1. I.Mirebeau, S.Petit, A. Gukasov, J.Robert, thesis S.Guitteny, Laboratoire Léon Brillouin, CEA-Saclay P.Bonville DSM/IRAMIS/SPEC, CEA-Saclay C.Decorse ICMMO, Université Paris XI H.Mutka, J.Ollivier, M.Boehm, P.Steffens Institut Laue Langevin, Grenoble A.Sazonov LLB, Aachen University Magnetic structures and anisotropic excitations in Tb 2 Ti 2 O 7 spin liquid

2. Tb 2 Ti 2 O 7 : a hot topic Why is Tb 2 Ti 2 O 7 (or TTO) so interesting ? 7 Posters at HFM’14 Kermarrec Malkin Fennel Hallas Kao Sazonov Yin

3. Tb 2 Ti 2 O 7 : a hot topic because nobody fully understands it! TTO quantum spin ice Spin liquid Antiferro- magnetic spin ice magneto- elastic liquid Spin Glass

4. Tb 2 Ti 2 O 7 : a hot topic More and more sophisticated experiments Influence of tiny defects Coupling with the lattice In the last 3 years Searching for a magnetization plateau : H //111 Probing dispersive excitations ½ ½ ½ structure Competing SRO structures : Spin glass like vs. mesoscopic order magneto-elastic mode Dynamic Jahn-Teller transition and/or interactions between quadrupolar moments Towards a more realistic description ?

5. Mc. Clarthy- Gingras Rev Modern Phys. ( Dipolar Spin ices: The Ising case R 2 Ti 2 O 7 pyrochlores R=Dy, Ho Effective interaction J eff = J+D dip > 0 Dipolar spin ice AF FeF 3 4in-4out Spin ice Den Hertog et al Phys. Rev. Lett. (1999) Bramwell et al Phys. Rev. Lett (2000) Tb Dy Ho Tb nearby the threshold Quantum fluctuations at play: « quantum spin ice » Molavian, Gingras, Canals, PRL (2008) Molavian, Clarthy, Gingras arxiv0912.2957 Mc. Clarthy- Gingras Rev Progress Physics 77 056501(2014) What about the Crystal field ?

6. The crystal field Δ = 200 – 300K Ho, Dy spin ices Δ = 10-20K (Tb) Tb 3+ is a non-Kramers ion Strong but finite anisotropy Δ ~ 1.5 meV No exchange fluctuations allowed within the GS doublet No intensity scattered by neutrons Gingras, PRB (2000) Bonville, IM, PRB( 2007) Bertin,Chapuis, JPCM(2012) Zhang, Fritsch, PRB (2014) Klekovina- Malkin J Opt. Phys. (2014) Cao et al PRL(2009)

7. Δ ~ 1.5 meV h: molecular field Splitting of the Ground state doublet In molecular field approach Δ ~ 1.5 meV d h Quantum mixing in the GS. 1st order perturbation0th order perturbation Simplest case: entangled wave functions (g j µ B h/  ) 2  (0.75/15) 2  2.10 -3 D: quantum mixing g J µ B /k B = 1 for Tb !

8. Δ ~ 1.5 meV h: molecular field Splitting of the Ground state doublet In molecular field approach Δ ~ 1.5 meV d h Quantum mixing in the GS. 1st order perturbation0th order perturbation Simplest case: entangled wave functions Virtual crystal field model Very small intensity associated with GS fluctuations (with resp. to CF ) Spin ice anisotropy: magnetization plateau Two singlet ground state each singlet is non magnetic : no static signal the transition has a large spectral weight Jahn-Teller distortion? Molavian, Gingras, Canals PRL(2007) Molavian, McClarthy, Gingras arxiv(2009) Bonville et al PRB(2011), PRB (2014)

9. Searching for a magnetization plateau Using Magnetization, susceptibility, MuSR : a controversial situation low field anomalies of the susceptibility: MuSR Baker PRB (2012) Legl et al PRL (2012) No plateau in the isothermal magnetization cross over regime in the dynamics Yin et al PRL(2013) Lhotel et al PRB-RC (2012) Spin glass-like freezing ? T F ~200-400 mK Fritsch, PRB(2014)

10. Searching for a magnetization plateau Using neutrons : magnetic structure for H//111 Exclude all-in all out structure Gradual reorientation of the Tb moments in the Kagome plane (keeping 1in- 3 out) without Kagome ice structure See poster A. Sazonov

11. Searching for a magnetization plateau No evidence for the 1/3 plateau at ~2µB expected at very small fields (down to 80mK) quantitative agreement with MF model assuming a dynamical JT distortion: 4 moment values and angles M(H) for H//100, 111, 110 Field Irreversibilities Spin glass like freezing? A. Sazonov et al PRB(2013) D=0 no mixing see poster A. Sazonov

12. Spin fluctuations at very low temperature Using unpolarized neutrons 2 components in the neutron cross section elastic (dominant) inelastic (low energy) elastic Pinch points diffuse maxima at ½ ½ ½ positions inelastic becomes structured at low T well accounted for by 2 singlet model + anisotropic exchange D=0.25K See also: Takatsu et al. JPCM (2011) Fritsch et al PRB(2013) Static character not reproduced by the 2 singlet model

13. diffuse scattering b = -0.13T/µ B ; D Q =0.25K Phase diagram P. Bonville et al Phys. Rev. B (2011) 3d-map Experiment Simulation 6T2 ( LLB) The main features of the diffuse scattering are reproduced Simulation with anisotropic exchange dipolar interactions CF JT distortion along equivalent 100, 010, 001 cubic axes.( preserves the overall cubic symmetry) Dynamical JT (average Structure factors and not intensities) Energy integrated intensity

14. S.Petit & al, PRB 86 (2012)T.Fennell & al, Science 326 (2009) Q dependence of the elastic scattering Pinch points in both compounds: Coulomb phase strong spectral weight at Q=0 no spectral weight at Q=0 ½ ½ ½ maxima : AF correlations

15. Analysis of the pinch points Strongly anisotropic correlations of algebric nature conservation law in TTO spin liquid analogous to the ice rules What are the spin component involved? S.Guitteny & al, PRL 111 (2013) T. Fennell et al PRL(2012)

16. Polarization analysis Fennell Science (2009) : Ho 2 Ti 2 O 7 PRL (2013) Tb 2 Ti 2 O 7 Longitudinal polarimetry separates spin components x Z //110 x// Q 1 2 34 1’ 2’ Neutron cross section Correlations along Q (or x) between spin components M ┴ Q Ho 2 Ti 2 O 7 NSF: correlations « up-down » 1-1’ or 2-2’: Weak (2 Spins, between T) SF: correlations « 2in-2 out » 1-2-3-4: Strong (4 spins, in a T) Q z y MzMz MyMy neutron polarization P// Z Non spin flip: N+ Spin Flip

17. Polarization +energy analysis Fennell Science (2009) : Ho 2 Ti 2 O 7 PRL (2013) Tb 2 Ti 2 O 7 Q z y MzMz MyMy x Z //110 x// Q 1 2 34 1’ 2’ Tb 2 Ti 2 O 7 Look at the dispersion Mz: « up-down » correlations: relaxing (Quasi-E) My: « 2 in-2out » correlations : dispersing (Inel.) T=50 mK Longitudinal polarimetry separates spin components Neutron cross section Correlations along Q (or x) between spin components M ┴ Q neutron polarization P// Z Non spin flip: N+ Spin Flip

18. Low energy excitations 18 In all directions Quasi-élastic Strong fluctuations My Along (h,h,h) quasi-élastic along (h,h,2-h) et (h,h,0) propagating excitation no gap (Δres = 0,07meV) Disperses up to 0,3 meV intensity varies like 1/ω First observation of a dispersive excitation in fluctuating disordered medium Mz S. Guitteny et al PRL(2013)

19. Nature of the static SRO? the ½ ½ ½ order ½ ½ ½ diffuse maxima Short range ~8-10 A below ~0.4K Vanish in a small field ( ~200G) Fennel PRL (2012) Fristch PRB(2012) Petit PRB (2012) In single crystals In powders ½ ½ ½ Mesoscopic structure Over 30-50A Associated with Cp anomaly tuned by minute defects in Tb content Taniguchi PRB RC(2011) Short range vs. mesoscopic order See also poster E. Kermarrec

20. powder samples Tb 2+x Ti 2-x O 7+y ½ ½ ½ ½ ½ 3/2 ½ ½ 5/2 3/2 3/2 1/2 X=0 Mesoscopic structure for x=0 and x=0.01 Difference pattern: I(50 mk)- I(1K) T=50mK N X=0 exp: P. Dalmas de Réotier 2  (deg) Neutron counts

21. Symmetry analysis 2 orbits with no common IR site 1 Sites 2-4 Nsite 10 0 0 2¾ ¼ ½ 3¼ ½ ¾ 4 ½ ¾ ¼ space group Fd-3M, K= ½ ½ ½ Champion, PRB (2001) Stewart, Wills JPCM(2004) Gd 2 Ti 2 O 7 No way to build a strong ½ ½ ½ peak for Ising spins! Needs to break either Ising anisotropy or cubic symmetry K // local axis no intensity at ½ ½ ½ No vectors of the IR along the local axes Contributions to ½ ½ ½ cancel by symmetry Systematic search of magnetic structures 1T cfc translations (cubic cell : a) K= ½ ½ ½ (magnetic unit cell: 2a)

22. The best structures (x=0) moments remain close to local axes (3-10 deg) M=1.9(4) µB/Tb; Lc =60 A (Y=1.4) X=0 Correlation length ~30 -50 A « Monopole layered structure » « AF -Ordered spin ice »

23. Ferrimagnetic piling of SI Tetrahedra moments remain close to local axes (<10 degs) Fritsch PRB (2012) The best structures (x=0) « AF -Ordered spin ice » « Monopole layered structure » AF packed OSI cubic cells, MZMZ Z//001 S. Guitteny (thesis) derived from Tb 2 Sn 2 O 7 I. M et al PRL (2005)

24. Ferrimagnetic piling of SI Tetrahedra separated by monopole layers moments remain close to local axes (<10 degs) Fritsch PRB (2012) The best structures (x=0) « AF -Ordered spin ice » « Monopole layered structure » AF packed OSI cubic cells, separated by SI tetrahedra with M  Full of monopoles, but compatible with a distortion No monopoles, but symmetry breaking at each cubic cell no possible LRO? MZMZ MZMZ Z//001

25. Calculated diffuse scattering In a single crystal, correlation length reduced to 2 cubic cells h, h, 0 0, 0, l h, h, 0 « Monopole layered structure » « AF -Ordered spin ice » Experiments Petit PRB (2013) Fennel PRL (2013) Fritsch PRB(2013) 1 2 3 4 12341234 1 2 3 4

26. The ½ ½ ½ order: summary ½ ½ ½ order cannot propagate without breaking the cubic symmetry different structures and/or K orientations may compete (in space, time) yielding: SRO (single crystal) mesoscopic orders (powders, tuned by x) Spin glass like irreversibilities : Yin (2013), Fritsch PRB (2014), Lhotel (2013) 2 physical mechanisms at play for the magnetic excitations Relaxation (quasielastic) Dispersive excitations Analog to the double dynamics in SP particles or quantum molecular magnets Magneto-elastic modes as a switching mechanism? Quasielastic or slow relaxations (thermally activated,QT) Inelastic modes

27. Probing the magneto-elastic coupling Interaction between 1st excited CF doublet and acoustic phonon branch Guitteny PRL(2013) see also: Fennel PRL(2013) this conf. M. Ruminy : next talk Other probes pressure induced magnetic order IM et al Nature 2002, PRL(2004) Elastic constants Klekovina-Malkin J. Phys. 2011, J. Opt. Phys. 2014 Thermal conductivity Li et al PRB(2013)

28. Summary: what is new in TTO? Quantum mixing in the GS doublet due to quadrupolar order: a necessary ingredient JT distortion « exchange » int. between quadrupolar moments Magnetoelastic coupling Non-Kramers character is crucial First observation of dispersive anisotropic excitations in a fluctuating disordered medium Two types of dynamics : relaxation, excitations Competing SI correlations with K=½ ½ ½ Not compatible with cubic symmetry Tuned by off-stoechiometries With different time and length scales Associated with glassy behaviour Gehring-Gehring (1985) Savary-Balents PRL(2012) Lee-Onoda-Balents PRB(2012) MF poster Malkin

29. x=0.01 coexistence of LRO and mesoscopic orders Mesoscopic: M= 1.3µ B /Tb LRO: M=0.3 µ B /Tb

30. I.M et al Nature (2002) Under pressure : a phase with larger unit cell is also stabilized Pressure induced structures