The intertwined order of underdoped cuprates. Jan Zaanen

Transcription

1 The intertwined order of underdoped cuprates. Jan Zaanen 1

2 Stripy charge order in cuprate superconductors. B. Keimer et al., Nature 518, 179 (2015) STS images, last word: A. Mesaros et al., PNAS 113, (2016) Also X-ray & neutron scattering, NMR, RIXS, 2

3 Plan 1. The prehistory: stripes in the 1990 s. 2. The revival of (fluctuating) stripe order in Electronic nematic order. 4. Stripes and pair density waves. 5. Varma s orbital currents. 3

4 Zaanen-Gunnarsson (1987) Unrestricted Hartree-Fock on 3-band model: Hole density on oxygen 4

5 Su-Schrieffer-Heeger polyacetelene Domain wall ( soliton ) mid gap state! 5

6 Large S holons Hartree-Fock :U n n U n i i n n n n n i i i i i i i i Uniform: Spin bag : Soliton : 6

7 Zaanen-Gunnarsson stripes Holons on a row Mid-gap band 7

8 Why Stripes? Xenon on graphite Discommensurations or `stripes Commensurate domains? 8

9 Why Stripes (cont.)? Mott insulator = commensurate charge density wave Doped Mott-insulator: incommensurate charge density wave => discommensuration lattice = STRIPES 9

10 Hartree-Fock Stripes are ubiquitous in doped Mott Insulators.. Nickelates Cobaltates Manganites 10

11 Static stripes in LTT 214. Tranquada et al., 375, 561 (1995) 11

12 Neutron scattering: the spin system (1995) Incommensurate antiferromagnet Spin waves Bragg peaks: Tranquada et al.: 1/8 system, Nature 1995, birth of stripes 12

13 Inelastic neutron scattering: the hourglass spin fluctuations Triplet excitons on ladders Interladder magnons 13

14 Quantum Fluctuating stripe order (1990 s) Fourier transform Neutron scattering: hour glass spin fluctuation spectrum Stripe spin wavevectors YBCO Tc=60K: Mook et al. JZ, Science 286, 251 (1999) 14

15 Spin fluctuations: weak- versus strong coupling? Quantum disordered fluctuating stripes or fermiology: RPA? 15

16 Plan 1. The prehistory: stripes in the 1990 s. 2. The revival of (fluctuating) stripe order in Electronic nematic order. 4. Stripes and pair density waves. 5. Varma s orbital currents. 16

17 Doping dependence of charge order periodicity Weak coupling: nesting interpretation. Comin, Damascelli, An. Rev. Cond.Mat.Phys. 7, 369 (2016) 17

18 STS: commensurate domains and discommensurations. Color: commensurate 4a domains. Grey:discommensurations. Mesaros et al, PNAS 113, (2016) 18

19 The BIG computational guns (2017). Simons collaboration : Hubbard model in 2D x=1/8 U/t = 8 arxiv:

20 Quantum Monte Carlo: fluctuating stripes (2017). Staggered spin correlations: Static stripes (DMRG T=0) Fluctuating QMC stripes at T = 1/12 ev QMC hourglass vs. experiment Three band Hubbard model: Devereaux group, arxiv:

21 Measuring dynamical charge correlations Information contained in dynamical charge susceptibility: Optical conductivity: To observe charge fluctuations at large momenta We need EELS or RIXS: 21

22 Electron Energy Loss EELS spectrometer Peter Abbomonte Charge response at charge ordering wavevector (BISCO) Relaxational? arxiv:

23 New ESRF ultrahigh resolution RIXS. Energy resolution: < 35 mev! 23

24 RIXS: observing the electronic stripe phonon in BISCO. Claim: RIXS measures density density dynamical susceptibility. Propagating mode with linear dispersion (large velocity: 0.6 ev A) emerging from stripe ordering wave vector. Fano resonance with regular lattice optical phonon. L. Chaix et al, preprint. 24

25 Plan 1. The prehistory: stripes in the 1990 s. 2. The revival of (fluctuating) stripe order in Electronic nematic order. 4. Stripes and pair density waves. 5. Varma s orbital currents. 25

26 Superconductors: quantum stripes J. Zaanen, Superconductivity: Quantum Stripe Search, Nature 440, 1118 (2006)

27 Correlated superconductors Ideal Bose-Einstein gas BEC cold atomic gas, BCS metallic superconductor Helium 4 superfluid High Tc superconductors Roton Strongly correlated fluid: locally like a solid. 27

28 Theoretical theoretical physics Quantum liquid crystals Kleinert s mathematics Nature 393, 550,

29 Gauge theory and quantum plasticity in Leiden Mukhin Nussinov Cvetkovic Kai Wu Beekman Ke Liu Slager Nissinen Review: A. Beekman et al., arxiv: (Physics Reports). See also: K. Liu et al., Phys. Rev. X 6, (2016), J. Nissinen et al., PRE 94, (2016); A. Beekman et al., QLC in 3D, arxiv:17xx.xxxx,. 29

30 Kosterlitz-Thouless 30

31 KTNHY: crystal melting in 2D Isolated dislocations Dislocations proliferate: hexatic liquid crystal Isolated disclination 31

32 The Topological defects Dislocation: Restores translational invariance Destroys shear rigidity Topological charge: Burgers vector Disclination: Restores rotational invariance Destroys curvature rigidity, like mass source in gravity (!) Topological charge: Franck scalar 32

33 Nelson and Halperin with hbar 33

34 Nematic order in cuprates Spontaneous anisotropy of the spin fluctuations Hinkov et al., Science 319, 597 (2008) Transport: Ando et al., PRL 88, (2002), Daou et al., Nature 463, 519 (2010) STS: Lawler et al., Nature 466, 347 (2010). 34

35 Stripy STS pictures: littered with dislocations! Mesaros et al., Science 333, 426 (2011) 35

36 The orderly limit Gas of Cooper pairs BCS superconductivity High Tc superconductors Crystal 36

37 Abelian Higgs duality in 2+1D Quantum phase dynamics (XY in 2+1D), sm Hubbard-Stratonovich auxiliary field Divide in smooth and multivalued field configurations, Relativistic short hand (magnon), H (n i ) 2 J cos( i j ) S 1 g ( ) 2 mod(2 ) i ij S g i sm MV S g i MV i sm g i MV i sm ( ) acts like Lagrange multiplier ==> conserved==>imposed by gauge field 0 A A Dual action: S gf F ia J V, Vortex current: 37

38 Disorder field theory S Theory describing vortex tangle: Long range interaction Core vs kinetic energy Hard cores dx d d ia 2 m 2 2 w 4 F F 38

39 Mott insulator = dual superconductor Superfluid/Coulomb phase Massless second sound mode = massless photon Coulomb phase Mott insulator/superconductor Holon, doublon = massive photons of relativistic superconductor. Longitudinal photon (small condensate velocity) 39

40 Dual shear superconductivity Dislocation: topological excitation uniquely associated with the restoration of translational order Shear rigidity: uniquely associated with translational order Phonons turn into stress photons = carriers of shear forces Dislocations Bose condense and they carry shear charge The orderly superconductor = the dual shear superconductor 40

41 Quantum-elasticity: basics Quantum elastic action, isotropic medium 1 /1 Shear modulus, Poisson ratio, compression modulus Describes transversal- (T) and longitudinal (L) phonon, 41

42 Dualize elasticity: stress gauge fields Quantum elasticity: S C ab u a u b S a 1 C ab b i a u a sm a i a a u MV u a Hubbard-Stratonovich = stress ( ) - strain ( ) duality, Integrate over smooth displacement fields a 0 a B a a u sm Conservation of stress imposed by stress photons B a Dual action: S ( B) C 1 ( B) ib a a J J a are dislocation currents, Burgers vector. 42

43 Phonons as stress photons Phonon action (elastic medium) in stress photons: Transversal phonon Longitudinal phonon Instantaneous stress Strain (phonon) propagator recovered through dual relation: a B a 43

44 Stress superconductors: the (dislocation) Higgs mechanism 44

45 Dynamics: glide principle Dislocation does not carry mass (or charge) Dislocation can only move along the Burger s vector: glide. Dislocation condensate decouples from compression: sound is massless and the system is a superfluid! 45

46 Collective modes of the quantum nematic Massive shear photons Transversal spectral fie Longitudinal spectral fie Rotational goldstone boson ( Torque mode ) Superfluid (second) sound. 46

47 Polariton spectrum of charged quantum smectic in 2D. Coupled transversal photon collective modes of superconducting quantum smectic arxiv:

48 Superconductivity: photons vs. stress-photons Charged elastic medium (bosonic Wigner crystal): couple in Electromagnetic gauge fields Dualize the stress-fields: L tot L AA L AB L BB Effective EM action: integrate out stress photons Shear length finite ; electromagnetic Meissner effect liberated Bare London penetration depth 1 L 2 n 2 ee 2 c 2 48

49 Adding electromagnetism: Screening current oscillations Novelty: magnetic ( )- vs shear ( ) length L L shear : Like normal gaseous superconductor L shear : Oscillating EM screening currents! 49

50 The electrically charged dual shear superconductor Quantum melt the bosonic Wigner Crystal ( preformed pairs ) The dislocation condensate is a EM Meissner phase: ODLRO of constituent bosons is not necessary! Universal form in the scaling limit of the longitudinal dielectric function (nematic): c g,l, : phonon velocities of Cooper pair crystal : plasmon frequency p : shear Higgs mass e c p : electrical screening length. : shear penetration depth 50

51 The dual shear photon and electron energy loss Weight shear photon Plasmon p 1eV Massive shear photon 30 mev 51

52 Plan 1. The prehistory: stripes in the 1990 s. 2. The revival of (fluctuating) stripe order in Electronic nematic order. 4. Stripes and pair density waves. 5. Varma s orbital currents. 52

53 The p=1/8 anomaly in cuprates 53

54 A 2D BKT superconducting transition in 3D!? La Ba CuO 4 Tranquada et al, PRL 99, (2007) 54

55 The pair density wave: frustrating the c-axis Josephson coupling e.g. Fradkin et al., arxiv:

56 Pair density waves: microscopic explanation? - Arises in weak coupling in a large magnetic field (FFLO) but a no-go in zero field: strong coupling phenomenon. - No go theorem for bosons: the order implies nodes in the vacuum wave function. Fermion signs are a necessary condition. No sign of it in big gun (Simons collaboration) numerics: these stripes do not even superconduct! 56

57 Direct observation of PDW using a Josephson tunneling tip. Davis group, Nature 523, 343 (2016) 57

58 Plan 1. The prehistory: stripes in the 1990 s. 2. The revival of (fluctuating) stripe order in Electronic nematic order. 4. Stripes and pair density waves. 5. Varma s orbital currents. 58

59 Spontaneous diamagnetic currents Varma Predicted by Chandra Varma, first measured by neutron scattering in 2005: see Bourges et al. arxiv: ,

60 Extra evidences: second harmonic generation. 60

61 Loop current excitations Greven et al., Nature 468, 283 (2010). 61

62 Loop currents versus pseudogap temperature Claim: onset loop current order coincident with pseudogap temperature (neutrons, ultrasound, SHG). 62

63 Loop currents intertwined with stripe order in La2-xSrxCuO4 Bourges et al., PRL 105, (2010) 63

64 Holographic current phases 64

65 Krikun s holographic stripes 65

66 Empty 66