Spin-1/2 Chains in Uniform and Staggered Fields

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

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

4. Magnetic Neutron Scattering The scattering cross section is proportional to the Fourier transformeddynamic spin correlation function NIST Center for Neutron Research

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

6. 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)

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

8. Neutron Scattering Stone et al. (2003). Exact two-spinon cross-section 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. Spinons in magnetized spin- ½ chain Broholm et al. (2002)

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

13. Uniform Spin ½ chain 8.7 T || ^ 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. 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

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

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

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

21. 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. 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)

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

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

25. 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=p and q=p-2pm • Neutron scattering data in excellent agreement with finite chain calculations Publications and viewgraphs at http://www.pha.jhu.edu/~broholm/homepage/

26. 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=p and p-2p 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/