1. Hole-Doped Antiferromagnets: Relief of Frustration Through Stripe Formation John Tranquada International Workshop on Frustrated Magnetism September 13 - 17, 2004 Montauk, New York

2. Outline Early ideas about La 2 CuO 4 : quantum spin liquid Reality: La 2 CuO 4 is a good antiferromagnet Hole doping frustrates commensurate Néel order Formation of charge stripes reduces magnetic frustration (and lowers KE) Are stripe correlations relevant to superconducting cuprates?

3. Anderson’s RVB proposal for La 2 CuO 4 PW Anderson, Science 235, 1196 (1987) “The oxide superconductors, particularly those … base on La 2 CuO 4, … tend … to occur near a metal-insulator transition …. This insulating phase is proposed to be the long-sought ‘resonating-valence-bond’ state or ‘quantum spin liquid’ hypothesized in 1973. This insulating magnetic phase is favored by low spin, low dimensionality, and magnetic frustration.” PW Anderson, Mat. Res. Bull. 8, 153 (1973) “Resonating Valence Bonds: A New Kind of Insulator” Proposal for S=1/2 on a triangular lattice

4. Local RVB singlets Kivelson, Rokhsar, and Sethna, PRB 35, 8865 (1987) Existence of a spin gap leads to Bose condensation of doped holes Requires dynamic modulation of superexchange by phonons Reality: Cu-O bonds are stiff

5. Frustration by AF next-nearest-neighbor exchange Sachdev and Read, Int. J. Mod. Phys. B 5, 219 (1991) spin-Peierls order

6. Reality: An isolated CuO 2 plane would order at T = 0 S(q 2D ) ~ 1 / [(q 2D ) 2 +  -2 ]  = spin-spin correlation length  -1 ~ exp(-  J/T) J = 135 meV ~ 1500 K Theory: Chakravarty, Halperin,+Nelson, PRB 39, 2344 (1989) Hasenfratz+Niedermayer, PL B 268, 231 (1991) Expt: Birgeneau et al., JPCS 56, 1913 (1995)    as T  0

7. Spin waves in La 2 CuO 4 : No sign of frustration J = 146 meV J c = 61 meV at T = 10K J’ = J’’ = 2 meV Coldea et al., PRL 86, 5377 (2001)

8. Typical Phase Diagram: La 2-x Sr x CuO 4

9. Doping kills LRO but not SRO Phase diagram for La 2-x Sr x CuO 4 and Y 1-2x Ca 2x Ba 2 Cu 3 O 6 p sh = x Local magnetic field at T = 1 K measured by muon spin rotation Niedermayer, Budnick, et al. PRL 80, 3843 (1998)

10. Magnetic dilution Destruction of LRO requires 40% dilution! Experimental results for La 2 Cu 1-z (Zn,Mg) z O 4 Vajk et al., Science 295, 1691 (2002)

11. Competing Interactions Motion of hole lowers kinetic energy but costs superexchange energy

12. One hole in an antiferromagnet Dispersion measured by angle-resolved photoemision in Sr 2 CuO 2 Cl 2 Wells et al., PRL 74, 964 (1995). Bandwidth for occupied states is ~ 2J << 4t

13. Hole segregation to antiphase domain walls 1D model 2D extrapolation

14. Charge and spin stripe order

15. Early stripe predictions Zaanen and Gunnarson Phys. Rev. B 40, 7391 (1989) Hubbard model Mean-field solution White and Scalapino, PRL 80, 1272 (1998) t-J model Density matrix renormalization group

16. Alternative: Frustrated Phase Separation Löw, Emery, Fabricius, and Kivelson, PRL 72, 1918 (1994) Competing interactions result in striped and checkerboard phases Analysis of t-J model by Emery and Kivelson: Holes tend to phase separate! t-J model lacks long-range part of Coulomb interaction Long-range Coulomb repulsion frustrates phase separation

17. Stripe ORDER seen only in special cases 1/8 problem LTT LTO

18. Antiferromagnetic “resonance” in SC cuprates T-dependent resonance observed by Keimer and coworkers in  YBa 2 Cu 3 O 6+x bilayer  Bi 2 Sr 2 CaCu 2 O 8+  bilayer  Tl 2 Ba 2 CuO 6+  single layer (But not in La 2-x Sr x CuO 4 ) YBa 2 Cu 3 O 7 Mook et al., PRL 70, 3490 (1993)

19. Spin fluctuations in YBCO do not look like spin waves Bourges et al., Science 288, 1234 (2000) YBa 2 Cu 3 O 6.85 Bourges et al., PRL 90, 147202 (2002) La 1.79 Sr 0.31 NiO 4

20. Large crystals of La 1.875 Ba 0.125 CuO 4 studied on MAPS Diameter = 8 mm Length = 140 mm Mass > 40 g MAPS spectrometer at ISIS Crystals grown at BNL by Genda Gu

21. Constant-energy slices through magnetic scattering Stripe-ordered La 1.875 Ba 0.125 CuO 4 T = 12 K T c < 6 K

22. 24 meV 34 meV 66 meV 105 meV h k La 2-x Ba x CuO 4 x = 1/8 Normal state with Stripe order YBa 2 Cu 3 O 6.6 Superconducting state Hayden et al., Nature 429, 531 (2004)

23. Comparison of LBCO and YBCO Magnetic excitation spectra look the same! (E LBCO ~ 1.5 E YBCO )  Implies same mechanism at work in both Excitations in LBCO associated with stripes  Suggests stripe correlations present in YBCO “Resonance peak” is just the most visible part of the spectrum  Present even in non-superconducting LBCO

24. How can we understand the stripe excitation spectrum?

25. Comparison with ladder model 2-leg, AF spin ladder J = 100 meV two domains

26. Evidence for spin gap

27. Better theoretical models Weakly-coupled stripes Vojta and Ulbricht cond-mat/0402377 Uhrig, Schmidt, and Grüninger cond-mat/0402659 included 4-spin cyclic exchange Mean-field stripe order + fluctuations Seibold and Lorenzana cond-mat/0406589 dispersion is more 2D-like

28. Universal Spectrum + Spin gap LSCO(?) YBCO(?)

29. Conclusions Stripes form due to competing interactions (frustration) Magnetic excitation spectrum of a stripe-ordered cuprate is same as in good superconductors  Suggests a universal spectrum Quantum spin gap of two-leg ladders may be important for hole pairing LBCO results: Nature 429, 534 (2004)

30. Collaborators BNL Hyungje Woo Genda Gu Guangyong Xu IMR, Tohoku Univ. Masa Fujita Hideto Goka Kazu Yamada ISIS Toby Perring

31. “Resonance” effects can be incommensurate LSCO x = 0.16 Christensen et al. cond-mat/0403439 Superconducting Normal state Effect of magnetic field in LSCO x=0.18 PRB 69, 174507 (2004)

32. Expected scattering patterns in reciprocal space

33. Single-domain YBa 2 Cu 3 O 6.85 Hinkov et al., Nature 430, 650 (2004) E = 35 meV E res = 41 meV