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Hastatic Order in URu Si : Hybridization with a Twist 2 2 Premala Chandra,1 Piers Coleman,1 and Rebecca Flint2 1 Center for Materials Theory, Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854 and 2 Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA (Dated: January 7, 2015) The broken symmetry that develops below 17.5K in the heavy fermion compound URu Si has 2 2 long eluded identification. Here we argue that the recent observation of Ising quasiparticles in URu Si resultsfromaspinorhybridizationorderparameterthatbreaksdoubletime-reversalsym- 2 2 metrybymixingstatesofintegerandhalf-integerspin. Such“hastaticorder”(hasta:[Latin]spear) hybridizes Kramers conduction electrons with Ising, non-Kramers 5f2 states of the uranium atoms toproduceIsingquasiparticles. Thedevelopmentofaspinorialhybridizationat17.5Kaccountsfor boththelargeentropyofcondensationandthemagneticanomalyobservedintorquemagnetometry. 5 Thispaperdevelopsthetheoryofhastaticorderindetail,providingthemathematicaldevelopment 1 of its key concepts. Hastatic order predicts a tiny transverse moment in the conduction sea, a 0 collosal Ising anisotropy in the nonlinear susceptibility anomaly and a resonant energy-dependent 2 nematicity in the tunneling density of states. n a PACSnumbers: J 6 Contents 2. ηz and η⊥ from Microscopics 21 ] C. Basal-Plane Moment 22 l e I. Introduction 1 - A. Experimental Motivation for Hastatic Order 2 VI. Discussion and Open Questions 23 r t B. Ground-State Configuration of the Uranium A. Broader Implications of Hastatic Order 23 s . Ion 3 B. Experimental Constraints and More Tests 25 t a C. Hybridization, Hastatic Order and C. Future Challenges 26 m Double-Time Reversal Symmetry 4 - D. Two-Channel Valence Fluctuation Model 5 References 26 d E. Structure of the Paper 6 n o c II. Landau theory: Pressure-induced I. INTRODUCTION [ antiferromagnetism 6 1 AB.. STohfetrmmooddeysnaamndicdsynamics 76 The heavy fermion superconductor URu2Si2 exhibits v a large specific heat anomaly at T0 =17.5K signalling 1 the development of long-range order with an associ- III. Microscopic Model: Two-Channel 28 Anderson Lattice 8 ated entropy of condensation, (cid:82)0T0 CTV dT ≈ 12Rln2 per 1 A. The Valence Fluctuation Hamiltonian 8 mole formula unit1,2. This sizable ordering entropy 0 1. The 5f1 model 8 in conjunction with the sharpness of the specific heat . 2. The 5f3 case 9 anomaly suggests underlying itinerant ordering; how- 1 0 B. Slave particle treatment 10 ever the anisotropic bulk spin susceptibility (cf. Fig. 5 C. Mean Field Theory for Hastatic Order 11 1) of URu2Si2 displays Curie-Weiss behavior down to 1 1. Mean Field Hamiltonian: a spinorial order T ∼ 70K indicative of local moment behavior. Initially, v: parameter 11 the hidden order was attributed to a spin density wave i 2. Mean field parameters 12 with a tiny c-axis moment3,4, which was later shown X D. Conduction and f-electron Green’s functions 13 to be extrinsic5. However, spin ordering in the form r E. Particle-Hole Symmetric Case 15 of antiferromagnetism is observed at pressures exceeding a F. Hybridization gaps 16 0.8GPa6–8. Despite thirty years of intense experimental effort,nolaboratoryprobehasyetcoupleddirectlytothe IV. Comparison to Experiment: Postdictions 17 orderparameterinURu2Si2 atambientpressure,though A. g-factor Anisotropy 17 there have been a wide variety of theoretical proposals B. Anisotropic Linear Susceptibility 17 for this “hidden order” (HO) problem9–21. We point the interestedreadertoarecentreviewonURu Si formore 2 2 V. Comparison to Experiment: Predictions 18 details.18 A. Resonant Nematicity in Scanning Probes 18 Expandingonourrecentproposal19–21 of“hastaticor- B. Anisotropy of the Nonlinear Susceptibility der” in URu Si , here we argue that the failure to ob- 2 2 Anomaly 20 serve the nature of its “hidden order” is not due to its 1. Landau Theory 20 intrinsiccomplexitybutinsteadresultsfromafundamen- 2 tally new kind of broken time-reversal symmetry associ- dependent magnetic field in the HO state; this magneti- ated with an order parameter of spinorial, half-integer zation is a periodic function of the ratio of the Zeeman spin character. Key evidence supporting this conjec- and the cyclotron energies, where the former is defined ture is the observation of quasiparticles with an Ising through an angle-dependent g-factor g(θ) anisotropy characteristic of integer spin f-moments22–25. Hastaticorderaccountsforthisunusualfeatureasacon- ∆E(θ)=g(θ)µ |B|. (1) B sequence of a spinor order parameter that coherently hybridizes the integer spin, Ising f-moments with half- Interference of Zeeman-split orbits in tilted fields leads integerspinconductionelectrons; theobservedquasipar- to spin zeroes in the quantum oscillation measurements ticle and the magnetic anisotropies thus have the same (cf. Fig. 2) satisfying the condition origin. m∗ g(θ ) =2n+1 (2) n m e wherenisapositiveintegerandθ isthe(indexed)angle n with respect to the c-axis. Sixteen such spin zeroes were identified in the HO state of URS22,23, and the experi- mentalists found that g 1 ⊥ < (3) g 30 c where g = g(θ ∼ π) and g = g(θ = 0), indicating ⊥ n 2 c n that ∆E(θ ) depends solely on the c-axis component of n FIG. 1: (a) Tetragonal structure of URu2Si2 (b) Tem- the applied magnetic field (Bc), namely that perature dependence of magnetic susceptibility in URu Si 2 2 after26. g(θ )=g∗cosθ (4) n n where g∗ =2.6 in contrast to the isotropic g =2 for free electrons. A. Experimental Motivation for Hastatic Order In these high-field measurements, the quasiparticle anisotropy manifests itself through the appearance of a Abovethehiddenordertransition,URu Si isaninco- rapid modulation in the amplitude of the dHvA oscilla- 2 2 herent heavy fermion metal, with an large, anisotropic, tions generated by the heavy α pockets of URu2Si2 as linear resistivity26,and a linear specific heat with γ = the magnetic field is tilted from the c-axis into the basal CV ∼200mJmol−1K−2 . The development of hidden or- plane. The same magnetic anisotropy is also observed in T der results in a significant reduction in the specific heat the angular dependence of the upper-critical field of the tloossγ0of=abCToVut∼tw60o-mthJimrdosl−of1Kth−e2h,ecaovryreFsperomndiisnugrftaoceth18e.tAhet stuurpeesrc(ocfn.dFuicgt.in2g)2s4t,2a5t.eWwhheicrheadsetvheelodpHsvaAtmloewasutermempeenrats- could in principle belong to a select region of the Fermi T = 1.5K, the remaining heavy quasiparticles go su- c surface, theupper-criticalfield, H (θ)issensitivetothe perconducting. Under a modest pressure of 0.8GPa, the c2 entire heavy fermion pair condensate, proving crucially hidden order ground-state of URu Si undergoes a first 2 2 that the Ising quasiparticle anisotropy pervades the en- order transition into an Ising antiferromagnet with an tire Fermi surface of hidden order state. We note that staggered ordered moment of order 0.4 µ aligned along B the c-axis7. de Haas-van Alphen (dHvA) shows that while Hc2(θ) matches the anisotropy of the g-factor for angles near the c-axis, where H is Pauli limited, when the quasiparticles in the hidden order phase form small, c2 thefieldisin-plane,H (θ)islargerthanexpected,likely highly coherent heavy electron pockets with an effective c2 massupto8.5m 22. Remarkably,thesesmallheavyelec- duetoorbitalcontributions. WenotethatsincethePauli e susceptibility χP scales with the square of the g-factor, tron pockets survive across the first order transition into the high-pressure antiferromagnetic phase, leading many theseresolution-limitedmeasurementsof gg⊥c suggestthat groups to conclude that the (commensurate) ordering wavevectors Q = (0,0,1) of the antiferromagnetic and χP χP(θ)=χP∗cos2θ c >900. (5) the HO phases are the same10,27–30. χP ⊥ Perhaps the most dramatic feature of these heavy electron pockets is the essentially perfect Ising mag- Such a large anisotropy should be observable in elec- netic anisotropy in the magnetic g-factors of the itin- tron spin resonance measurements that probe the Pauli erant heavy f-electrons in the HO state of URS23. This susceptibility directly in contrast to bulk susceptibility Ising quasiparticle anisotropy has been determined by measurements where Van Vleck contributions are also measuring the Fermi surface magnetization in an angle- present. 3 metryofthetetragonalcrystal. However,thepresenceof a perfect Ising anisotropy requires an Ising selection rule (cid:104)Γ |J |Γ (cid:105)=0 (8) ± ± ∓ that, in the absence of fine-tuning of the coefficients a , n leadstotheconditionthat−(J +4n(cid:48))(cid:54)=(J +4n)±1,or z z J (cid:54)=2(n−n(cid:48))± 1, requiring J ∈Z must be an integer. z 2 z Foranyhalf-integerJ ,correspondingtoaKramersdou- z blet, the selection rule is absent and the ion will develop a generic basal-plane moment. As an aside we note that FIG.2: (a)Anisotropyofthemeasuredg-factor23plotted(a) the fine-tuned case will produce an Ising Kramers dou- inpolarco-ordinatesderivedfromspinzerosinquantumoscil- blet,butcorrespondstothecompleteabsenceoftetrago- lation measurements and the anisotropy of the upper critical nal mixing, highly unlikely in a tetragonal environment. field(b)versussineoftheangleoutofthebasalplane,showing that the data23 requires a Pauli susceptibility anisotropy Let us apply this argument specifically to the case of in excess of a thousand. URu2Si2 . We first suppose that the U ion is in a 5f3 (J = 9) configuration, predominantly in a |± 7(cid:105) state. 2 2 Thepresenceoftetragonalsymmetryresultsinacrystal- B. Ground-State Configuration of the Uranium Ion field ground-state Bulk susceptibility measurements (see Fig. 1) of 7 1 9 |±(cid:105)=a|± (cid:105)+b|∓ (cid:105)+c|∓ (cid:105) (9) URu2Si2 reveal that the magnetic U ions have an Ising 2 2 2 anisotropy, with a magnetic moment that is consistent either with a U3+ (5f3) configuration or a non-Kramers so that |(cid:104)−|J |+(cid:105)|2 = 5b2 + 6ac and perfect Ising + U4+ (5f2) ion1,31. A natural explanation for the quasi- anisotropyisonlyachievedwithfine-tuningofthetetrag- particleIsinganisotropyisthattheIsingcharacterofthe onal mixing coefficients such that the condition 5b2 + uranium (U) ions has been transferred to the quasipar- 6ac = 0 is satisfied. By contrast, for U ion in a 5f2 ticles via hybridization, and this is a key element of the (J = 4) configuration, its ground-state may be a non- hastatic proposal. The giant anisotropy in g⊥, places a Kramers Γ doublet gc 5 strong constraint on the energy-splitting ∆ between the two Ising states |±(cid:105)=a|±3(cid:105)+b|∓1(cid:105), (10) 2∆ 1 < (6) (g µ B ) 30 where Ising anisotropy exists for arbitrary mixing be- c B dHvA tween the |±3(cid:105) and |∓1(cid:105) states since this tetragonally- requiring that in a transverse field, BdHvA = 11T the stabilized Γ5 state is dipolar in the c-direction and U ion is doubly degenerate to within ∆ ∼ 1K. Fur- quadrupolar in the a-b plane. Because the phase space ther support for a very small ∆ comes from the di- associated with the non-Kramers doublet is significantly lute limit, UxTh1xRu2Si2 (x ≤ .07), where the Curie- largerthanthatforitsfinely-tunedKramerscounterpart, like single-ion behavior crosses over to a critical loga- we take the Γ doublet to be the more natural ground- 5 rithmictemperature-dependence9 below10K, logT/TK, state configuration of the U ion in URS. However we where TK ≈ 10K. This physics has been attributed to have proposed a direct benchtop test to distinguish be- two channelKondocriticality, againrequiringasplitting tween these two single-ion U ground-state configurations ∆ (cid:28) 10K. Such a degeneracy is of course natural for inURu Si usingthebasal-planenonlinearsusceptibility 2 2 in a Kramers U (5f3) ion, containing an odd-number to check this key assumption in the hastatic proposal33. of f-electrons. However it can also occur in a 5f2 “non- The combination of the observed Ising anisotropy and Kramersdoublet”withatwo-foldorbitaldegeneracypro- tetragonal symmetry are crucial towards pointing us to tected by tetragonal symmetry9,32. thenon-KramersΓ doublet. Bycontrastinahexagonal 5 Motivated and constrained by the bulk spin suscepti- system, like CeAl 34, the six-fold symmetry mixes terms 3 bility and the quantum spin zeroes, we therefore require that differ by 6(cid:126) units of angular momentum, so a pure the U ion to be in an Ising doublet in the tetragonal en- doublet|±M(cid:105)mixeswith|±M(cid:48)(cid:105)statesonlyifM(cid:48) =M− vironment of URS (cf. Fig. 1); such a magnetic doublet 6n (n ∈ Z). For J < 7/2, the maximum M −M(cid:48) is 5 − of URu Si has the form 2 2 2 (−5)=5 meaning there is no valid choice of M and M(cid:48); 2 (cid:88) thus there are two Ising doublets for the Ce (J = 5/2) |Γ (cid:105)= a |±(J −4n)(cid:105), (7) ± n z case: |±5/2(cid:105) and |±3/2(cid:105). These Kramers doublets can n undergo a single channel Ising Kondo effect34,35that will where the addition and subtraction of angular momen- differ substantially from the two-channel Kondo physics tum in units of 4(cid:126) is a consequence of the four-fold sym- associated with a non-Kramers doublet36. 4 C. Hybridization, Hastatic Order and an order parameter that breaks double time-reversal Double-Time Reversal Symmetry symmetry, a requirement that leads us to conclude that the order parameter has a spinorial quality. (See Fig. 3) The observation of heavy quasiparticles with Ising anisotropyinthetetragonalenvironmentofURSimplies anunderlyinghybridizationofhalf-integerspinelectrons with integer-spin doublets that has important symmetry implications for the nature of the hidden order. More specifically such hybridization requires a quasiparticle mixing terms in the low-energy fixed-point Hamiltonian of the form H=|kσ(cid:105)V (k)(cid:104)α|+H.C. (11) σα where |kσ(cid:105) and |α(cid:105) refer to the half-integer spin conduction and the integer-spin doublet states respectively and H.C. is the Hermitian conjugate; here k and σ are the momentum and the spin components respectively. Be- cause time-reversal, Θˆ, is an anti-unitary quantum operator, it has no associated eigenvalue. However double time-reversalΘˆ2,equivalenttoa2πrotation,isaunitary FIG. 3: Schematic of (a) conventional (scalar) vs (b) spino- operator whose quantum number is the Kramers index, rial hybridization where the hybridization is a) a crossover K =(−1)2J where J refers to the total angular momen- and b) breaks spin-rotational and time-reversal symmetries tum of the quantum state; K defines the phase factor and thus develops discontinuously as a phase transition. acquired by its wavefunction after two successive time- reversals: The formation of heavy f-bands in heavy fermion sys- Θˆ2|ψ(cid:105)=|ψ2π(cid:105)=K|ψ(cid:105). (12) tems involves the formation of f-resonances within the conduction sea. Hybridization between these many- For an integer-spin state |α(cid:105), K =1 since |α2π(cid:105)=+|α(cid:105), body resonances and the conduction electrons produces indicating that it is unchanged by a 2π rotation. By chargedheavyf-electronsthatinheritthemagneticprop- contrast, for conduction electrons with half-integer spin ertiesofthelocalmoments. Whenthisprocessinvolvesa states, |kσ(cid:105), |kσ2π(cid:105) = −|kσ(cid:105), so that K = −1. There- Kramers doublet (the usual case), the hybridization can fore, by mixing half-integer and integer spin states, the develop without any broken symmetry and thus is asso- quasiparticlehybridizationHdoesnotconserveKramers ciated with a crossover. At first sight, the most straight- index. Indeedapplicationoftwosuccessivetime-reversals forward explanation of hidden order is to attribute it to to H yields the formation of a “heavy density wave” within a pre- formed heavy electron fluid. Since there is no observed Θˆ2(V|kσ(cid:105)(cid:104)α|)=V2π|kσ2π(cid:105)(cid:104)α2π|=−V2π|kσ(cid:105)(cid:104)α|. (13) magnetic moment or charge density observed in the hidden order phase, such a density wave must necessarily SincethemicroscopicHamiltonianmustbetime-reversal involve a higher order multipole of the charge or spin invariant, it follows that degrees of freedom and various theories in this category V =−V2π (14) have indeed been advanced. In each of these scenarios, the heavy electrons develop coherence via cross-over at so the hybridization transforms as a half-integer spin higher temperatures, and the essential hidden order is state and is therefore a spinor. It then follows that this then a multipolar charge or spin density wave. However spinorial hybridization breaks both single- and double- such multipolar order can not account for the emergence timereversalsymmetriesdistinctfromconventionalmag- of heavy Ising quasiparticles. The essential point here, netism where Kramers index is conserved; we call this isthatconventionalquasiparticleshavehalf-integerspin, new state of matter “hastatic (Latin: spear) order”. lackingthelacktheessentialIsingprotectionrequiredby Before we proceed to discuss the theory of spinorial experiment. hybridization and its consequences, let us pause briefly Moreover, in URu2Si2 both optical37 and to elaborate on the distinction between spinors and vec- tunnelling38–40 probes suggest that hybridization tors, expanding the previous discussion with emphasis develops abruptly at the HO transition, leading to on time-reversal symmetry properties. Quantum me- proposals14,15 that the hybridization is an order param- chanically, the non-relativistic evolution of a wavefunc- eter. The associated global broken symmetry and phase tion ψ(x,t) is described by the Schrödinger equation transition is naturally described within the hastatic Hψ =i(cid:126)∂ψ so that proposal. As we now describe, hybridization with ∂t a non-Kramers doublet requires the development of Θψ(x,t)=ψ∗(x,−t) (15) 5 where Θ is the time-reversal operator; this is an anti- corresponding to time-reversed configurations on alter- unitary operation. For vector spins, S(cid:126) → −S(cid:126). The spin nating layers A and B, leading to a staggered Ising mo- 1 wavefunction is a spinor ment. For the HO state, the spinor points in the basal 2 plane (cid:18) (cid:19) ψ Ψ= ψ↑ (16) 1 (cid:18)e−iφ/2(cid:19) 1 (cid:18)−e−iφ/2(cid:19) ↓ Ψ ∼ √ , Ψ ∼ √ , (22) A 2 eiφ/2 B 2 eiφ/2 so that (cid:18)−ψ∗(x,−t)(cid:19) where again, ΨB = ΘΨA. This state is protected from ΘΨ(x,t)= ψ∗↓(x,−t) (17) developing large moments by the pure Ising character of ↑ the 5f2 ground-state. and (cid:18)−ψ∗(x,t)(cid:19) Θ2Ψ(x,t)= ↑ =−Ψ(x,t) (18) D. Two-Channel Valence Fluctuation Model −ψ∗(x,t) ↓ Bosons, with integer spins, are vectors with Θ2 = +1, We next present a model that relates hastatic order to the particular valence fluctuations in URu Si , based whereas fermions with half-integer spins have K = −1. 2 2 on a two-channel Anderson lattice model. The uranium Because of its spinorial character, we can think of ground state is a 5f2 Ising Γ doublet9, which then fluc- hastatic order as “hybridization with a twist” since this 5 tuates to an excited 5f3 or 5f1 state via valence fluctu- is a simple way of visualization its behavior under 2π ations. The lowest lying excited state is most likely the (inverted) and 4π (restored) rotations. 5f3 (J =9/2) state, but for simplicity here we take it to Hybridization in heavy fermion compounds is usu- be the symmetry equivalent 5f1 state, and assume that allydrivenbyvalencefluctuationsmixingaground-state fluctuations to the 5f3 are suppressed - in this sense, we Kramers doublet and an excited singlet (cf. Fig. 1a). In take an infinite-U two-channel Anderson model. thiscase,thehybridizationamplitudeisascalarthatde- Aswecanwritethef2Γ doublet,|±(cid:105)=a|±3(cid:105)+b|∓1(cid:105) velops via a crossover, leading to mobile heavy fermions. 5 However valence fluctuations from a 5f2 ground-state in terms of combinations of two J = 5/2 f-electrons in the three tetragonal orbitals, Γ± and Γ , create excited states with an odd number of electrons 7 6 and hence a Kramers degeneracy (cf. Fig. 1b). Then |+(cid:105) = (pf† f† +qf† f† +sf† f† )|Ω(cid:105) the quasiparticle hybridization has two components, Ψσ, Γ−7↓ Γ+7↓ Γ6↑ Γ+7↑ Γ6↑ Γ−7↑ that determine the mixing of the excited Kramers dou- |−(cid:105) = (pf† f† +qf† f† +sf† f† )|Ω(cid:105), blet into the ground-state. These two amplitudes form a Γ−7↑ Γ+7↑ Γ6↓ Γ+7↓ Γ6↓ Γ−7↓ (23) spinor defining the “hastatic” order parameter (cid:18) (cid:19) ifweassumea5f1 Γ+ excitedstate, wecannowreadoff Ψ= Ψ↑ . (19) thevalencefluctuatio7nmatrixelementsdirectly. Valence Ψ ↓ fluctuations occur in two orthogonal conduction electron Loosely speaking, the hastatic spinor is the square root channels, Γ− and Γ , and we find 7 6 of a multipole H (j) = V c† (j)|Γ+±(cid:105)(cid:104)Γ ±| Ψ∼(cid:112)multipole. (20) VF 6 Γ6± 7 5 + V c† (j)|Γ+∓(cid:105)(cid:104)Γ ±|+H.c.. (24) 7 Γ7∓ 7 5 Moreover, the presence of distinct up/down hybridiza- where ± denotes the “up” and “down” states of the tion components indicates that Ψ carries a half-integer coupled Kramers and non-Kramers doublets. The field spin quantum number; its development must now break (cid:104) (cid:105) double time-reversal and spin rotational invariance via a c† (j)=(cid:80) Φ†(k) c† e−ik·Rj createsaconduction Γσ k Γ kτ phase transition. electron at uranium sσitτe j with spin σ, in a Wannier or- Underpressure,URu2Si2 undergoesafirst-orderphase bital with symmetry Γ∈{6,7}, while V6 and V7 are the transition from the hidden order (HO) state to an an- corresponding hybridization strengths. The full model is tiferromagnet (AFM)7. These two states are remark- then written ably close in energy and share many key features27–29 (cid:88) (cid:88) including common Fermi surface pockets; this motivated H = (cid:15) c† c + [H (j)+H (j)] (25) k kσ kσ VF a therecentproposalthatdespitethefirstordertransition kσ j separating the two phases, they are linked by “adiabatic (cid:80) continuity,”27 correspondingtoanotionalrotationofthe while Ha(j) = ∆E ±|Γ7±,j(cid:105)(cid:104)Γ7±,j| is the atomic HO in internal parameter space10,30. In the magnetic Hamiltonian. phase, this spinor points along the c-axis To encompass the Hubbard operators in a field-theory description, we factorize them as follows (cid:18) (cid:19) (cid:18) (cid:19) 1 0 ΨA ∼ 0 , ΨB ∼ 1 (21) X =|Γ+σ(cid:105)(cid:104)Γ α|=Ψˆ†χ . (26) σα 7 5 σ α 6 E. Structure of the Paper To recap, here we are arguing that the observation of an anisotropic conduction fluid in URu Si indicates the 2 2 coherent admixture of spin 1 electrons with integer spin 2 doublets, leadingustoproposethattheorderparameter in URu Si is spinorial hybridization that breaks both 2 2 single- and double-time reversal. In conventional heavy fermionmaterials,hybridizationisdrivenbyvalencefluc- tuations between a Kramers doublet and an excited singlet in a single channel. The hybridization carries no quantum numbers and develops as a crossover resulting FIG. 4: Showing conduction electron self-energy Σ . Hy- inheavymobileelectrons. Howeveriftheground-stateis c bridization with spinorial order parameter (cid:104)Ψσ(cid:105) permits the a non-Kramers doublet, the Kondo effect occurs via an development of a Γ5 Ising resonance inside the conduction excited state with an odd number of elections that is a sea, represented by the above Feynman diagram. Kramers doublet. The quasiparticle hybridization then carries a global spin quantum number and has two distinct amplitudes that form a spinor defining the hastatic Here |Γ α(cid:105) = χ†|Ω(cid:105) is the non-Kramers doublet, rep- order parameter 5 α resented by the pseudo-fermions χ†, while Ψˆ† are slave (cid:18) (cid:19) α σ ψ bosons41 representing the excited f1 doublet |Γ+σ(cid:105) = Ψ= ↑ . (28) 7 ψ Ψˆ†|Ω(cid:105). Hastatic order is realized as then condensation ↓ σ of this bosonic spinor Theonsetofhybridizationmustbreakspinrotationalin- variance in addition to double time-reversal invariance Ψ†χ →(cid:104)Ψˆ (cid:105)χ , (27) via a phase transition; we note that optical, spectro- σ α σ α scopic and tunneling probes in URu Si indicate the hy- 2 2 bridizationoccursabruptlyatthehiddenordertransition generating a hybridization between the conduction elec- in contrast to the crossover behavior observed in other trons and the Ising 5f2 state while also breaking double heavy fermion systems. time reversal (see Fig. 4). These slave bosons play a We now describe the structure of this paper. The mi- dualroleincapturingthehybridizationwhilealsoacting croscopicbasisofhastaticorderispresentedusingatwo- as Schwinger bosons describing a 5f1 magnetic moment channel Anderson lattice model in section II, along with with a reduced amplitude, 2−nf. The tensor product the mean field solution. In section III, we develop the Q ≡Ψ Ψ† describesthedevelopmentofcompositeor- Landau-Ginzburg theory of hastatic order, including the αβ α β der between the non-Kramers doublet and the spin den- appearance of pressure induced antiferromagnetism and sity of conduction electrons. Composite order has been the nonlinear susceptibility, while in section IV, we de- consideredbyseveralearlierauthorsinthecontextoftwo rive and discuss a number of experimental consequences channel Kondo lattices42–44 in which the valence fluctu- of hastatic order, showing the consistency of hastatic or- ationshavebeenintegratedout. However, byfactorizing der with a number of experiments, including the large the composite order in terms of the spinor Ψ , we are entropyofcondensationandtetragonalsymmetrybreak- α able to directly understand the development of coherent ing observed in torque magnetometry, and then making Ising quasiparticles and the broken double time-reversal. a number of key predictions, including a tiny staggered basalplanemomentintheconductionelectrons. Weend There are two general aspects of this condensation with discussion and future implications in section V. that deserve special comment. First, the two-channel Anderson impurity model is known to possess a non- Fermi liquid ground-state with an entanglement entropy of 1k ln245. The development of hastatic order in the II. LANDAU THEORY: PRESSURE-INDUCED 2 B ANTIFERROMAGNETISM lattice liberates this zero-point entropy, accounting naturally for the large entropy of condensation. As a slave boson, Ψ carries both the charge e of the electrons and A. Thermodynamics the local gauge charge Q =Ψ†Ψ +χ†χ of constrained j j j j j valence fluctuations, its condensation gives a mass to Hastatic order captures the key features of the ob- their relative phase via the Higgs mechanism46. But served pressure-induced first-order phase transition in as a Schwinger boson, the condensation of Ψ this pro- URu Si between the hidden order and an Ising anti- 2 2 cess breaks the SU (2) spin symmetry. In this way the ferromagnetic (AFM) phases. The most general Landau hastatic boson can be regarded as a magnetic analog of functional for the free energy density of a hastatic state the Higgs boson. withaspinorialorderparameterΨas afunctionofpres- 7 sure and temperature is IfP <P ,thenγ <0andtheminimumofthefreeenergy c occurs for θ = π/2, corresponding to the hidden order f[Ψ]=α(T −T)|Ψ|2+β|Ψ|4−γ(Ψ†σ Ψ)2 (29) c z state ordered state. By contrast, if P > P , then γ > 0 c andtheminimumofthefreeenergyoccursatθ ={0,π}, whereγ =δ(P−P )isapressure-tunedanisotropyterm c correspondingtotheantiferromagnet. The“spinflop”in and θ at P =P corresponds to a first order phase transition c (cid:18)cos(θ/2)eiφ/2 (cid:19) between the hidden order and antiferromagnet (See Fig. Ψ=r , (30) sin(θ/2)e−iφ/2 5). where θ is the disclination of Ψ†(cid:126)σΨ from the c-axis, and |Ψ|2 = r2. Experimentally the TAFM(P) line is almost B. Soft modes and dynamics vertical, indicating by the Clausius-Clapeyron relation that the there will be negligible change in entropy be- The adiabatic continuity between the hastatic and an- tween the HO and the AFM states. Indeed these two tiferromagnetic phases allows for a simple interpretation phases share a number of key features, including com- of the soft longitudinal spin fluctuations that have been mon Fermi surface pockets; this has prompted the pro- observed to develop in the HO state47,48, and even to go posal that they are linked by “adiabatic continuity”, as- soft, but not critical upon approaching T 49. These HO sociated by a notational rotation in the space of internal longitudinal modes can be thought of as incipient Gold- parameters. This is easily accommodated with a spinor stone excitations between the two phases30. Specifically, order parameter; for the AFM phase (P > P ), there is c intheHOstate,rotationsofthehastaticspinoroutofthe a large staggered Ising f-moment with basal plane will lead to a gapped Ising collective mode (cid:18) (cid:19) (cid:18) (cid:19) liketheoneobserved. WithinourLandautheory,wecan 1 0 ΨA ∝ 0 , ΨB ∝ 1 (31) study the evolution of this mode with pressure. In order to study the soft modes of the hastatic or- correspondingtotime-reversedspinconfigurationsonal- der, we need to generalize the Landau theory to a time- ternating layers A and B. For the HO state (P < P ), dependent Landau Ginzburg theory for the action, with c the spinor points in the basal plane action S =(cid:82) Ldtd3x, where the Lagrangian 1 (cid:18)e−iφ/2(cid:19) 1 (cid:18)−e−iφ/2(cid:19) (cid:16) (cid:17) Ψ ∝ √ , Ψ ∝ √ (32) −L[Ψ]=f[Ψ]+ρ |∇Ψ|2−c−2|Ψ˙|2 , A 2 eiφ/2 B 2 eiφ/2 where Ψ = ΘΨ and there is no Ising f-moment, con- and ρ is the stiffness. Expanding Ψ around its equilib- B b sistent with experiment, but Ising fluctuations do exist. rium value Ψ0, we take φ=0 for convenience and write According to the above expression for Ψ, (Ψ†σ Ψ) = r2cosθsothattheLandaufunctionalcanbereexpzressed Ψ(x,t)=Ψ eiδθ(x)σy/2 =(1+i/2(cid:88)δθ(q)eiq(cid:126)·(cid:126)x−ωtσ )Ψ . 0 y 0 as q (34) f =−α(T −Tc)r2+(β−γcos2θ)r4. (33) This gives rise to a change in Ψ†(cid:126)σΨ = xˆ|Ψ0|2 + δθ(x)zˆ|Ψ |2 corresponding to a fluctuation in the longi- 0 tudinalmagnetization. ThisrotationinΨdoesnotaffect thefirsttwoisotropictermsinf[Ψ]. Thevariationinthe action is then given by (cid:88) (cid:18) ω2 2δ (cid:19) δS =ρ|Ψ |2 |θ(q)|2 (cid:126)q 2− + (P −P)|Ψ |2 0 c2 ρ c 0 q (35) The dispersion is therefore ω2 =(cq)2+∆2 (36) where 2δ(P −P) ∆2 = c |Ψ |2, (37) ρ 0 so that even though the phase transition at P = P is c first order, the gap for longitudinal spin fluctuations is FIG. 5: Global phase diagram predicted by Landau theory. ∆∝|Ψ |(cid:112)P −P. 0 c 8 Since dP /dT is finite, close to the transition, where √ c c √ (cid:112) P −P ≈ dP /dT (T −T ), and ∆ ∝ T −T . In- c c c c c |5f1 :Γ+±(cid:105)=η|±5/2(cid:105)+δ|∓3/2(cid:105) (42) elastic neutron scattering experiments can measure this 7 gap a function of temperature at a fixed pressure where is the excited Kramers doublet. The following section there is a finite temperature first order transition. The willshowhowparticle-holesymmetrycanbeusedtofor- iron-dopedcompound,URu Fe Si canprovideanat- 2−x x 2 mulatetheequivalentmodelwithfluctuationsintoa5f3 tractive alternative to hydrostatic pressure, as iron dop- Kramers doublet. ing acts as uniform chemical pressure and tunes the hid- To evaluate the matrix elements for valence fluctua- den order state into the antiferromagnet50. tions we need to express the 5f2 state in terms of one- However, higher order terms, like λ(Ψ†σ Ψ)2|Ψ|2 are z particle states. The Γ state can be rewritten as a prod- alsoallowedbysymmetry,andwillintroduceaweakdis- 5 uct of the one particle J = 5/2 f-orbitals, |5/2,m(cid:105) ≡ continuity of order |Ψ |4, 0 f†|Ω(cid:105), using the Clebsch Gordan decomposition m 2δ(P −P) 2λ ∆2 = c |Ψ |2+ |Ψ |4+O(|Ψ |6). (38) (cid:32)(cid:114) (cid:114) (cid:33) ρ 0 ρ 0 0 |∓1(cid:105) = 5f† f† + 2f† f† |Ω(cid:105), 7 ±1/2 ∓3/2 7 ±3/2 ∓5/2 So there will likely always be a discontinuity in the longitudinal spin fluctuation mode at the first order phase |±3(cid:105) = f† f† |Ω(cid:105). (43) ±5/2 ∓1/2 transition, although it will be of a lower order (|Ψ |4) 0 than a generic first order transition (|Ψ2|). Next we decompose the one-particle f-states in terms of 0 theone-particlecrystalfieldeigenstates,|Γ±(cid:105)≡f† |Ω(cid:105); Γ,± writing f† =f† (cid:104)Γβ|m(cid:105), or more explicitly, III. MICROSCOPIC MODEL: TWO-CHANNEL m Γβ ANDERSON LATTICE f† = f† ±1/2 Γ6± Hastatic order emerges as a spinorial hybridization f† = δf† −ηf† ∓3/2 Γ+± Γ−± between a non-Kramers doublet ground state and a f† = ηf†7 +δf†7 (44) Kramers doublet excited state. In our picture of ±5/2 Γ+± Γ−± 7 7 URu Si , a lattice of 5f2 (J = 4) U4+ ions provide the 2 2 the non-Kramer’s doublet can be written in the form non-Kramers doublet (Γ ), which are then surrounded 5 by a sea of conduction electrons that facilitate valence |Γ +(cid:105) = (pf† f† +qf† f† +sf† f† )|Ω(cid:105) fluctuations between the 5f2 non-Kramers doublet and 5 Γ−7↓ Γ+7↓ Γ6↑ Γ+7↑ Γ6↑ Γ−7↑ aAn5dfe1rsoorn5fla3tteixcecitmedodKelrahmasertshrdeoeucbolemt.poTnheenttsw:o-channel |Γ5−(cid:105) = (pfΓ†−7↑fΓ†+7↑+qfΓ†6↓fΓ†+7↓+sfΓ†6↓fΓ†−7↓)|Ω((cid:105)4,5) (cid:88) H =H + [H (j)+H (j)], (39) (cid:113) (cid:113) (cid:113) c VF a where p = b 2, q = bδ 5 − aη, s = −bδ 5 − aη. j 7 7 7 Valencefluctuationsfromthegroundstate(5f2Γ )tothe a conduction electron term, H , a valence fluctuation 5 c excited state (5f1 Γ+) are described by a a one-particle termcapturinghowtheconductionelectronshoponand 7 Anderson model with an on-site hybridization term offtheUsite,H andanatomicHamiltoniancapturing VF the different energy levels of the U ion, H . (cid:88) (cid:104) (cid:105) a H (j)= v c† (j)f (j)+H.c , (46) VF Γ Γσ Γσ Γ=Γ6,Γ±7;σ A. The Valence Fluctuation Hamiltonian wherethev arethehybridizationmatrixelementsinthe Γ threeorthogonalcrystalfieldchannelsandc† (j)creates 1. The 5f1 model Γσ a conduction electron in a Wannier state with symmetry Γσ on site j. The 5f2 Γ non-Kramers doublet is given by 5 Now we need to project this Hamiltonian down into |5f2 :Γ5±(cid:105)=a|±3(cid:105)+b|∓1(cid:105), (40) the reduced subspace of the ground-state |5f2 : Γ5α(cid:105) and |5f1 :Γ−σ(cid:105) excited state doublets, replacing 7 and all energies are measured relative to the energy of (cid:32) (cid:33) this isolated doublet. In principle, valence fluctuations (cid:88) may either occur to 5f1 (J = 5/2) or to 5f3 (J = 9/2), fΓσ(j)→ |Γ+7σ(cid:48)(cid:105) (cid:104)Γ+7σ(cid:48)|fΓσ(j)|Γ5α(cid:105) (cid:104)Γ5α|(47) and in fact the lowest lying excited state is likely to be σ(cid:48),α=± the 5f3 state. However, for technical simplicity, we take in (46). Using (45), the only non-vanishing matrix ele- the lowest lying valence fluctuation excitation to be the 5f1 Γ+ excited state, ments between the 5f2 and 5f1 states are 7 |5f2 :Γ5±(cid:105)(cid:10)|5f1 :Γ+7±(cid:105)+e− (41) (cid:104)Γ+7 ∓|fΓ−7∓|Γ5±(cid:105) = p, 9 (cid:104)Γ+±|f |±(cid:105) = q. (48) To further develop the valence fluctuation term, we 7 Γ6± need to determine the form factors relating the conduc- Matrix elements for the Γ+ channel identically vanish tion electron Wannier states in terms of Bloch waves, 7 (cid:104)Γ+σ|f |σ(cid:48)(cid:105) = 0, so the third term in (45) does not 7 Γ+± contribu7te to the projected Hamiltonian. The final pro- c† =(cid:88)c† [Φ ] e−ik·Rj. (52) Γα kβ Γk βα jected model is then written k (cid:88) (cid:88) H = (cid:15)kc†kσckσ+ [HVF(j)+Ha(j)] (49) For a single site interacting with a plane wave, these are kσ j given by [ΦΓk]αβ =yαΓβ(k), where where c† creates a conduction electron of momentum k kσ 5/2 spin σ, with energy (cid:15)k, and yαΓβ(k) = (cid:88) Y3m−α2(kˆ)(cid:104)3m− α2,21α2|5/2m(cid:105)(cid:104)m|Γ, β(cid:105) H (j) = V c† (j)|Γ+±(cid:105)(cid:104)Γ ±| m=−5/2 VF + V67cΓ†Γ6−7±∓(j)|Γ7+7∓(cid:105)(cid:104)Γ55±|+H.c. (50) = α (cid:88)5/2 (cid:114)12 − m7α Y3m−α2(kˆ)(cid:104)m|Γβ(cid:105). m=−5/2 describesthevalencefluctuationsbetweentheΓ5doublet (53) andtheexcitedΓ+Kramersdoublet. HereV =v qand 7 6 Γ6 V7 =vΓ−p while where α,β ∈ ±. However, in URu2Si2, the uranium 7 atoms are located on a body centered tetragonal(bct) (cid:88) Ha(j)=∆E |Γ7±(cid:105)(cid:104)Γ7±| (51) latticeatrelativelocations, RNN =(±a/2,±a/2,±c/2), and the correct form factor must respective the lat- ± tice symmetries. Our model treats the conduction band is the atomic Hamiltonian for the excited 5f1 : Γ+ as a single band of s-electrons located at the U sites. 7 Kramers doublet. Notice that, as we have projected out Moreover, we assume that the f-electrons hybridize via mostofthepossibleUstates, weareworkingintermsof electron states at the nearby silicon atoms located at the Hubbard operators, |Γ ±(cid:105) and |Γ ±(cid:105) to describe the a = (±a/2,±a/2,±z) where z = .371c is the height 5 7 NN allowed U states. of the silicon atom above the U atom51. The form-factor is then [Φ ] = (cid:88) e−ik·(RNN−aNN)×e−ik·aNNyΓ (a )= (cid:88) e−ik·RNNyΓ (a ). (54) Γk αβ αβ NN αβ NN {RNN,aNN} {RNN,aNN} Here, the term e−ik·aNNyαΓβ(aNN) is the amplitude to 2. The 5f3 case hybridize with the silicon site at site a , while the NN prefactor e−ik·(RNN−aNN) is the additional phase fac- For simplicity we have discussed the two channel An- tor for hopping from the silicon site a to the U atom, NN dersonmodelinvolvingfluctuationsfroma5f2Γ ground R directly above it. This form of the hybridization 5 NN state to 5f1 (J =5/2). However, the more realistic case can also be derived using Slater-Koster52 methods, un- involvesfluctuationsto5f3,whoselowenergystateshave der the assumption that the important part of the hy- J =9/2, and are split into five Kramers doublets by the bridization potential is symmetric about the axis be- tetragonal crystal field, tween the U and Si atom. Notice that this function has the following properties: [Φ ] = [Φ ] and Γk+G αβ Γk αβ [ΦΓk+Q]αβ =−[ΦΓk]αβ. Weshouldnotethatthismodel ||ΓΓλ61±±(cid:105)(cid:105) == caλ||±±59//22(cid:105)(cid:105)++bd|∓λ|3±/21(cid:105)/2(cid:105)+eλ|∓7/2(cid:105), of the hybridization is overly simplified, in that the U 7 |Γ2±(cid:105) = −b|±5/2(cid:105)+a|∓3/2(cid:105), (55) most likely hybridizes with the d-electrons sitting on the 7 Ru site. Such a d-f hybridization can be treated in a similar fashion, and is the subject of future work. whereλ∈(1,2,3)labelsthethreeΓ Kramer’sdoublets. 6 There are two generic situations: either a Γ doublet is 7 lowest in energy, and the valence fluctuations are then determined by the overlap, |Γ ±(cid:105)=αψ† |Γ ±(cid:105)+βψ† |Γ ∓(cid:105), (56) 7 6∓ 5 7∓ 5 10 or alternatively, a Γ doublet is lowest in energy, with We may interpret this replacement as a kind of multi- 6 the relevant overlap, particlecontractionofthemanybodyHubbardoperator into a single fermionic bound-state. Once this bound- |Γ ±(cid:105)=αψ† |Γ ±(cid:105)+βψ† |Γ ∓(cid:105), (57) stateforms,asymmetry-breakinghybridizationdevelops 6 7∓ 5 6∓ 5 betweentheconductionelectronsandtheIsing5f2 state. where the form-factors are as above. In both cases fluc- The dual Schwinger/slave character of the boson Ψˆ† σ tuationswillinvolveconductionelectronsinbothΓ6 and representingtheoccupationof5f1 meanswhenthisfield Γ7 symmetries. When the lowest excited state is a Γ7, condenses, it not only breaks the local U(1) gauge sym- the valence fluctuation Hamiltonian is given by, metry, but also the global SU(2) spin symmetry. How- ever, it breaks the U(1) gauge symmetry as a slave bo- H (j) = V (ψ† |Γ ±(cid:105)(cid:104)Γ ±|+H.c) VF3 6 jΓ6∓ 5 7 son,whichhavebeenwellstudiedinthecontextofheavy + V (ψ† |Γ ∓(cid:105)(cid:104)Γ ±|+H.c.). (58) fermions. TheU(1)phaseofthelocalgaugesymmetryis 7 jΓ7∓ 5 7 subject to the Anderson-Higgs mechanism, in which the The only difference between the 5f3 and the 5f1 model difference between the electromagnetic gauge field and isthattheexcitedstaterequiresaddingaparticle,sothe theinternalU(1)gaugefieldacquiresamass46. Itisthis valence fluctuation term here is the particle-hole conju- process that gives the χ fermions a physical charge. The gate of the 5f1 case. combination of the global and local symmetry breaking processes means that the hastatic order parameter can be thought of as a magnetic Higgs boson. B. Slave particle treatment With this slave particle factorization, we can reformu- late Hubbard operators cannot be directly treated with (cid:88) (cid:88) quantum field theory techniques since they do not sat- Ha(j)→∆E Ψ†σ(j)Ψσ(j). (64) isfy Wick’s theorem. We follow the standard approach j j,σ=± and introduce a slave particle factorization of the Hub- bard operators that permits a field theoretic treatment, The valence fluctuation term at each site takes the form |Γ+σ(cid:105)(cid:104)Γ α|=Ψˆ†χ . (59) H (j) = V c† (j)Ψ† (j)χ (j)+V c† (j)Ψ† (j)χ (j) 7 5 σ α VF 6 Γ6± ± ± 7 Γ−7∓ ∓ ± + H.c.. (65) where We now rewrite this expression in terms of Bloch waves |Γ α(cid:105)=χ†|Ω(cid:105) (60) 5 α by absorbing the momentum dependent Wannier form- factors into the spin-dependent hybridization matrix, in- is the non-Kramers doublet, represented by the pseudo- fermion χ†, while Ψˆ† is a slave boson41 representing the troducing the operator α σ excited f1 doublet Vˆ(k,j)=V Φ B†+V Φ Bˆ†σ . (66) |Γ+σ(cid:105)=Ψˆ†|Ω(cid:105) (61) 6 Γ6 j 7 Γ−7 j 1 7 σ where the matrix that carries a positive charge and a spin quantum num- btheer.hCasotnadtiecnsoartdieornpoafrtahmeestperin-12 boson then gives rise to Bˆj† =(cid:32)Ψˆ0†j↑ Ψˆ0† (cid:33) (67) j↓ (cid:18)(cid:104)Ψˆ (cid:105)(cid:19) Ψ= (cid:104)Ψˆ↑(cid:105) . (62) contains the hastatic boson. The valence fluctuation ↓ term then becomes THhuibsbcaornddoepnesraattioonr Xpˆrσoαcebsys tahsaitngwleebmoauyndn-oswtarteepflearcmeitohne HVF(j)=(cid:88)c†kσVˆσα(k,j)χα(j)e−ik·Rj +H.c. (68) k Ψ†χ →(cid:104)Ψˆ (cid:105)χ . (63) σ α σ α Putting these results all together, our model for URu Si is given by 2 2 H = (cid:88)(cid:15) c† c +(cid:88)(cid:104)c† (cid:16)V Φ B†+V Φ Bˆ†σ (cid:17)χ (j)e−ik·Rj +H.c.(cid:105)+∆E (cid:88) Ψ†(j)Ψ (j) k kσ kσ kσ 6 Γ6 j 7 Γ−7 j 1 α σ σ kσ k j,σ=± (cid:32) (cid:33) (cid:88) (cid:88) (cid:88) + λ Ψ†(j)Ψ (j)+ χ†(j)χ (j)−1 . (69) j σ σ α α j σ α