1. Probing Matter with X-Rays and Neutrons Tallahassee, May 10-12, 2005 Magnetic order refinement in high field Outline Magnetic field as a source of Luttinger liquid –alternate route to “quantum” criticality Enhancing weak antiferromagnetism in coupled Haldane chains Magnetic order refinement in high field: challenges and caveats Igor Zaliznyak Neutron Scattering Group, Brookhaven National Laboratory

2. Probing Matter with X-Rays and Neutrons Tallahassee, May 10-12, 2005 Haldane chain in magnetic field. 3,5,…-particle continuum H=0H~Hc H>Hc ? particles holes particles Macroscopic quantum phase in the string operator at H>Hc results in the shift in q-space between fermions and magnons. Haldane(Quantum) Critical Luttinger Liquid

3. Probing Matter with X-Rays and Neutrons Tallahassee, May 10-12, 2005 Haldane chain in magnetic field. L.P. Regnault, I. Zaliznyak, J.P. Renard, C. Vettier, PRB 50, 9174 (1994). Luttinger Liquid

4. Probing Matter with X-Rays and Neutrons Tallahassee, May 10-12, 2005 Coupled Haldane chains in (Cs,Rb)NiCl 3 : weak antiferromagnetic order in zero field CsNiCl 3 : J = 2.3 meV = 26 K J  = 0.03 meV = 0.37 K = 0.014 J D = 0.002 meV = 0.023 K = 0.0009 J 3D magnetic order below T N = 4.84 K ≈ 1  B ≈ 1  B

5. Probing Matter with X-Rays and Neutrons Tallahassee, May 10-12, 2005 Coupled Haldane chains in (Cs,Rb)NiCl 3 in magnetic field Field along easy axis: spin-flop + increase in magnetic order Field perpendicular to easy axis: no spin-flop, just increase in magnetic order

6. Probing Matter with X-Rays and Neutrons Tallahassee, May 10-12, 2005 Coupled Haldane chains: magnetic field enhances antiferromagnetic order. Hc Spin-flop

7. Probing Matter with X-Rays and Neutrons Tallahassee, May 10-12, 2005 Measuring the field dependence of magnetic Bragg peaks: challenges and caveats. Equivalent “Friedel” reflections have different intensities –non-uniform illumination of absorbing sample is a source of the dominant systematic error –sample/wavelength optimization is vital Realignment of spins in the spin-flop process greatly impacts intensities –very sensitive to magnetic field orientation with respect to crystallographic “easy” axis –sensitive to sample mosaicity –different bias for different reflections

8. Probing Matter with X-Rays and Neutrons Tallahassee, May 10-12, 2005 Spin-flop is nothing new and is well understood J. W. Lynn, P. Heller, N. A. Lurie, PRB 16 (1977). ψ is misalignment of the magnetic field from the easy axis φ is corresponding misalignment of staggered magnetization Eq. (14) is a venerable expression with long history dating back to L. Neel (J. Lynn et. al.) It also is general: goes beyond simple quasiclassical approximation

9. Probing Matter with X-Rays and Neutrons Tallahassee, May 10-12, 2005 Spin realignment: powder in magnetic field

10. Probing Matter with X-Rays and Neutrons Tallahassee, May 10-12, 2005 Brave attempt: refine on powder Red: H = 6.8 T Black: H = 0 T Red: H = 6.8 T Black: H = 1 T

11. Probing Matter with X-Rays and Neutrons Tallahassee, May 10-12, 2005 The right way: do the real thing 15 T magnet on D23 @ ILL (courtesy B. Grenier)

12. Probing Matter with X-Rays and Neutrons Tallahassee, May 10-12, 2005 Cavalry approach: just follow the Bragg peaks Not satisfactory! H perpendicular to the easy axis single-domenization

13. Probing Matter with X-Rays and Neutrons Tallahassee, May 10-12, 2005 Full refinement in mangetic field

14. Probing Matter with X-Rays and Neutrons Tallahassee, May 10-12, 2005 Full refinement in mangetic field Haldane gap in CsNiCl 3

15. Probing Matter with X-Rays and Neutrons Tallahassee, May 10-12, 2005 Compare to LCO superconductors B. Khaykovich, Y. S. Lee, et. al., PRB 66 (2002). E. Demler, S. Sachdev, and Y. Zhang, PRL 87 (2001).

16. Probing Matter with X-Rays and Neutrons Tallahassee, May 10-12, 2005 Summary and conclusions Magnetic field brings about fascinating new phases –Luttinger-liquid (quantum) critical state –tunes antiferromagnetism in weakly ordered systems Refining field dependence of magnetic order is a challenging experimental task –field-dependent variation of intensity is often smaller than systematic (not statistical!) errors –only one reciprocal lattice (hkl) plane is typically available –spin realignment is often a complication: serious science requires serious refinement This work was carried out under Contract DE-AC02-98CH10886, Division of Materials Sciences, US Department of Energy. The work on SPINS was supported by NSF through DMR-9986442

17. Probing Matter with X-Rays and Neutrons Tallahassee, May 10-12, 2005 Acknowledgements: thanks go to S. V. Petrov B. Grenier and L.-P. Regnault R. Erwin and C. Quang C. Broholm A. Savici / U. Maryland

18. Probing Matter with X-Rays and Neutrons Tallahassee, May 10-12, 2005 What is quantum spin liquid? What is liquid? − no shear modulus − no elastic scattering = no static density-density correlation ‹ρ q (0)ρ -q (t)› → 0 t → ∞ What is quantum liquid? − all of the above at T → 0 (i.e. at temperatures much lower than interactions between the particles in the system) Quantum liquid state for a system of Heisenberg spins H = J ||  S i S i +  || + J   S i S i    D  (S i z ) 2 no static spin correlations ‹ S q α (0) S - β q (t)› → 0, i.e. ‹S q α (0)S - β q (t)› = 0 hence, no elastic scattering (e.g. no magnetic Bragg peaks) t → ∞ J || /J  >> 1 (<<1) parameterize quasi-1D (quasi-2D) case

19. Probing Matter with X-Rays and Neutrons Tallahassee, May 10-12, 2005 What would be a “spin solid” S >> 1 Heisenberg antiferromagnet with classical spins, S >> 1 − and quasiparticles that are gapless Goldstone magnons  (q) = 2J(S(S+1)) 1/2 sin (q)  (q)/J/(S(S+1)) 1/2 − has Neel-ordered ground state with elastic Bragg scattering at q=π

20. Probing Matter with X-Rays and Neutrons Tallahassee, May 10-12, 2005 1D quantum spin liquid: Haldane spin chain − short-range-correlated “spin liquid” Haldane ground state S = 1 Heisenberg antiferromagnetic chain with S = 1  (q)/J/(S(S+1)) 1/2 − quasiparticles with a gap  ≈ 0.4J at q=π  2 (q) =  2 + (cq) 2  Quantum Monte-Carlo for 128 spins. Regnault, Zaliznyak & Meshkov, J. Phys. C (1993)

21. Probing Matter with X-Rays and Neutrons Tallahassee, May 10-12, 2005 weak interaction  2D quantum spin liquid: a lattice of frustrated dimers M. B. Stone, I. Zaliznyak, et. al. PRB (2001) (C 4 H 12 N 2 )Cu 2 Cl 6 (PHCC) − singlet disordered ground state − gapped triplet spin excitation strong interaction 

22. Probing Matter with X-Rays and Neutrons Tallahassee, May 10-12, 2005 How do neutrons measure quasiparticles. Typical geometry of a scattering experiment, (a) elastic, (b) inelastic. I. A. Zaliznyak and S.-H. Lee, in Modern Techniques for Characterizing Magnetic Materials, Ed. Y. Zhu, Springer (2005)

23. Probing Matter with X-Rays and Neutrons Tallahassee, May 10-12, 2005 Spin-quasiparticles in Haldane chains in CsNiCl 3 J = 2.3 meV = 26 K J  = 0.03 meV = 0.37 K = 0.014 J D = 0.002 meV = 0.023 K = 0.0009 J 3D magnetic order below T N = 4.84 K unimportant for high energies q 0π

24. Probing Matter with X-Rays and Neutrons Tallahassee, May 10-12, 2005 Spin-quasiparticles in Haldane chains in CsNiCl 3

25. Probing Matter with X-Rays and Neutrons Tallahassee, May 10-12, 2005 Spectrum termination point in CsNiCl 3 I. A. Zaliznyak, S.-H. Lee, S. V. Petrov, PRL 017202 (2001)

26. Probing Matter with X-Rays and Neutrons Tallahassee, May 10-12, 2005 Quasiparticle spectrum termination line in PHCC max{E 2-particle (q)} min{E 2-particle (q)} E 1-particle (q) Spectrum termination line

27. Probing Matter with X-Rays and Neutrons Tallahassee, May 10-12, 2005 Summary and conclusions Quasiparticle spectrum termination at E > 2  is a generic property of the quantum Bose (spin) fluids –observed in the superfluid 4 He –observed in the Haldane spin chains in CsNiCl 3 –observed in the 2D frustrated quantum spin liquid in PHCC A real physical alternative to the ad-hoc “excitation fractionalization” explanation of scattering continua Implications for the high-Tc cuprates: spin gap induces disappearance of the coherent quasiparticles at high E This work was carried out under Contract DE-AC02-98CH10886, Division of Materials Sciences, US Department of Energy. The work on SPINS was supported by NSF through DMR-9986442