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Heavy Fermions: a DMFT Perspective G. Kotliar

Heavy Fermions: a DMFT Perspective G. Kotliar. Work with Kristjan Haule and Jihoon Shim at Rutgers University. Supported by the National Science Foundation. . July 27 th 2008, Ohio State University .

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Heavy Fermions: a DMFT Perspective G. Kotliar

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  1. Heavy Fermions: a DMFT PerspectiveG. Kotliar Work with Kristjan Haule and Jihoon Shim at Rutgers University. Supported by the National Science Foundation. July 27th 2008, Ohio State University

  2. DMFT: trick to sum an infinite diagrams Lattice Model, i, j,k,l site indices DMFT sums an infinite number of graphs. One can provide a on perturbative definition for the sum, useful tricks for carrying with high precision, a simple picture in terms of impurity models

  3. But how accurate is it ? Ulrich Schneider’s talk But…..one band model, relatively high temperatures, how else can we test (and therefore improve ) the method ? A different (tried and true) approach, compare against experiment in a wide set of materials, explore chemical trends.

  4. How do we know that the electrons are heavy ? Heavy Fermions: intermetallics containing 4f elements Cerium, and 5f elements Uranium. Broad spd bands + atomic f open shells.

  5. 300 CeAl3 -1 (emu/mol)-1 200 UBe13 100 0 0 100 200 T(K) Heavy Fermion Metals

  6. A Very Selected Class of HF

  7. LDA+DMFT. V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). Treat the local correlations of the f shell using DMFT. Treat the non local correlations and the spd bands using LDA. Determine energy and and S self consistently from extremizing a functional Chitra and Kotliar (2001) . Savrasov and Kotliar (2001) Full self consistent implementation 12

  8. X In Ce LDA+DMFT CeIrIn5 (115) Local f spectral function vs T • At 300K, only Hubbard bands • At low T, very narrow q.p. peak • (width ~3meV) SO • SO coupling splits q.p.: +-0.28eV • Redistribution of weight up to very high • frequency J. H. Shim, KH, and G. Kotliar Science 318, 1618 (2007).

  9. Buildup of coherence in single impurity case Very slow crossover! coherent spectral weight TK T T* Buildup of coherence coherence peak scattering rate Slow crossover compared to AIM Crossover around 50K

  10. Optical conductivity in LDA+DMFT Shim, HK Gotliar Science (2007)‏ D. Basov et.al. K. Burch et.al. • At 300K very broad Drude peak (e-e scattering, spd lifetime~0.1eV) • At 10K: • very narrow Drude peak • First MI peak at 0.03eV~250cm-1 • Second MI peak at 0.07eV~600cm-1

  11. 10K eV In Ce In Structure Property Relation: Ce115’s Optics and Multiple hybridization gaps J. Shim et. al. Science non-f spectra 300K • Larger gap due to hybridization with out of plane In • Smaller gap due to hybridization with in-plane In

  12. Conclusions Ce 115’s • Accounts for many of the observed features of Ce based heavy fermions. • Crossover is slower than in single impurity because of the self consistency condition feedback. • Structure Property Relation. Out of plane In site controls hybridization. Confirmed by NMR. • Predictions for ARPES currently being tested. • Validates renormalized band theory at T=0 , explains • why it is not a good guide to experiments at most • temperatures. • Accounts for Co 3d –Rh 4d -Ir5d (Haule et. al. 2009)

  13. A Very Selected Class of HF

  14. U Ru Si Hidden Order in URu2Si2 dark matter problem. T. T. M. Palstra et.al. PRL 55, 2727 (1985) D. A. Bonn et al. PRL (1988).

  15. “Adiabatic continuity” between HO & AFM phase • Similar T0 and TN • Almost identical thermodynamic • quantities (jump in Cv) E. Hassinger et.al. PRL 77, 115117 (2008)

  16. Two Broken Symmetry Solutions Hidden Order LMA K. Haule and GK

  17. DMFT excitonicorder parameter Order parameter: Different orientation gives different phases: “adiabatic continuity” explained. Hexadecapole order testable by resonant X-ray In the atomic limit:

  18. DMFT “STM” URu2Si2 T=20 K Ru Si Si U Fano lineshape: q~1.24, G~6.8meV, very similar to exp Davis

  19. Orbitally resolved DOS

  20. Simplified toy model phase diagram mean field theory Mean field Exp. by E. Hassinger et.al. PRL 77, 115117 (2008)

  21. U Ru Si A Hidden Order The CMT dark matter problem. (A. Schofield) URu2Si2: T. T. M. Palstra, A. A. Menovsky, J. van den Berg, A. J. Dirkmaat, P. H. Kes, G. J. Nieuwenhuys and J. A. Mydosh Physical Review Letters 55, 2727 (1985)

  22. 500 400 300 200 100 1 0 0 5 10 15 20 25 Q = (1,0,0) Intensity (arb.unit) Mason Fåk Honma T (K) Neutron Scattering. Specific heat vs. magnetic Bragg-peak intensity. Tc’s. URu2Si2 Smag ~ 0.2 R ln 2 C5f / T (mJ/K2mol) Tc To mord ~ 0.01 - 0.04 mB 0 xc ~ 100 Å xa ~ 300 Å Type-I AF

  23. Hidden order • Moment is tiny • (likely small admixture of AFM phase) • Large loss of entropy can not be reconciled • with small moment • Other primary symmetry breaking. • WHICH?

  24. 500 U 400 Ru 300 Si 200 100 0 1 10 100 1000 Small Effect at T0. Resistivity decreases as T decrease. ThCr2Si2 bct - type ( I4/mmm ) URu2Si2 a = 4.127 (Å) c = 9.570 (Å) Heavy fermion at high T,low T HO +SC T.T.M. Palstra et al.(1985) W. Schlabitz et al.(1986) M.B. Maple et al.(1986) I // a To ~ 17.5 K r (mWcm ) I // c Tc ~ 1.2 K T (K)

  25. Pseudo-gap opens at Tc. URu2Si2 measured through optical conductivity, D. A. Bonn et al. PRL (1988).

  26. Hall effect as function of temperature in different external fields, Y.S. Oh et al. PRL 98, 016401(2007). Fermi surface reconstruction in zero and small fields. Very large fields polarized Fermi liquid.

  27. Bernal et. al. PRL,2002. P. Chandra P. Coleman et. al. (orbital antiferromagnetism) time reversal symmetry breaking. Si NMR Spectra T = 20 K See however the later experiments in unstrained samples that do not show the effect. T = 4.2 K URu2Si2

  28. Some Proposals for hidden order in the literature • Lev. P. Gorkov: 1996: • -Three spin correlators. • • Chandra et al., Nature’02 - Incommensurate Orbital Antiferromagnetism • •Mineev & Zhitomirsky, PRB ’05 - SDW with tiny moment. • • Varma & Zhu, PRL’06 - Helical Order, Pomeranchuk instability of the Fermi surface ? • • Elgazaar, & Oppeneer, Nature Materials’08- DFT: with weak antiferromagneticorder parameter • Santini and Amoretti PRL 04 • -Quadrupolar ordering. • Fazekas and Kiss PRB 07 • -Octupolar ordering.

  29. Neutron scattering under hydrostatic pressure H. Amitsuka, M. Sato, N. Metoki, M. Yokoyama, K. Kuwahara, T. Sakakibara, H. Morimoto, S. Kawarazaki, Y. Miyako, and JAM PRL 83 (1999) 5114

  30. URu2Si2 Stress in ab plane Large moment when stress in ab plane No moment when stress in c plane M Yokoyama, JPSJ 71, Supl 264 (2002). Further Japanese work showed that NMR in unstrained samples did not broaden below T0

  31. P – T phase diagram H. Amitsuka et al.,JMMM310, 214(2007). LMAF Little change in bulk properties with const. P when crossings into HO(T0) or LMAF(TN) phases, e.g. opening of similar gaps: Adiabatic Continuity.

  32. Phase diagram T vs P based upon resistivity and calorimetric experiments under pressure. E. Hassinger et al. PRB 77, 115117(2008). Similar to Amitsuka’sT – P phase diagram

  33. “Adiabatic continuity” between HO & AFM phase • Similar T0 and TN • Almost identical thermodynamic • quantities (jump in Cv) E. Hassinger et.al. PRL 77, 115117 (2008)

  34. ARPES does not agree with LDA J.D. Denlinger et.al., 2001

  35. Comparison of low-field bulk properties - pure vs.4%Rh Y.S.Oh, K.H.Kim, N. Harrison, H.Amitsuka & JAM, JMMM 310,855(2007).

  36. Effects of RhDopiong. U(Ru1-x,Rhx)2Si2 Rh doping removes HO state to make HF groundstates HF HO HO+AF Rh x 0.0 0.04 T HO state in URu2Si2 develops a gap in FS below 17K Tcoh~56 K THO=17.5 K TC1.5 K 0K M. Jaime et al. PRL (2002) N. Harrison PRL (2003) K. H. Kim et al., PRL (2003) K. H. Kim et al., PRL (2004)

  37. URu2Si2, at high temperatures is not too different from a garden variety heavy fermion. ARPES does not agree with LDA at 30 K. HO can be totally destroyed by H and Rh-x HO converts to LMAF by P thru a first order line. HO and LMAF are remarkably similar (“Mydosh’s adiabatic continuity”) HO opens some form of a gap in optics. HO likely involves an electronic topological transition [Hall Effect, also Nernst] HO exhibits two INS modes:(100)@2meV and (1.400)@5meV of longitudinal fluctuations/excitations. HO (but not LAMF) turns into superconductivity at 1.7 K. Comments concerning Hidden Order

  38. Heavy Femions Early Theoretical Work. Early work : variational wave functionsVarma, C. M. and Yafet, Y., Phys. Rev. B13, 295 (1975); SlaveBosons MFT , 1/ N , A. Auerbach and K. Levin. Phys. Rev. Lett. 57 (1986), p. 877 A. Millis and P.A. Lee. Phys. Rev. B 35, 3394 (1987) Simple description of high and low T regimes. Very simple Renormalized band theory at T=0. G.Kotliar A. Ruckenstein extensions to finite U

  39. High temperature • Ce-4f local moments Early workGeneralized Anderson Lattice Model • Low temperature – • Itinerant heavy bands 6

  40. Dynamical Mean Field Theory. Cavity Construction. A(w) 10

  41. A(w) A. Georges, G. Kotliar (1992) 11

  42. Dynamical Mean Field Theory • Exact in the limit of large coordination (Metzner and Vollhardt 89) . • Can treat arbitrary broken symmetry solutions delta is site , spin, orbital, etc. dependent. • Extension to real materials (Anisimov and Kotliar 1997, Kotliar et. al. RMP 2006). • DMFT equations are still hard to analyze and solve. 12 DFT+DMFT

  43. DMFT “STM” URu2Si2 T=20 K Ru Si Si U Fano lineshape: q~1.24, G~6.8meV, very similar to exp Davis

  44. At T=20 small effects on spd larger gapping of the f’s. PES-DMFT Partial DOS Ground state atomic multiplet of f2 configuration in tetragonal field High T Low T J=4 Only 35K! |state>==|J=4,Jz>

  45. DMFT allows two broken symmetry states at low T. Look for two sublattice structure. J=5/2 Density matrix for U 5f state the J=5/2 subspace Large moment phase: J=5/2 Moment free phase: tetragonal symmetry broken-> these terms nonzero

  46. Valence histogram point of view. The DMFT density matrix has mostly weight in two singlet f^2 configurations Definitely f^2, “Kondo “ limit, J=4, two low lying singlets Test via photoemission [Denlinger and Allen ] Therefore there are two singlests relevant at low energies but they are not non Kramer doublets. Conspiracy between cubic crystal field splittings and tetragonal splittings bring these two states close. This is why URu2Si2 is sort of unique. .

  47. DMFT excitonicorder parameter Order parameter: Different orientation gives different phases: “adiabatic continuity” explained. Hexadecapole order testable by resonant X-ray In the atomic limit:

  48. Simplified toy model phase diagram mean field theory Mean field Exp. by E. Hassinger et.al. PRL 77, 115117 (2008)

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