Evidence for a vestigial nematic state in the cuprate pseudogap phase Sourin Mukhopadhyaya,b, Rahul Sharmab,c, Chung Koo Kimc, Stephen D. Edkinsd, Mohammad H. Hamidiane, Hiroshi Eisakif, Shin-ichi Uchidaf,g, Eun-Ah Kimb, Michael J. Lawlerb, Andrew P. Mackenzieh, J. C. Séamus Davisi,j,1, and Kazuhiro Fujitac aDepartment of Physics, Indian Institute of Space Science and Technology, 695547 Thiruvananthapuram, India; bLaboratory of Atomic and Solid State Physics, Department of Physics, Cornell University, Ithaca, NY 14853; cCondensed Matter Physics and Material Science Department, Brookhaven National Laboratory, Upton, NY 11973; dDepartment of Applied Physics, Stanford University, Stanford, CA 94305; eDepartment of Physics, Harvard University, Cambridge, MA 02138; fNanoelectronics Research Institute, Institute of Advanced Industrial Science and Technology, 305-8568 Tsukuba, Ibaraki, Japan; gDepartment of Physics, University of Tokyo, 113-0033 Tokyo, Japan; hMax Planck Institute for Chemical Physics of Solids, D-01187 Dresden, Germany; iDepartment of Physics, University College Cork, T12 R5C Cork, Ireland; and jClarendon Laboratory, University of Oxford, OX1 3PU Oxford, United Kingdom Contributed by J. C. Séamus Davis, April 27, 2019 (sent for review December 21, 2018; reviewed by Riccardo Comin and Laimei Nie) The CuO2 antiferromagnetic insulator is transformed by hole- Already this is a conundrum: While the onset temperature TNEðpÞ doping into an exotic quantum fluid usually referred to as the of this broken symmetry tracks the temperature TpðpÞ at which pseudogap (PG) phase. Its defining characteristic is a strong sup- pseudogap opens (Fig. 1C), within mean-field theory of elec- pression of the electronic density-of-states D(E) for energies jEj < Δ*, tron fluids ordering at Q = 0 it cannot open an energy gap on the where Δ* is the PG energy. Unanticipated broken-symmetry phases Fermi surface. The second ordered state within the pseudogap have been detected by a wide variety of techniques in the PG regime, phase breaks translational symmetry with axial wavevectors most significantly a finite-Q density-wave (DW) state and a Q = 0ne- Q = ðQ,0Þ; ð0, QÞ parallel to the CuO2 axes (17–27). [Here we matic (NE) state. Sublattice-phase-resolved imaging of electronic do not discuss the density-wave (DW) state generated by very high structure allows the doping and energy dependence of these distinct magnetic fields (26).] This zero-field DW state has been reported broken-symmetry states to be visualized simultaneously. Using this to occur in La2−xBaxCuO4,YBa2Cu3O7−x, Bi2Sr2CaCu2O8+x, PHYSICS approach, we show that even though their reported ordering tem- Bi2Sr2CuO6+x,HgBa2CuO4+x,andCa2− xNaxCuO2Cl2. peratures TDW and TNE are unrelated to each other, both the DW and In YBa Cu O − ,BiSr CuO + ,BiSr CaCu O + ,and NE states always exhibit their maximum spectral intensity at the same 2 3 7 x 2 2 6 x 2 2 2 8 x Ca − Na CuO Cl samples, the DW exhibits a predominantly energy, and using independent measurements that this is the PG 2 x x 2 2 d-symmetry form factor (28–32). The DW phenomenology reveals a energy Δ*. Moreover, no new energy-gap opening coincides with second puzzle: Although a DW with finite Q should open a the appearance of the DW state (which should theoretically open mean-field energy gap spanning the Fermi energy (while folding an energy gap on the Fermi surface), while the observed PG open- ing coincides with the appearance of the NE state (which should the band structure due to the larger electronic unit cell), no energy gap is reported to open specifically at the cuprate DW theoretically be incapable of opening a Fermi-surface gap). We ð Þ ð Þ demonstrate how this perplexing phenomenology of thermal onset temperature TDW p .Moreover,TDW p is manifestly un- p ð Þ transitions and energy-gap opening at the breaking of two highly relatedtothetemperatureT p at which the pseudogap opens distinct symmetries may be understood as the natural conse- (Fig. 1C). Thus, the essence of the broken-symmetry riddle of quence of a vestigial nematic state within the pseudogap phase of Bi2Sr2CaCu2O8. Significance cuprate | pseudogap | broken symmetry | density wave | vestigial nematic In the cuprate pseudogap phase, an energy gap of unknown mechanism opens, and both an electronic nematic phase (NE) he cuprate pseudogap energy Δp is defined to be the energy and a density-wave (DW) phase appear. Perplexingly, the DW, Trange within which significant D(E) suppression occurs in which should generate an energy gap, appears without any underdoped cuprates (1, 2). Fig. 1A shows measured D(E) for new gap opening; and the NE, which should be incapable of hole-density P = 0.06 in Bi2Sr2CaCu2O8+δ, with arrow indicating opening an energy gap, emerges coincident with the pseudo- a typical example of the feature in D(E) that occurs at E = Δp. gap opening. Recently, however, it was demonstrated theo- Fig. 1A (Inset) shows the measured evolution of D(E) and this retically that a disordered unidirectional DW can generate a signature of Δp with p. Fig. 1B shows the consequent doping vestigial nematic (VN) phase. If the cuprate pseudogap phase were in such a VN state, the energy gap of the NE and DW dependence for this sequence of Bi2Sr2CaCu2O8+δ samples. This exemplifies the well-known correspondence (1) between ΔpðpÞ should be identical to each other and to the pseudogap. We p and the temperature T (p) at which the pseudogap phenome- report discovery of such a phenomenology throughout the nology appears; both fall linearly with increasing p until phase diagram of underdoped Bi2Sr2CaCu2O8. reaching zero at approximately p ≈ 0.19. In the last decade, extensive evidence has emerged for electronic symmetry Author contributions: S.M., C.K.K., S.D.E., M.H.H., and K.F. performed experiments; S.M., p C.K.K., and K.F. analyzed data; E.-A.K. and M.J.L. provided theoretical guidance; H.E. and breaking at or below T (p) as shown schematically in Fig. 1C.For p S.-i.U. synthesized the crystals; A.P.M., J.C.S.D., and K.F. supervised the project and wrote T < T ðpÞ and p < pC ≈ 0.19, two distinct classes of broken- the paper with key contributions from S.M., R.S., C.K.K., S.D.E., and M.H.H. symmetry states are widely reported. The first is a nematic Reviewers: R.C., Massachusetts Institute of Technology; and L.N., University of Chicago. Q = state; it occurs at wavevector 0 and generally exhibits Conflict of interest statement: The authors declare a conflict of interest. H.E. and R.C. are breaking of 90° rotational (C4) symmetry (3–16); sometimes coauthors on a 2015 article, “Symmetry of charge order in cuprates,” Nat. Mater. 14, 796– additional Q = 0 broken symmetries are detected depending on the 800 (2015). experimental technique. These phenomena have been reported Published under the PNAS license. 1 using multiple techniques in YBa2Cu3O7−x, Bi2Sr2CaCu2O8+x, To whom correspondence may be addressed. Email: [email protected]. Bi2Sr2CuO6+x,andHgBa2CuO4+δ, and are observed to occur This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. below a characteristic temperature TNEðpÞ, which appears to 1073/pnas.1821454116/-/DCSupplemental. fall linearly with increasing p along the trajectory shown in Fig. 1C. www.pnas.org/cgi/doi/10.1073/pnas.1821454116 PNAS Latest Articles | 1of6 Downloaded by guest on October 2, 2021 electronic structure imaging of the CuO2 plane is required (8, AB – ðr Þ 7 5 120 28 30). The tunnel-current I , E and tip-sample differential 4 p=0.06 dI 6 tunneling conductance ðr, E = eVÞ ≡ gðr, EÞ are first measured 3 100 p=0.08 dV p=0.10 5 2 80 p=0.14 1 dI/dV (arb. unit) dI/dV p=0.17 4 ∆* 0 -0.15 -0.1 -0.5 0 0.5 0.1 0.15 60 A V (V) (meV) (arb. unit) 3 bias * p=0.06 ∆ 40 Low High Ox p=0.08 * dI/dV 2 p=0.10 ∆ p=0.14 a O 1 p=0.17 20 0 = 2 /Q y p = 0.06 0 0 Cu -0.2 -0.1 0 0.1 0.2 0 0.05 0.1 0.15 0.2 0.25 0.3 Vbias (V) p (hole density) φ C x(r) 400 Q=0 [3] : ARPES φ (r) [4] : NS y [5] : NS 350 [6] : NS T * [7] : PKE 10 Å [9] : ARPES,PKE,TRR [10] : NS 300 [11] : RUS [12] : ORA [13] : NS 250 PG [14] : NS [15] : MT Q≠0 [20] : XRD 200 [21] : RXS [22] : RXS y T(K) [23] : XRD [24] : RXS 150 [25] : RXS [26] : XRD-FEL [27] : SμXRD 100 dFFDW T [32] : XRD x Z(r, 94 meV) SC Low High 50 B dSC AF dSC + dFFDW p = 0.06 0 0 0.05 0.1 0.15 0.2 0.25 0.3 (0, 1) 2 /a0 (1, 0) 2 /a0 p (hole density) Fig. 1. Doping evolution of cuprate symmetry breaking and pseudogap. (A) Measured D(E)atP = 0.06 and T = 2.2K in Bi2Sr2CaCu2O8+δ. This dI=dVðE = eVÞ ≡ gðEÞ curve is a typical tunneling spectrum of underdoped cuprates with empty-state density of states peak at energy Δ*. (Inset) Characteristic gðEÞ for the range of Bi2Sr2CaCu2O8+δ samples (P = 0.06, 0.08, 0.10, 0.14, and 0.17) studied. The evolution of Δ*(p) is indicated by the black arrows. (B) Doping dependence of the measured average value of Δ*(P = 0.06, 0.08, 0.10, 0.14, and 0.17, respectively) from the set of Bi2Sr2CaCu2O8+δ samples studied.