1. Supported by: US National Science Foundation, Research Corporation, NHMFL, & University of Florida The effect of anisotropy on the Bose-Einstein condensation of magnons in BaCuSi2O6 Stephen Hill, Sung Su Kim and Anthony Wilson Department of Physics, University of Florida, Gainesville, FL 32611, USA Stanford - Ian Fisher Cambridge - Suchitra Sebastian LANL - Cristian Batista, Marcello Jaime, Neil Harrison, Paul Goddard, Vivian Zapf and Ross McDonald Sao Paolo, Brazil - Armando Paduan-Filho NHMFL, Tallahassee - Stan Tozer Collaborators:

2.  Introduction to giant spin approximation – why use it?  Mn 12 SMM as an example.  A model system: a tetranuclear nickel complex  Evaluation of giant spin Hamiltonian parameters  Excellent application of high-field EPR  Evaluation of single-ion zero-field splitting tensors  Origin of fourth- and higher-order zfs interactions  Assessment of the giant spin approximation  Some consequences  Summary and conclusions

3. Sparta & Roth, Acta Cryst. B60, 491 (2004). i.e. quasi-2D square lattice of weakly-connected, vertical, symmetric dimers T > 610 K: I4/mmm (No. 139), a = 7.1104 Å, c = 11.175 Å T < 610 K: I4 1 /acd (No. 142), a = 10.0091 Å, c = 22.467 Å Ba Cu Si 2 O 6 c ab The BaCuSi 2 O 6 structure

4. Body-centered tetragonal magnetic lattice  [Cu 2+ ] 2 dimer J J'J' J'J' JfJf Each Cu 2+ provides a spin-½ Intra-dimer separation: 2.74 Å NN inter-dimer distance: 7 Å NNN inter-dimer distance: ~10 Å All J s are antiferromagnetic Intra-dimer J = 4.45 meV (36 cm  1 ) J' = 0.51 meV (4 cm  1 ) J f < J' is frustrating interaction a b c To lowest order, treat as independent spin-½ dimersTo lowest order, treat as independent spin-½ dimers [Cu 2+ ] 2 Hamiltonian has perfect cylindrical [U(1)] symmetry[Cu 2+ ] 2 Hamiltonian has perfect cylindrical [U(1)] symmetry

5. Properties of the isolated dimer Triplet (T ) Singlet (S ) TTTT TTTT S TTTT Magnetic field JHeisenberg: EnergyZeeman:

6. exp 1/T J = 33 K, g = 2.00 Properties of the isolated dimer Triplet Singlet TTTT TTTT TTTT S J J = 4.45 meV Low energy degrees of freedom B = 0B > 20 T At low fields: TTTT S  Effective two-level system: pseudospin

7. Low-field properties of BaCuSi 2 O 6 : evidence for a spin gap J = 4.45 meV ≡ 51 K g a = 2.01, g c ~ 2.31 Y. Sasago et al., Phys. Rev. B 55, 8357 (1997). S. Sebastian et al., cond-mat/0606244. Question: what happens for weakly interacting spin dimers?

8. F. Mila, Euro Phys. J. B. 6, 201 (1998). T. Giamarchi & A. M. Tsvelik, PRB 59, 11398 (1999). For J > J', treat perturbatively; basis of | S  and | T  states Exchanges triplets and singlets on adjacent sitesExchanges triplets and singlets on adjacent sites Describes delocalized band of bosonic excitations (triplons)Describes delocalized band of bosonic excitations (triplons) This term represents kinetic energy of the triplonsThis term represents kinetic energy of the triplons Insight from the two leg ladder J J'J' i = 12 3 4 5.....

9. F. Mila, Euro Phys. J. B. 6, 201 (1998). T. Giamarchi & A. M. Tsvelik, PRB 59, 11398 (1999). Insight from the two leg ladder J J'J' i = 12 3 4 5..... K.E.P.E.C.P.

10. Insight from the two leg ladder J J'J' i = 12 3 4 5..... Energy Momentum J J  J' T S  In 1D In 2D

11. Magnetic field Magnetization B c1 B c2 M sat Paramagnet Ferro- magnet ? S TTTT Effect of a magnetic field

12. Magnetic field Magnetization B c2 M sat Finite temperature (smooth evolution) Maxwell-Boltzmann Effect of a magnetic field S TTTT

13. Sebastian et al., cond-mat/0606244 9 Jun 2006 Nothing exotic yet – all explainable in classical termsNothing exotic yet – all explainable in classical terms Continuous evolution of magnetization from high to low TContinuous evolution of magnetization from high to low T

14. M. Jaime et al., Phys. Rev. Lett. 93, 087203 (2004). Heat capacity and magnetocaloric effect: (Marcelo Jaime, NHMFL) Implies magnetic ordering Could this be a Bose-Einstein condensation? Clear phase transition for B c1 > B > B c2 -anomaly

15. T > T c Momentum Energy  ±  TT S Thermal population of singlet (S ) and triplet (T ) statesThermal population of singlet (S ) and triplet (T ) states Corresponds to an incoherent mixture of S and TCorresponds to an incoherent mixture of S and T A paramagnet Maxwell-Boltzmann

16. Momentum Energy  ±  S TT Macroscopic occupation at (  ±  ) pointsMacroscopic occupation at (  ±  ) points All spins condense forming a coherent superposition of S and T statesAll spins condense forming a coherent superposition of S and T states T < T c A canted XY antiferromagnet dB > d TT Bose-Einstein

17. a b c Development of M z does not break U(1) symmetry Canted XY antiferromagnet a b Coherence and broken symmetry XY antiferromagnetic order does break the U(1) symmetry BEC universality for d ≥ 2

18. T = 20 K, f = 92.5 GHz