Spinons, Solitons, and Breathers in Quasi-One-Dimensional Magnets

1. Spinons, Solitons, and Breathers in Quasi-One-Dimensional Magnets Collin Broholm Johns Hopkins University NIST Center for Neutron Research PPHMF-IV

2. condensed matter Standard model New Physics • Single energy scale controls all • Simple Linear Response to • Impurities • Electric fields • Magnetic fields • Temperature • No phase transition below • melting point • Low energy degrees of freedom • Non-linear Response to • Impurities • Electric fields • Magnetic fields • Temperature • Phase transitions • below the melting point

3. Spin degrees of freedom Magnetism Ti V Cr Mn Fe Co Ni Cu Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm U Np Pu Am Cm

4. No interactions Magnetic “Ideal gas” 2S+1 levels FeBr(C44H28N4) Susceptibility data for paramagnetic salt

5. Coulomb + Pauli = Heisenberg S=1/2 S=1/2 S=1/2 S=1/2 |J| |J| Coulomb interactions plus Pauli principle split 4-fold spin degeneracy The level scheme is reproduced by Heisenberg Exchange Hamiltonian Tripletgnd. State:J < 0 Singlet gnd. State:J > 0

6. Interactions cooperative phenomena Ferromagnetic EuO Antiferromagnetic KNiF3

7. Overview • Introduction • Why we like magnets • Neutron Scattering • Magnetic spaghetti • Strange excitations in a spin chain • The experimental result • The theoretical explanation: Spinons • Are spinons for real? • Theoretical expectations in a field • High field neutron scattering • Can we tie them together? • Experimental observation of bound spinons • Quantum sine-Gordon model • Conclusions and outlook

8. Acknowledgements D. H. Reich JHU G. Aeppli UCL C. P. Landee Clarke University M. M. Turnbull Clarke University M. Kenzelmann JHU & NIST M. B. Stone Penn State University Y. Chen LANL D. C. Dender NIST Y. Qiu NIST & Univ. Maryland K. Lefmann Risø National Lab C. Rische Univ. of Copenhagen

9. Inelastic Magnetic Neutron Scattering • We can measure dispersion relations • We determine structure through transition rate

10. pi pf ħQ SPINS cold neutron spectrometer at NCNR

11. Spin waves in antiferromagnet From A. Zheludev’s web page La2CuO4 Coldea et al. PRL (2001)

12. Can quantum fluctuations break order? 1. Assume Neel order, derive spin wave dispersion relation 2. Calculate the reduction in staggered magnetization due to quantum fluctuations 3. If then Neel order is an inconsistent assumption There can be no Neel order in one dimension

13. Copper pyrazine dinitrate C/T (J/mol/K2) //a T2 (K2) Hammar et al. (1999) Cu(C4H4N2)(NO3)2

14. Neutron Scattering from Spin-1/2 chain Stone et al., PRL (2003)

15. Disintegration of a spin flip Spinon Spinon

16. Fermions in spin ½ chain Uniform spin-1/2 chain (XY case for simplicity) Jordan-Wigner transformation Diagonalizes H|| e/J Non interacting fermionic lattice gas q (p)

17. From band-structure to bounded continuum J e/J w h Q (p) q (p)

18. Neutron Scattering Stone et al. (2003). Exact two-spinon cross-section Karbach et al. 2000

19. Neutron Data & Two-Spinon Cross section 1.0 Stone et al., PRL (2003)

20. Are spinons for real?

21. Spinons in magnetized spin- ½ chain Broholm et al. (2002)

22. Uniform Spin ½ chain 0.0 T Stone et al. (2003)

23. Uniform Spin ½ chain 8.7 T || ^ Stone et al. (2003)

24. Neutron Scattering Pentium Scattering Stone et al. (2003)

25. Can we tie them together?

26. Why staggered field yields bound states Zero field state quasi-long range AFM order Without staggered field distant spinons don’t interact With staggered field solitons separate “good” from “bad” domains, which leads to interactions and bound states

27. Spin-½ chain with two spins per chain unit Landee et al. (1986) CuCl2.2(dimethylsulfoxide) Oshikawa and Affleck (1997) The staggered field is given by

28. 3 2 ħw (meV) 1 0 0 0.2 0.4 0.6 0.8 1 H=0 T Kenzelmann et al. (2003)

29. 3 2 ħw (meV) 1 0 0 0.2 0.4 0.6 0.8 1 H=11 T Kenzelmann et al. (2003)

30. Bound states from 2-spinon continuum Kenzelmann et al. (2003)

31. Sine-Gordon mapping of spin-1/2 chain Effective staggered + uniform field spin hamiltonian Spin operators are represented through a phase field relative to incommensurate quasi-long-range order with Lagrangian density • This is sine-Gordon model with interaction term proportional to hs • Spectrum consists of • Solitons, anti-solitons • Breather bound states Oshikawa and Affleck (1997)

32. Bound states from 2-spinon continuum Breathers n=1,2 and possibly 3 Soliton, M Kenzelmann et al. (2003)

33. Testing sine-Gordon predictions Theory by Essler-Tsvelik (1998) Cu-Benz Dender et al. (1997). Neel order Neel order 0 Kenzelmann et al. (2003)

34. What we learned about spin-1/2 chains • A quantum liquid without spin order at T=0 • Fundamental excitations are spinon pairs • Spinons form a fermionic liquid with field dependent chemical potential • A staggered field confines spinons • “masses” and structure factors consistent with sine-Gordon solitons and breathers Publications and viewgraphs at http://www.pha.jhu.edu/~broholm/homepage/

35. Are there quantum liquids for D>1 What is special about D=1? Order in one part of lattice does not constrain surroundings Maybe, when there is frustration and/or low connectivity

36. Better Instruments at Existing Sources MACS Cold Neutron Spectrometer at NCNR To be completed June 2005

37. New and Brighter Neutron Sources US 1.4 MW Spallation Neutron Source To be completed in 2006