Hidden order in URu 2. Si 2. hybridization with a twist. Rebecca Flint. Iowa State University. Hasta: spear (Latin) C/T (mj/mol K 2 ) T (K)

Transcription

1 Hidden order in URu 2 Si 2 : Rebecca Flint Iowa State University hybridization with a twist C/T (mj/mol K 2 ) T (K) P. Chandra, P. Coleman and R. Flint, arxiv: (2015) P. Chandra, P. Coleman and R. Flint, Nature 493, 611 (2013) R. Flint, P. Chandra and P. Coleman PRB 86, (2012) Hasta: spear (Latin)

2 Acknowledgements Collaborators: Piers Coleman (Rutgers) Premi Chandra (Rutgers) Funding: Simons Foundation Useful discussions: Laura Greene Neil Harrison Yuji Matsuda John Mydosh Philip Niklowitz Gabriel Kotliar Patrick Lee Senthil P. Chandra, P. Coleman and R. Flint, arxiv: (2015) P. Chandra, P. Coleman and R. Flint, Nature 493, 611 (2013) R. Flint, P. Chandra and P. Coleman PRB 86, (2012)

3 Phase transitions and broken symmetries Landau 1937 Temperature Critical Temperature 32 F, 0 C Breaks rotational symmetry Symmetry breaking measured by order parameter

4 Phase transitions and broken symmetries Example: Ferromagnetism (iron) Broken symmetry: Time-reversal, spin rotation Order parameter: magnetization Temperature

5 Phase transitions and broken symmetries Example: Ferromagnetism (iron) Broken symmetry: Time-reversal, spin rotation Order parameter: magnetization Magnetization Temperature T C

6 Phase transitions and broken symmetries Example: Ferromagnetism (iron) Broken symmetry: Time-reversal, spin rotation Order parameter: magnetization Orr and Chipman (1967) Structural phase transitions Specific heat Curie point (T C ) Temperature Specific heat has discontinuities/divergences at phase transitions

7 Phase transitions and broken symmetries Example: Antiferromagnetism (UPd 2 Al 3 ) Broken symmetry: Time-reversal, spin rotation Order parameter: Staggered magnetization Temperature

8 Phase transitions and broken symmetries Example: Antiferromagnetism (UPd 2 Al 3 ) Broken symmetry: Time-reversal, spin rotation Order parameter: Staggered magnetization Specific heat T N T SC 0 Temperature

9 Phase transitions and broken symmetries Example: Antiferromagnetism (UPd 2 Al 3 ) Broken symmetry: Time-reversal, spin rotation Order parameter: Staggered magnetization Neutron Scattering Specific heat 0 T N T SC Temperature Staggered magnetization

10 Phase transitions and broken symmetries URu 2 Si Matsuda et al Broken symmetry:??? Order parameter:??? Heat capacity/temperature Temperature Palstra et al 1985

11 Phase transitions and broken symmetries URu 2 Si Matsuda et al Broken symmetry:??? Order parameter:??? C/T (mj/mol K 2 ) Looks straightforward, right? T (K) Palstra et al 1985

12 Phase transitions and broken symmetries URu 2 Si 2 Matsuda et al 2008 Broken symmetry:??? Order parameter:??? Entropy C/T (mj/mol K 2 ) Looks straightforward, right? T (K) Palstra et al 1985

13 Phase transitions and broken symmetries URu 2 Si 2 Matsuda et al 2008 Broken symmetry:??? Order parameter:??? Entropy Expect large order parameter C/T (mj/mol K 2 ) Looks straightforward, right? T (K) Palstra et al 1985

14 Phase transitions and broken symmetries URu 2 Si Matsuda et al Broken symmetry:??? Order parameter:??? antiferromagnetism structural transition charge order Looks straightforward, right?

15 Phase transitions and broken symmetries URu 2 Si Matsuda et al Broken symmetry:??? Order parameter:??? antiferromagnetism structural transition charge order Looks straightforward, right?

16 Phase transitions and broken symmetries URu 2 Si Matsuda et al Broken symmetry:??? Order parameter:??? antiferromagnetism structural transition charge order quadrupolar order (Santini+Amoretti '94, Harima et al '98) quadrupolarγ 5 order (Amitsuka+Sakakibara '94) octupolar order (Kiss+Fazekas '04 and others) hexadecapolar order (Haule+Kotliar '09) triakontadipolar order (Cricchi et al '09 spin density wave (Mineev+Zhitomirsky '04) unconventional spin density wave (Maki+Dora '03) quadrupolar density wave (Ramirez et al '92) d-density wave (Ikeda+Ohashi '98) chiral d-density wave (Kotetes et al '10) orbital antiferromagnetism (Tripathi et al '02) dimerization (Kasuya '97) Looks straightforward, so what are we missing? two spin correlators (Gorkov+Sokol '92) three spin correlators (Barzykin+Gorkov '93) helicity order (Varma+Zhu '05) spin nematic (Fujimoto '11) rank 5 nematic (Ikeda et al '12) topological spin nematic (Das '12) dynamical symmetry breaking (Elgazzar et al '06) modulated spin liquid (Pepin,Burdin,Norman '10) rank 5 pseudo-spin vector (Rau+Kee '12) hybridization wave (Dubi+Balatsky '10) mixed valence (Barzykin+Gorkov '93) duality (Okuno+Miyake, Sikkema '98)

17 Phase transitions and broken symmetries URu 2 Si 2 Matsuda et al 2008 Broken symmetry:??? Order parameter:??? antiferromagnetism structural transition charge order quadrupolar order (Santini+Amoretti '94, Harima et al '98) quadrupolarγ 5 order (Amitsuka+Sakakibara '94) octupolar order (Kiss+Fazekas '04 and others) hexadecapolar order (Haule+Kotliar '09) triakontadipolar order (Cricchi et al '09 spin density wave (Mineev+Zhitomirsky '04) unconventional spin density wave (Maki+Dora '03) quadrupolar density wave (Ramirez et al '92) d-density wave (Ikeda+Ohashi '98) chiral d-density wave (Kotetes et al '10) orbital antiferromagnetism (Tripathi et al '02) dimerization (Kasuya '97) Hidden order Looks straightforward, so what are we missing? two spin correlators (Gorkov+Sokol '92) three spin correlators (Barzykin+Gorkov '93) helicity order (Varma+Zhu '05) spin nematic (Fujimoto '11) rank 5 nematic (Ikeda et al '12) topological spin nematic (Das '12) dynamical symmetry breaking (Elgazzar et al '06) modulated spin liquid (Pepin,Burdin,Norman '10) rank 5 pseudo-spin vector (Rau+Kee '12) hybridization wave (Dubi+Balatsky '10) mixed valence (Barzykin+Gorkov '93) duality (Okuno+Miyake, Sikkema '98)

18 Phase transitions and broken symmetries URu 2 Si Matsuda et al Broken symmetry: single and double time-reversal Order parameter: hybridization spinor Our proposal: A fundamentally new way to break time-reversal symmetry Hastatic order Hasta: spear (Latin) Looks straightforward, so what are we missing?

19 Phase transitions and broken symmetries URu 2 Si Matsuda et al Broken symmetry: single and double time-reversal Order parameter: hybridization spinor Our proposal: A fundamentally new way to break time-reversal symmetry Hastatic order Hasta: spear (Latin) Looks straightforward, so what are we missing?

20 Outline Time-reversal symmetry (spinor) Heavy fermion basics (hybridization) The hidden order problem A few relevant experiments Spinorial hybridization Experimental consequences of hastatic order Future directions

21 Time-reversal symmetry Spins invert under time-reversal: Spin ½ wave-function is a spinor Complex conjugation

22 Time-reversal symmetry Spins invert under time-reversal: Spin ½ wave-function is a spinor Complex conjugation

23 Time-reversal symmetry A spinor is like the square root of a vector (Like Vector Spinor Double time-reversal: Fermions, half-integer spins Bosons, integer spins

24 Time-reversal symmetry A spinor is like the square root of a vector (Like Vector Spinor Double time-reversal: Fermions, half-integer spins Bosons, integer spins

25 Time-reversal symmetry A spinor is like the square root of a vector (Like Vector Spinor Double time-reversal: Fermions, half-integer spins Bosons, integer spins

26 Time-reversal symmetry Consequence: Kramers theorem Kramers doublet A state and its time-reversed twin are orthogonal But degenerate in energy doublet protected by time-reversal Double time-reversal: Fermions, half-integer spins Bosons, integer spins

27 Time-reversal symmetry Consequence: Kramers theorem Kramers doublet A state and its time-reversed twin are orthogonal But degenerate in energy doublet protected by time-reversal Non-Kramers doublet Not protected by time-reversal (can be broken by electric fields) Can be magnetic, but doesn't have to be (quadrupolar) Double time-reversal: Fermions, half-integer spins Bosons, integer spins

28 Broken time-reversal symmetry Spin order Ferromagnetism, antiferromagnetism

29 Broken time-reversal symmetry Spin order Ferromagnetism, antiferromagnetism And higher multipoles (octupolar, dotriakontapolar, ) Spin chirality Complex (p+ip) superconductors (Sr 2 RuO 4?) Orbital currents (CuO) / toroidal moments Cu O Scagnoli et al 2011 All of these are vector order parameters

30 Heavy fermion ingredients In search of

31 Localization vs. itineracy 4f Magnetic Moments 5f 3d Metallic Increasing itineracy Increasing localization Kmetko and Smith, 1983

32 Example: high temperature superconductors Increasing localization Increasing itineracy

33 Localization vs. Itineracy In the cuprates, this competition is induced by doping In heavy fermions, this competition is built into the atom 4f 5f Magnetic Moments 3d Metallic Increasing itineracy Increasing localization Kmetko and Smith, 1983

34 Heavy fermion physics Local moments Conduction electrons

35 Kondo physics Magnetic impurity in a metal (e.g. - Fe in Au) J. Kondo Spin flip scattering Electron spin density Impurity spin

36 Kondo physics Magnetic impurity in a metal (e.g. - Fe in Au) J. Kondo Moment screened below the Kondo temperature - - Electron spin density Impurity spin

37 Kondo physics Magnetic impurity in a metal (e.g. - Fe in Au) J. Kondo Moment screened below the Kondo Quenched temperature moment - - Free moment The local moment is asymptotically free Quenched moment T

38 Kondo physics in the lattice At high temperatures: Local moments Conduction electrons At lower temperatures: Heavy electrons x more massive!

39 Momentum space picture Conduction electrons Energy Local moments (f-electrons) Momentum

40 Momentum space picture Energy Heavy electrons Momentum Conduction and f-electrons hybridize at low temperatures No broken symmetries, develops as crossover

41 Momentum space picture Broken symmetry Phase transition Magnetization Energy Hybridization No broken symmetry Crossover T* Heavy electrons Temperature T C Temperature Momentum Conduction and f-electrons hybridize at low temperatures No broken symmetries, develops as crossover

42 Momentum space picture Energy Heavy electrons Momentum Much flatter band (means heavier) Conduction and f-electrons hybridize at low temperatures No broken symmetries, develops as crossover

43 Momentum space picture Energy Heavy electrons Momentum Hybridization gap Conduction and f-electrons hybridize (mix) at low temperatures No broken symmetries, develops as crossover

44 What is hybridization really? or how do you flip a spin? Initial -+

45 What is hybridization really? Initial or how do you flip a spin? Virtual valence fluctuations -+

46 What is hybridization really? Initial or how do you flip a spin? Virtual valence fluctuations Final -+ -+

47 Hidden order in URu 2 Si 2 Matsuda et al 2008 Palstra et al 1985 A twenty seven year old mystery Mean field phase transition But what order parameter? Large entropy But no large moments

48 Hidden order in URu 2 Si 2 Local moments (f-electrons) Matsuda et al 2008 Magnetic susceptibility z-axis in plane Palstra et al 1985

49 Hidden order in URu 2 Si 2 Matsuda et al 2008 Magnetic susceptibility z-axis in plane free Ising moments Palstra et al 1985

50 Hidden order in URu 2 Si 2 Heavy fermion metal Matsuda et al 2008 Magnetic susceptibility z-axis in plane free Ising moments Palstra et al 1985 Ising moments quench to form heavy Fermi liquid

51 Proximity to antiferromagnetism First order transition to local moment antiferromagnetic phase Villaume et al 2008 Broholm et al 1991 Jo et al 2007 Kotliar + Haule 2009 Niklowitz et al 2011 Same Fermi surface in both phases same ordering vector Q = [001] Longitudinal spin fluctuation mode (INS); gets soft, but not critical at T HO Pseudo-Goldstone mode? Can t relate phases with different time-reversal properties!

52 Measuring the hybridization gap Quasiparticle interference reveals band structure new insights from spectroscopy STM-STS on URu 2 Si 2 E (mv) E (mv) Theory q q Schmidt et al 2010

53 Measuring the hybridization gap Quasiparticle interference reveals band structure new insights from spectroscopy STM-STS on URu 2 Si 2 E (mv) E (mv) Theory Experiment q q Schmidt et al 2010

54 Measuring the hybridization gap Quasiparticle interference reveals band structure new insights from spectroscopy STM-STS on URu 2 Si 2 E (mv) E (mv) Experiment q q Schmidt et al 2010

55 Measuring the hybridization gap Quasiparticle interference reveals band structure new insights from spectroscopy STM-STS on URu 2 Si 2 E (mv) E (mv) Gap q q Schmidt et al 2010

56 Measuring the hybridization gap Quasiparticle interference reveals band structure new insights from spectroscopy STM-STS on URu 2 Si 2 E (mv) E (mv) Gap q Tunneling density of states reveals mean field gap matches specific heat Normalized density of states Gap Gap (mev) q Schmidt et al 2010 Aynajian et al 2010

57 Measuring the hybridization gap Quasiparticle interference reveals band structure new insights from spectroscopy Optical spectroscopy confirms (bulk) STM-STS on URu 2 Si 2 E (mv) E (mv) Gap q Tunneling density of states reveals mean field gap matches specific heat Normalized density of states Gap Gap (mev) q Schmidt et al 2010 Aynajian et al 2010

58 Hybridization as an order parameter Quasiparticle interference reveals band structure new insights from spectroscopy STM-STS on URu 2 Si 2 E (mv) E (mv) Gap Hybridization gap is the order parameter?! q Tunneling density of states reveals mean field gap matches specific heat Normalized density of states Gap Gap (mev) q Schmidt et al 2010 Aynajian et al 2010

59 z Broken symmetries? new insights from torque magnetometry y U Tetragonal symmetry: x Two-fold component of in-plane torque: Broken tetragonal symmetry: R. Okazaki, et al., Science 331, 439 (2011)

60 z Broken symmetries? new insights from torque magnetometry y x T < T HO T > T HO T = 6 K T = 8 K T = 10 K T = 14 K T = 18 K Tetragonal symmetry: Two-fold component of in-plane torque: Broken tetragonal symmetry: R. Okazaki, et al., Science 331, 439 (2011)

61 z Broken symmetries? new insights from torque magnetometry y x T < T HO T > T HO T = 6 K T = 8 K T = 10 K T = 14 K T = 18 K Tetragonal symmetry: Broken tetragonal symmetry: T HO R. Okazaki, et al., Science 331, 439 (2011)

62 Broken tetragonal symmetry No transverse f-electron response must be conduction electrons scattering off the hidden order T HO Hidden order breaks tetragonal symmetry R. Okazaki, et al., Science 331, 439 (2011)

63 Broken tetragonal symmetry No transverse f-electron response must be conduction electrons scattering off the hidden order T HO Resonant f-electron scattering Spin dependent t-matrix: Spin nematic Fujimoto 2010 (all conduction electrons) Broken time-reversal

64 Phase transitions and broken symmetries URu 2 Si Matsuda et al Broken symmetry: magnetic tetragonal symmetry, time-reversal Order parameter: hybridization gap (???) How can we connect these?

65 Giant Ising anisotropy Uranium moments are severely Ising: Magnetic susceptibility c-axis in plane Non-Kramers doublet (Integer spin eg - 5f 2 ) Palstra et al 1985 What about the conduction electrons?

66 Giant Ising anisotropy What about the conduction electrons? How can we measure the conduction electron anisotropy? Measure Fermi surface magnetization in field de Haas van Alphen

67 Giant Ising anisotropy What about the conduction electrons? How can we measure the conduction electron anisotropy? Measure Fermi surface magnetization in field de Haas van Alphen

68 Giant Ising anisotropy What about the conduction electrons? How can we measure the conduction electron anisotropy? Measure Fermi surface magnetization in field de Haas van Alphen Spin zero condition

69 Giant Ising anisotropy What about the conduction electrons? How can we measure the conduction electron anisotropy? Measure Fermi surface magnetization in field de Haas van Alphen Altarawneh 2011 Ohkuni 1999 Spin zero condition Heavy electron with perfect Ising symmetry Hybridizing with 5f 2

70 Consequences of Ising hybridization Kramers index Local moment (integer spin): Two successive time-reversals Θ 2 = 2π rotation

71 Consequences of Ising hybridization Kramers index Local moment (integer spin): Two successive time-reversals Θ 2 = 2π rotation Conduction electron (spin 1/2):

72 Consequences of Ising hybridization Kramers index Local moment (integer spin): Two successive time-reversals Θ 2 = 2π rotation Conduction electron (spin 1/2): Hybridization:

73 Consequences of Ising hybridization Kramers index Local moment (integer spin): Two successive time-reversals Θ 2 = 2π rotation Conduction electron (spin 1/2): Hybridization:

74 Consequences of Ising hybridization Kramers index Local moment (integer spin): Two successive time-reversals Θ 2 = 2π rotation Conduction electron (spin 1/2): Hybridization: V breaks time-reversal symmetry! (like a spinor so double time-reversal symmetry too)

75 Consequences of Ising hybridization Kramers index Local moment (integer spin): Two successive time-reversals Θ 2 = 2π rotation Conduction electron (spin 1/2): Hybridization: V breaks time-reversal symmetry! (like a spinor so double time-reversal symmetry too) In other words: mixing spin ½ with spin 1 hybridization must carry spin 1/2

76 Review: Normal hybridization Kramers doublet flips its spin by fluctuating to a singlet excited state Kramers doublet One channel Kondo effect - excited singlet carries no quantum numbers - hybridization breaks no symmetries - develops as a crossover Hybridization T* Temperature

77 Spinorial hybridization Kramers doublet Non-Kramers doublet flips its spin by fluctuating to a Kramers doublet excited state Non-Kramers doublet Two channel Kondo effect - excited Kramers doublet carries magnetic quantum number - hybridization breaks time-reversal - hybridization develops as a phase transition Hybridization T HO Temperature

78 Spinorial hybridization Kramers doublet Non-Kramers doublet flips its spin by fluctuating to a Kramers doublet excited state Non-Kramers doublet Two channel Kondo effect - excited Kramers doublet carries magnetic quantum number - hybridization breaks time-reversal, develops as a phase transition Disordered Paramagnet

79 Spinorial hybridization Kramers doublet Non-Kramers doublet flips its spin by fluctuating to a Kramers doublet excited state Non-Kramers doublet Two channel Kondo effect - excited Kramers doublet carries magnetic quantum number - hybridization breaks time-reversal, develops as a phase transition Disordered Ordered Staggered Q = [001] Paramagnet Antiferromagnet Hidden (hastatic) order

80 Spinorial hybridization Landau theory

81 Spinorial hybridization Landau theory T HO Like a spin-flop transition

82 Spinorial hybridization Landau theory T HO Gap to longitudinal spin fluctuations:

83 Nonlinear susceptibility anisotropy Nonlinear susceptibility: Nonlinear magnetic susceptibility c-axis T HO T(K) 5 in plane Ramirez et al 1992

84 Nonlinear susceptibility anisotropy Landau theory Ising f-states couple only to B z

85 Nonlinear susceptibility anisotropy Landau theory Ising f-states couple only to B z Nonlinear magnetic susceptibility c-axis T HO in plane T(K)

86 Nonlinear susceptibility anisotropy Landau theory Ising f-states couple only to B z > 1000-fold anisotropy predicted Nonlinear magnetic susceptibility c-axis T HO in plane T(K)

87 Spinorial hybridization Kramers doublet Non-Kramers doublet flips its spin by fluctuating to a Kramers doublet excited state Non-Kramers doublet Two channel Kondo effect - excited Kramers doublet carries magnetic quantum number - hybridization breaks time-reversal, develops as a phase transition Staggered Q = [001] Disordered (paramagnet) Antiferromagnet Hidden (hastatic) order

88 Microscopic theory Non-Kramers doublet (5f 2, J = 4): Γ 5 Protected by tetragonal symmetry Magnetic along, quadrupolar in-plane Amitsuka + Sakakibara 1994

89 Microscopic theory Non-Kramers doublet (5f 2, J = 4): Γ 5 Protected by tetragonal symmetry Magnetic along, quadrupolar in-plane Constrains valence fluctuations: Γ 5 Amitsuka + Sakakibara 1994 Cox+Jarrell 1996 Cox+Zawadowskii 2002

90 Microscopic theory Non-Kramers doublet (5f 2, J = 4): Γ 5 Protected by tetragonal symmetry Magnetic along, quadrupolar in-plane Constrains valence fluctuations: Conduction electron at site j, with symmetry Γ 5

91 Microscopic theory Non-Kramers doublet (5f 2, J = 4): Γ 5 Protected by tetragonal symmetry Magnetic along, quadrupolar in-plane Constrains valence fluctuations: Conduction electron at site j, with symmetry Γ 5

92 Microscopic theory Two channel Anderson model Γ 5 Introduce slave bosons/fermions to represent doublets: (fermion) (boson)

93 Microscopic theory Two channel Anderson model Γ 5 Introduce slave bosons/fermions to represent doublets: Moments: Local (5f 2 ): Mixed valent (5f 1 ): Conduction electron: (fermion) (boson)

94 Microscopic theory Two channel Anderson model Γ 5 Introduce slave bosons/fermions: (fermion) (boson)

95 Microscopic theory Two channel Anderson model Γ 5 Hidden order Ansatz: Hidden (hastatic) order

96 Microscopic theory Two channel Anderson model Γ 5 Hidden order Ansatz: Redefine Hidden (hastatic) order

97 Microscopic theory Two channel Anderson model Γ 5 uniform staggered uniform staggered

98 Microscopic theory Two channel Anderson model Γ 5 uniform staggered uniform Breaks time-reversal staggered

99 Experimental consequences Moments: Local (5f 2 ): Mixed valent (5f 1 ): Conduction electron: Non-zero moments?

100 Experimental consequences Moments: Local (5f 2 ): Mixed valent (5f 1 ): Conduction electron: Non-zero moments?

101 Experimental consequences Moments: Local (5f 2 ): Mixed valent (5f 1 ): Conduction electron: Non-zero moments? Kondo effect enforces small moment: Clogston-Anderson 1961

102 Experimental consequences Moments: Local (5f 2 ): Mixed valent (5f 1 ): Conduction electron: Non-zero moments? Kondo effect enforces small moment: 20% mixed valency Clogston-Anderson 1961

103 Experimental consequences Moments: Local (5f 2 ): Mixed valent (5f 1 ): Conduction electron: Non-zero moments? Magnetic (m z ) form factor No f-electron (Γ 5 ) moments Quadrupolar form factor

104 Experimental consequences Broken tetragonal symmetry No quadrupolar moments, so no structural transition But anisotropic spin response: Data from Okazaki 2011 Mean-field calculation

105 Experimental consequences Broken tetragonal symmetry No quadrupolar moments, so no structural transition But anisotropic spin response: And anisotropic hybridization gap: Data from Okazaki 2011 Meanfield calc.

106 Experimental consequences Broken tetragonal symmetry No quadrupolar moments, so no structural transition But anisotropic spin response: And anisotropic hybridization gap: Fermi surface splits below T HO : de Haas-van Alphen splitting (Ohkuni et al '99) Cyclotron resonance (Tonegawa et al '12) Small x-ray signals (second order effect) Energy dependent nematicity Data from Okazaki 2011 Meanfield calc. Riggs et al, arxiv:

107 Experimental consequences Broken tetragonal symmetry No quadrupolar moments, so no structural transition But anisotropic spin response: And anisotropic hybridization gap: Energy dependent nematicity (Scanning tunneling microscopy) 10m

108 Experimental consequences g-factor anisotropy: Other consequences data from Altarawneh 2011 calculation Nonlinear susceptibility (χ 3 ) anomaly anisotropy: > 1000-fold anisotropy predicted Longitudinal spin fluctuations:

109 Consistency Absence of large moments Aynajian et al 2010 Hybridization gap as an order parameter Ising conduction electrons Broken tetragonal symmetry Susceptibility Fermi surface (dhva, cyclotron resonance) Pseudo-Goldstone mode (neutron scattering) Predictions: Small basal plane moment Longitudinal spin fluctuations 10 3 anisotropy in nonlinear susceptibility Resonant nematicity

110 What about above the hidden order? Beyond mean field theory Hastatic order likely melts via phase fluctuations: uniform staggered Breaks no symmetries Breaks symmetries

111 What about above the hidden order? Hastatic order likely melts via phase fluctuations: Hastatic pseudogap Beyond mean field theory uniform staggered Breaks no symmetries Breaks symmetries

112 What about above the hidden order? Hastatic order likely melts via phase fluctuations: Hastatic pseudogap Beyond mean field theory uniform staggered Breaks no symmetries Breaks symmetries Forms incoherent heavy Fermi liquid Heavy mass seen in thermodynamics, but no hybridization gap

113 Conclusions Ising quasiparticles indicate hybridization between conduction electrons and a non-kramers doublet Γ 5

114 Conclusions Ising quasiparticles indicate hybridization between conduction electrons and a non-kramers doublet Hybridization mixes half-integer and integer spin states, must be spinorial Unlike magnetism, this spinorial hybridization breaks both single and double time-reversal Hastatic order How can a (bosonic) order parameter behave like a spinor? Spin statistics theorem requires relativity Explains many features of hidden order in URu 2 Si 2 Implies tiny staggered transverse moment Neutrons have now ruled out moments larger than.0007µ B NMR and µsr are both consistent with moments ~ 10-4 µ B So the moments might be a lot smaller than expected Hasta: spear (Latin)

115 Open questions What about the superconductivity at 1.5K? What is its origin? Is it d+id? Or can pairing hastatic quasiparticles explain the Kerr effect? Other examples of hastatic order? Non-Kramers doublets in other Pr, U compounds? Why doesn t PrInAg order down to.1t*?

116 Open questions What about superconducting analogues? UBe 13 (1974): an even older mystery U Ott et al 1974 Resistivity Never forms heavy Fermi liquid Triplet superconductivity Temperature

117 Open questions What about superconducting analogues? UBe 13 (1974): an even older mystery U Non-Kramers doublet Kramers doublet Never forms heavy Fermi liquid Triplet superconductivity Cox 1988

118 Thank you!

119 What do we know about URu 2 Si 2? Villaume et al 2008 Kim et al 2003 At high T > 70K, looks like a Kondo lattice, with Ising moments Moments quench ~ 70K, electrons get modestly heavy, plenty of entropy left Hidden order develops at 17.5K mean-field like With pressure, becomes AF (1 st order dhva same FS, Q = [001]) In field, HO favored over AF, but slowly killed by 35T (QCP?) Doping (on Ru) can mimic pressure (Fe) or induce FM (Re) Heavy quasiparticles inherit Ising g-factor Breaks tetragonal symmetry Torque magnetometry shows χ xy develops at T HO, cyclotron resonance/dhva suggest FS splitting Small (δ ~ 10-6 ) orthorhombic signature Superconductivity at 1.5K Probably d-wave singlet pairing, but indications of time-reversal symmetry breaking Pairing of heavy quasiparticles (H C2 shows Ising anisotropy)

120 What do we know about URu 2 Si 2? Broken tetragonal symmetry And broken time-reversal symmetry? Adiabatic continuity -- HO and AF related by rotation in parameter space Longitudinal spin fluctuations get soft (but not critical) at T HO Pseudo-goldstone mode? HO must also break time-reversal χ xy also suggests broken time-reversal Due to conduction electrons scattering off HO Spin dependent t-matrix Large magnitude (χ xy /χ xx ~10) explained by resonant scattering off f-electrons Villaume 2008; Broholm 1991; Jo 2007 Kotliar + Haule 2009; Niklowitz et al 2011 Broken time-reversal T HO

121 What do we know about URu 2 Si 2? Schmidt et al 2010 Aynajian et al 2010 E (mv) Broken tetragonal symmetry. And broken time reversal? Gaps: E (mv) Thermodynamics/transport indicate that most of the FS gaps out at T HO (like a density wave - gap magnitude ~ 4meV from specific heat) STM-STS quasiparticle interference shows a hybridization gap developing at T HO - also ~4meV q q E (mv) q q Gap Normalized density of states Gap

122 What do we know about URu 2 Si 2? Schmidt et al 2010 Aynajian et al 2010 Broken tetragonal symmetry. And broken time-reversal? Gaps: Thermodynamics/transport indicate that most of the FS gaps out at T HO (like a density wave - gap magnitude ~ 4meV from specific heat) STM-STS quasiparticle interference shows a hybridization gap developing at T HO - also ~4meV Optical conductivity sees gap developing at T HO, also at 50K Point contact spectroscopy sees gaps developing between 17.5K and 60K, depending Pseudogap proposed by Balatsky at 25K (consistent with Greene s PCS gap at 27K) Multiple gaps developing at multiple temperatures Hidden order gap appears to be a hybridization gap (!) Various indications from ARPES that the coherent FL develops at T HO

123 Big open issues (esp. for microscopic theorists) Local or itinerant physics? (or both) Local physics must further resolve: f-electron valence: 5f 2 or 5f 3? Crystal field levels Older theories: quadrupolar, octupolar order Two singlets hexadecapolar order (Haule + Kotliar 2008) Γ 5 doublet (originally proposed by Amitsuka) hastatic order Itinerant physics Various density waves Rank 5 order (Ikeda 2012) Duality between local and itinerant pictures?

124 ARPES Hole pockets at Γ, M, Z points Quasiparticle band crossing FS Claim to see hybridization developing at T HO * Also see hybridization developing above T HO Ito et al PRB 1999 Santander-Syro Nat. Phys. 2009

125 Time-resolved ARPES