1. Vivien Zapf National High Magnetic Field Laboratory Los Alamos National Lab Bose-Einstein Condensation and Magnetostriction in NiCl 2 -4SC(NH 2 ) 2

2. National High Magnetic Field Laboratory Pulsed magnets to 75 T M(H),  (H), magnetostriction, ESR, optics, etc 300 T single-turn (  s pulse) Coming soon: 60 T and 100 T long pulse magnets (2 s)

3. 300 T single-turn nondestructive magnet Chuck Mielke, Ross McDonald, et al Capacitor bank pulses a short (  s) mega-amp current pulse to achieve ultra high magnetic fields. 10 mm long. 10mm ID.  2  s rise time,  dB/dt ≈ 10 8 Ts -1 3 orders of magnitude faster than standard short pulse magnets at the NHMFL. Low inductance capacitor bank. L = 18 nH, C = 144  F, V = 60 kV, E = 259kJ. Single turn magnet coil, L = 7 nH. Peak current 4 MA. 1st megagauss shot (February 8th 2005)

4. Ni Cl J plane /k B = 0.17 K c a a Ni S = 1 Cl NiCl 2 -4SC(NH 2 ) 2 (DTN) No Haldane gap Other BEC compounds All Cu spin-1/2 dimers TlCuCl 3 (wrong symmetry) BaCuSi 2 O 6 (3D -> 2D crossover) CsCuCl 4 J chain /k B = 1.72 K

5. H c2 = 12.6 TH c1 = 2.1 T k z =0 (FM) k z =  (AFM) Antiferromagnetic exchange NiCl 2 -4SC(NH 2 ) 2 (DTN) Spin-orbit coupling Zeeman term S z = +1 H D~8K E S z = 0 S z = -1 Ni 2+ S = 1: Triplet split by spin-orbit coupling Tetragonal lattice S z = +1 S z = 0 S z = -1 D~8K H c SzSz a S=1 M. Kenzelmann et al

6. Ni spins M z (x10 3 emu/mol) A. Paduan-Filho et al, Phys. Rev. B 69, 020405(R) (2004) H c a BEC/AFM H c1 H c2

7. spin language boson language H ~ M z ~ N (# of bosons) S z = 1 D H || c E or T(K) XY AFM order AFM FM b or N 0 1 Boson filling fraction/Number of bosons Occupied bosonEmpty site Note: Approximate theory neglects S z = -1 state. Complete theory: H.-T. Wang and Y. Wang, Phys. Rev. B 71, 104429 (2005) K.-K. Ng and T.-K. Lee, cond-mat/0507663 Example: treat upper state as an energetically unfavorable double occupancy state S z = -1

8. AFM exchange Spin-orbit coupling Zeeman term Repulsion (2 nd order in N) hoppingnumber operator Spin language Hamiltonian Boson language Hamiltonian (neglecting S z = -1 term) S + -> b + (boson creation operator) (Hardcore Constraint: One boson per site)

9. SpinsBosons |S z = 1> stateoccupied boson |1> |S z = 0> stateunoccupied boson |0> Order parameter:Order parameter: Staggered magnetization M x Boson creation operator b † = S + Magnetic fieldChemical potential (H ~ N ~  ) Boson mapping This also works for S=1/2: |↑> = occupied, |↓> = unoccupied Spins are prevented from obeying fermion statistics since no real- space overlap between states allowed

10. Why do the bosons obey number conservation? Tetragonal symmetry of crystal: U(1) symmetry of spins. Symmetric under rotation by  in a-b plane Hamiltonian must be independent of  Rotation by  : b † → b † e i   and  b → be -i  H can only contains b † b or bb † terms ( = number operator) (b † e i   be -i    b † b = N Hamiltonian commutes with boson number operator Boson number is conserved. a b |S z = 1> = occupied boson |S z = 0> = unoccupied boson b † = boson creation operator H

11. X Energy landscape of spins S  Symmetry in plane X Ni H XY model Ising model E(  ) Ni H Boson number only conserved in statistical average Spin fluctuations, lattice fluctuations limit super current lifetimes Small corrections to XY model: DM interactions, dipole- dipole, and spin-orbit coupling Limitations

12. Experimental Tests for BEC S z = 1 D H || c E or T(K) XY AFM order 0 N MzMz where

13. H T H c1 H c2 Quantum Phase Transition (BEC) amplitude driven (d=3, z=2) XY AFM Thermal phase transition (XY AFM) phase driven (d=3, z=1) Quantum Phase Transition: universality class of a BEC BaCuSi 2 O 6 (3D -> 2D crossover) CsCuCl 4 NiCl 2 -4SC(NH 2 ) 2 3D BEC:  = 3/2 3D Ising:  = 2 2D “BEC”:  = 1

14. Sapphire platform C=Q/  T C=  /  Quasi-AdiabaticThermal Relaxation Time Measuring Specific Heat

15. C/T (mJ/mol K 2 ) Specific Heat 0 2 4 6 8 10 00.40.81.21.6 T (K ) H = 10 T 0 0.2 0.4 0.6 0.8 1 1.2 02468101214 Magnetocaloric effect Specific heat H (T) V. S. Zapf, D. Zocco, B. R. Hansen, M. Jaime, N. Harrison, C. D. Batista, M. Kenzelmann, C. Niedermayer, A. Lacerda, and A. Paduan-Filho, Phys. Rev. Lett., 96, 077204 (2006)

16. H – H c  T N  H – H c  T N    2 (3D Ising magnet) 3/2 (3D BEC) 1 (2D “BEC”)

17. H-H c1 = aT   = 1.47 ± 0.10  = 1.5 BEC)  Ising magnet 2D BEC 3D BEC Windowing Technique: see V. S. Zapf, et al, Phys. Rev. Lett., 96, 077204 (2006) S. Sebastian et al, Phys. Rev. B 72, 100404(R) (2005) 0 0.5 1 1.5 2 2.5 00.20.40.60.81 T max /1.2 K A. Paduan Filho, unpublished

18. T c (K) Predictions (3-level system): Inelastic Neutron Diffraction and magnetization : C. D. Batista, M. Tsukamoto, N. Kawashima, in progress V. S. Zapf et al, Phys. Rev. Lett. 96, 77204 (2006). predicted H c1 predicted H c2 Spin wave theory

19. Magnetostriction Capacitance Titanium Dilatometer (design by G. Schmiedeshoff) CuBe spring V. Correa, V.S. Zapf, T. Murphy, E. Palm, S. Tozer, A. Lacerda, A. Paduan-Filho  L/L (%) H c1 H c2 T = 25 mK H || c LcLc LaLa H c a

20. J J J J J J J J J = 1.7 K J = 0.17 K  L/L (%) Magnetostriction H c1 H c2 LcLc LaLa Ni ++ c a H

21.  L/L (%) Magnetostriction H c1 H c2 LcLc LaLa predicted H c1 predicted H c2 T c (K) Summary BEC confirmed experimentally via H-H c1 ~ T  and M ~ T  Magnetostriction effect distorts phase diagram with increasing magnetic field

22. ? Future work: Frustration-induced symmetry change?

23. Acknowledgements LANL Cristian Batista (T-11 theory group) NHMFL-LANL Diego Zocco Marcelo Jaime Neil Harrison Alex Lacerda NHMFL-Tallahassee Victor Correa (Magnetostriction) Tim Murphy Eric Palm Stan Tozer Universidade de Sao Paulo, Brazil Armando Paduan-Filho (Crystal growth and Magnetization) Paul Scherrer Institute and ETH, Zürich, Switzerland B. R. Hansen M. Kenzelmann (Neutron scattering) University of Tokyo Mitsuaki Tsukamoto Naoki Kawashima (Monte Carlo Simulations) Occidental College George Schmiedeshoff (dilatometer design) NSF NHMFL DOE