1. Collin Broholm * Johns Hopkins University and NIST Center for Neutron Research Y. Chen LANL M. Kenzelmann JHU & NIST C. P. LandeeClarke University K. Lefmann Risø National Lab Y. QiuNIST & Univ. Maryland D. H. ReichJHU C. Rische Univ. of Copenhagen M. B. StonePenn State University M. M. TurnbullClarke University Spin-1/2 Chains in Uniform and Staggered Fields * Supported by the National Science Foundation

2. SCTMC 8/5/03 Spin-1/2 chain preliminaries  Simple Hamiltonian; complex properties  Good model materials and experimental tools  Integrable through Bethe Ansatz: –Ground state energy –Equal time correlation function  Quantum critical  Exact results for dynamic spin correlations

3. SCTMC 8/5/03 Copper pyrazine dinitrate Hammar et al. (1999) T 2 (K 2 ) C/T (J/mol/K 2 ) //a Cu(C 4 H 4 N 2 )(NO 3 ) 2

4. SCTMC 8/5/03 Magnetic Neutron Scattering The scattering cross section is proportional to the Fourier transformed dynamic spin correlation function NIST Center for Neutron Research

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

6. SCTMC 8/5/03 Fermions in spin ½ chain Jordan-Wigner transformation Uniform spin-1/2 chain (XY case for simplicity) Diagonalizes H || Non interacting fermionic lattice gas q (  )  /J

7. SCTMC 8/5/03 From band-structure to bounded continuum q (  )  /J Q (  ) J  

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

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

10. Spin-½ chain in uniform field

11. SCTMC 8/5/03 Spinons in magnetized spin- ½ chain Broholm et al. (2002)

12. SCTMC 8/5/03 0.0 T Uniform Spin ½ chain Stone et al. (2003)

13. SCTMC 8/5/03 8.7 T Uniform Spin ½ chain Stone et al. (2003)  ||

14. Diagonalization of spin-½ chain in a field + Stone et al. (2003)

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

16. Spin-½ chain in staggered field

17. SCTMC 8/5/03 Spin-½ chain with two spins per chain unit The staggered field is given by Landee et al. (1986) Oshikawa and Affleck (1997) CuCl 2. 2(dimethylsulfoxide)

18. H=0 T 0 0.2 0.4 0.6 0.8 1 0 1 2 3 ħ   meV) Kenzelmann et al. (2003)

19. H=11 T 0 0.2 0.4 0.6 0.8 1 0 1 2 3 ħ   meV) Kenzelmann et al. (2003)

20. SCTMC 8/5/03 Bound states from 2-spinon continuum Kenzelmann et al. (2003)

21. SCTMC 8/5/03 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

22. SCTMC 8/5/03 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 h s Spectrum consists of Solitons, anti-solitons Breather bound states Oshikawa and Affleck (1997)

23. SCTMC 8/5/03 Bound states from 2-spinon continuum Soliton, M Breathers n=1,2 and possibly 3 Kenzelmann et al. (2003)

24. SCTMC 8/5/03 Testing sine-Gordon predictions Kenzelmann et al. (2003) Neel order Cu-Benz Dender et al. (1997). 0 Neel order Theory by Essler-Tsvelik (1998)

25. SCTMC 8/5/03 Conclusions: S=½ Chain in Uniform Field  H=0: data well described by exact two-spinon continuum scattering  H>0: –Incommensurate correlations from shifted fermi points –Gapless excitation at q=  and q=  -2  m –Neutron scattering data in excellent agreement with finite chain calculations Publications and viewgraphs at http://www.pha.jhu.edu/~broholm/homepage/

26. SCTMC 8/5/03 Conclusions: S=½ Chain in Staggered Field  Staggered g-tensor and DM interaction inherent to multi- atom cell and produce effective staggered field  Staggered field yields bound states  Features described by sine-Gordon model: –Relative energies of bound states at q=  and  -2  m –Relative intensities of breather excitations –Field dependent incommensurability  Excellent experimental realization of quantum sine-Gordon model Publications and viewgraphs at http: // www.pha.jhu.edu / ~broholm / homepage /