Vivien Zapf National High Magnetic Field Laboratory Los Alamos National Lab

1. Bose-Einstein Condensation and Magnetostriction in NiCl2-4SC(NH2)2 Vivien ZapfNational High Magnetic Field LaboratoryLos Alamos National Lab

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

3. 300 T single-turn nondestructive magnetChuck 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. 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) • 2 s rise time, • dB/dt ≈ 108 Ts-1 3 orders of magnitude faster than standard short pulse magnets at the NHMFL.

4. NiCl2-4SC(NH2)2 (DTN)No Haldane gap c a a Ni Cl Ni S = 1 • Other BEC compoundsAll Cu spin-1/2 dimers • TlCuCl3 (wrong symmetry) • BaCuSi2O6(3D -> 2D crossover) • CsCuCl4 Cl Jchain/kB = 1.72 K Jplane/kB = 0.17 K

5. Zeeman term Antiferromagnetic exchange E kz=0 (FM) Sz = -1 Sz = +1 D~8K kz=p (AFM) H Sz = 0 Hc1 = 2.1 T Hc2 = 12.6 T Sz = -1 Sz = +1 D~8K c H Sz = 0 NiCl2-4SC(NH2)2 (DTN) Sz Spin-orbitcoupling a Ni2+ S = 1: Triplet split by spin-orbit coupling Tetragonal lattice S=1 M. Kenzelmann et al

6. BEC/AFM Mz (x103 emu/mol) c H a Hc1 Hc2 Ni spins A. Paduan-Filho et al, Phys. Rev. B 69, 020405(R) (2004)

7. Note: Approximate theory neglects Sz = -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 spin language boson language Empty site Occupied boson Sz = -1 E or T(K) H ~ Mz ~ N (# of bosons) Sz = 1 D FM AFM XY AFM order H || c b or N 1 0 Boson filling fraction/Number of bosons

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

9. Boson mapping Spins Bosons |Sz = 1> state occupied boson |1> |Sz= 0> state unoccupied boson |0> Order parameter: Order parameter:Staggered magnetization Mx Boson creation operator b† = S+ Magnetic field Chemical potential (H ~ N ~ m) 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. H Why do the bosons obey number conservation? |Sz = 1> = occupied boson |Sz = 0> = unoccupied bosonb† = boson creation operator b Tetragonal symmetry of crystal: U(1) symmetry of spins. Symmetric under rotation by fin a-b plane a • Hamiltonian must be independent of f • Rotation by f: b† → b† eifand b → be-if • H can only contains b†b or bb† terms ( = number operator) • (b† eif) (be-if) = b† b = N • Hamiltonian commutes with boson number operator • Boson number is conserved.

11. S f Symmetry in plane Energy landscape of spins Ni X XY model E(f) H Ni Ising model X H Limitations • 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

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

13. Quantum Phase Transition: universality class of a BEC Thermal phase transition (XY AFM) phase driven (d=3, z=1) T XY AFM H Hc2 Hc1 Quantum Phase Transition (BEC)amplitude driven (d=3, z=2) • BaCuSi2O6(3D -> 2D crossover) • CsCuCl4 • NiCl2-4SC(NH2)2 3D BEC: a = 3/2 3D Ising: a = 2 2D “BEC”: a = 1

14. Sapphire platform Measuring Specific Heat C=t/k C=Q/DT Quasi-Adiabatic Thermal Relaxation Time

15. Magnetocaloric Effect Specific Heat 10 0.44 increasing H H = 10 T decreasing H 8 0.43 C/T (mJ/mol K2) 0.42 6 T (K) inflec. point 0.41 4 0.4 2 0.39 0 10 11 12 13 14 0 0.4 0.8 1.2 1.6 B(T) T (K) 1.2 1 0.8 0.6 Magnetocaloric effect 0.4 Specific heat 0.2 0 0 2 4 6 8 10 12 14 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 – Hc TNa H – Hc TNaa = 2 (3D Ising magnet) 3/2 (3D BEC) 1 (2D “BEC”)

17. 2.5 Ising magnet 2 1.5 3D BEC a 1 H-Hc1 = aTa a = 1.47 ± 0.10 (a = 1.5 BEC) 2D BEC 0.5 0 0 0.2 0.4 0.6 0.8 1 Tmax/1.2 K A. Paduan Filho, unpublished 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)

18. predicted Hc2 predicted Hc1 Spin wave theory Inelastic Neutron Diffractionand magnetization : Tc (K) Predictions (3-level system): C. D. Batista, M. Tsukamoto, N. Kawashima, in progress V. S. Zapf et al, Phys. Rev. Lett. 96, 77204 (2006).

19. H T = 25 mK H || c Hc2 Lc DL/L (%) Hc1 La Magnetostriction c a Capacitance CuBe spring Titanium Dilatometer (design by G. Schmiedeshoff) V. Correa,V.S. Zapf,T. Murphy, E. Palm, S. Tozer, A. Lacerda, A. Paduan-Filho

20. J = 1.7 K J = 0.17 K Ni++ Hc2 Magnetostriction J J J J J J J J DL/L (%) Hc1 Lc La c H a

21. Hc2 Magnetostriction DL/L (%) Tc (K) Hc1 Lc La predicted Hc2 predicted Hc1 Summary BEC confirmed experimentally via H-Hc1 ~ Taand M ~ Ta Magnetostriction effect distorts phase diagram with increasing magnetic field

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

23. 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) Acknowledgements NSF NHMFL DOE