XY Rare-Earth Pyrochlore Oxides an Order-by-Disorder Playground

Transcription

1 SPICE-Workshop on "Computational Quantum Magnetism, Mainz May 2015 XY Rare-Earth Pyrochlore Oxides an Order-by-Disorder Playground Michel Gingras Department of Physics & Astronomy, University of Waterloo, & Canadian Institute for Advanced Research/Quantum Materials Program

2 Collaborators Er 2 Ti 2 O 7 : Zhihao Hao (UW) Paul McClarty Pawel Stasiak Behnam Javanparast (UW) Alex Day Jaan Oitmaa Rajiv Singh Behnaz Bagheri Anson Wong Yb 2 Ti 2 O 7 : Ludovic Jaubert (OIST) Zhihao Hao (UW) Jeff Rau (UW) Owen Benton Nic Shannon Rajiv Singh Jaan Oitmaa Robert Hayre Ryan Applegate Taoran Lin Anson Wong Alex Day

3 Outline 1. Rare-earth pyrochlore oxides 2. Order-by-disorder (ObD) in Er 2 Ti 2 O 7 3. Why should one care about ObD? 4. Yb 2 Ti 2 O 7 : what is the hidden order? 5. Conclusion

4 Outline 1. Rare-earth pyrochlore oxides 2. Order-by-disorder (ObD) in Er 2 Ti 2 O 7 3. Why should one care about ObD? 4. Yb 2 Ti 2 O 7 : what is the hidden order? 5. Conclusion

5 Message of Talk 1. Order-by-disorder (ObD) appears quite likely in Er 2 Ti 2 O 7 2. Yb 2 Ti 2 O 7 is puzzling: Some form of hidden order in Yb 2 Ti 2 O 7? Are (i) competing orders and (ii) the consequential ObD the missing links in Yb 2 Ti 2 O 7?

6 kagome hyperkagome pyrochlore garnet

7 kagome pyrochlore hyperkagome garnet

8 kagome hyperkagome pyrochlore garnet

9 A 2 B 2 O 7 A=Pr,Gd,Tb,Dy,Ho,Er,Yb B=Ti,Sn, Zr,Sn a ~ 10 Å

10 R 2 Ti 2 O 7 R=Gd, Tb, Dy, Ho, Er, Yb For example: Ho 3+ : J=L+S = 6+2 = 8! e ± crystal electric field lifts 2J+1 degeneracy! a ~ 10 Å! 0 ±! 0 ± " m J = C mj J,m J

11 crystal electric field splits 2J+1 degeneracy! e ±!! 0 ±! 0 ± " = C mj J,m J m J Parallel to cubic [111] direction

12 1 mev = 11 K Crystal field energy levels in R 2 Ti 2 O 7 Δ~600K Δ~300K Narrow doublet-doublet gap of 18 K Δ~70K Non-magnetic singlet

13 Local [111] direction g! g! Ising Ho 2 Ti 2 O 7 Dy 2 Ti 2 O 7 Tb 2 Ti 2 O 7 XY Yb 2 Ti 2 O 7 Er 2 Ti 2 O 7 Local XY plane We know that symmetry plays an important role in the statistical mechanics of conventional (i.e. nonfrustrated) magnets what happens in frustrated ones? Heis. Gd 2 Sn 2 O 7 Gd 2 Ti 2 O 7

14 Effective Hamiltonian Method H = H cef + H int H int =! i> j! K,K '! Q,Q' H eff = PH int P + PH int QH int P +!!! I QQ' KK ' O Q K (J i )O Q' K ' (J j ) P = α α Q = α P Large (CEF gap Δ) denominator compared to the energy scale of H int %!$P!! E 0 " # E 0! H eff = PH int P + PH int QH int P +!

15 Effective Hamiltonian Method H = H + H H int = I QQ' KK ' O Q K (J i )O Q' K ' (J j ) cef int! i> j! K,K '! Q,Q' H eff = PH int P + PH int QH int P +!!! P = α α Q = α P Large (CEF gap Δ) denominator compared to the energy scale of H int %!$P!! E 0 " # E 0! Note: Fine for Dy, Ho, Er, Yb; not Tb H eff = PH int P + PH int QH int P +!

16 " " ( ) H eff = J zz S i z S j z! J ± S i + S j! + S i! S j + i, j i, j z +J z± S i # ij S + j +# *! "% ( ij S ) & j + i $ j' ( + J ±± ) ij S + i S + j + ) * ij S!! "( i S ) j S = 1 2 operator i, j K.A. Ross, L. Savary, B.D. Gaulin, and L. Balents; PRX 1, (2011) z H eff = J Ising S i z i S j!! j + J iso Si i S!!! j + J DM dij i S! i " S! j i, j i, j For Δ/J zz >>1, ground doublet projection is good enough i, j! i, j! d ij ( )! + J! pd Si i S! j # 3 S! i i ˆr ij ˆr ij i S! j + H LR_dip i, j P.A. McClarty, S.H. Curnoe, and M J.P. Gingras, J. Phys.: Conf. Ser. 145, (2009) ( S! 3 H LR_dip = Dr i i S! j! 3 S! i i!! r ij S j i! J.D. Thompson et al. Phys. Rev. Lett.106, (2011) r ij ) " nn 3 i> j r ij

17 Frustrated Rare-earth Pyrochlore Oxides Examples of phenomena (Ho,Dy) 2 (Ti,Sn,Ge) 2 O 7 : spin ice Yb 2 Ti 2 O 7 : quantum spin ice? Tb 2 Ti 2 O 7 : spin liquid / QSI? Gd 2 Ti 2 O 7 : multiple transitions Er 2 Ti 2 O 7 : order-by-disorder J.S. Gardner, M.J.P. Gingras, and J.E. Greedan, Magnetic Pyrochlore Oxides, Rev. Mod. Phys. 82, 53 (2010)

18 " " ( ) H = J! S i z S j z! J ± S i + S j! + S i! S j + i, j i, j z +J z± S i # ij S + j +# *! "% ( ij S ) & j + i $ j' ( + J ±± ) ij S + i S + j + ) * ij S!! "( i S ) j S = 1 2 operator i, j i, j Savary & Balents, PRL 108, (2012). a) Kramers b) non-kramers CSI Savary & Balents, PRL 108, (2012). CSI Lee, Onoda & Balents, PRB 86, (2012). QSI is expected to be a U(1) quantum spin liquid Gingras & McClarty, Rep. Prog. Phys. 77, (2014)

19 There is an ongoing search for quantum spin ice materials

20 Frustrated Rare-earth Pyrochlore Oxides Examples of phenomena (Ho,Dy) 2 (Ti,Sn,Ge) 2 O 7 : spin ice!yb 2 Ti 2 O 7 (XY) : quantum spin ice? Tb 2 Ti 2 O 7 : spin liquid / QSI? Gd 2 Ti 2 O 7 : multiple transitions! Er 2 Ti 2 O 7 (XY) : order-by-disorder J.S. Gardner, M.J.P. Gingras, and J.E. Greedan, Magnetic Pyrochlore Oxides, Rev. Mod. Phys. 82, 53 (2010)

21 Outline 1. Rare- earth pyrochlore oxides 2. Order-by-disorder (ObD) in Er 2 Ti 2 O 7 3. Why should one care about ObD? 4. Yb 2 Ti 2 O 7 : what is the hidden order? 5. Conclusion

22 Order-by-disorder ObD is a phenomenon via which thermal fluctuations and/or quantum fluctuations lift an accidental classical T=0 ground state degeneracy by (i) selecting the subset of states of the degenerate around which the system fluctuate the most which (ii) then allows for a spontaneous symmetry breaking and long-range order

23 Order-by-Disorder Degeneracy parameter E = E classical (!) = E classical (independent of!) # k E(!) = E classical + 1 2!" k (!) Thermal contribution to ObD - either at T=0 + - or at T~T c Quantum contribution to ObD F = E $ TS(!) E = E + # 1!" (!) classical 2 k k

24 Order-by-disorder does it occur? The concept of (thermal/quantum) order-bydisorder (ObD) is compelling (and presumably correct) for a number of theoretical models described by some minimal Hamiltonian. "But does it really occur in nature, in real materials? "Evidence for ObD is strong in Er 2 Ti 2 O 7

25 The problem of order in XY-like Er 2 Ti 2 O 7 T c ~1.2K Champion et al., PRB (RC) 68, (2003) Champion and Holdsworth, JPCM 16, S665 (2004) 50 mk Er 3+ behave as local S=1/2 XY spin Poole et al., JPCM 19, (2007) ψ 2 ψ 3

26 Everywhere in this talk, x, y and z are local axes x z y AF / ψ 2! q = 0 order 1. Champion et al. found in a classical pyrochlore XY antiferromagnet toymodel a selection of ψ 2 proceeding via thermal order-by-disorder (ObD) 2. They further speculated that quantum fluctuations could proceed similarly and that a ψ 2 ground state could be stabilized via quantum ObD

27 Degeneracy of Classical Ground State!

28 Degeneracy of Classical Ground State! Bramwell, Gingras, Reimers, J, Appl. Phys. 75, 5523 (1994) Champion et al. PRB 68, R (2003) Champion, Holdsworth, J. Phys. Cond Matt. 16, S665 (2004) Stasiak, McClarty, Gingras, Phys. Rev. B 89, (2014) Zhitomirsky et al., Phys. Rev. Lett. 109, (2012) No bilinear spin-spin interaction of arbitrary range and anisotropy can lift the classical degeneracy of the ψ 2 /ψ 3 manifold Savary, Ross, Gaulin, Ruff, and Balents; Phys. Rev. Lett. 109, (2012).

29 Proposed Solution to the Er 2 Ti 2 O 7 Problem Order by Quantum Disorder in Er 2 Ti 2 O 7 L. Savary, K.A. Ross, B.D. Gaulin, J.P.C. Ruff, and L. Balents; Phys. Rev. Lett. 109, (Published 15 October 2012) " " ( ) H = J zz S i z S j z! J ± S i + S j! + S i! S j + i, j i, j z +J z± S i # ij S + j +# *! "% ( ij S ) & j + i $ j' ( + J ±± ) ij S + i S + j + ) * ij S!! "( i S ) j S = 1 2 operator i, j i, j

30 What are (classical) long-range ordered phases? AF / ψ 4 (PC) AF / ψ 2 m AF / ψ 3 SF

31 More about the splayed-ferromagnet phase m - Canted spin ice ( 2-in/2-out ) - Splayed ferromagnet (SF) - Net magnetization along <100>

32 Stability of ψ 2 for Er 2 Ti 2 O 7 against long-range dipolar interaction Er 2 Ti 2 O 7 SF = Splayed ferromagnet Ground state phase diagram of generic XY pyrochlore magnets with quantum fluctuations; Anson W.C. Wong, Zhihao Hao, and M.J.P. Gingras, Phys. Rev. B 88, (2013)

33 Phase Transition and Thermal Order-by-Disorder in the Pyrochlore Quantum Antiferromagnet Er 2 Ti 2 O 7 : a High-Temperature Series Expansion Study ; J. Oitmaa, R.P. Singh, A.G.R. Day, B.V. Bagheri, M.J.P. Gingras; Phys. Rev. B (RC) 88, (2013). 6 th order order-parameter susceptibility shows that all terms in the 1/T expansion are larger term by term, starting at order 1/T 4, for ψ 2 than for ψ 3! demonstration of thermal ObD operating at T c in H Savary

34 Outline 1. Rare- earth pyrochlore oxides 2. Order-by-disorder (ObD) in Er 2 Ti 2 O 7 3. Why should one care about ObD? 4. Yb 2 Ti 2 O 7 : what is the hidden order? 5. Conclusion

35 sharp peak Blöte et al. Physica 43, 549 (1969) broad bump rods at T<1K Hodges et al. Phys. Rev. Lett. 88, (2002) No peaks Ross et al. PRL 103, (2009) I(0.11K)-I(7K)

36 ( ) H = J zz S z i S z j! J ± S + i S! j + S! + z " " i S j + J z± S i # ij S + j +# *! "% & ij S j ' ( + J ±± ) ij S + i S + j + ) * ij S!! " i S j i, j i, j i, j ( ) + i $ j i, j ( )

37 Numerical linked-cluster expansion (NLC)

38 Where is Yb 2 Ti 2 O 7? Er 2 Ti 2 O 7 m SF = Splayed ferromagnet Yb 2 Ti 2 O 7 Ground state phase diagram of generic XY pyrochlore magnets with quantum fluctuations; Anson W.C. Wong, Zhihao Hao, and M.J.P. Gingras, Phys. Rev. B 88, (2013)

39 Yasui et al., J.Phys. Soc. Jpn 72, 3014 (2003) q=0 peaks + ferromag.-like bulk at 0.03K < T c ~ 0.24K But not confirmed by neutron depolarization experiment [Gardner et al., Phys. Rev. B 70, (2004).

40 Chang et al., Nature Communications 3:992 DOI: /ncomms1989 SF NSF TOT T=0.3K q=0 peak

41 D Ortenzio et al., Phys. Rev. B 88, (2013) powders Ross et al., Phys. Rev. B. 84, (2011 crystals Chang et al., Phys. Rev. B 89, (2014) Chang et al., Nat. Comm. DOI: /ncomms1989

42 D Ortenzio et al., Phys. Rev. B 88, (2013) Ross et al., Phys. Rev. B. 84, (2011 Is this some kind of hidden order problem? e.g. URu Si 2 2 crystals powders Chang et al., Phys. Rev. B 89, (2014) Chang et al., Nat. Comm. DOI: /ncomms1989

43 Crucial hint: Lhotel et al. Physical Review B 89, (2014)

44 Summary of Yb 2 Ti 2 O 7 Experiments Large sensitivity to sample preparation (powders vs single crystals) Clear tendency towards ferromagnetic-like state A few (powder and single crystal) samples show a 5-25% difference in rise of magnetization vs highest-temperature magnetic specific heat peak " Suggests that the material is in the vicinity of a ferromagnetic-like state and a competing order

45 Jaubert, Benton, Rau, Oitmaa, Singh, Shannon, Gingras; Are multiphase competition & order-by-disorder the keys to understanding Yb 2 Ti 2 O 7? ; arxiv: By combining: - Classical Monte Carlo - Classical and quantum spin wave calculations - Exact diagonalization - Cluster mean-field theory - Numerical linked-cluster expansion - High-temperature series expansion - Given the uncertainty on the exchange parameters determined by Ross, Savary, Gaulin and Balents; Phys. Rev. X 1, (2011) - and the computed quantum shift/renormalization of the SFM-AF boundary! We found that Yb 2 Ti 2 O 7 is very close to the SFM-AF boundary, and

46

47 High-T peak in C(T) No spontaneous magnetization

48 High-T peak in C(T) No spontaneous magnetization Low-T peak in C(T) Spontaneous magnetization

49 Yb 2Ti 2O 7

50 Message of Talk 1. Order-by-disorder (ObD) appears quite likely in Er 2 Ti 2 O 7 2. Yb 2 Ti 2 O 7 is puzzling: The competition of energetically versus entropically stabilized orders are likely (the) essential missing ingredients to understand Yb 2 Ti 2 O 7.

51 Alighiero Fabrizio Boetti,