lattice of we the spacings will Fermi present at pockets low an hole unrestricted responsible densities. Hartree-Fock-BCS fortions the in termsquantum of their on symmetriesa simple lattice of rotation model about of a the metal Cu with antiferromagnetic spin formation of the Fermi pockets responsible for theMoreover, quantum thisoscillations. densitya simple wave lattice Some order is ofmodel co-incident the experiments of a with metal regions with [3, 5, antiferromagnetic 6,lattice 8, 11] points. are also In our spin approach,corrections. the “d-density We will wave” make state no of direct reference to the contin- oscillations. Some of the experiments [3, 5, 6, 8, 11]of the are phase also sensitive diagramcorrections. where to the quantum electronic We will oscillations make microstructure no [13] direct were reference of theRef. density [22] to actually the wave: contin- has a puumx,y symmetry limit or about to the the lattice hot spots, points and will retain full momen- observed in YBa Cu O , strongly supporting the hypothesis [17], and so we will identify it by the p label. sensitive to the electronic microstructure of the density wave: these2uum indicate3 limity that or to there the is hot negligible spots, and modulation will retain of the fullcharge momentum dependencex,y of all variables across the Brillouin zone. [7, 14–16] that the density wave plays a central role in the Here we will present an unrestricted Hartree-Fock-BCS on these indicate that there is negligible modulation offormation the charge ofdensity thetum Fermi on dependence pockets the Cu responsible sites. of all Instead, for variables the quantumit appears across toa the simple be Brillouin primarily lattice model zone. a ofWe a metal will with find, antiferromagnetic as expected, spin that the dominant instability of density on the Cu sites. Instead, it appears to be primarilyoscillations. a SomebondWe ofdensity the will experiments find, wave, as with expected, [3, 5, modulations 6, 8, 11] that are the also in dominant spin-singletcorrections. instability observ- We will of makethe no metal direct is reference always to towards the contin- superconductivity with d-wave bond density wave, with modulations in spin-singletsensitive observ- to theables electronicthe associated metal microstructure is always with a Cu-Cu of towards the density link, superconductivity wave: such as theuum average limit with or elec- tod-wave the hotpairing. spots, and The will leading retain full sub-dominant momen- instability is towards a ables associated with a Cu-Cu link, such as the averagethese indicate electron thatpairing. there kinetic is negligible energy. The leading modulation sub-dominant of the charge instabilitytum dependence is towards of all a spin-singlet variables across charge the Brillouin ordering zone. closely related to that proposed tron kinetic energy. density on the Cuspin-singlet sites. Instead, charge it appears ordering to be primarily closely a relatedWe to will that find, proposed as expected,in Ref. that 17. the For dominant a range instability of small of wavevectors, including those bond density wave,This with paper modulations will examine in spin-singlet a model observ- for the charge ordering in Ref. 17. For a range of small wavevectors,the metal including is always those towardsobserved superconductivity in the experiments with d-wave [1–6, 8–11]. the charge order has This paper will examine a model for the chargeables ordering associatedproposed with a Cu-Cu in Ref. link, [17]. such They as the studied average a elec- two-dimensionalpairing. The metal leading sub-dominant instability is towards a proposed in Ref. [17]. They studied a two-dimensionaltron kinetic metal energy.withobserved antiferromagnetic in the experiments spin correlations, [1–6, 8–11]. using thespin-singlet charge a continuum order charge has orderinga predominant closely relatedd symmetry to that proposed of rotations about the Cu lattice a predominant d symmetry of rotations about the Cu lattice sites. The predominant d symmetry implies that the charge with antiferromagnetic spin correlations, using a continuumThis paperlimit will examine which focused a model on for particular the charge ‘hot ordering spots’ onin Ref. the Fermi 17. For sur- a range of small wavevectors, including those limit which focused on particular ‘hot spots’ on theproposed Fermi sur- in Ref.face [17].sites. [18]. They The In studied this predominant limit, a two-dimensional Ref.d [17]symmetry showed metal implies thatobserved theory that in the had the charge experiments an order [1–6, is primary 8–11]. the located charge order on the has Cu-Cu bonds, and there is lit- face [18]. In this limit, Ref. [17] showed that theorywith had antiferromagnetic an emergentorder spin is pseudospin primary correlations, located symmetry, using on a the continuum and Cu-Cu consequently bonds,a predominant and the there metald issymmetry little modulation of rotations about of the the charge Cu lattice density on the Cu sites. sites. The predominant d symmetry implies that the charge limit which focusedhadtle two on modulation particular degenerate ‘hot of spots’ theinstabilities: charge on the density Fermi to sur- superconductivity on the Cu sites. with Method. We will examine the following Hamiltonian of emergent pseudospin symmetry, and consequently the metal order is primary located on the Cu-Cu bonds, and there is lit- had two degenerate instabilities: to superconductivityface [18]. with Ind this-wave limit,Method pairing, Ref.. [17] We and showed will to a examine that density theory wave the had following an which had Hamiltonian a local d- of electrons on a square lattice with annihilation operators ck,↵, emergent pseudospin symmetry, and consequently the metal tle modulation of the charge density on the Cu sites. electrons on a square lattice with annihilation operators ck,↵, where k is a crystal momentum and ↵ = , is a spin label: d-wave pairing, and to a density wave which hadhad a local two degenerated- wave symmetry instabilities: of to rotations superconductivity about each with lattice point.Method The. We den- will examine the following Hamiltonian of " # sitywhere wavek instabilityis a crystal had momentum a small incommensurate and ↵ = ,electronsis awavevector spin on a label: square lattice with annihilation operators c ,↵, wave symmetry of rotations about each lattice point.d-wave The den- pairing, and to a density wave which had a local d- " # k1 sity wave instability had a small incommensurate wavevectorwave symmetryQ of= rotations(Q, Q about), where eachQ latticeis determined point. The den- by thewhereHartree-Fock positionsk is a of crystal the momentumcomputationH = and ↵ ="on(k, ) latticeisc† a spinc label:model (q) S~( q) S~(q). (1) ± 1 " # k,↵ k,↵ 2V · Q = , sity wave instabilityhot spots. had a= Lattice small" incommensurate corrections† break wavevector the pseudospin~ ~ symme-. k q (Q Q), where Q is determined by the positions of the H (k) ck,↵ck,↵ (q) S ( q) S (q) (1) 1 X X ± Q = (Q, Q), where Q is determined by the positions 2V of the H =· "(k) c† c (q) S~( q) S~(q). (1) hot spots. Lattice corrections break the pseudospin symme-± try, and the superconductingk instabilityq becomes stronger thank,↵ k,↵ X X 2V q · try, and the superconducting instability becomes strongerhot spots. than Latticethe charge corrections order break in the the full pseudospin theory. symme- Metzner and collaboratorsXk where thereX is an implicit sum over spin indices. Here S~(q) = try, and the superconducting instability becomes stronger than [19,where 20] studied there is this an implicit charge ordering sum over using spin indices. functional Here renor-S~(q) = k c† ~ ↵ c is the electron spin density (~ are the Pauli the charge order in the full theory. Metzner and collaboratorsthe charge order in the full theory. Metzner and collaborators where there is an implicit sumk over+q,↵ spin indices.k, Here S~(q) = [19, 20] studied this charge ordering using functional renor- malizationk c† group~ ↵ methods,c is the and electron called spin it an density‘incommensuratec† (~ are~ thec Pauliis thematrices), electron spin and densityV is ( the~ are system the Pauli volume. For the electronic dis- [19, 20] studied this chargek+q,↵ orderingk, using functional renor- k k+q,↵ ↵ k, P malization group methods, and called it an ‘incommensuratemalization groupnematic’matrices), methods,. They and and alsocalledV is found it the an ‘incommensurate system similar volume. instabilities Formatrices), withthe electronic wavevec- and V is dis- the systempersion, volume. we assume For the electronic a tight-binding dis- model which provides a P P nematic’. They also found similar instabilities withnematic’ wavevec-. Theytors alsopersion,Q found= (Q similar we, 0) assume, instabilities(0, Q), a corresponding tight-binding with wavevec- model to thosepersion, which seen we provides in assume the a a tight-bindinggood fit to photoemissionmodel which provides data a tors Q = (Q, 0), (0, Q), corresponding to those seentors Q in= the(Qexperiments, 0),good(0, Q), fit corresponding to [1–6, photoemission 8–11]. to And those data recently seen in the Efetovgoodet al. fit to[21] photoemission for- data experiments [1–6, 8–11]. And recently Efetov et al. [21] for- experiments [1–6, 8–11]. And recently Efetov et al. [21] for- mulated these instabilities in terms of a non-linear model "(k) = 2t1 cos(kx) + cos(ky) 4t2 cos(kx) cos(ky) mulated these instabilities in terms of a non-linear model "(k) = 2t1 cos(kx) + cos(ky) 4t2 cos(kx) cos(ky) on pseudospin"(k) = space,2t1 andcos( emphasizedkx) + cos(ky their) 4 importancet2 cos(kx) cos( fork they) mulated these instabilities in terms of a non-linearon pseudospinmodel space, and emphasized their importance for the ⇣ + ⌘ µ. 2t ⇣cos(2k ) + cos(2k⌘ ) 2µ.t3 cos(2kx) (2)cos(2ky) (2) on pseudospin space, and emphasized their importance for the physics of the pseudo-gap; they labelled the above charge or-3 x y physics of the pseudo-gap; they labelled2t3 ⇣cos(2 the abovekx) + chargecos(2 or-k⌘y) µ. (2) der as a ‘quadrupolar’ order. ⇣ ⌘ ⇣ ⌘ physics of the pseudo-gap; they labelled the aboveder charge as a ‘quadrupolar’ or- order. The interactions betweenThe the interactions electrons couple between their spin the den- electrons couple their spin den- ⇣ ⌘ der as a ‘quadrupolar’ order. It should be notedItThe should that interactions above be noted the charge-ordering between that above the the is electrons distinct charge-ordering couplesity via their is a susceptibility distinct spin density(q) which via a susceptibility is peaked near the(q antifer-) which is peaked near the antifer- It should be noted that above the charge-ordering is distinct sity via a susceptibility (q) which is peaked near the antifer-

S. Sachdev and R. La Placa, arXiv:1303.2114 Saturday, March 23, 13 Charge ordering in metals with antiferromagnetic spin correlations Charge ordering in metals with antiferromagnetic spin correlations Rolando La Placa and Subir Sachdev Rolando La Placa and Subir Sachdev Department of Physics, Harvard University, Cambridge MA 02138 Department of Physics, Harvard University, Cambridge MA 02138 (Dated: February 28, 2013) (Dated: February 28, 2013) Metals with antiferromagnetic spin correlations have an instability to the superconductivity of spin-singlet Metals with antiferromagnetic spin correlations have an instabilityCooper to the pairs superconductivity with d symmetry of spin-singlet (for the Fermi surface of the cuprates). Metlitski et al. (Phys. Rev. B 82, 075128 Cooper pairs with d symmetry (for the Fermi surface of the cuprates).(2010)) Metlitski notedet al. that(Phys. in two Rev. dimensions, B 82, 075128 in the low energy continuum theory, this superconductivity is degenerate (2010)) noted that in two dimensions, in the low energy continuumCharge theory,with ordering a this charge superconductivity in density metals wave with ordering is antiferromagnetic degenerate which has a local spind symmetry correlations under rotations about the lattice points. We with a charge density wave ordering which has a local d symmetry underpresent rotations an unrestricted about the Hartree-Focklattice points. computation We on a simple lattice model, and find that the d symmetry is Rolando La Placa and Subir Sachdev present an unrestricted Hartree-Fock computation on a simple latticedominant model, and for find a range that of the smalld symmetry ordering iswavevectors, including those observed in recent experiments. We note implicationsDepartment for of Physics,the phase Harvard diagram University, of the cuprates. Cambridge MA 02138 dominant for a range of small ordering wavevectors, including those observed in recent experiments.(Dated: February We 28, note 2013) 2 implications for the phase diagram of the cuprates. Metals with antiferromagneticPACS numbers: spin correlations have an instability to the superconductivity of spin-singlet Cooper pairs with d symmetry (for the Fermi surface of the cuprates). Metlitski et al. (Phys. Rev. B 82, 075128 PACS numbers: romagnet wavevector; we assume the simple form while the polarizabilities are (2010)) noted that in two dimensions, in the low energy continuum theory, this superconductivity is degenerate with a charge density wave ordering which has a local d symmetry under rotations about the lattice points. We Introduction. A remarkable series of experimentsq = [1–12] from the0 time-reversal-symmetry breaking “d-density wave” 1 2 f ("(k)) present an unrestricted Hartree-Fock computation on( a simple) lattice model,2 and find that the d symmetry is (3) Introduction. A remarkable series of experiments [1–12] havefrom shed the new time-reversal-symmetry light on the enigmatic breaking underdoped “d-density region⇠ + wave”2(2 of cos(stateq [22],x K andx) thecos( confusionqy Ky)) in the nomenclature is⇧ unfortu-S (k) = dominant for a range of small ordering wavevectors, includingK those observed in recent experiments. We note 2"(k) have shed new light on the enigmatic underdoped region of theimplicationsstate cuprate [22], for high the and phase temperature the diagram confusion of the superconductors. cuprates. in the nomenclature TheseX is exper- unfortu- nate. As will become clear from our presentation below, it f ("(k + Q/2)) f ("(k Q/2)) iments consistently detect a bi-directionalwhere density⇠ is the wave antiferromagnetic with is useful correlation to formulate length. the various The sum charge ordering configura- the cuprate high temperature superconductors. These exper- PACSnate. numbers: As will become clear from our presentation below, it ⇧Q(k) = (8) iments consistently detect a bi-directional density wave with a periodis useful of about to formulate 3-4 lattice the various spacingsover charge at low wavevectors ordering hole densities. configura- extends overtionsK in= terms(⇡, of⇡(1 their symmetries)), (⇡(1 of rotation about the Cu "(k Q/2) "(k + Q/2) ± ± a period of about 3-4 lattice spacings at low hole densities. Moreover,tions in this terms density of their wave symmetries order is co-incident), of⇡), rotation and withwe about will regions theuse Cu bothlattice the commensurate points. In our approach, case = the0 or “d-density wave” state of Introduction. A remarkable series of experiments [1–12] from the time-reversal-symmetry breaking “d-density wave” with f the Fermi function. Note that the kernels in Eq. (7) Moreover, this density wave order is co-incident with regions of thelattice phase points. diagram In our where approach, quantum thethe oscillations“d incommensurate-density wave” [13] were state case of Ref.= 1 [22]/4. actually We will has only a px need,y symmetry mod- about the lattice points have shed new light on the enigmatic underdoped region of state [22], and the confusion in the nomenclature is unfortu- have an identical form: this is a consequence of the spin- of the phase diagram where quantum oscillationsthe [13] cuprate were highobserved temperatureRef. [22] in YBa actually superconductors.2Cu3 hasOy, a stronglypx,y Thesesymmetry supporting exper-erate about valuesnate. the the Asofhypothesis lattice⇠ willfor become points our main clear[17], conclusions, from and our so we presentation will and identify the below, e↵ itects byit the we px,y label. spin interaction in H; density-density interactions yield dis- observed in YBa2Cu3Oy, strongly supporting the hypothesisiments consistently[7,[17], 14–16] detect and a that so bi-directional we the will density identify density wave itwave by playsdiscuss withthe apx central,y islabel. are useful present role to in formulate the even at the⇠ variousHere= 1. we charge We will do ordering present not include configura- an unrestricted an ex- Hartree-Fock-BCS on tinct terms in the pairing and charge channels. Note further [7, 14–16] that the density wave plays a central rolea period in the of aboutformation 3-4Here lattice of we the spacings will Fermi present at pockets low an hole unrestricted responsible densities.plicit Hartree-Fock-BCS density-density fortions the in termsquantum of their interaction on symmetriesa simple between lattice of rotation model the about electrons, of a the metal Cu and with antiferromagnetic spin formation of the Fermi pockets responsible for theMoreover, quantum thisoscillations. density wave Some order is of co-incident the experiments with regions [3, 5, 6,lattice 8, 11] points. are also In our approach,corrections. the “d-density We will wave” make state no of direct referencethat for to the dispersions contin- with "(k + Q) = "(k) the two kernels a simple lattice model of a metal withassume antiferromagnetic its main e↵ect spin is to renormalize the dispersion. oscillations. Some of the experiments [3, 5, 6, 8, 11]of the are phase also sensitive diagramcorrections. where to the quantum electronic We will oscillations make microstructure no [13] direct were reference of theRef. density [22] to actually the wave: contin- has a puumx,y symmetry limit or about to the the lattice hot spots, points and will retainequal full each momen- other: Ref. [17] pointed out that such a relation- observed in YBa Cu O , strongly supporting the hypothesisWe will[17], perform and so we an will unrestricted identify it by the Hartree-Fock-BCSp label. analy- sensitive to the electronic microstructure of the density wave: these2uum indicate3 limity that or to there the is hot negligible spots, and modulation will retain of the fullcharge momentum dependencex,y of all variables across theship Brillouin holds close zone. to the hot spots of a generic Fermi surface [7, 14–16] that the density wave plays a central role insis the of H.Here We need we will the present best an variational unrestricted estimate Hartree-Fock-BCS for the on mean- these indicate that there is negligible modulation of the charge densitytum on dependence the Cu sites. of all Instead, variables it appears across to the be Brillouin primarily zone. a We will find, as expected, that the dominantfor certain instabilityQ, and of this was a key ingredient in the emergent formation of the Fermi pockets responsible for the quantumfield Hamiltoniana simple lattice model of a metal with antiferromagnetic spin density on the Cu sites. Instead, it appears to be primarilyoscillations. a SomebondWe ofdensity the will experiments find, wave, as with expected, [3, 5, modulations 6, 8, 11] that are the also in dominant spin-singletcorrections. instability observ- We will of makethe no metal direct is reference always to towards the contin- superconductivitypseudospin with d symmetry;-wave we will see the same Q emerge in our bond density wave, with modulations in spin-singletsensitive observ- to theables electronicthe associated metal microstructure is always with a Cu-Cu of towards the density link, superconductivity wave: such as theuum average limit with or elec- tod-wave the hotpairing. spots, and The will leading retain full sub-dominant momen- instabilitycomputation is towards below. a ables associated with a Cu-Cu link, such as the averagethese indicate electron thatpairing. there kinetic is negligible energy. The leading modulation sub-dominant of the charge instabilityHMFtum= dependence is towards"(k) ofc† all acspin-singlet variables+ S across(k) charge✏↵ theck Brillouin, ordering↵c k + zone.H.c. closely related to that proposed 2 k,↵ k,↵ From Eq. (6) we see that the linear instability of the metal tron kinetic energy. density on the Cuspin-singlet sites. Instead, charge it appears ordering to be primarily closely a relatedWe to will thatk find," proposed as expected,in Ref. that 17. the For dominant a range instability of small of wavevectors, including those bond density wave,This with paper modulations will examine in spin-singlet a model observ- for thethe charge metalX ordering is always towards superconductivity with d-wave occurs via condensation in the eigenmodes of the operators This paper will examine a model for the charge ordering proposedin Ref. in 17. Ref. For [17]. a range They of studied small wavevectors, a two-dimensionalromagnet including wavevector; metal those weobserved assume the in simple the experiments form [1–6, 8–11].while the the charge polarizabilities order has are ables associated with a Cu-Cu link, such as the average elec- pairing. The+ leading k sub-dominant† instability, is towards a S,Q(k, k0) with the lowest eigenvalues. We have chosen the observed in the experiments [1–6, 8–11]. the charge order hasQ( a) c predominantk+Q/2,↵ck Qd/2symmetry,↵ of rotations(4) about the Cu lattice proposed in Ref. [17]. They studied a two-dimensionaltron kinetic metal energy.with antiferromagnetic spin correlations, usingspin-singlet a continuum charge ordering closely0 related to that proposed M q = Q specific forms1 of2 f the("(k kernels)) in Eq. (7) so that we need only a predominant d symmetry of rotations about( ) the Cu2 lattice sites. The predominant# d symmetry(3) implies that the charge with antiferromagnetic spin correlations, using a continuumThis paperlimit will examine which focused a model on for particular the charge ‘hot ordering spots’ onin Ref. the Fermi 17. For⇠X sur- a+ range2(2 ofcos( smallqx wavevectors,Kx) cos(q includingy Ky)) those ⇧S (k) = K order is primary located on the Cu-Cu bonds,solve and the there following is lit-2"(k eigenvalue) problem limit which focused on particular ‘hot spots’ on theproposed Fermi sur- in Ref.face [17].sites. [18]. They The In studied this predominant limit, a two-dimensional Ref.d [17]symmetry showed metaloptimized implies thatobserved theory over thatX in the the had the spin-singlet charge experiments an [1–6, superconducting 8–11]. the charge pairing order has func- order is primary located on the Cu-Cuwhere bonds,a predominant⇠ andis the there antiferromagneticd issymmetry little modulation of rotations correlation about of length. the the charge Cu The lattice density sum on the Cu sites. f ("(k + Q/2)) f ("(k Q/2)) face [18]. In this limit, Ref. [17] showed that theorywith had antiferromagnetic an emergent spin pseudospin correlations, symmetry, using a continuum andtion consequently (k), the and metal the density wave orders (k) at the ⇧Q(k) = (8) Ssites. The predominant d symmetry= implies⇡, ⇡ that the,Q charge⇡ 3 "(k Q/2) "(k + Q/2) limit which focusedhadtle two on modulation particular degenerate ‘hot of spots’ theinstabilities: charge on the density Fermi to sur- superconductivity onover the Cu wavevectors sites. with extends overMethodK .( We(1 will)) examine( (1 the following Hamiltonian⇧ , (k of) (k k0) ⇧ , (k ) , (k0) = , , (k) emergent pseudospin symmetry, and consequently the metal wavevectorsorder isQ primary. Note located that on both the Cu-Cu orders± bonds, are and characterized there± is lit- by S Q S Q 0 S Q S Q S Q face [18]. In this limit,Method Ref.. [17] We showed will examine that theory the had following an ), ⇡), and Hamiltonian we will use of bothelectrons the commensurate on a square case lattice = with0 or annihilationV operators ck,↵, had two degenerate instabilities: to superconductivity with d-wave pairing, and to a density wave which had a local d- with f thek0 Fermi function. Note that the kernels in Eq. (7) emergent pseudospin symmetry, and consequently the metalarbitrarythetle incommensurate functions modulation of of the casek chargeextending = density1/4. We over on the will the Cu only sites. Brillouin need mod- zone, X q q d-wave pairing, and to a density wave which had a local d- waveelectrons symmetry on of a square rotations lattice about with each annihilation lattice point. operators The den-ck,↵, where k is a crystal momentum and ↵have= , anis identical a spin label: form: this is a consequence of the spin- had two degenerate instabilities: to superconductivityand with theseerateMethod values will of. be We⇠ determinedfor will our examine main conclusions, by the a following functional and Hamiltonian the minimization e↵ects ofwe " # sitywhere wavek instabilityis a crystal had momentum a small incommensurate and ↵ = ,electronsis awavevector spin on a label: square lattice with annihilation operators c ,↵, spin interactionfor the minimum in H; density-density eigenvalues interactionsS,Q and yield corresponding dis- eigen- wave symmetry of rotations about each lattice point.d-wave The den- pairing, and to a density wave which had a localofd the- discuss" free# are energy. present Here, even at we⇠ = only1. We consider do not include the cases ank1 ex- where Q = , where k is a crystal momentum= and ↵ =" , is† a spin label: tinct~ termsvectors in~ theS,. pairingQ(k), and and charge their channels. structure Note is independent further of the sity wave instability had a small incommensurate wavevectorwave symmetry of rotations(Q Q about), where eachQ latticeis determined point. The den- by theplicitHartree-Fock positions density-density of the interactioncomputationH between on"(k# the) latticeck, electrons,↵ck,↵ model and (q) S ( q) S (q) (1) ± 1 either S (k) or Q(k) are non-zero, but not both; 2V in the · Q = , sity wave instabilityhot spots. had a= Lattice small" incommensurate corrections† break wavevector the pseudospin~ ~ symme-. k q that foroverall dispersions strength with of"(k the+ Q interaction) = "(k) the0. two The kernels order parameters (Q Q), where Q is determined by the positions of the H (k) ck,↵ck,↵ (qassume) S ( q its) mainS (q) e↵ect(1) is to renormalize1 X the dispersion. X ± = , 2V case of co-existing =· " order,† HMF must also~ include~ . terms likeequal each other: Ref. [17] pointed out that such a relation- Q (Q Q),try, where andQ theis determined superconductingk by the positions instability ofq the becomes strongerH ( thank) ck,↵ck,↵ (q) S ( q) S (q) (1) characterizing the condensed phases will then be (k) hot spots. Lattice corrections break the pseudospin symme-± X X We will perform an unrestricted 2V Hartree-Fock-BCS · analy- S,Q hot spots. Lattice corrections break the pseudospin symme-✏↵ ck+Q/ ,↵ckk+Q/ ,, and we deferq this case to future work.ship holds close to~ the hot spots of a generic Fermi surface / try, and the superconducting instability becomes stronger than the charge order in the full theory. Metznersis and of H collaborators.2 WeX need the2 bestwhere variational thereX estimate is an implicit for the sum mean- over spin indices. , (k Here)/ ⇧S (q, ) (=k). Our principal numerical results below try, and the superconducting instability becomes stronger⇠ than for certainS Q Q, and thisS Q was a key ingredient in the emergent Fermi statistics requires~ =S ( ck†) = ~↵S (ck), whileis the electron hermiticity spin density (~ are the Pauli the charge order in the full theory. Metzner and collaborators [19,where 20] studied there is this an implicit charge ordering sum over using spinfield indices. functionalwhere Hamiltonian there Here renor- is anS ( implicitq) sumk k over+q,↵ spin indices.k, Here S~(q) = are on the Q dependence of Q, and on the k dependence the charge order in the full theory. Metzner and collaborators = pseudospin symmetry;p we will see the same Q emerge in our [19, 20] studied this charge ordering using functional renor- malizationk c† group~ ↵ methods,c is the and electron called spinrequires it an density‘incommensurateQ⇤c† (~(kare)~ thec PauliQis(k the).matrices), electron For time-reversal spin and densityV is ( the~ symmetryare system the Pauli volume. to be For the electronic dis- [19, 20] studied this chargek+q,↵ orderingk, using functional renor- k k+q,↵ ↵ k, P computationof S,Q below.(k) so obtained. We diagonalized the kernels after dis- malization group methods, and called it an ‘incommensuratemalization groupnematic’matrices), methods,. They and and alsocalledV is found it the an ‘incommensurate system similar volume. instabilitiespreserved Formatrices),H withtheMF we electronic= wavevec- need and V"(Qisk dis-)( theck†k, system↵)persion,ck=,↵ + volume.Q(Sk we(k),) ✏ assumebut For↵ ck thewe,↵c electronic will ak tight-binding+ notH.c. dis- impose model which provides a 2 P P Fromcretizing Eq. (6) the we see Brillouin that the zone linear to instabilityL points of the with metalL up to 80, with nematic’. They also found similar instabilities withnematic’ wavevec-. Theytors alsopersion,Q found= (Q similar we, 0) assume, instabilities(0, Q), a corresponding tight-binding with wavevec-this model to as those apersion, constraint, which seen we providesk in assume" andthe so a a tight-binding willgood allow fit to photoemissionmodel for the which breaking provides data of a time- X occurst1 via= 1, condensationt2 = 0.32, in and thet eigenmodes3 = 0.16 for of a the range operators of values of T, µ, tors Q = (Q, 0), (0, Q), corresponding to those seentors Q in= the(Qexperiments, 0),good(0, Q), fit corresponding to [1–6, photoemission 8–11]. to And those data recently seen in the Efetovgoodet al. fit to[21] photoemission for- data reversal. + (k) c† c , (4) S,Q(k, k⇠0) with the lowest eigenvalues. We have chosen the experiments [1–6, 8–11]. And recently Efetov et al. [21] for- Q k"+(Qk/)2,↵=k Q2/2t,↵ cos(k ) + cos(k ) M4t andcos(k. ) cos(k ) experiments [1–6, 8–11]. And recently Efetov et al. [21] for- mulated these instabilities in terms of aThe non-linear free energy model ofQH obeys the variational 1 principlex y specific2 formsx of they kernels in Eq. (7) so that we need only mulated these instabilities in terms of a non-linear model "(k) = 2Xt1 cos(kx) + cos(ky) 4t2 cos(# kx) cos(ky) Results. For the full range of parameters examined, we con- on pseudospin"(k) = space,2t1 andcos( emphasizedkx) + cos(ky their) 4 importancet2 cos(kx) cos( fork they) mulated these instabilities in terms of a non-linearon pseudospinmodel space, and emphasized their importance for the 2t ⇣cos(2k ) + cos(2k⌘ solve) µ. the following eigenvalue(2) problem optimized over the2t spin-singlet⇣cos(2k ) + cos(2 superconductingk⌘ ) µ.3 pairingx func-(2) y sistently found that S was the minimal eigenvalue, and the on pseudospin space, and emphasized their importancephysics for of the thephysics pseudo-gap; of the they pseudo-gap; labelled the above they labelled charge or- the above chargeF or-3 FMF +x H HyMF MF (5) 2t3 ⇣cos(2kx) + cos(2k⌘y) tionµ. (k), and the(2) densityh wave ordersi (k) at the der as a ‘quadrupolar’ order. S ⇣ ⌘ ⇣ Q 3⌘ corresponding eigenvector S (k) was well approximated by physics of the pseudo-gap; they labelled the aboveder charge as a ‘quadrupolar’ or- order. The interactions between the electrons couple⇧ theirS,Q(k spin) (k den-k0) ⇧S,Q(k ) S,Q(k0) = S,QS,Q(k) wavevectorsThe interactionsQ. Note between that the both electrons orders couple are characterized their spin den- by V 0 der as a ‘quadrupolar’ order. ⇣ where⌘ the average is over a thermal ensemble defined by HMF k the d-wave form (cos kx cos ky) (specific results for the ac- It should be notedItThe should that interactions above be noted the charge-ordering between that above the the is electrons distinct charge-ordering couplesity via their is a susceptibility distinct spin den-k sity(q) which via a susceptibility is peaked near the(q antifer-) which is peaked0 q near the antifer-⇠q at a temperaturearbitrary functionsT. We of computeextending the over right the hand Brillouin side in zone, powers Xcuracy of this eigenvector appear below in Table I). So d-wave It should be noted that above the charge-ordering is distinct sity via a susceptibility (q) which is peakedand these near will the be antifer- determined by a functional minimization for the minimum eigenvalues S,Q and corresponding eigen- of Sof(k the) and free energy.Q(k), and Here, replace we only the consider inequality the cases by an where inequal- superconductivity is the primary instability. S. Sachdev and R. La Placa, arXiv:1303.2114vectors S,Q(k), and their structure is independent of the ity.Saturday, Toeither March 23, quadratic 13 S (k) or orderQ(k in) are the non-zero, order parameters, but not both; we in write the the We also examined the structure of the leading charge order- overall strength of the interaction 0. The order parameters resultcase in termsof co-existing of hermitian order, H functionalMF must also operators include terms on the like Bril- ing instability, and show the Q dependence of Q in Fig ??. characterizing the condensed phases will then be S,Q(k) louin zone✏↵ ck as+Q/2,↵c k+Q/2,, and we defer this case to future work. The minimum is at a wavevector along the/ diagonal with ⇠ S,Q(k)/ ⇧S,Q(k). Our principal numerical results below Fermi statistics requires S ( k) = S (k), while hermiticity are onQ the= Q(Qdependence, Q), with ofQ closeQ, and to on the the valuek dependence specified by the hot- requires= ⇤ (⇤k)k= ⇧Q(kk). For time-reversalk, k0 ⇧ symmetryk k0 to be p F 2 QS ( ) S ( ) S ( ) S ( 0) S ( ) (6)of S,spotQ(k) computation so obtained. We of diagonalized Ref. [17]. the There kernels are after also dis- notable station- preserved we need Q( k)M= Q(k), but we will not impose 2 kX,k0 cretizingary the points Brillouin in the zone eigenvalues to L points at with (Q,L0),up (0 to, 80,0) and with (⇡, ⇡), whose this as a constraint,p and so will allow forp the breaking of time- t = 1, t = 0.32, and t = 0.16 for a range of values of T, µ, 1 nature2 will become3 clearer after consideration of the eigen- +reversal.Q⇤ (k) ⇧Q(k) Q(k, k0) ⇧Q(k0)Q(k0) + ...and ⇠. The free energy of H obeysM the variational principle vectors below. kX,k0,Q q q Results. For the full range of parameters examined, we con- sistentlyWe found found that S thatwas we the minimal could characterize eigenvalue, and the the eigenvectors where the kernels areF FMF + H HMF MF (5)  h i correspondingS,Q(k)e eigenvectorciently byS (k an) was expansion well approximated in simple basis by functions, where the average is over a thermal ensemble defined by HMF the d-wave k form (cos kx cos ky) (specific results for the ac- 3 ( ) of the⇠ square lattice space group. For this, our specific at aS temperature(k, k0) =T.k We,k compute+ (k the rightk0) hand⇧S side(k)⇧ inS powers(k0) curacy of this eigenvector appear below in Table I). So d-wave M 0 V parameterization of the charge order Q(k) in Eq. (4) we im- of (k) and (k), and replace the inequality by an inequal- superconductivity is the primary instability. S Q 3 p portant, in which we identified Q and k as the center-of-mass ity. Tok quadratic, k0 = order in+ the orderk parameters,k0 ⇧ wek write⇧ k the We also examined the structure of the leading charge order- Q( ) k,k0 ( ) Q( 0) Q( ) (7) and relative momenta of the particle-hole pair. Then we can Mresult in terms of hermitianV functional operators on the Bril- ing instability, and show the Q dependence of Q in Fig ??. louin zone as q The minimum is at a wavevector along the diagonal with Q = (Q, Q), with Q close to the value specified by the hot- F = 2 ⇤ (k) ⇧ (k) (k, k0) ⇧ (k ) (k0) (6) S S MS S 0 S spot computation of Ref. [17]. There are also notable station- k,k X0 p p ary points in the eigenvalues at (Q, 0), (0, 0) and (⇡, ⇡), whose nature will become clearer after consideration of the eigen- + Q⇤ (k) ⇧Q(k) Q(k, k0) ⇧Q(k0)Q(k0) + ... M vectors below. k,k0,Q q q X We found that we could characterize the eigenvectors where the kernels are S,Q(k)eciently by an expansion in simple basis functions, 3 (k) of the square lattice space group. For this, our specific (k, k0) = k,k + (k k0) ⇧ (k)⇧ (k ) MS 0 V S S 0 parameterization of the charge order Q(k) in Eq. (4) we im- 3 p portant, in which we identified Q and k as the center-of-mass (k, k0) = , + (k k0) ⇧ (k )⇧ (k) (7) MQ k k0 V Q 0 Q and relative momenta of the particle-hole pair. Then we can q Charge ordering in metals with antiferromagnetic spin correlations Charge ordering in metals with antiferromagnetic spin correlations Rolando La Placa and Subir Sachdev Rolando La Placa and Subir Sachdev Department of Physics, Harvard University, Cambridge MA 02138 Department of Physics, Harvard University, Cambridge MA 02138 (Dated: February 28, 2013) (Dated: February 28, 2013) Metals with antiferromagnetic spin correlations have an instability to the superconductivity of spin-singlet Metals with antiferromagnetic spin correlations have an instabilityCooper to the pairs superconductivity with d symmetry of spin-singlet (for the Fermi surface of the cuprates). Metlitski et al. (Phys. Rev. B 82, 075128 Cooper pairs with d symmetry (for the Fermi surface of the cuprates).(2010)) Metlitski notedet al. that(Phys. in two Rev. dimensions, B 82, 075128 in the low energy continuum theory, this superconductivity is degenerate (2010)) noted that in two dimensions, in the low energy continuumCharge theory,with ordering a this charge superconductivity in density metals wave with ordering is antiferromagnetic degenerate which has a local spind symmetry correlations under rotations about the lattice points. We with a charge density wave ordering which has a local d symmetry underpresent rotations an unrestricted about the Hartree-Focklattice points. computation We on a simple lattice model, and find that the d symmetry is Rolando La Placa and Subir Sachdev present an unrestricted Hartree-Fock computation on a simple latticedominant model, and for find a range that of the smalld symmetry ordering iswavevectors, including those observed in recent experiments. We note implicationsDepartment for of Physics,the phase Harvard diagram University, of the cuprates. Cambridge MA 02138 dominant for a range of small ordering wavevectors, including those observed in recent experiments.(Dated: February We 28, note 2013) implications for the phase diagram of the cuprates. Metals with antiferromagneticPACS numbers: spin correlations have an instability to the superconductivity of spin-singlet Cooper pairs with d symmetry (for the Fermi surface of the cuprates). Metlitski et al. (Phys. Rev. B 82, 075128 PACS numbers: (2010)) noted that in two dimensions, in the low energy continuum theory, this superconductivity is degenerate withIntroduction a charge density. A wave remarkable ordering which series has a local of experimentsd symmetry under [1–12] rotations aboutfrom the the lattice time-reversal-symmetry points. We breaking “d-density wave” present an unrestricted Hartree-Fock computation on a simple lattice model, and find that the d symmetry is Introduction. A remarkable series of experiments [1–12] havedominantfrom shed for the a new range time-reversal-symmetry light of small on ordering the enigmatic wavevectors, breaking underdoped including “d those-density region observed wave” of in recentstate experiments. [22], and We the note confusion in the nomenclature is unfortu- have shed new light on the enigmatic underdoped region of theimplicationsstate cuprate [22], for high the and phase temperature the diagram confusion of the superconductors. cuprates. in the nomenclature These is exper- unfortu- nate. As will become clear from our presentation below, it iments consistently detect a bi-directional density wave with is useful to formulate the various charge ordering configura- the cuprate high temperature superconductors. These exper- PACSnate. numbers: As will become clear from our presentation below, it iments consistently detect a bi-directional density wave with a periodis useful of about to formulate 3-4 lattice the various spacings charge at low ordering hole densities. configurations in terms of their symmetries of rotation about the Cu a period of about 3-4 lattice spacings at low hole densities. Moreover,tions in this terms density of their wave symmetries order is co-incident of rotation with about regions the Cu lattice points. In our approach, the “d-density wave” state of Introductionof. the A remarkable phase diagram series of where experiments quantum [1–12] oscillationsfrom the [13] time-reversal-symmetry were Ref. [22] breaking actually “d has-density a p wave”symmetry about the lattice points Moreover, this density wave order is co-incident withhave regions shed new lightlattice on the points. enigmatic In our underdoped approach, region the “ ofd-densitystate [22], wave” and state the confusion of in the nomenclature is unfortu-x,y of the phase diagram where quantum oscillationsthe [13] cuprate were highobserved temperatureRef. [22] in YBa actually superconductors.2Cu3 hasOy, a stronglypx,y Thesesymmetry supporting exper- aboutnate. the the Ashypothesis lattice will become points clear[17], from and our so we presentation will identify below, it byit the px,y label. observed in YBa2Cu3Oy, strongly supporting the hypothesisiments consistently[7,[17], 14–16] detect and a that so bi-directional we the will density identify density wave itwave by plays withthe apx central,y islabel. useful role to in formulate the the variousHere we charge will ordering present configura- an unrestricted Hartree-Fock-BCS on [7, 14–16] that the density wave plays a central rolea period in the of aboutformation 3-4Here lattice of we the spacings will Fermi present at pockets low an hole unrestricted responsible densities. Hartree-Fock-BCS fortions the in termsquantum of their on symmetriesa simple lattice of rotation model about of a the metal Cu with antiferromagnetic spin formation of the Fermi pockets responsible for theMoreover, quantum thisoscillations. densitya simple wave lattice Some order is ofmodel co-incident the experiments of a with metal regions with [3, 5, antiferromagnetic 6,lattice 8, 11] points. are also In our spin approach,corrections. the “d-density We will wave” make state no of direct reference to the contin- oscillations. Some of the experiments [3, 5, 6, 8, 11]of the are phase also sensitive diagramcorrections. where to the quantum electronic We will oscillations make microstructure no [13] direct were reference of theRef. density [22] to actually the wave: contin- has a puumx,y symmetry limit or about to the the lattice hot spots, points and will retain full momen- observed in YBa Cu O , strongly supporting the hypothesis [17], and so we will identify it by the p label. sensitive to the electronic microstructure of the density wave: these2uum indicate3 limity that or to there the is hot negligible spots, and modulation will retain of the fullcharge momentum dependencex,y of all variables across the Brillouin zone. [7, 14–16] that the density wave plays a central role in the Here we will present an unrestricted Hartree-Fock-BCS on these indicate that there is negligible modulation offormation the charge ofdensity thetum Fermi on dependence pockets the Cu responsible sites. of all Instead, for variables the quantumit appears across toa the simple be Brillouin primarily lattice model zone. a ofWe a metal will with find, antiferromagnetic as expected, spin that the dominant instability of density on the Cu sites. Instead, it appears to be primarilyoscillations. a SomebondWe ofdensity the will experiments find, wave, as with expected, [3, 5, modulations 6, 8, 11] that are the also in dominant spin-singletcorrections. instability observ- We will of makethe no metal direct is reference always to towards the contin- superconductivity with d-wave bond density wave, with modulations in spin-singletsensitive observ- to theables electronicthe associated metal microstructure is always with a Cu-Cu of towards the density link, superconductivity wave: such as theuum average limit with or elec- tod-wave the hotpairing. spots, and The will leading retain full sub-dominant momen- instability is towards a ables associated with a Cu-Cu link, such as the averagethese indicate electron thatpairing. there kinetic is negligible energy. The leading modulation sub-dominant of the charge instabilitytum dependence is towards of all a spin-singlet variables across charge the Brillouin ordering zone. closely related to that proposed 2 tron kinetic energy. density on the Cuspin-singlet sites. Instead, charge it appears ordering to be primarily closely a relatedWe to will that find, proposed as expected,in Ref. that 17. the For dominant a range instability of small of wavevectors, including those bond density wave,This with paper modulations will examine in spin-singlet a model observ- for thethe charge metal ordering is always towards superconductivity with d-wave in Ref. 17. For a range of small wavevectors,romagnet including wavevector; those weobserved assume the in simple the experiments form [1–6, 8–11].while the the charge polarizabilities order has are This paper will examine a model for the chargeables ordering associatedproposed with a Cu-Cu in Ref. link, [17]. such They as the studied average a elec- two-dimensionalpairing. The metal leading sub-dominant instability is towards a observed in the experiments [1–6, 8–11]. the charge order has a predominant d symmetry of rotations about the Cu lattice proposed in Ref. [17]. They studied a two-dimensionaltron kinetic metal energy.with antiferromagnetic spin correlations, usingspin-singlet a continuum charge ordering closely0 related to that proposed q = 1 2 f ("(k)) a predominant d symmetry of rotations about( ) the Cu2 lattice sites. The predominant d symmetry(3) implies that the charge with antiferromagnetic spin correlations, using a continuumThis paperlimit will examine which focused a model on for particular the charge ‘hot ordering spots’ onin Ref. the Fermi 17. For⇠ sur- a+ range2(2 ofcos( smallqx wavevectors,Kx) cos(q includingy Ky)) those ⇧S (k) = K 2"(k) limit which focused on particular ‘hot spots’ on theproposed Fermi sur- in Ref.face [17].sites. [18]. They The In studied this predominant limit, a two-dimensional Ref.d [17]symmetry showed metal implies thatobserved theory thatX in the had the charge experiments an order [1–6, is primary 8–11]. the located charge order on the has Cu-Cu bonds, and there is lit- a predominant⇠ d symmetrytle modulation of rotations about of the the charge Cu lattice density on the Cu sites. f ("(k + Q/2)) f ("(k Q/2)) face [18]. In this limit, Ref. [17] showed that theorywith had antiferromagnetic an emergentorder spin is pseudospin primary correlations, located symmetry, using on a the continuum and Cu-Cu consequentlywhere bonds, andis the the there metal antiferromagnetic is lit- correlation length. The sum ⇧Q(k) = (8) limit which focusedtle on modulation particular ‘hot of spots’ the charge on the density Fermi sur- onover thesites. Cu wavevectors sites. The predominant extends overd MethodsymmetryK = .( implies⇡ We, ⇡(1 will that)) examine the, charge(⇡(1 the following Hamiltonian" of(k Q/2) "(k + Q/2) emergent pseudospin symmetry, and consequently the metal had two degenerate instabilities: to superconductivity with ± ± face [18]. In this limit, Ref. [17] showed that theory had an ), ⇡order), and is primary we will located use bothelectrons on the the commensurate Cu-Cu on bonds, a square and case lattice there = is with0 lit- or annihilation operators c , had two degenerate instabilities: to superconductivity with d-waveMethod pairing,. We and will to a examine density wave the following which had Hamiltonian a local d- of with f the Fermi function.k,↵ Note that the kernels in Eq. (7) emergent pseudospin symmetry, and consequently the metal thetle incommensurate modulation of the case charge = density1/4. We on the will Cu only sites. need mod- d-wave pairing, and to a density wave which had a local d- waveelectrons symmetry on of a square rotations lattice about with each annihilation lattice point. operators The den-ck,↵, where k is a crystal momentum and ↵have= , anis identical a spin label: form: this is a consequence of the spin- had two degenerate instabilities: to superconductivity with erateMethod values of. We⇠ for will our examine main conclusions, the following and Hamiltonian the e↵ects ofwe " # sitywhere wavek instabilityis a crystal had momentum a small incommensurate and ↵ = ,electronsis awavevector spin on a label: square lattice with annihilation operators c ,↵, spin interaction in H; density-density interactions yield dis- wave symmetry of rotations about each lattice point.d-wave The den- pairing, and to a density wave which had a local d- discuss" # are present even at ⇠ = 1. We do not include ank1 ex- sity wave instability had a small incommensurate wavevectorwave symmetryQ of= rotations(Q, Q about), where eachQ latticeis determined point. The den- by thewhereHartree-Fock positionsk is a of crystal the momentumcomputationH = and ↵ ="on(k, ) latticeisc† a spinc label:model tinct(q) S~ terms( q) inS~ the(q). pairing(1) and charge channels. Note further ± 1 plicit density-density interaction between" # thek, electrons,↵ k,↵ 2V and · Q = , sity wave instabilityhot spots. had a= Lattice small" incommensurate corrections† break wavevector the pseudospin~ ~ symme-. k q that for dispersions with "(k + Q) = "(k) the two kernels (Q Q), where Q is determined by the positions of the H (k) ck,↵ck,↵ (qassume) S ( q its) mainS (q) e↵ect(1) is to renormalize1 X the dispersion. X ± Q = (Q, Q), where Q is determined by the positions 2V of the H =· "(k) c† c (q) S~( q) S~(q). (1) equal each other: Ref. [17] pointed out that such a relation- hot spots. Lattice corrections break the pseudospin symme-± try, and the superconductingk instabilityq becomesWe will stronger perform than ank,↵ unrestrictedk,↵ 2V Hartree-Fock-BCS · analy- hot spots. Lattice correctionsX break the pseudospin symme-X k q ship holds close to~ the hot spots of a generic Fermi surface try, and the superconducting instability becomes stronger than the charge order in the full theory. Metznersis and of H collaborators. WeX need the bestwhere variational thereX estimate is an implicit for the sum mean- over spin indices. Here S (q) = try, and the superconducting instability becomes stronger than for certain Q, and this was a key ingredient in the emergent the charge order in the full theory. Metzner and collaborators [19,where 20] studied there is this an implicit charge ordering sum over using spinfield indices. functional Hamiltonian Here renor-S~(q) = k c† ~ ↵ c is the electron~ spin density (~ are the Pauli the charge order in the full theory. Metzner and collaborators where there is an implicit sumk over+q,↵ spin indices.k, Here S (q) = pseudospin symmetry; we will see the same Q emerge in our [19, 20] studied this charge ordering using functional renor- malizationk c† group~ ↵ methods,c is the and electron called spin it an density‘incommensuratec† (~ are~ thec Pauliis thematrices), electron spin and densityV is ( the~ are system the Pauli volume. For the electronic dis- [19, 20] studied this chargek+q,↵ orderingk, using functional renor- k k+q,↵ ↵ k, P computation below. malization group methods, and called it an ‘incommensuratemalization groupnematic’matrices), methods,. They and and alsocalledV is found it the an ‘incommensurate system similar volume. instabilities Formatrices),H withtheMF electronic= wavevec- and V"(isk dis-) theck† , system↵persion,ck,↵ + volume.S we(k) ✏ assume For↵ ck the,↵c electronic ak tight-binding+ H.c. dis- model which provides a P From Eq. (6) we see that the linear instability of the metal nematic’ P = , , , persion, wek assume" a tight-bindinggood fit to photoemissionmodel which provides data a nematic’. They also found similar instabilities with wavevec-. Theytors alsopersion,Q found(Q similar we0) assume instabilities(0 Q), a corresponding tight-binding with wavevec- model to those which seenX provides in the a occurs via condensation in the eigenmodes of the operators tors Q = (Q, 0), (0, Q), corresponding to those seentors Q in= the(Qexperiments, 0),good(0, Q), fit corresponding to [1–6, photoemission 8–11]. to And those data recently seen in the Efetovgoodet al. fit to[21] photoemission for- data + (k) c† c , (4) S,Q(k, k0) with the lowest eigenvalues. We have chosen the experiments [1–6, 8–11]. And recently Efetov et al. [21] for- Q k"+(Qk/)2,↵=k Q2/2t,↵ cos(k ) + cos(k ) M4t cos(k ) cos(k ) experiments [1–6, 8–11]. And recently Efetov et al. [21] for- mulated these instabilities in terms of a non-linear modelQ 1 x y specific2 formsx of they kernels in Eq. (7) so that we need only mulated these instabilities in terms of a non-linear model "(k) = 2Xt1 cos(kx) + cos(ky) 4t2 cos(# kx) cos(ky) mulated these instabilities in terms of a non-linear model on pseudospin"(k) = space,2t1 andcos( emphasizedkx) + cos(ky their) 4 importancet2 cos(kx) cos( fork they) solve the following eigenvalue problem on pseudospin space, and emphasized their importance for the 2t3 ⇣cos(2kx) + cos(2k⌘y) µ. (2) optimizedExpand overF to the2t3 second spin-singlet⇣cos(2kx) order+ cos(2 superconductingk⌘ iny) µ.S(k) pairing and func-(2)Q(k), on pseudospin space, and emphasized their importancephysics for of the thephysics pseudo-gap; of the they pseudo-gap; labelled the above they labelled charge or- the above charge or- 2t3 ⇣cos(2kx) + cos(2k⌘y) tionµ. (k), and the(2) density wave orders (k) at the der as a ‘quadrupolar’ order. and obtainS lowest⇣ eigenvalues⌘ S and⇣ Q Q and 3⌘ physics of the pseudo-gap; they labelled the aboveder charge as a ‘quadrupolar’ or- order. The interactions between the electrons couple⇧ theirS,Q(k spin) (k den-k0) ⇧S,Q(k ) S,Q(k0) = S,QS,Q(k) wavevectorsThe interactionsQ. Note between that the both electrons orders couple are characterized their spin den- by V 0 der as a ‘quadrupolar’ order. ⇣ ⌘corresponding eigenvectors S(k) and Q(k). k It should be notedItThe should that interactions above be noted the charge-ordering between that above the the is electrons distinct charge-orderingarbitrary couplesity via their functions is a susceptibility distinct spin den- of k sityextending(q) which via a susceptibility is over peaked the near Brillouin the(q antifer-) zone, which is peakedX0 q near the antifer- q It should be noted that above the charge-ordering is distinct sity via a susceptibility (q) which is peakedand these near will the be antifer- determined by a functional minimization for the minimum eigenvalues S,Q and corresponding eigen- of the free energy. Here, we only consider the cases where S. Sachdev and R. La Placa, arXiv:1303.2114vectors S,Q(k), and their structure is independent of the Saturday,either March 23, 13 S (k) or Q(k) are non-zero, but not both; in the overall strength of the interaction 0. The order parameters case of co-existing order, HMF must also include terms like characterizing the condensed phases will then be S,Q(k) ✏↵ ck+Q/2,↵c k+Q/2,, and we defer this case to future work. / ⇠ S,Q(k)/ ⇧S,Q(k). Our principal numerical results below Fermi statistics requires S ( k) = S (k), while hermiticity are on the Q dependence of Q, and on the k dependence requires ⇤ (k) = Q(k). For time-reversal symmetry to be p Q of S,Q(k) so obtained. We diagonalized the kernels after dis- preserved we need ( k) = (k), but we will not impose 2 Q Q cretizing the Brillouin zone to L points with L up to 80, with this as a constraint, and so will allow for the breaking of time- t = 1, t = 0.32, and t = 0.16 for a range of values of T, µ, 1 2 3 reversal. and ⇠. The free energy of H obeys the variational principle Results. For the full range of parameters examined, we consistently found that S was the minimal eigenvalue, and the F FMF + H HMF MF (5)  h i corresponding eigenvector S (k) was well approximated by where the average is over a thermal ensemble defined by H the d-wave form (cos kx cos ky) (specific results for the ac- MF ⇠ at a temperature T. We compute the right hand side in powers curacy of this eigenvector appear below in Table I). So d-wave of S (k) and Q(k), and replace the inequality by an inequal- superconductivity is the primary instability. ity. To quadratic order in the order parameters, we write the We also examined the structure of the leading charge order- result in terms of hermitian functional operators on the Bril- ing instability, and show the Q dependence of Q in Fig ??. louin zone as The minimum is at a wavevector along the diagonal with Q = (Q, Q), with Q close to the value specified by the hot- F = 2 ⇤ (k) ⇧ (k) (k, k0) ⇧ (k ) (k0) (6) S S MS S 0 S spot computation of Ref. [17]. There are also notable station- k,k X0 p p ary points in the eigenvalues at (Q, 0), (0, 0) and (⇡, ⇡), whose nature will become clearer after consideration of the eigen- + Q⇤ (k) ⇧Q(k) Q(k, k0) ⇧Q(k0)Q(k0) + ... M vectors below. k,k0,Q q q X We found that we could characterize the eigenvectors where the kernels are S,Q(k)eciently by an expansion in simple basis functions, 3 (k) of the square lattice space group. For this, our specific (k, k0) = k,k + (k k0) ⇧ (k)⇧ (k ) MS 0 V S S 0 parameterization of the charge order Q(k) in Eq. (4) we im- 3 p portant, in which we identified Q and k as the center-of-mass (k, k0) = , + (k k0) ⇧ (k )⇧ (k) (7) MQ k k0 V Q 0 Q and relative momenta of the particle-hole pair. Then we can q Hartree-Fock computation on lattice model

Charge-ordering eigenvalue Q. S. Sachdev and R. La Placa, arXiv:1303.2114 Saturday, March 23, 13 3

(k) Q = Q = Q = Q = S (k) tov, H. Meier, W. Metzner, and C. Pepin´ useful discus- (1.15,1.15) (1.15, 0) (0,0) (⇡, ⇡) sions. This research was supported by the NSF under Grant s 1 0 -0.231 0 0 0 DMR-1103860, and by the U.S. Army Research Oce Award s0 cos k + cos k 0 0.044 0 0 0 x y W911NF-12-1-0227. s00 cos(2kx) + cos(2ky) 0 -0.046 0 0 0 d cos k cos k 0.993 0.963 0.997 0 0.997 x y d0 cos(2k ) cos(2k ) - 0.069 -0.067 -0.057 0 -0.056 x y d00 2 sin kx sin ky 0 0 0 0 0 px p2 sin kx 0 0 0 0.706 0 [1] J. E. Ho↵man et al., Science 295, 466 (2002). py p2 sin ky 0 0 0 -0.706 0 [2] M. Vershinin, S. Misra, S. Ono, Y. Abe, Yoichi Ando, and g (cos kx cos ky) -0.009 0 0 0 0 A. Yazdani, Science , 1995 (2004). p8 sin k sin k 303 ⇥ x y [3] Y. Kohsaka, et al., Science 315, 1380 (2007). [4] W. D. Wise et al. Nature Physics 4, 696 (2008). TABLE I: Values of cQ, in the expansion in Eq. (9) for various val- [5] M. J. Lawler et al., Nature 466, 374 (2010). ues Q and . The last column shows the coecients in the corre- [6] A. Mesaros et al., Science 333, 426 (2011). sponding expansion for (k). We used µ = 1.2, ⇠ = 2, T = 0.06, [7] T. Wu et al., Nature 477, 191 (2011). S and L = 80. [8] Y. Kohsaka, et al., Nature Physics 8, 534 (2012). [9] G. Ghiringhelli et al., Science 337, 821 (2012). [10] J. Chang et al., Nature Phys. 8, 871 (2012). writeHartree-Fock computation on lattice model [11] A. J. Achkar et al., Phys. Rev. Lett. 109, 167001 (2012). [12] D. LeBoeuf, S. Kramer,¨ W. N. Hardy, Ruixing Liang, D. A. Bonn, and C. Proust, Nature Physics , 79 (2013). Q(k) = cQ, (k) (9) 9 [13] N. Doiron-Leyraud et al., Nature 447, 565 (2007). 3 X [14] N. Harrison and S. E. Sebastian, Phys. Rev. Lett. 106, 226402 where labels various orthonormal basis functions, and cQ, (2011). = = = = are numerical(k) coecientsQ that weQ determine.Q ThusQ we haveS (k) tov,[15] S. H. E. Meier, Sebastian, W. N. Metzner, Harrison and and G. C. G. Pepin´ Lonzarich, useful Rep. discus- Prog. (1.15,1.15) (1.15, 0) (0,0) (⇡, ⇡) = sions.Phys. This75 research, 102501 (2012). was supported by the NSF under Grant thes s basis1 function s(k0) 1,-0.231 the extended0 s0 function0 = + = DMR-1103860,[16] B. Vignolle, and D. by Vignolles, the U.S. Army M.-H. Research Julien, and O C.ce Award Proust, ss00 (k)coscosk +kcosx kcos ky, the0 d function0.044 d(k)0 cos0kx cos0 ky x y W911NF-12-1-0227.C. R. Physique 14, 39 (2013). ands00 cos(2 so on,kx) as+ showncos(2ky in) Table0 I. Depending-0.046 upon0 the0 symmetry0 [17] M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075128 ofd Q (incos particular,kx cos ky the little0.993 group0.963 of the wavevector0.997 0 Q0.997) and (2010). d0 cos(2k ) cos(2k ) - 0.069 -0.067 -0.057 0 -0.056 of the eigenvector,x somey of the cQ, may be exactly zero. But [18] Ar. Abanov and A. V. Chubukov, Phys. Rev. Lett. 84, 5608 ford00 a generic2 sin kx sinQ,ky only time-reversal0 0 constrains0 the0 values0 of (2000). px p2 sin kx 0 0 0 0.706 0 [19] T. Holder and W. Metzner, Phys. Rev. B 85, 165130 (2012). cQ,, and we are allowed to have an admixture of many basis [1] J. E. Ho↵man et al., Science 295, 466 (2002). py p2 sin ky 0 0 0 -0.706 0 [20] C. Husemann and W. Metzner, Phys. Rev. B 86, 085113 (2012). functions. Nevertheless, we will see that only a small number [2] M. Vershinin, S. Misra, S. Ono, Y. Abe, Yoichi Ando, and g (cos kx cos ky) -0.009 0 0 0 0 [21] K. B. Efetov, H. Meier, and C. Pepin,´ arXiv:1210.3276. of basis functions have appreciable coecients, and so Eq. (9) A. Yazdani, Science , 1995 (2004). p8 sin k sin k [22] S. Chakravarty, R. B.303 Laughlin, D. K. Morr, and C. Nayak, x y [3] Y. Kohsaka, et al., Science , 1380 (2007). represents⇥ a useful expansion. Phys. Rev. B 63, 094503 (2001).315 AcknowledgmentsCharge-ordering. We thank for eigenvector A. Chubukov, K. Efe- [4] W. D. Wise et al. Nature Physics 4, 696 (2008). [5] M. J. Lawler et al., Nature 466, 374 (2010). TABLE I: Values of cQ, in the expansion inS. Eq.Sachdev (9) and for R. various La Placa, arXiv:1303.2114 val- Saturday,ues March 23,Q 13 and . The last column shows the coecients in the corre- [6] A. Mesaros et al., Science 333, 426 (2011). sponding expansion for (k). We used µ = 1.2, ⇠ = 2, T = 0.06, [7] T. Wu et al., Nature 477, 191 (2011). S and L = 80. [8] Y. Kohsaka, et al., Nature Physics 8, 534 (2012). [9] G. Ghiringhelli et al., Science 337, 821 (2012). [10] J. Chang et al., Nature Phys. 8, 871 (2012). write [11] A. J. Achkar et al., Phys. Rev. Lett. 109, 167001 (2012). [12] D. LeBoeuf, S. Kramer,¨ W. N. Hardy, Ruixing Liang, D. A. Bonn, and C. Proust, Nature Physics , 79 (2013). Q(k) = cQ, (k) (9) 9 [13] N. Doiron-Leyraud et al., Nature 447, 565 (2007). X [14] N. Harrison and S. E. Sebastian, Phys. Rev. Lett. 106, 226402 where labels various orthonormal basis functions, and cQ, (2011). are numerical coecients that we determine. Thus we have [15] S. E. Sebastian, N. Harrison and G. G. Lonzarich, Rep. Prog. the s basis function (k) = 1, the extended s function Phys. 75, 102501 (2012). s [16] B. Vignolle, D. Vignolles, M.-H. Julien, and C. Proust, (k) = cos k + cos k , the d function (k) = cos k cos k s0 x y d x y C. R. Physique 14, 39 (2013). and so on, as shown in Table I. Depending upon the symmetry [17] M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075128 of Q (in particular, the little group of the wavevector Q) and (2010). of the eigenvector, some of the cQ, may be exactly zero. But [18] Ar. Abanov and A. V. Chubukov, Phys. Rev. Lett. 84, 5608 for a generic Q, only time-reversal constrains the values of (2000). cQ,, and we are allowed to have an admixture of many basis [19] T. Holder and W. Metzner, Phys. Rev. B 85, 165130 (2012). functions. Nevertheless, we will see that only a small number [20] C. Husemann and W. Metzner, Phys. Rev. B 86, 085113 (2012). [21] K. B. Efetov, H. Meier, and C. Pepin,´ arXiv:1210.3276. of basis functions have appreciable coecients, and so Eq. (9) [22] S. Chakravarty, R. B. Laughlin, D. K. Morr, and C. Nayak, represents a useful expansion. Phys. Rev. B 63, 094503 (2001). Acknowledgments. We thank for A. Chubukov, K. Efe- 3

Incommen- surate d-wave bond order

Charge-ordering eigenvalue Q. S. Sachdev and R. La Placa, arXiv:1303.2114 Saturday, March 23, 13 3

Ising- nematic order

Charge-ordering eigenvalue Q. S. Sachdev and R. La Placa, arXiv:1303.2114 Saturday, March 23, 13 3

“d-density wave” orbital- currents

Charge-ordering eigenvalue Q. S. Sachdev and R. La Placa, arXiv:1303.2114 Saturday, March 23, 13 3

Incommensurate d(+s) wave bond order

Charge-ordering eigenvalue Q. S. Sachdev and R. La Placa, arXiv:1303.2114 Saturday, March 23, 13 energy with respect to the site charge density N(1 Evidence bond order is along (1,0), (0,1) directions2 − bi )= ni and the complex bond pairing amplitude " #αβ 2 in low T superconducting phase NQ = c† c† /(b b )(whereb is the hole density ij "J iα jβ# i j i at site i and denotes the Sp(2N)-invariant antisym- metric tensor),J while maintaining certain local and global constraints. There have been a number of related earlier p 4 mean-ﬁeld studies17,buttheyhaveall(withtheexcep-

2 PHYSICAL REVIEW B 77, 094504 ͑2008͒

Superconducting d-wave stripes in cuprates: Valence bond order coexisting with nodal quasiparticles

Matthias Vojta and Oliver Rösch Institut für Theoretische Physik, Universität zu Köln, Zülpicher Straße 77, 50937 Köln, Germany ͑Received 8 January 2008; revised manuscript received 10 January 2008; published 6 March 2008͒ We point out that unidirectional bond-centered charge-density-wave states in cuprates involve electronic order in both s- and d-wave channels, with nonlocal Coulomb repulsion suppressing the s-wave component. The resulting bond-charge-density wave, coexisting with superconductivity, is compatible with recent photoemission and tunneling data and as well as neutron-scattering measurements, once long-range order is destroyed by slow ﬂuctuations or glassy disorder. In particular, the real-space structure of d-wave stripes is consistent with the scanning-tunneling-microscopy measurements on both underdoped Bi2Sr2CaCu2O8+␦ and Ca2−xNaxCuO2Cl2 of Kohsaka et al. ͓Science 315, 1380 ͑2007͔͒.

DOI: 10.1103/PhysRevB.77.094504 PACS number͑s͒: 74.20.Mn, 74.25.Dw, 74.72.Ϫh

I. INTRODUCTION underdoped cuprates.16–18 The concept of fluctuating stripes, i.e., almost charge-ordered states, has been discussed early A remarkable aspect of the copper-oxide high-Tc super- 1,3,16,18,19 conductors is that various ordering phenomena apparently on. This concept, appropriate for compounds with- compete, including commensurate and incommensurate out static long-range order, assumes the existence of a nearby magnetisms, superconducting pairing, and charge-density- stripe-ordered state, with physical observables being influ- wave formation. ͑More exotic states have also been pro- enced by the low-lying collective modes associated with posed, but not verified experimentally beyond doubt. While charge-ordering instability. Following this idea, we have re- ͒ 20 commensurate magnetism and superconductivity are com- cently calculated the spin excitation spectrum of slowly mon phases in essentially all families of cuprates, the role of fluctuating ͑or disordered͒ stripes. We were able to show that other instabilities for the global features of the phase diagram fluctuating stripes give rise to an “hour-glass” magnetic is less clear. spectrum, very similar to that observed in neutron-scattering A particularly interesting role is taken by charge-density experiments both on La2−xBaxCuO4 ͑Ref. 21͒ and Evidence bond22 order,23 is along (1,0), (0,1) directions waves. Such states break the discrete lattice translation sym- YBa2Cu3O6+␦. metry, with examples being stripe, checkerboard, and The focusin low of thisT superconducting paper is on the electronicphase structure of valence-bond order. In the compounds, La2−xBaxCuO4 and stripe states. WePHYSICAL introduce REVIEW the B 77 concept, 094504 ͑2008 of͒ “d-wave stripes”: La2−xSrxCuO4 ͑with Nd or Eu codoping͒ evidence for stripe- Here, the modulation of charge densities has primarily a like spin and charge modulations with static long-range orderSuperconductingd-wave formd-wave factor, stripes i.e., in lives cuprates: more Valence on the bond bonds order than coexisting on the were detected,1–4 in particular, near one-eighth doping. ͑This sites of the squarewith lattice, nodal leaving quasiparticles nodal QP unaffected. We is supported, e.g., by strong phonon anomalies seen in illustrate that a pictureMatthias Vojta of such and Oliver bond-centered Rösch charge order, neutron-scattering experiments.5͒ While in other cuprate Institutcoexisting für Theoretische with Physik, superconductivity Universität zu Köln, Zülpicher͑this Straße state 77, 50937 may Köln, be Germany dubbed ͑Received 8 January 2008; revised manuscript received 10 January 2008; published 6 March 2008͒ families similar long-range order has not been found, signa- “valence-bond supersolid”͒, is consistent with various fea- We point out that unidirectional bond-centered charge-density-wave states in cuprates involve electronic tures of short-range charge order, likely pinned by impurities,order in bothturess- and seend-wave in channels, both with ARPES nonlocal and Coulomb STM repulsion measurements. suppressing the s-wave In component. par- have been observed in scanning-tunneling microscopyThe resultingticular, bond-charge-density the real-space wave, coexisting pattern with superconductivity, of d-wave is stripes compatible͑ withFig. recent1͒ photo-is emission and tunneling data and as well as neutron-scattering measurements, once long-range order is de- ͑STM͒ on underdoped Bi2Sr2CaCu2O8+␦ ͑Refs. 6–9͒ andstroyed bystrikingly slow fluctuations similar or glassy to disorder. the STM In particular, results the real-space of Ref. structure9, obtained of d-wave stripes on is 9,10 Ca2−xNaxCuO2Cl2. The low-energy electronic structure inconsistentunderdoped with the scanning-tunneling-microscopy Bi2Sr2CaCu2O measurements8+␦ and Ca on both2−xNa underdopedxCuO Bi2Cl2Sr22CaCu. 2O8+␦ and the presence of charge order turns out to be remarkable: InCa2−xNaxCuO2Cl2 of Kohsaka et al. ͓Science 315, 1380 ͑2007͔͒. DOI: 10.1103/PhysRevB.77.094504 PACS number͑s͒: 74.20.Mn, 74.25.Dw, 74.72.Ϫh La15/8Ba1/8CuO4, angle-resolved photoemission spectroscopy ͑ARPES͒ indicated a quasiparticle ͑QP͒ gap with d-wavelike form, i.e., charge order coexists with gapless nodal QP in I. INTRODUCTION underdoped cuprates.16–18 The concept of fluctuating stripes, ͑ ͒ i.e., almost charge-ordered states, has been discussed early A remarkable aspect of the copper-oxide high-Tc super- 1,3,16,18,19 the ͑1,1͒ direction ͓while antinodal QP near ͑conductors0,␲͒ are is that various ordering phenomena apparently on. This concept, appropriate for compounds with- 11 compete, including commensurate and incommensurate out static long-range order, assumes the existence of a nearby gapped͔. STM data on both underdoped Bi2Sr2CaCu2O8+␦ magnetisms, superconducting pairing, and charge-density- stripe-ordered state, with physical observables being influ- and Ca2−xNaxCuO2Cl2 and show QP interferencewave arising formation. ͑More exotic states have also been pro- enced by the low-lying collective modes associated with posed, but not verified experimentally beyond doubt.͒ While charge-ordering instability. Following this idea, we have re- from coherent low-energy states near the nodes, whereas cently calculated20 the spin excitation spectrum of slowly commensurateSaturday, March 23, magnetism 13 and superconductivity are com- electronic states at higher energy and wave vectormon close phases to in essentiallyFIG. all families 1. Schematic of cuprates, the real-space role of fluctuating structure͑or disordered of a stripe͒ stripes. state We were with able to show that other instabilities for the global features of the phase diagram fluctuating stripes give rise to an “hour-glass” magnetic the antinode are dominated by the real-space modulation of primarily d-wave character andspectrum, a 4 veryϫ1 similar unit to that cell, observed i.e., inQ neutron-scattering 7,9,12 is less clear. the short-range charge order. This dichotomy in momen-A particularly interesting=͑Ϯ␲ role/2,0 is͒ taken. Cu by lattice charge-density sites areexperiments shown as both circles, on with La2−xBa theirxCuO size4 ͑Ref. 21͒ and waves. Such states break the discrete lattice translation sym- YBa Cu O .22,23 tum space has also been found in ARPES experiments in representing the on-site hole densities.2 3 The6+␦ line strengths indicate 13 14 metry, with examples being stripe, checkerboard, and The focus of this paper is on the electronic structure of La2−xSrxCuO4, Bi2Sr2CaCu2O8+␦, and Ca2−xNavalence-bondxCuO2Cl2 order. Inthe the amplitude compounds, of La2 bond−xBaxCuO variables4 and suchstripe asstates. kinetic We introduce and magnetic the concept ener- of “d-wave stripes”: ͑Ref. 15͒ where well-defined nodal and ill-definedLa antinodal2−xSrxCuO4 ͑with Nd or Eu codoping͒ evidence for stripe- Here, the modulation of charge densities has primarily a like spin and charge modulationsgies. The with modulation static long-range in the order sited charge-wave form densities factor, i.e., is lives small, more whereas on the bonds than on the QP are frequently observed. were detected,1–4 in particular,the one near in the one-eighth bond doping. densities͑This is largesites of and the of squared-wave lattice, type leaving͑Ref. nodal24 QP͒. unaffected. We These results suggest that momentum-space differentia-is supported, e.g., byNote strong the phonon similarity anomalies of the seen bond in modulationillustrate that a with picture the of such STM bond-centered data of charge order, neutron-scattering experiments.5͒ While in other cuprate coexisting with superconductivity ͑this state may be dubbed tion and tendencies toward charge ordering are commonfamilies similar to long-rangeRef. order9. has not been found, signa- “valence-bond supersolid”͒, is consistent with various features of short-range charge order, likely pinned by impurities, tures seen in both ARPES and STM measurements. In par- have been observed in scanning-tunneling microscopy ticular, the real-space pattern of d-wave stripes ͑Fig. 1͒ is 1098-0121/2008/77͑9͒/094504͑5͒ ͑STM͒ on094504-1 underdoped Bi2Sr2CaCu2O8+␦ ͑Refs. 6–9͒ and©2008strikingly The American similar to the Physical STM results Society of Ref. 9, obtained on 9,10 Ca2−xNaxCuO2Cl2. The low-energy electronic structure in underdoped Bi2Sr2CaCu2O8+␦ and Ca2−xNaxCuO2Cl2. the presence of charge order turns out to be remarkable: In La15/8Ba1/8CuO4, angle-resolved photoemission spectroscopy ͑ARPES͒ indicated a quasiparticle ͑QP͒ gap with d-wavelike form, i.e., charge order coexists with gapless ͑nodal͒ QP in the ͑1,1͒ direction ͓while antinodal QP near ͑0,␲͒ are 11 gapped͔. STM data on both underdoped Bi2Sr2CaCu2O8+␦ and Ca2−xNaxCuO2Cl2 and show QP interference arising from coherent low-energy states near the nodes, whereas electronic states at higher energy and wave vector close to FIG. 1. Schematic real-space structure of a stripe state with the antinode are dominated by the real-space modulation of primarily d-wave character and a 4ϫ1 unit cell, i.e., Q 7,9,12 the short-range charge order. This dichotomy in momen- =͑Ϯ␲/2,0͒. Cu lattice sites are shown as circles, with their size tum space has also been found in ARPES experiments in representing the on-site hole densities. The line strengths indicate 13 14 La2−xSrxCuO4, Bi2Sr2CaCu2O8+␦, and Ca2−xNaxCuO2Cl2 the amplitude of bond variables such as kinetic and magnetic ener- ͑Ref. 15͒ where well-defined nodal and ill-defined antinodal gies. The modulation in the site charge densities is small, whereas QP are frequently observed. the one in the bond densities is large and of d-wave type ͑Ref. 24͒. These results suggest that momentum-space differentia- Note the similarity of the bond modulation with the STM data of tion and tendencies toward charge ordering are common to Ref. 9.

(k) Q = Q = Q = Q = S (k) with Q = ( Q, 0), (0, Q). (1.15,1.15) (1.15, 0) (0,0) (⇡, ⇡) ± ± s 1 0 -0.231 0 0 0 Acknowledgments. We thank for A. Chubukov, K. Efe- 4 s0 cos kx + cos ky 0 0.044 0 0 0 tov, H. Meier, W. Metzner, and C. Pepin´ useful discus- s00 cos(2kx) + cos(2ky) 0 -0.046 0 0 0 sions. This research was supported by the NSF under Grant d cos k cosFinally,k note0.993 that Fig.0.963 2 also0.997 has a broad0 0.997 local minimum found that the spin-singlet, d character persists to the exper- x y DMR-1103860, and by the U.S. Army Research Oce Award d0 cos(2k ) nearcos(2Qk )= (-⇡ 0.069, ⇡). Here-0.067 (k)-0.057 is found to0 have-0.056 the odd-parity imentally observed wavevectors. This leads to our proposal x y Q W911NF-12-1-0227. d00 2 sin kx sinpx,ykyform in Table0 I. Condensation0 0 of0 this mode0 will break for charge ordering in the underdoped cuprates, summarized px p2 sintime-reversalkx 0 symmetry,0 and lead0 to0.706 the state0 with sponta- in Eq. (9). py p2 sinneousky orbital currents0 proposed0 by0 Chakravarty-0.706 0et al. [25]. The charge order here, and its connection to spin order, g (cos kx cosWeky) now-0.009 study the electronic0 spectral0 0 function0 in the Q = should be distinguished from that of the “stripe” model [31]; p8 sin k(xQsin, 0)ky charge-ordered state. We will work with the state with ⇥ 0 [1] J. E. Hoin↵ theman latteret al., model, Science the295, charge 466 (2002). order is tied to the square bi-directional charge order [13]; in our theory the degeneracy[2] M. Vershinin,of incommensurate S. Misra, S. spin Ono, order, Y. Abe, and occurs Yoichi Ando,at twice and the spin- between the uni-directional and bi-directional charge orderedA. Yazdani, Science 303, 1995 (2004). TABLE I: Values of cQ, in the expansion in Eq. (9) for various val- ordering wavevector. Instead, in our model, bond order ues Q and .state The last is broken column only shows by the terms coe quarticcients in in the the corre-Q, and we[3] have Y. Kohsaka,appearset in al., a Science regime315 of, “quantum-disordered” 1380 (2007). antiferromag- sponding expansionnot accounted for (k). for We these used here.µ = Choosing1.2, ⇠ = 2, theT = largest0.06, 2 compo-[4] W. D.netism Wise et [26]. al. Nature This Physics is consistent4, 696 with (2008). Ref. [7], which showed S and L = 80. nents from Table I, we have the order parameter [5] M. J. Lawlerthat theet spin al., Nature order466 and, 374 charge (2010). order are in distinct doping [6] A. Mesaros et al., Science 333, 426 (2011). regimes in YBa2Cu3Oy, with the charge-ordering regime co- s + d(cos kx cos ky) , Q = ( Q0, 0) [7] T. Wu et al., Nature 477, 191 (2011). Q(k) = ± (9) inciding with regime of quantum oscillations [16, 17]. s d(cos kx cos ky) , Q = (0, Q0) [8] Y. Kohsaka, et al., Nature Physics 8, 534 (2012). write ( ± [9] G. GhiringhelliWe alsoet presented al., Science computations337, 821 (2012). of the spectral function of with s/d = 0.234. We computed the imaginary part of [10] J. Changtheet charge-ordered al., Nature Phys. metal,8, 871 and (2012). showed that it contains “Fermi the single-electronQ(k) = Green’scQ, ( function,k) ImGk,k(! (9)+ i⌘), and the Electron spectral function [11] A. J. Achkararc” featureset al., Phys. across Rev. the Lett. diagonals109, 167001 of the Brillouin (2012). zone. It was results are shown in Fig. 4. The stability of the Fermi arc in X [12] D. LeBoeuf,argued by S. Harrison Kramer,¨ and W. Sebastian N. Hardy, [15] Ruixing that the Liang, same Fermi D. A.arcs Bonn, can and be C. combined Proust, Nature to explain Physics the9 quantum, 79 (2013). oscillations. where labels various orthonormal basis functions, and cQ, 4 [13] N. Doiron-Leyraud et al., Nature , 565 (2007). are numerical coecients that we determine. Thus we have In our computations here,447 the strongest instability to charge [14] N. Harrison and S. E. Sebastian, Phys. Rev. Lett. 106, 226402 Finally, note that Fig. 2 also has a broad local minimumordering wasfound at wavevector that the spin-singlet,(Q0, Q0d);character but notice persists also the to the exper- the s basis function s(k) = 1, the extended s function (2011). ± ± near Q = (⇡, ⇡). Here Q(k) is found to have the odd-parityblue “valleys”imentally in Fig. observed 2 extending wavevectors. from this This wavevector leads to our to proposal s0 (k) = cos kx + cos ky, the d function d(k) = cos kx cos ky [15] S. E. Sebastian, N. Harrison and G. G. Lonzarich, Rep. Prog. p form in Table I. Condensation of this mode will( breakQ0, 0), (0for, Q charge0). Other ordering approaches in the underdoped to charge cuprates, ordering summarized and so on, as shown in Tablex I.,y Depending upon the symmetry Phys. 75±, 102501 (2012).± due to antiferromagnetic interactions [23, 26], which include of Q (in particular, the littletime-reversal group of the symmetry, wavevector andQ lead) and to the[16] state B. with Vignolle, sponta- D.in Vignolles, Eq. (9). M.-H. Julien, and C. Proust, neous orbital currents proposed by Chakravarty et al. strong-coupling[25]. e↵ects, do find ordering along the (1, 0) and of the eigenvector, some of the c may be exactly zero. But C. R. Physique 14, 39The (2013). charge order here, and its connection to spin order, WeQ now, study the electronic spectral function in the(0,Q1)= directions. Specifically, we expect that extending our for a generic Q, only time-reversal constrains the values of [17] M. A. Metlitski andshould S. Sachdev, be distinguished Phys. Rev. from B that82 of, 075128 the “stripe” model [31]; (Q0, 0) charge-ordered state. We will work with(2010). the statepresent with computationin the latter to include model, the pairing charge e↵ects order will is tied enhance to the square c , and we are allowed to have an admixture of many basis Q, bi-directional charge order [13]; in our theory[18] the Ar. degeneracy Abanovcharge and order A.of correlations V. incommensurate Chubukov, along Phys. spin the Rev. (1 order,, 0) Lett. and and (084 occurs,,1) 5608 directions, at twice the spin- functions. Nevertheless, webetween will see the that uni-directional only a small and number bi-directional charge(2000).as ordered was the caseordering in the wavevector. computations Instead, of Ref. [26]. in our It would model, also bond order of basis functions have appreciablestate is broken coecients, only by and terms so quartic Eq. (9) in the[19]Q, T. and Holder webe interestinghave and W. Metzner,appears to include in Phys. a regime an Rev. applied B of85, “quantum-disordered” magnetic 165130 (2012). field in such antiferromag- an represents a usefulFIG. 4: expansion.Im ElectronG(knot, spectral! accounted+ i⌘ density) for these inlog the here. phase ImG Choosing with(k, bidirectional! + the[20]i largest⌘) C. Husemann 2extension. compo- and W.netism Metzner, [26]. Phys. This Rev. is consistent B 86, 085113 with (2012). Ref. [7], which showed nents from Table I, we have the order parameter[21] K. B. Efetov, H. Meier, and C. Pepin,´ arXiv:1210.3276. charge order at Q = (Q0, 0) and (0, Q0) with Q0 = 4⇡/11. The For the phasethat the diagram spin order of the and hole-doped charge order cuprates, are in distinct our doping left panel show ImGk,k(! + i⌘) at ! = 0 and ⌘ = 0.02; the[22] right S. Chakravarty, R.regimes B. Laughlin, in YBa D.2Cu K.3O Morr,y, with and the C. charge-ordering Nayak, regime co- s + d(cos kx cos ky) , Q = ( Q0,model0) has a quantum-critical point near optimal doping as- c† panelc shows log ImQG(kkk,k)(==! + i⌘) fork the samek parameters, as a wayPhys.± Rev.(9) B 63, 094503 (2001). k Q/2,↵ k+Q/2,↵ Q( ) 0(cosx(coscosk ycos) k ) , Q = (0, Qsociated) withinciding disappearance with regime of bond of quantum order oscillations [7, 26] described [16, 17]. of enhancing the/ low intensities.( s Thed dashed x line is they underlying ± 0 ⇥ ⇤ by Eq. (9). AnWe important also presented challenge computations is to use of such the spectral a critical function of D Fermi surfaceE ofwith the metals/ withoutd = 0 charge.234. order.We computed The charge the order imaginary is part of point to describethe charge-ordered the evolution metal, of the and Fermi showed surface that [16], it contains and “Fermi as in Eq. (9) withthed single-electron= 0.3, and other Green’s parameters function, as in Fig. ImG 2.k,k(! + i⌘), and the S. Sachdev and R. La Placa, arXiv:1303.2114the ‘strange’arc” metal. features across the diagonals of the Brillouin zone. It was Saturday, March 23, 13 results are shown in Fig. 4. The stability of the Fermi arc in argued by Harrison and Sebastian [15] that the same Fermi Acknowledgments. We thank for A. Chubukov, D. Chowd- ‘nodal’ region (k k ) is enhanced [30] because of the weak arcs can be combined to explain the quantum oscillations. x ⇡ y hury, J. C. Davis, E. Demler, K. Efetov, D. Hawthorn, coupling to the charge order, arising from the predominant In our computations here, the strongest instability to charge P. Hirschfeld, J. Ho↵man, H. Meier, W. Metzner, C. Pepin,´ d character of Eq. (9). However, there is strong coupling in ordering was at wavevector (Q , Q ); but notice also the and L. Taillefer useful discussions. This± research0 ± 0 was sup- the anti-nodal region, and any Fermi surfaces appearing in the blue “valleys” in Fig. 2 extending from this wavevector to ported by the NSF under Grant DMR-1103860, and by the latter region should be easily broadened by impurity-induced ( Q0, 0), (0, Q0). Other approaches to charge ordering U.S. Army Research± O±ce Award W911NF-12-1-0227. phase-shifts in the charge ordering. due to antiferromagnetic interactions [23, 26], which include Discussion. We have described the features of a simple strong-coupling e↵ects, do find ordering along the (1, 0) and model of the underdoped cuprates. We began with a metal (0, 1) directions. Specifically, we expect that extending our with antiferromagnetic spin correlations. Exchange of the an- present computation to include pairing e↵ects will enhance charge order correlations along the (1, 0) and (0, 1) directions, tiferromagnetic fluctuations leads to an attractive force in the [1] J. E. Ho↵man et al., Science 295, 466 (2002). spin-singlet d channel of the particle-particle sector, and a cor- [2] M. Vershinin,as was S. the Misra, case in S. the Ono, computations Y. Abe, Yoichi of Ref. Ando, [26]. and It would also responding instability to superconductivity. Ref. [20] noted A. Yazdani,be interesting Science 303 to, 1995 include (2004). an applied magnetic field in such an that the sameFIG. antiferromagnetic 4: Electron spectral fluctuations density in also the lead phase to with an bidirectional[3] Y. Kohsaka,extension.et al., Science 315, 1380 (2007). charge order at Q = (Q , 0) and (0, Q ) with Q = 4⇡/11. The enhancement in the spin-singlet d channel0 of the particle-hole0 0 [4] W. D. WiseForet al. theNature phase Physics diagram4, 696 of (2008). the hole-doped cuprates, our left panel show ImGk,k(! + i⌘) at ! = 0 and ⌘ = 0.02; the[5] right M. J. Lawlermodelet has al., Nature a quantum-critical466, 374 (2010). point near optimal doping as- sector, and a sub-dominantpanel shows log instabilityImG (! to+ i bond⌘) for order. the same Here parameters, we as a way k,k [6] A. Mesarossociatedet al., with Science disappearance333, 426 (2011). of bond order [7, 26] described have studied theof enhancing momentum-space the low intensities. structure The of dashed the latter line isin- the underlying ⇥ ⇤ [7] T. Wu etby al. Eq., Nature (9).477 An, 191 important (2011). challenge is to use such a critical stability acrossFermi the entire surface Brillouin of the metal zone, without without charge any order. assump- The charge order[8] Y. is Kohsaka, et al., Nature Physics 8, 534 (2012). = . point to describe the evolution of the Fermi surface [16], and tions of particle-holeas in Eq. symmetry (9) with d or0 the3, and continuum other parameters limit, as and in Fig. 2.[9] G. Ghiringhelli et al., Science 337, 821 (2012). the ‘strange’ metal. Acknowledgments. We thank for A. Chubukov, D. Chowd- ‘nodal’ region (k k ) is enhanced [30] because of the weak x ⇡ y hury, J. C. Davis, E. Demler, K. Efetov, D. Hawthorn, coupling to the charge order, arising from the predominant P. Hirschfeld, J. Ho↵man, H. Meier, W. Metzner, C. Pepin,´ d character of Eq. (9). However, there is strong coupling in and L. Taillefer useful discussions. This research was sup- the anti-nodal region, and any Fermi surfaces appearing in the ported by the NSF under Grant DMR-1103860, and by the latter region should be easily broadened by impurity-induced U.S. Army Research Oce Award W911NF-12-1-0227. phase-shifts in the charge ordering. Discussion. We have described the features of a simple model of the underdoped cuprates. We began with a metal with antiferromagnetic spin correlations. Exchange of the antiferromagnetic fluctuations leads to an attractive force in the [1] J. E. Ho↵man et al., Science 295, 466 (2002). spin-singlet d channel of the particle-particle sector, and a cor- [2] M. Vershinin, S. Misra, S. Ono, Y. Abe, Yoichi Ando, and responding instability to superconductivity. Ref. [20] noted A. Yazdani, Science 303, 1995 (2004). that the same antiferromagnetic fluctuations also lead to an [3] Y. Kohsaka, et al., Science 315, 1380 (2007). enhancement in the spin-singlet d channel of the particle-hole [4] W. D. Wise et al. Nature Physics 4, 696 (2008). [5] M. J. Lawler et al., Nature 466, 374 (2010). sector, and a sub-dominant instability to bond order. Here we [6] A. Mesaros et al., Science 333, 426 (2011). have studied the momentum-space structure of the latter in- [7] T. Wu et al., Nature 477, 191 (2011). stability across the entire Brillouin zone, without any assump- [8] Y. Kohsaka, et al., Nature Physics 8, 534 (2012). tions of particle-hole symmetry or the continuum limit, and [9] G. Ghiringhelli et al., Science 337, 821 (2012). Do we finally have a resolution to the low energy electronic structure of underdoped YBCO?

N. Harrison and S. E. Sebastian Phys. Rev. Lett. 106, 226402 (2011). 54 54 Saturday, March 23, 13 Outline 1. The “modern era” of cuprate experiments

2. Antiferromagnetism in metals: d-wave superconductivity

3. Low energy theory, emergent pseudospin symmetry, and bond order

4. Unrestricted Hartree-Fock-BCS

5. Quantum Monte Carlo without the sign problem

Saturday, March 23, 13 Outline 1. The “modern era” of cuprate experiments

2. Antiferromagnetism in metals: d-wave superconductivity

3. Low energy theory, emergent pseudospin symmetry, and bond order

4. Unrestricted Hartree-Fock-BCS

5. Quantum Monte Carlo without the sign problem

Saturday, March 23, 13 Low energy theory for critical point near hot spots

Saturday, March 23, 13 QMC for the onset of antiferromagnetism

Hot spots in a single band model

Saturday, March 23, 13 QMC for the onset of antiferromagnetism

E. Berg, M. Metlitski, and S. Sachdev, K Science 338, 1606 (2012).

Hot spots in a two band model

Saturday, March 23, 13 QMC for the onset of antiferromagnetism

Faithful realization of the E. Berg, M. Metlitski, and generic S. Sachdev, K Science 338, 1606 universal (2012). low energy theory for the onset of antiferromagnetism. Hot spots in a two band model

Saturday, March 23, 13 QMC for the onset of antiferromagnetism

E. Berg, M. Metlitski, and S. Sachdev, K Science 338, 1606 (2012).

Hot spots in a two band model

Saturday, March 23, 13 QMC for the onset of antiferromagnetism

Sign E. Berg, problem is M. Metlitski, and S. Sachdev, absent as K Science 338, 1606 long as K (2012). connects hotspots in distinct bands

Hot spots in a two band model

Saturday, March 23, 13 QMC for the onset of antiferromagnetism

Sign E. Berg, problem is M. Metlitski, and S. Sachdev, absent as K Science 338, 1606 long as K (2012). connects Requires only hotspots in time-reversal symmetry. distinct Particle-hole or bands point-group symmetries or commensurate densities not Hot spots in a two band model required !

Saturday, March 23, 13 QMC for the onset of antiferromagnetism

Electrons with dispersion "k interacting with ﬂuctuations of the antiferromagnetic order parameter ~' .

= c ~' exp ( ) Z D ↵D S Z @ = d⌧ c† " c S k↵ @⌧ k k↵ Z Xk ✓ ◆ 1 r + d⌧d2x ( ~' )2 + ~' 2 + ... 2 rx 2 Z  x d⌧ ~' ( 1) i c† ~ c i · i↵ ↵ i i Z X

Saturday, March 23, 13 QMC for the onset of antiferromagnetism (x) (y) Electrons with dispersions "k and "k interacting with fluctuations of the antiferromagnetic order parameter ~' . E. Berg, (x) (y) = c↵ c↵ ~' exp ( ) M. Metlitski, and Z D D D S S. Sachdev, Z Science 338, 1606 (x) @ (x) (x) (2012). = d⌧ c † " c S k↵ @⌧ k k↵ Z Xk ✓ ◆ (y) @ (y) (y) + d⌧ c † " c k↵ @⌧ k k↵ Z Xk ✓ ◆ 1 r + d⌧d2x ( ~' )2 + ~' 2 + ... 2 rx 2 Z  x (x) (y) d⌧ ~' ( 1) i c †~ c + H.c. i · i↵ ↵ i Z i Saturday, March 23, 13 X QMC for the onset of antiferromagnetism (x) (y) Electrons with dispersions "k and "k interacting with fluctuations of the antiferromagnetic order parameter ~' . E. Berg, (x) (y) = c↵ c↵ ~' exp ( ) M. Metlitski, and Z D D D S S. Sachdev, Z Science 338, 1606 (x) @ (x) (x) (2012). = d⌧ c † " c S k↵ @⌧ k k↵ Z Xk ✓ ◆ (y) @ (y) (y) + d⌧ c † " c k↵ @⌧ k k↵ Z Xk ✓ ◆ 1 r No sign problem ! + d⌧d2x ( ~' )2 + ~' 2 + ... 2 rx 2 Z  x (x) (y) d⌧ ~' ( 1) i c †~ c + H.c. i · i↵ ↵ i Z i Saturday, March 23, 13 X QMC for the onset of antiferromagnetism (x) (y) Electrons with dispersions "k and "k interacting with fluctuations of the antiferromagnetic order parameter ~' . E. Berg, (x) (y) = c↵ c↵ ~' exp ( ) M. Metlitski, and Z D D D S S. Sachdev, Z Science 338, 1606 (x) @ (x) (x) (2012). = d⌧ c † " c S k↵ @⌧ k k↵ k Applies without Z X ✓ ◆ (y) @ (y) (y) changes to the + d⌧ c † " c k↵ @⌧ k k↵ microscopic band Z Xk ✓ ◆ structure in the 1 r iron-based + d⌧d2x ( ~' )2 + ~' 2 + ... 2 rx 2 superconductors Z  x (x) (y) d⌧ ~' ( 1) i c †~ c + H.c. i · i↵ ↵ i Z i Saturday, March 23, 13 X QMC for the onset of antiferromagnetism (x) (y) Electrons with dispersions "k and "k interacting with fluctuations of the antiferromagnetic order parameter ~' . E. Berg, (x) (y) = c↵ c↵ ~' exp ( ) M. Metlitski, and Z D D D S S. Sachdev, Z Science 338, 1606 (x) @ (x) (x) (2012). = d⌧ c † " c S k↵ @⌧ k k↵ Z Xk ✓ ◆ Can integrate out ~' to (y) @ (y) (y) obtain an extended + d⌧ c † " c k↵ @⌧ k k↵ Hubbard model. The Z k ✓ ◆ interactions in this model X only couple electrons in 1 2 r + d⌧d2x ( ~' ) + ~' 2 + ... separate bands. 2 rx 2 Z  x (x) (y) d⌧ ~' ( 1) i c †~ c + H.c. i · i↵ ↵ i Z i Saturday, March 23, 13 X QMC for the onset of antiferromagnetism

E. Berg, M. Metlitski, and S. Sachdev, K Science 338, 1606 (2012).

Hot spots in a two band model

Saturday, March 23, 13 QMC for the onset of antiferromagnetism 1 a)

E. Berg, 0.5 M. Metlitski, and S. Sachdev, Science 338, 1606 (2012). π

/ K y 0 k

−0.5

−1 Center Brillouin zone at (π,π,)

Saturday, March 23, 13 QMC for the onset of antiferromagnetism 1 1 a) b)

E. Berg, 0.5 M. Metlitski,0.5 and S. Sachdev, Science 338, 1606 (2012). π K π / / y 0 y 0 k k

−0.5 −0.5

−1 −1 Move−1 one of the Fermi0 surface by (π1,π,) −1 0 1 Saturday, March 23, 13 k /π k /π x x QMC for the onset of antiferromagnetism 1 a)

E. Berg, 0.5 M. Metlitski, and S. Sachdev, Science 338, 1606 (2012). π / y 0 k

−0.5

−1 Now hot spots are at Fermi surface intersections

Saturday, March 23, 13 QMC for the onset of antiferromagnetism 1 1 a) b)

E. Berg, 0.5 0.5 M. Metlitski, and S. Sachdev, Science 338, 1606 (2012). π K π / / y 0 y 0 k k

−0.5 −0.5

−1 −1 −1 0 1 Expected−1 Fermi surfaces 0in the AFM ordered1 phase k /π Saturday, March 23, 13 k /π x x QMC for the onset of antiferromagnetism

r = −0.5 r = 0 r = 0.5 1 1 1

0.5 0.5 0.5 1.5 π /

y 0 0 0 1 k

−0.5 −0.5 −0.5 0.5

−1 −1 −1 −1 0 1 −1 0 1 −1 0 1 k / k / k / x π x π x π

Electron occupation number nk as a function of the tuning parameter r

E. Berg, M. Metlitski, and S. Sachdev, Science 338, 1606 (2012).

Saturday, March 23, 13 QMC for the onset of antiferromagnetism

a) b) 0.6 L=8 0.4 L=10 0.3 ) L=12 β 0.4 2 L=14 0.2 /(L φ χ 0.2 0.1 Binder cumulant 0 0 −0.5 0 0.5 1 −0.5 0 0.5 1 r r

AF susceptibility, ', and Binder cumulant as a function of the tuning parameter r

E. Berg, M. Metlitski, and S. Sachdev, Science 338, 1606 (2012).

Saturday, March 23, 13 QMC for the onset of antiferromagnetism

x 10−4 10 _ 8 P _+ L = 10 P_

) 6

max L = 12

↑ x

( 4 | ± P L = 14 2 r c 0

−2 −2 −1 0 1 2 3 r

s/d pairing amplitudes P+/P as a function of the tuning parameter r

E. Berg, M. Metlitski, and S. Sachdev, Science 338, 1606 (2012).

Saturday, March 23, 13 Conclusions Metals with antiferromagnetic spin correlations have nearly degenerate instabilities: to d-wave superconductivity, and to a charge density wave with a d-wave form factor.

New sign-problem-free quantum Monte Carlo for studying such metals. Obtained (ﬁrst ?) convincing evidence for unconventional superconductivity at strong coupling.

Good prospects for studying competing charge orders, and non-Fermi liquid physics at non-zero temperature.

New sign-problem-free quantum Monte Carlo for studying such metals. Obtained (ﬁrst ?) convincing evidence for unconventional superconductivity at strong coupling.

Good prospects for studying competing charge orders, and non-Fermi liquid physics at non-zero temperature.

New sign-problem-free quantum Monte Carlo for studying such metals. Obtained (ﬁrst ?) convincing evidence for unconventional superconductivity at strong coupling.

Good prospects for studying competing charge orders, and non-Fermi liquid physics at non-zero temperature.

Saturday, March 23, 13