Evidence for a Vestigial Nematic State in the Cuprate Pseudogap Phase

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 opening 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) 3 bias (meV) * p=0.06 ∆ 40 Low High Ox p=0.08 * dI/dV (arb. unit) 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. (C) Colored background represents schematic phase diagram of hole-doped CuO . Two classes of broken-symmetry states have been q q 2 y x widely reported therein. The first indicated by circles occurs at wavevector

Q = 0 and typically exhibits breaking of C4 symmetry within the CuO2 unit cell that appears at the pseudogap temperature scale T*. The second state indicated by diamonds breaks translational symmetry with wavevector D Q = ðQ,0Þ; ð0, QÞ exhibiting a predominantly dFF, and often appears at a temperature scale far lower than T*.

cuprates is that the DW state that could produce an energy gap p in D(E) is only detectable at temperatures TDW ðpÞ << T ðpÞ and |Z(q, 94 meV)| without the opening of a new energy gap, while the nematic (NE) state that appears at the temperature T ðpÞ ≈ TpðpÞ where the NE Low High pseudogap opens should be incapable of opening a Fermi surface energy gap. No comprehensive explanation has been proposed to Fig. 2. Q = 0 and Q ≠ 0 broken-symmetry states imaged simultaneously. (A) address these linked mysteries. Real-space phase-resolved SI-STM image of dFF DW modulations in Zðr, E = 94 meVÞ of a Bi2Sr2CaCu2O8+δ sample with P = 0.06 measured at T = Symmetry Breaking in the Cuprate Pseudogap Regime 2.2 K. Transparent overlay shows the correspondences between the schematic Both types of broken-symmetry states can be studied simulta- dFF DW and the real-space phase-resolved electronic structure. Intraunit cell C4 symmetry is broken in virtually every CuO unit cell, and it is the modulation of neously in the same field of view (FOV) using high-resolution 2 this electronic inequivalence between Ox and Oy sites that generates the dFF spectroscopic imaging scanning tunneling microscopy (SI-STM) DW state. (B) Typical Fourier transform image Zðq, EÞ from underdoped QB QB (33). To do so, Fourier analysis of sublattice phase-resolved Bi2Sr2CaCu2O8+δ. Locations of the Bragg peaks x , y are indicated by circles.

2of6 | www.pnas.org/cgi/doi/10.1073/pnas.1821454116 Mukhopadhyay et al. Downloaded by guest on October 2, 2021 Z B B p = 0.06 N ðEÞ ≡ ReZ Q E − ReZ Q E [1] A p = 0.06 1 B B p = 0.08 1 y , x , , Re Z (Q x ) Re Z (Q B )

q Re Z q x B Re y x Re Z (Q ) B 0.8 y 0.8 Re Z (Q ) Z B B ( 0.2 ~ y ( Q Q Q

Re Z(q,72 meV) Q N(E) where and are the two Bragg wavevectors along the y axis/ _ 0.3 y x )

+ ) B N(E) 0.6 0.6 B (

( x axis, respectively. Any spurious effects of STM tip anisotropy arb. unit arb. unit 0.2 Z

0.4 arb. unit) 0.4 ð Þ (arb. unit) 0.1 ( on N E are ruled out as described in SI Appendix, section 1.If N Z

E ) ð Þ max 0.1 ) the measured N E is then nonzero, breaking of C4 symmetry is N(E) 0.2 N(E) N 0.2 E max occurring due to inequivalence on the average of electronic 0 0 0 0 0 25 50 75 100 125 150 175 200 0 25 50 75 100 125 150 structure at the two oxygen sites within each CuO2 cell (8, 28– C Energy (meV) D Energy (meV) 33).Fig.3A–E shows the measured energy dependence 1 p = 0.14 1 B B Z Re Z (Q B ) p = 0.10 Re Z (Q B ) ðQ Þ ðQ Þ ð Þ x x of ReZ y , E , ReZ x , E , and their difference N E for

B B Re

Re Z (Q ) Re Z Re Z (Q ) = y 0.8 y 0.8 Bi2Sr2CaCu2O8+x at P 0.06, 0.08, 0.10, 0.14, and 0.17. Fig. Z (

N(E) ( N(E) Q

Q ) 3F then summarizes the doping dependence of the energy, ) 0.6 0.6 B 0.3 B N Z

( ð Þ arb. unit) (arb. unit) 0.6 Emax,atwhichN E achieves its maximum intensity. Thus, 0.2 0.4 0.4 N 0.4 the characteristic energy of maximum spectral intensity of this Emax (arb. unit) (arb. unit) 0.1 0.2 0.2 measure of cuprate nematicity diminishes approximately linearly 0.2 N E N(E) N(E) max with increasing p. 0 0 0 0 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 Energy (meV) Energy (meV) E F 120 p = 0.17 1 p=0.06 Re Z (Q B ) p = 0.06 3 nm p = 0.06 p = 0.06 x 100 p=0.08 B p=0.10 A F K q Re Z (Q ) 0.8 Re 3 q y p=0.14 q 4 q 5 Z 80 2 N(E) ( p=0.17 0.6 Q ) q q 0.4 B 60 1 6

(arb. unit q 7

0.3 0.4 N max

E (meV) 40 q q y q y q (arb. unit) 0.2 N x x

E ) max 0.2 20 y x 0.1 N(E) Z 0 0 0 Z(r,E), E= 84meV |D (q,E)| 2 , E=84 meV |S ´ Z (q,E)| 2 , E= 10 meV

0 10 20 30 40 50 60 70 80 90 0 0.05 0.1 0.15 0.2 0.25 0.3 PHYSICS Energy (meV) p (hole density) B p = 0.08 3 nm G p = 0.08 L p = 0.08 Fig. 3. Doping- and energy evolution of the Q = 0 broken-symmetry (NE)

state. (A, Inset) Measured ReZðq, E = 72 meVÞ for Bi2Sr2CaCu2O8+δ with P = = Z ð Þ ≡ ðQB Þ − ðQB Þ 0.06 at T 2.2 K. The magnitude of N E ReZ y , E ReZ x , E q q q q is measured as the difference in real intensity of the two Bragg peaks y x y x QB QB – y x at x , y (indicated by diamond and star shapes). (A E) Measured B B ReZðQ , EÞ, ReZðQ , EÞ, and NðEÞ versus hole density (P = 0.06, 0.08, 0.10, Z Z y x Z(r,E), E= 72meV |D (q,E)| 2 , E=72 meV |S ´ (q,E)| 2 , E= 12 meV 0.14, and 0.17, respectively) in the Bi Sr CaCu O +δ samples at T = 2.2 K 2 2 2 8 p = 0.10 3 nm p = 0.10 p = 0.10 Z ð Þ N Z ð Þ C H M reported here. For each N E curve, an energy Emax where N E achieves its N ð Þ = maximum is indicated by a vertical arrow.(F) Plot of measured Emax p for P 0.06, 0.08, 0.10, 0.14, and 0.17. This shows how the energy associated with maximum spectral intensity of the NE state diminishes approximately linearly with increasing p. q q q q y x y x y x

Z Z(r,E), E= 56 meV |D (q,E)| 2 , E=56 meV |S ´ Z (q,E)| 2 , E= 14 meV at bias voltage V = E/e and with spatial resolution of typically 7 × 7 D p = 0.14 3 nm I p = 0.14 N p = 0.14 pixels within each CuO2 unit cell. To suppress serious systematic “set-point” errors (33) we evaluate Zðr, EÞ = gðr, EÞ=gðr, − EÞ,a function for which distances, wavelengths, spatial phases, and

energy magnitudes of electronic structure can be measured q q q q y x y x accurately. The necessary spatial-phase accuracy in gðr, EÞ and y x

ðr Þ Z Z , E is achieved using a picometer-scale transformation that Z(r,E), E= 52meV |D (q,E)| 2 , E=52 meV |S ´ Z (q,E)| 2 , E= 12 meV renders the topographic image T(r)perfectlya -periodic, and 0 E p = 0.17 3 nm J p = 0.17 O p = 0.17 is then applied to gðr, EÞ to register all of the electronic- structure data to the ideal CuO2 lattice (8, 28–30). The ne- cessity of these advanced SI-STM techniques to measure the cuprate broken-symmetry phases can be seen directly in Fig. q q q q y x y x y x 2A which shows a typical Zðr, EÞ, with local breaking of C4 Z 2 Z 2 symmetry within the CuO2 cell (NE) and its periodic modu- Z(r,E), E= 46 meV |D (q,E)| , E=46 meV |S ´ (q,E)| , E= 16 meV lations (DW); a schematic overlay is used to identify the lo- 0.25 1.5 0 50 10 35 cations of Cu and O orbitals. Fig. 2B is the Fourier transform of a larger-FOV Zðr, E ∼ 100 meVÞ image of Bi2Sr2CaCu2O8+x at Fig. 4. Doping evolution of the Q ≠ 0 broken-symmetry (DW) state. (A–E) p ∼ 0.06 containing the two basic signatures of electronic symmetry Real-space map of phase-resolved DW modulations in Zðr, EÞ measured at B T = 2.2 K for doping P = 0.06, 0.08, 0.10, 0.14, and 0.17 at E = 84, 72, 56, 52, breaking, the Bragg peaks at Q = ðð1,0Þð0,1ÞÞ2π=a0 and the peaks and 46 meV, respectively. (F–J) Corresponding q-space DW power-spectral D 2 near Q ≈ ðð0.75, 0Þð0,0.75ÞÞ2π=a0 due to the d-symmetry form densities DZ ðq, EÞ from Eq. 2 for dopings P = 0.06, 0.08, 0.10, 0.14, and factor DW (28–30). 0.17, respectively. Red circles represent the relevant regions of interest for QD ≅ ðð Þð ÞÞ π= The Q = 0 rotational symmetry-breaking state (NE) is de- dFF DW at wavevectors 0.25, 0 0,0.25 2 a0. Locations of the Bragg B B B “+” – ðQ Þ peaks Qx , Qy are indicated by sign. (K O) The power-spectral densities of tected by resolving both the real ReZ , E and imaginary ’Z ðq Þ2 = B Bogoliubov quasiparticle S , E from Eq. 2 for dopings P 0.06, 0.08, ImZðQ , EÞ components of the Bragg amplitudes of Zðq, EÞ,to 0.10, 0.14 and 0.17, respectively, measured at lower energy on the same B B “+” form a direct measure of intraunit cell nematicity (8) sample set. Locations of the Bragg peaks Qx , Qy are indicated by sign.

Mukhopadhyay et al. PNAS Latest Articles | 3of6 Downloaded by guest on October 2, 2021 The translational symmetry-breaking (DW) state exhibits a A predominantly d-symmetry form factor (dFF) because the DW is 1 p = 0.06 based on modulating the inequivalence between the Ox and Oy p = 0.08 sites within each CuO unit cell (8, 28–33) at wavevector QD,as p = 0.10 2 x 0.8 p = 0.14 discussed in SI Appendix, section 2. Such modulations are de- D ðrÞ = ðrÞ ðϕðrÞ + ϕ ðrÞÞ ϕðrÞ = QD · r Emax p = 0.17 scribed by A D Cos 0 , where x is the DW spatial phase at location r and DðrÞ is the magnitude 0.6 ðrÞ = = ðrÞ = − = of the dFF. D AD when r rOx, D AD when r rOy, and DðrÞ = 0 otherwise, where AD is a real number for the am- 0.4 plitude. To study this state, the sublattice-phase-resolved Zðr, EÞ r D images are separated into three: Cu( ), containing only Cu sites, and O (r); O (r), containing only the x/y axis oxygen sites. 0.2 x y z Phase-resolved Fourier transforms of the Ox(r,E) and Oy(r,E)

|D ( Q , E )| (norm.) (arb. unit) ~ ~ ~ ~ sublattice images, Oxðq, EÞ = ReOxðq, EÞ + iImOx ðq, EÞ; Oyðq, EÞ = 0 ~ ðq Þ + ~ ðq Þ 0 25 50 75 100 125 150 175 200 ReOy , E iImOy , E , are then used to determine the FF q Energy (meV) symmetry for modulations at any 120 120 ~ ~ B DZðq, EÞ = O ðq, EÞ − O ðq, EÞ , [2a] p = 0.06 x y p = 0.08 100 p = 0.10 100 ~ ~ S’Zðq, EÞ = O ðq, EÞ + O ðq, EÞ . [2b] p = 0.14 x y p = 0.17 p 80 80 Fig. 4 A–E show our typical images of Zðr, E ∼ Δ Þ while Fig. 4 E (meV) – Zðq ∼ ΔpÞ2 max F J show D , E the dFF, power spectral density Four- D ier transforms derived from Fig. 4 A–E. The peaks identified by 60 60 N – max red dashed circles are the signature of the dFF DW state (28 32). E (meV) At atomic scale, these features in DZðq, E ∼ ΔpÞ represent two pop- 40 40 ulations of nanoscale unidirectional dFF DW domains (29). Evolution of Broken-Symmetry Spectral Weight with Hole Density. 20 20 Our second key objective is then to measure the doping de- D ZðQD Þ pendence of the energy, Emax, at which the DW order D , E achieves its maximum spectral intensity. Fig. 5A shows the 0 0 measured DZðQD, EÞ integrated within the dashed circles in Fig. 0 20 40 60 80 100 120 4 F–J for P = 0.6, 0.8, 0.10, 0.14, and 0.17 plotted versus energy * (meV) Δ E. The dFF is negligible for modulations in the low-energy range C of Bogoliubov quasiparticles (Fig. 4 K–O), but DZðQD, EÞ rapidly becomes intense at higher energy before finally diminishing quickly at highest E. Evidently, for each p there is a specific D ZðQD Þ VN energy Emax at which a maximum in D , E occurs. The T c D values of Emax are indicated by the black arrows in Fig. 5A, and these we associate with the energy gaps of the disordered DW (SI Appendix, section 3). Fig. 5B then summarizes the doping D dependence of measured Emax, revealing that that the characteristic energy of maximum spectral intensity of the DW state VN also diminishes approximately linearly with increasing p. Temperature To appreciate the implications of these observations, it is important to realize that the order parameters NZðEÞ in Eq. 1 n Thre o sh Z ti o ðq Þ 2a q c ld SC and D , E in Eq. use no common -space data whatsoever, e t T e DW c despite having been measured in the identical real-space field of D view. The key experimental result of these studies is then shown N ð Þ D ð Þ in Fig. 5B: a combined plot of measured Emax p , Emax p versus AFM - Mott Insulating p the measured values of Δ ðpÞ for underdoped Bi Sr CaCu O + p 2 2 2 8 x p c samples with 0.06 ≤ p ≤ 0.17.The striking fact it reveals is that the two characteristic energies EN and ED , where the distinct Fig. 5. Relationship of characteristic energies of NE, DW states, and the max max PG. (A) Measured energy variation of normalized amplitude for dFF DW versus doping. The corresponding Gaussian fit to the data, used to extract the peak positions, is shown in solid curves. The dFF DW amplitude is the DW states diminishes approximately linearly with increasing p, and that 2 extracted by integrating DZ ðq, EÞ over the primary DW wavevectors at they are always virtually identical to each other and to the PG energy. D Q ≅ ðð0.25, 0Þð0,0.25ÞÞ2π=a0 (red circles in Fig. 4 F–J). The data for hole Dashed blue line is a guide to the eye. (C) A schematic diagram of the hy- = density p 0.06, 0.08, 0.10, 0.14, and 0.17 in the Bi2Sr2CaCu2O8+δ samples are pothesized VN state within the cuprate phase diagram (37). The VN hy- Z ðq Þ2 D reported here. For each D , E curve, an energy Emax for maximum pothesis appears consistent with all of the data and analysis presented N ð Þ spectral weight is indicated by a vertical arrow. (B) Plot of measured Emax p throughout this paper. Note that the antiferromagnetic Mott insulator and D ð Þ Δ*ð Þ = and Emax p versus p for values of p 0.06, 0.08, 0.10, 0.14, and 0.17. This superconducting regions shown here are not included in the analysis of refs. shows how the energy for maximum spectral intensity of both the NE and 37 and 38.

4of6 | www.pnas.org/cgi/doi/10.1073/pnas.1821454116 Mukhopadhyay et al. Downloaded by guest on October 2, 2021 broken-symmetry states NE and DW have maximal spectral in- opening, as indicated schematically by dashed contours in Fig. 5C. tensity, are indistinguishable from each other throughout this Here, the horizontal axis is carrier density but, because strong wide region of the underdoped phase diagram. Even more no- spatial correlations exist between dopant atom distributions and N ð Þ D ð Þ tably, both Emax p and Emax p follow a trajectory with doping electronic disorder in Bi2Sr2CaCu2O8+x (39), it also corresponds that is indistinguishable within error bars from that of the in- with disorder strength (37, 38) for that material. dependently measured pseudogap energy ΔpðpÞ (Fig. 1 A and B). VN theories cast a fresh light on the perplexing phenome- Therefore, we find that the characteristic energy of maximal nology of the NE (3–16) and DW (17–27) symmetry breaking in spectral intensity for both cuprate broken-symmetry states is the cuprate pseudogap phase (Figs. 2–5). If this phase is a VN p always the pseudogap energy ΔpðpÞ. These results add yet an- state, then a highly disordered DW opens a gap Δp at T but other conundrum to our list: How could the characteristic en- without long-range DW order, so that only a VN phase appears. ergies of NE and DW states be virtually identical to each other The VN model then implies that all three energies: the pseu- p and to the PG energy Δ (Fig. 5B) over this wide range of doping, dogap, the characteristic energy gap of the DW, and the ap- while the onset temperatures TNE(p) and TDW(p) of these phases parent energy gap of the NE state, should all coincide at every appear unrelated to each other and to vary quite differently with doping. This is because all three signatures derive from the same doping (Fig. 1C)? Combined with the question of why the DW microscopic phenomenon: the appearance of the unidirectional state (which should generate an energy gap) appears without any N ð Þ but disordered DW state. Coincidence of measured Emax p , ð Þ p new gap opening at TDW p , while the NE state (which should be ED ðpÞ, and Δ ðpÞ is precisely what we report in Fig. 5B, for incapable of opening an energy gap) emerges at the pseudogap max ð Þ ≈ pð Þ underdoped Bi2Sr2CaCu2O8+x samples over a wide doping range opening temperature TNE p T p (Fig. 1C), presents an in- 0.06 ≤ p ≤ 0.17. Moreover, in a VN state with phase transition at tricate puzzle. TNE(p), the onset temperatures TNE(p) and TDW(p) should be unrelated to each other and vary independently with doping, Analysis. In pursuit of its resolution, we note that electronic liquid provided T (p) is an instrumentation-limited threshold for crystal states of a doped Mott insulator are naturally expected to DW detection of the disordered charge modulations rather than a exhibit distinct NE and DW broken-symmetry phases (34–36). phase transition. In that case, detection of DW phenomena Within this context, the critical temperatures and characteristic ð Þ electronic energies of these two highly distinct phases are not should not coincide with opening of a new gap at TDW p because the DW gap has already opened at T ðpÞ ≈ TpðpÞ when the VN required to be related, except for the fact that the NE phase NE PHYSICS should appear first with falling temperature. Recently, however, phase transition occurs. Thus, VN theory (37, 38) appears highly the effects of quenched disorder on the quasi-2D version of this consistent with the experimental observations reported herein situation have been discovered (37, 38). The remarkable con- (Fig. 5B), and more generally with the intricate phenomenology clusion is that, while long-range order of a unidirectional in- of thermal transitions, energy-gap opening, and distinct sym- – commensurate finite-Q DW cannot exist in the presence of metry breaking in underdoped cuprates (3 32) (Fig. 1C). We quenched disorder, its short-range relict survives up to a certain note, however, that neither VN theory nor our studies here yet critical disorder strength but in the form of a Q = 0 broken discriminate whether the fundamental DW state with wavevector Q h † i rotational-symmetry state. This state was dubbed a vestigial ne- is a conventional charge-DW with orderD parameterE ckck+Q or matic (37, 38) (VN). Note that, as with the unidirectional s- † † is a pair-DW (40) with order parameter c c− + . symmetry FF DW state considered in ref. 37, a unidirectional k k Q dFF DW state (as in underdoped cuprates) also breaks local – ACKNOWLEDGMENTS. S.M. acknowledges support from NSF Grant DMR- translational and rotational symmetries (28 30), and should also 1719875 to the Cornell Center for Materials Research. S.-i.U. and H.E. yield a VN state in the presence of strong disorder. Several dis- acknowledge support from a Grant-in-Aid for Scientific Research from the tinguishing consequences are predicted within such VN theories. Ministry of Science and Education (Japan) and the Global Centers of First, the energy scale of the NE state for which rotational symmetry Excellence Program for Japan Society for the Promotion of Science. K.F. is broken globally should always be the same as that of the DW state and C.K.K. acknowledge salary support from the US Department of Energy, Office of Basic Energy Sciences, under Contract DEAC02-98CH10886. E.-A.K. in which translational symmetry is broken locally, since the two acknowledges Simons Fellow in Theoretical Physics Award 392182 and from effects stem from the identical microscopic electronic phenomenon the US Department of Energy, Office of Basic Energy Sciences, under Grant (unidirectional DW with disorder). Second, as temperature dimin- DE-SC0018946. S.D.E. acknowledges studentship funding from the Engineer- ishes below the phase transition where the VN state appears and its ing and Physical Sciences Research Council under Grant EP/G03673X/1. M.H.H., S.D.E., and J.C.S.D. acknowledge support from the Gordon and Betty gap opens, the intensity of the short-range DW underpinning the Moore Foundation’s EPiQS Initiative through Grant GBMF4544. J.C.S.D. ac- VN should increase continuously as thermal fluctuations are re- knowledges support from Science Foundation Ireland under Award SFI 17/ duced but with no new phase transition and no additional gap RP/5445 and from the European Research Council under Award DLV-788932.

1. J. L. Tallon, J. W. Loram, The doping dependence of T*–What is the real high-Tc phase 10. Y. Li et al., Magnetic order in the pseudogap phase of HgBa2CuO4+δ studied by spin- diagram? Physica C 349,53–68 (2001). polarized neutron diffraction. Phys. Rev. B Condens. Matter Mater. Phys. 84, 224508 (2011). 2. M. R. Norman, D. Pines, C. Kallin, The pseudogap: Friend or foe of high Tc? Adv. Phys. 11. A. Shekhter et al., Bounding the pseudogap with a line of phase transitions in

54, 715–733 (2005). YBa2Cu3O6+δ. Nature 498,75–77 (2013). 3. A. Kaminski et al., Spontaneous breaking of time-reversal symmetry in the pseudogap 12. L. Zhao et al., A global inversion-symmetry-broken phase inside the pseudogap re-

state of a high-Tc superconductor. Nature 416, 610–613 (2002). gion of YBa2Cu3Oy. Nat. Phys. 13, 250–254 (2017). 4. B. Fauqué et al., Magnetic order in the pseudogap phase of high-Tc superconductors. 13. L. Mangin-Thro, Y. Sidis, A. Wildes, P. Bourges, Intra-unit-cell magnetic correlations Phys. Rev. Lett. 96, 197001 (2006). near optimal doping in YBa2Cu3O6.85. Nat Commun 6, 7705 (2015). 5. H. A. Mook, Y. Sidis, B. Fauqué, V. Balédent, P. Bourges, Observation of magnetic 14. L. Mangin-Thro, Y. Li, Y. Sidis, P. Bourges, a−b anisotropy of the intra-unit-cell

order in a superconducting YBa2Cu3O6.6 single crystal using polarized neutron scat- magnetic order in YBa2Cu3O6.6. Phys. Rev. Lett. 118, 097003 (2017). tering. Phys. Rev. B Condens. Matter Mater. Phys. 78, 020506(R) (2008). 15. Y. Sato et al., Thermodynamic evidence for a nematic phase transition at the onset of

6. Y. Li et al., Unusual magnetic order in the pseudogap region of the superconductor the pseudogap in YBa2Cu3Oy. Nat. Phys. 13, 1074–1078 (2017).

HgBa2CuO4+δ. Nature 455, 372–375 (2008). 16. S. Nakata, et al., Nematicity in the pseudogap state of cuprate superconductors re-

7. J. Xia et al., Polar Kerr-effect measurements of the high-temperature YBa2Cu3O6+x vealed by ARPES. arxiv 1811.10028. Accessed 5 December 2018. superconductor: Evidence for broken symmetry near the pseudogap temperature. 17. J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura, S. Uchida, Evidence for stripe cor- Phys. Rev. Lett. 100, 127002 (2008). relations of spins and holes in copper oxide superconductors. Nature 375, 561–563 (1995).

8. M. J. Lawler et al., Intra-unit-cell electronic nematicity of the high-T(c) copper-oxide 18. M. Hücker et al., Stripe order in superconducting La2−xBaxCuO4 (0.095≤x≤0.155). Phys. pseudogap states. Nature 466, 347–351 (2010). Rev. B Condens. Matter Mater. Phys. 83, 104506-1–104506-16 (2011). 9. R.-H. He et al., From a single-band metal to a high-temperature superconductor via 19. Y. J. Kim, G. D. Gu, T. Gog, D. Casa, X-ray scattering study of charge density waves in

two thermal phase transitions. Science 331, 1579–1583 (2011). La2-xBaxCuO4. Phys. Rev. B Condens. Matter Mater. Phys. 77, 064520 (2008).

Mukhopadhyay et al. PNAS Latest Articles | 5of6 Downloaded by guest on October 2, 2021 20. J. Chang et al., Direct observation of competition between superconductivity and 30. A. Mesaros et al., Topological defects coupling smectic modulations to intra-unit-cell

charge density wave order in YBa2Cu3O6.67. Nat. Phys. 8, 871–876 (2012). nematicity in cuprates. Science 333, 426–430 (2011). 21. G. Ghiringhelli et al., Long-range incommensurate charge fluctuations in 31. R. Comin et al., Symmetry of charge order in cuprates. Nat. Mater. 14, 796–800 (2015). (Y,Nd)Ba2Cu3O(6+x). Science 337, 821–825 (2012). 32. E. M. Forgan et al., The microscopic structure of charge density waves in underdoped 22. A. J. Achkar et al., Distinct charge orders in the planes and chains of ortho-III-ordered YBa2Cu3O6.54 revealed by X-ray diffraction. Nat Commun 6, 10064 (2015). YBa2Cu3O(6+δ) superconductors identified by resonant elastic x-ray scattering. Phys. 33. K. Fujita et al., Spectroscopic imaging scanning tunneling microscopy studies of Rev. Lett. 109, 167001 (2012). electronic structure in the superconducting and pseudogap phases of cuprate high-Tc 23. E. Blackburn et al., X-ray diffraction observations of a charge-density-wave order in superconductors. J. Phys. Soc. Jpn. 81, 011005 (2011). superconducting ortho-II YBa2Cu3O6.54 single crystals in zero magnetic field. Phys. 34. S. A. Kivelson, E. Fradkin, V. J. Emery, Electronic liquid-crystal phases of a doped Mott Rev. Lett. 110, 137004 (2013). insulator. Nature 393, 550–553 (1998). 24. R. Comin et al., Charge order driven by Fermi-arc instability in Bi Sr( La( CuO( +δ . 2 2-x) x) 6 ) 35. V. J. Emery, S. A. Kivelson, J. M. Tranquada, Stripe phases in high-temperature su- Science 343, 390–392 (2014). perconductors. Proc. Natl. Acad. Sci. U.S.A. 96, 8814–8817 (1999). 25. S. Blanco-Canosa et al., Resonant x-ray scattering study of charge-density wave cor- 36. E. Fradkin, S. A. Kivelson, J. M. Tranquada, Colloquium: Theory of intertwined orders relations in YBa Cu O + . Phys. Rev. B Condens. Matter Mater. Phys. 90, 054513 2 3 6 x – (2014). in HTSC. Rev. Mod. Phys. 87, 457 482 (2015). 26. S. Gerber et al., Three-dimensional charge density wave order in YBa Cu O at high 37. L. Nie, G. Tarjus, S. A. Kivelson, Quenched disorder and vestigial nematicity in the 2 3 6.67 – magnetic fields. Science 350, 949–952 (2015). pseudogap regime of the cuprates. Proc. Natl. Acad. Sci. U.S.A. 111, 7980 7985 (2014). 27. G. Campi et al., Inhomogeneity of charge-density-wave order and quenched disorder 38. L. Nie, A. V. Maharaj, E. Fradkin, S. A. Kivelson, Vestigial nematicity from spin and/or in a high-Tc superconductor. Nature 525, 359–362 (2015). charge order in the cuprates. Phys. Rev. B 96, 085142 (2017). 28. K. Fujita et al., Direct phase-sensitive identification of a d-form factor density wave in 39. K. McElroy et al., Atomic-scale sources and mechanism of nanoscale electronic- underdoped cuprates. Proc. Natl. Acad. Sci. U.S.A. 111, E3026–E3032 (2014). disorder in Bi2Sr2CaCu2O8+δ. Science 309, 1048–1052 (2005). 29. M. H. Hamidian et al., Atomic-scale electronic structure of the cuprate d-symmetry 40. M. H. Hamidian et al., Detection of a Cooper-pair density wave in Bi2Sr2CaCu2O8+x. form factor density wave state. Nat. Phys. 12, 150–156 (2016). Nature 532, 343–347 (2016).

6of6 | www.pnas.org/cgi/doi/10.1073/pnas.1821454116 Mukhopadhyay et al. Downloaded by guest on October 2, 2021