Abstract
Complex electronic matter shows subtle forms of selforganization, which are almost invisible to the available experimental tools. One prominent example is provided by the heavyfermion material URu_{2}Si_{2}. At high temperature, the 5f electrons of uranium carry a very large entropy. This entropy is released at 17.5 K by means of a secondorder phase transition^{1} to a state that remains shrouded in mystery, termed a ‘hidden order’ state^{2}. Here, we develop a firstprinciples theoretical method to analyse the electronic spectrum of correlated materials as a function of the position inside the unit cell of the crystal and use it to identify the lowenergy excitations of URu_{2}Si_{2}. We identify the order parameter of the hiddenorder state and show that it is intimately connected to magnetism. Below 70 K, the 5f electrons undergo a multichannel Kondo effect, which is ‘arrested’ at low temperature by the crystalfield splitting. At lower temperatures, two brokensymmetry states emerge, characterized by a complex order parameter ψ. A real ψ describes the hiddenorder phase and an imaginary ψ corresponds to the largemoment antiferromagnetic phase. Together, they provide a unified picture of the two brokensymmetry phases in this material.
Main
URu_{2}Si_{2} crystallizes in the bodycentred tetragonal structure shown in Fig. 1a. Starting from a localized point of view for U5f electrons, that is, treating them as core states in density functional theory (DFT) calculations, one obtains a set of wide bands, entirely of s p d character, as shown in Fig. 1b. In an itinerant picture, there are f states close to the Fermi level, which hybridize with the s p d bands to form itinerant states of predominantly f character, with a bandwidth of the order of 1 eV. This is shown in Fig. 1c. This situation is realized in Ce and Yrbased heavyfermion materials, the electronic states of which are well described by narrowing the bands of the DFT by a factor of 10–1,000, to account for the heavy mass^{3}.
For URu_{2}Si_{2}, we propose a new scenario for the transfer of atomic f weight to the itinerant carriers, which we name the ‘arrested Kondo effect’. At high temperatures, above the characteristic coherence temperature T^{*}∼70 K, the U5f electrons are localized and do not participate in forming the lowenergy bands. The U atoms settle in the 5f^{2} configuration, for which the crystal environment chooses the nondegenerate atomic ground state. However, the first excited state, which is also nondegenerate, is only Δ=35 K above the ground state, as first observed in polarized neutron scattering experiments^{4}. Hence, in the temperature range Δ<T<T^{*}, the ground state seems doubly degenerate, and hence the Kondo effect develops, leading to the formation of very narrow states near the Fermi energy and a narrow peak in the density of states. The Kondo effect is partially arrested below T<Δ, because the crystalfield splitting between the two singlet states induces a partial gapping at the Fermi level. The situation is shown in Fig. 1c. A nondispersing slab of f spectral weight is pushed to roughly 8 meV away from the Fermi energy. Although only a small fraction of the f spectral weight is present at the Fermi level, it has a significant effect, resulting in a mass enhancement factor that is over 200 at T=19 K. For temperatures above the crystalfield splitting energy, T∼Δ, the electronic scattering rate is anomalously large. This is a signature of multichannel Kondo physics, whereby the two singlets have the role of the spin in the Kondo problem that scatters degenerate bands. The relevance of the twochannel Kondo physics to Ubased impurity models was first proposed by Cox^{5}. Indeed experimental data on dilute U_{x}Th_{1−x}Ru_{2}Si_{2} systems are well described by the twochannel Kondo model^{6}. It is remarkable that the Fermiliquid regime in URu_{2}Si_{2} can never be reached. Once the twochannel Kondo effect is arrested by the crystalfield splitting, longrange order, driven by intersite effects, preempts the formation of a Fermiliquid state with a low coherent scale.
The oneelectron spectral function A(r,ω) represents the quantum mechanical probability for adding or removing an electron with energy ω at a point r in real space. We calculate it using the combination of the DFT and dynamical mean field theory^{7} (DMFT)
where χ_{α}(r) are the basis functions and G_{α β} is the electron Green’s function expressed in the basis of χ. More details of the implementation of the DMFT method for this problem are given in the Supplementary Information.
These computational studies are the theoretical counterpart of the scanning tunnelling spectroscopy technique, which has been very useful in describing the properties of numerous correlated materials^{8} and was very recently applied to URu_{2}Si_{2} (ref. 9).
The evolution of the electron spectral function A(r,ω) along the cut above the U atom in the unit cell, as depicted in Fig. 1a, is shown in Fig. 2. The curves have an asymmetric line shape of the type
where Γ is the width and q measures the asymmetry of the line shape. These curves were first introduced by Fano^{10}, to describe scattering interference between a discrete state and a degenerate continuum of states. The continuum of states in scanning tunnelling microscopy on URu_{2}Si_{2} is provided by itinerant s p d bands and the narrow discrete states are the U5f electronic states, shown with bright colours in Fig. 1d. Notice that the line shape has the most characteristic Fano shape when the position r is on the U or Si atom, with positive asymmetry q>0 on the U atom and negative asymmetry q<0 on the Si atom. With the exception of the small peak, marked with the black arrow in Fig. 2, the line shape on the U atom can be well fitted to the Fano line shape with the parameters q=1.24 and Γ=6.82 meV.
The main features of our calculations, including the characteristic line shape, the width and the strength of the asymmetry of the Fano line shape, as well as the extra small peak around 6 meV, were recently measured in scanning tunnelling experiments by the J.C. Seamus Davis group^{9}, the first experiment of this type on a heavyfermion system.
To understand the origin of the small peak at 6.8 meV, we integrate the spectrum over the whole unit cell and resolve it in the different angular momentum channels of the different atoms. These orbitally resolved spectra at low energies are shown in Fig. 3a.
Although there are many Si3p and Ru4d states at the Fermi level, these states are only weakly energy dependent in the 200 meV interval. On the other hand, the heavy U5f electron states have a dip and a peak at the same energy of 6.8 meV, which is clearly visible in the realspace spectra in Fig. 2. Therefore, it is the U5f nonLorentzian shape of the electronic density of states that is responsible for the extra peak and the asymmetry variation in the unit cell. This dip–peak line shape is also the key to unravelling the puzzle of the electronic structure of the material, because it shows that instead of a regular Lorentzian Kondo resonance, we have a double peak at low temperature, with a pseudogap at zero energy. This is the signature of the Kondo effect, losing its strength at temperature T below the crystalfield splitting energy T<Δ. Below the ordering temperature (the ordering will be discussed below), the pseudogap in the partial U5f spectrum is considerably enhanced, as shown in Fig. 3a. Note that owing to mixing of s p d and f states, there is a finite admixture of U5f states at the Fermi level, giving some ‘heaviness’ to the quasiparticle even below the ordering temperature.
The electronic states can also be resolved in momentum space by computing the spectral function A(k,ω). The active state at the Fermi level, given by peaks of A(k,ω=0), determines the Fermi surface of the material, which is shown in Fig. 3b.
For many Cebased heavyfermion compounds, the Fermi surface of the itinerant DFT calculation, as well as that of the DMFT calculation, accounts for the experimentally measured de Haas–van Alphen frequencies. On the other hand, the DMFT Fermi surface of URu_{2}Si_{2}, shown in Fig. 3b, is qualitatively similar to the localized DFT calculation, with 5f states excluded from the valence bands^{11}. Note that Luttinger’s theorem, which counts the number of electrons modulo 2, does not constrain the Fermi surface of the material. Hence, the Fermi surface does not need to resemble the itinerant DFT result.
The firstprinciples DMFT calculations demonstrate that for URu_{2}Si_{2} the 5f^{2} configuration has the dominant weight. This has important consequences, because it allows the physical Fermi surface to have the same volume as a system with no f electrons, such as ThRu_{2}Si_{2} (ref. 12).
Finally, the DMFT Fermi surface is holelike, and the large hole Fermi surface centred at Z (where Z is (1,0,0)a^{*}) shows characteristic wave vectors (0.6,0,0)a^{*} and (1.4,0,0)a^{*} shown in Fig. 3b. Recent neutron scattering experiments show lowenergy spectral weight at these incommensurate wave vectors^{13,14,15}.
In the longrange order phase, the Fermi surface substantially reconstructs and multiple small electron and hole pockets appear, making this system a compensated metal, with the same number of hole and electron carriers.
The U5f^{2} configuration has the total angular momentum J=4 and is split into five singlets and two doublets in the tetragonal crystal environment of URu_{2}Si_{2}. The relative energy of these crystalfield levels is still an open problem and several sequences have been proposed. For a recent review, see ref. 16.
The wavefunctions of the U5f^{2} configuration, with the largest weight in the DMFT density matrix, are
Here, J_{z}〉=J=4,J z〉 is a twoparticle state of the J=4 multiplet. We obtained φ∼0.23π. The separation between the two singlets is of the order of 35 K. The probability for the atomic ground state, and the first excited state, 1〉, at 20 K is 0.54 and 0.1, respectively.
The transition into ordered states requires an understanding of the collective excitations, which are bound states of particle–hole pairs. The identification of the lowlying singlets leads us to consider the following order parameter for URu_{2}Si_{2}
where is the Hubbard operator, which measures the excitonic mixing between the two lowestlying U5f singlets at lattice site R_{i}.
This order parameter is complex. Its real part is proportional to the hexadecapole operator of the A_{2g} irreducible representation of the tetragonal symmetry, Reψ ∝ 〈(J_{x}J_{y}+J_{y}J_{x})(J_{x}^{2}−J_{y}^{2})〉, introduced in the context of nuclear physics long ago. Its imaginary part is proportional to the magnetization along the z axis, Imψ ∝ 〈J_{z}〉, which is the only direction allowed within this crystalfield set of states. Experimentally, the moment indeed points in the z direction^{4}.
At low temperatures, we found two different stable DMFT solutions, describing ordered states with nonzero staggered 〈ψ〉 with wave vector Q=(0,0,1). The first solution has 〈ψ〉 purely real and describes the hiddenorder phase of URu_{2}Si_{2}. This solution has zero magnetic moment, does not break timereversal symmetry and has a nonzero hexadecapole. The second DMFT solution has a purely imaginary 〈ψ〉 and we associate this phase with the largemoment antiferromagnet phase, which is experimentally realized at pressures larger than 0.7 GPa (refs 17, 18, 19, 20). Hence, the microscopic approach succeeds in unifying two very distinct brokensymmetry states in a single complex order parameter. The existence of a solution with either purely real or purely imaginary ψ, but without mixtures, indicates a firstorder phase transition between these two phases, as observed in the pressure experiments of refs 17, 18, 19, 20.
It is useful to visualize the meaning of this order parameter in a limiting case of a simple atomic wavefunction, where it takes the form . The average magnetic moment of the ground state is hence 〈g sJg s〉=4cos(φ)sin (2θ)sin(ϕ)*(0,0,1). If ϕ is π/2, we have a phase with large magnetic moment and if ϕ vanishes, the moment vanishes.
Our theory provides a natural explanation for a large number of experiments, which are very puzzling, when examined from other perspectives and suggests new experiments.
First, it has been advocated phenomenologically that even though the hiddenorder phase and the largemoment phase have distinct order parameters, the behaviour of many observables across the transition is remarkably similar. The term adiabatic continuity has been used to describe this situation^{17,21}, but it is not justified on theoretical grounds because the two phases are separated by a firstorder phase transition. The proposed order parameter ψ_{i}, which unifies the nomoment and largemoment phase, explains why even though the two phases are separated by a firstorder phase transition, they are in many respects very similar, for example in the critical temperature and entropy change across the transition.
Second, at the hiddenorder transition, a small gap of the order of 10 meV opens in the 5f quasiparticle spectrum, as seen in optical conductivity^{22}, specific heat^{23}, thermal conductivity^{24} and relaxation rate measurements^{25,26}. On the other hand, the d.c. resistivity continues to decrease at low temperatures^{27}, because it is dominated by the itinerant s p d carriers. This fact is hard to understand in a simple itinerant density wave picture. The fstate gap at low energies is also seen in neutron scattering experiments^{4,14,15}.
Third, unlike DFT calculations^{28}, our approach described a strongly correlated normal state with large entropy and can account for the large specificheat coefficient in the paramagnetic state of the material.
It has recently been proposed that one can detect hexadecapole order using the resonant Xray technique^{29}. This would be a direct test of our proposed order parameter. Highresolution angleresolved photoemission spectroscopy can detect the small kink in the very lowenergy spectra (<10 meV) of Fig. 1d, which is a unique signature of the arrested Kondo effect. It would be interesting to control the crystalfield splitting energy Δ. A slight decrease (increase) of Δ, will increase (decrease) the hiddenorder transition temperature.
Finally, our results set the stage for understanding the mysterious superconducting transition that takes place at the much lower temperatures (T_{c}=0.8 K). Hidden order is fertile ground for superconductivity, whereas the largemoment antiferromagnetic phase completely eliminates this instability. Our results suggest that Cooper pairing can take place only when the electrons propagate in the timereversal symmetric background, but unravelling the precise origin of the superconductivity in this material will require further investigation.
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Acknowledgements
We are grateful to J. Allen, J. Denlinger and J.C. Seamus Davis for fruitful discussion and for sharing unpublished work with us. K.H. was supported by grant NSF DMR0746395 and an Alfred P. Sloan fellowship. G.K. was supported by NSF DMR0906943.
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K.H. and G.K. both developed the LDA+DMFT methodology and the physical interpretation of the results.
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Haule, K., Kotliar, G. Arrested Kondo effect and hidden order in URu_{2}Si_{2}. Nature Phys 5, 796–799 (2009). https://doi.org/10.1038/nphys1392
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DOI: https://doi.org/10.1038/nphys1392
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