Abstract
Exotic electronic states resulting from entangled spin and orbital degrees of freedom are hallmarks of strongly correlated felectron systems. A spectacular example is the socalled hiddenorder (HO) phase transition^{1} in the heavyelectron metal URu_{2}Si_{2}, which is characterized by the huge amount of entropy lost at T_{HO}=17.5 K (refs 2, 3). However, no evidence of magnetic/structural phase transition has been found below T_{HO} so far. The origin of the HO phase transition has been a longstanding mystery in condensedmatter physics. Here, on the basis of a firstprinciples theoretical approach, we examine the complete set of multipole correlations allowed in this material. The results uncover that the HO parameter is a rank5 multipole (dotriacontapole) order with nematic E^{−} symmetry, which exhibits staggered pseudospin moments along the [110] direction. This naturally provides comprehensive explanations of all key features in the HO phase including anisotropic magnetic excitations, the nearly degenerate antiferromagneticordered state and spontaneous rotationalsymmetry breaking.
Main
In the rareearth and actinide compounds, f electrons behave like welllocalized moments at high temperatures. As the temperature is lowered, f electrons begin to delocalize owing to the hybridization with conduction electron wavefunctions. At yet lower temperatures the f electrons become itinerant, forming a narrow conduction band with heavy effective electron mass, which is largely enhanced from the freeelectron mass. Notable manybody effects within the narrow band lead to a plethora of fascinating physical phenomena including multipole order, quantum phase transition and unconventional superconductivity. Among them, perhaps the appearance of a HO state in URu_{2}Si_{2} is one of the most mysterious phenomena. Identification of the microscopic order parameter and mechanism that drives the HO transition continues to be a central question in strongly correlated felectron systems^{1}.
There are several unique features that seem to be clues for understanding the HO in URu_{2}Si_{2}. In the paramagnetic state above T_{HO}, the magnetic susceptibility exhibits the Isinglike anisotropy^{2,4}. In the HO state below T_{HO}, an electronic excitation gap is formed on a large portion of the Fermi surface^{5,6} (FS) and most of the carriers disappear^{7,8}. Closely related to this, the gap formation also occurs in the magnetic excitation spectra at commensurate and incommensurate wave numbers, Q_{C}=(0 0 1) and Q_{IC}=(0.6 0 0), respectively, as revealed by the neutron inelastic scattering^{9,10,11}. The HO ground state changes to the largemoment antiferromagnetic (AFM) state with the ordering vector Q_{C} on applying hydrostatic pressure^{12,13}, but the FS has a striking similarity between these different phases^{14,15}, implying that the HO is nearly degenerate with the AFM order. The magnetic torque measurements reveal the nematicity, which breaks the inplane rotational (tetragonal) symmetry in the HO^{16}. The challenge for the theory has been to identify the order parameter that explains all of the above key features.
The theories that have been proposed to describe the HO state can be divided into two prevailing approaches; one is based on the localized 5felectron model^{17,18,19,20,21,22,23} and the other the itinerant one^{24,25,26,27,28,29}. Recent angleresolved photoemission spectroscopy results clearly demonstrate that all 5f electrons are itinerant^{30} and the crystalline electric field, which is a signature of the localized nature, has never been observed. Moreover, the nuclear magnetic resonance measurements^{31} show a formation of the coherent heavyelectron state well above T_{HO}. Therefore, it is natural to discuss the electronic structure on the basis of the itinerant picture. However, reliable calculation of the physical quantities by taking into account the complicated band structure is a difficult task. For this purpose, we use a stateoftheart ab initio downfolding (Supplementary Section SI) and dissect the electronic structure obtained from the densityfunctional theory (DFT) calculations. The obtained tightbinding Hamiltonian is constructed from 56 orbitals of U 5f, U 6d, Ru 4d and Si 3p. Introducing the onsite Coulomb interactions between 5f electrons, we obtain a realistic itinerant model, that is, a 56band Andersonlattice model including the spin–orbit interaction. On the basis of this realistic model Hamiltonian, magnetic and multipole correlations are analysed by the randomphase approximation (RPA) and beyond. To account for the mass renormalization effect in the Fermiliquid theory, the energy and temperature scale is reduced by a factor of 10 throughout this study^{2,14,15}, which makes comparisons to the experiments straightforward.
Figure 1 shows the paramagnetic FS and the band structure near the Fermi level, respectively. The energy bands crossing the Fermi level have mainly the total angular momentum j=5/2 multiplet of U 5f. Each j_{z} component of j=5/2 multiplet is coloured by weight. It turns out that each separated FS is mainly composed of a rather specific j_{z} component without large mixing, except for the outer FS around the Z point (Fig. 1). Such a j_{z} component map is quite useful in that we are able to capture valuable information such as which parts of the FS play an essential role for the HO formation. Indeed, the disentanglement of FS orbital characters has also been an important theoretical advance to understand the electronic properties in ironpnictide superconductors^{32}.
First we discuss the RPA analysis of rank1 (dipole) correlation, which is the conventional static magnetic correlation. The regime with j_{z}=±5/2, shown in red, in the outer FS around the Z point is well nested with the outer FS around the Γ point by the vector Q_{C}, as indicated by the arrow in Fig. 1 (ref. 33). This nesting gives rise to a sharp peak of the correlation parallel to the c axis (dipole J_{z}) at Z (0 0 1) shown in Fig. 2a. Another salient feature is the hump structure at around (0.6 0 0) and the equivalent points, whose Qvectors coincide with Q_{IC}. We point out that these peak and hump structures in the paramagnetic phase are directly related to the magnetic excitation gap at Q_{C} and Q_{IC} in the HO phase^{11} because the gap opening occurs at the nested parts of the FS. We also note that the inplane magnetic correlations, J_{x} and J_{y}, are not enhanced in contrast to J_{z} (Fig. 2b), which is also consistent with the Isinglike magnetic susceptibility^{2,4} and polarizedneutron measurements^{10}.
Next we examine the higherrank multipole correlations. According to the group theory, there are 36 multipole moments up to the fifth rank in j=5/2 subspace (Supplementary Table S1). Figure 2c–f shows the correlations between the basis functions belonging to rank 2 (quadrupole), 3 (octupole), 4 (hexadecapole) and 5 (dotriacontapole), respectively. What is remarkable is that as in the case for the dipole J_{z}, the Q_{C} correlation at the Z point is strongly enhanced in some cases such as O_{20} (rank 2), H_{x(y)b} (rank 4), D_{4} (rank 5) and so on. Generally, these bases are mixed in the tetragonal symmetry, as shown by finite offdiagonal terms (red lines). The multipole correlations obtained by the diagonalization are depicted in Fig. 2g, in which each correlation at Q_{C} is classified by the irreducible representations and the dominant component is denoted in parentheses (Supplementary Section SIII). At low temperatures, A_{2}^{−}(J_{z}), E^{−}(D_{x(y)}) and A_{1}^{−}(D_{4}) symmetries exhibit the first, second and third strongest enhancement. The first and the last two correspond to the AFM and dotriacontapole states, respectively. Within the RPA, the AFM state always overcomes the dotriacontapole states. To go beyond the RPA, we consider a staggered particle–hole pairing (generalized multipole orders at the Z point) mediated by the RPA fluctuation from an analogy with unconventional superconductivity (Supplementary Section SIV). This corresponds to the inclusion of the mode–mode coupling. From an analogue of the superconducting gap equation for the staggered ordering, we calculate the maximum eigenvalue λ. The corresponding multipole correlation increases proportionally to ∼1/(1−λ), and then λ=1 provides the transition temperature. The temperature dependence of λ of each symmetry is shown in Fig. 3a. As the temperature is lowered, λ of E^{−}(D_{x(y)}) is most strongly enhanced and the condition λ=1 is fulfilled at finite temperature, indicating a phase transition to the E^{−}(D_{x(y)})state. We emphasize that E^{−}(D_{x(y)}) symmetry breaks the inplane fourfold symmetry, which naturally accounts for the nematicity observed in the magnetic torque results^{16,34}. In addition, the E^{−}(D_{x(y)}) state breaks the timereversal symmetry, which is consistent with the nuclear magnetic resonance measurements^{31}. These lead us to conclude that the HO is E^{−}(D_{x(y)}) dotriacontapole order.
The present calculations also reproduce well other key features of the HO, that is, near degeneracy of the HO and AFM states and the anisotropic temperature dependence of the uniform susceptibility. Figure 3a demonstrates that λ of E^{−}(D_{x(y)}) is very close to that of A_{2}^{−}(J_{z}). This indicates that both states are nearly degenerate and a small perturbation can change the HO to the AFM state. Indeed, we can construct a phase diagram by tuning the interactions (Supplementary Fig. S4), which is consistent with the pressure–temperature phase diagram (Fig. 3b). The temperature dependence of the uniform susceptibility χ_{c}(0) parallel to the c axis exhibits a broad maximum at around 40 K, whereas χ_{ab}(0) perpendicular to the c axis is smaller and nearly temperature independent (Fig. 3c), in good agreement with experiments^{2,4}. The lowtemperature decrease of χ_{c}(0) arises from the deep dip structure in the density of states near the Fermi level (Supplementary Fig. S1b). The Isinglike susceptibility including its temperature dependence has been discussed in terms of the crystalline electric field excitations of the localized 5f electrons so far. However, the present results demonstrate that the susceptibility can be well accounted for by the itinerant scenario.
Why is such a highrank multipole state (rank 5) with E^{−}(D_{x(y)}) symmetry realized in URu_{2}Si_{2}? Similar highrank multipole states have been proposed already; a rank4 A_{2}^{+} state by the DFT+DMFT (dynamical mean field theory) method^{20}, and a rank5 A_{2}^{−} state by the DFT+U method^{27}. However, these states obtained in the strong correlation limit are inconsistent with the nematic behaviour^{34}. Detailed comparisons with other proposed HO parameters are listed in Supplementary Table S2. In our approach from the itinerant limit, the FS nesting with the Q_{C} vector plays an essential role in the multipole fluctuations. What is crucially important is that the FS regions connected by this Q_{C} are dominated by the ±5/2 components, as shown in Fig. 1. In this situation, we can consider a subspace consisting of only two components j_{z}=5/2 and −5/2, which allows us to map j_{z}=±5/2 to pseudospin ↑ and ↓. Then the dipole J_{z} is described by the Pauli matrix σ_{z} spanned in the pseudospin space, as it has only diagonal elements corresponding to . In the same way, D_{x(y)} is given by σ_{x(y)}, representing offdiagonal components describing the transition (Supplementary Section SV), which accompanies the angular momentum change of 5ℏ allowed only in rank 5. In this pseudospin space, the staggered J_{z} state corresponds to the Néel order along the caxis. On the other hand, the D_{x(y)} state corresponds to the inplane order breaking the rotational symmetry, where inplane pseudospin moments are antiferromagnetically coupled along the c axis (Fig. 3d). Thus, the pressureinduced firstorder transition from the HO to AFM state can be explained by the pseudospin staggered moment flip from the inplane to the outofplane direction. The experimentally observed nematicity along the [110] direction corresponds to the linear combination of the twofold degenerate D_{x} and D_{y}. The HO parameter is then represented by
where f_{k α} is an annihilation operator for an f electron with momentum k and pseudospin α. It should be noted that under inplane 180° rotation, the pseudospins change their direction, which discriminates this state from a nematic phase in the strict sense. However, its staggered nature leads to the twofold nematic symmetry of the bulk susceptibility as observed experimentally.
Figure 4a shows the FS in the HO and AFM states, which is calculated by applying the effective multipole field so as to open the gap of 4 meV observed by scanning tunnelling microscopy^{5,6}. The lattice doubling in the AFM phase with Q_{C} also occurs in the HO phase. Most of the FS having j_{z}=±5/2 components disappears as a result of the gap opening at the nested parts of the paramagnetic FS. Around the Γpoint, small electron and large hole (α) pockets, the FS with a cagelike structure and four electron pockets (β) exist. The FS in the HO phase bears a striking resemblance to that in the AFM state (Supplementary Section SVII), consistent with the quantum oscillation measurements. However, the broken fourfold symmetry in the HO state can be seen clearly in the FS with a cagelike structure (Fig. 4b), in sharp contrast to the AFM state.
The present approach based on the firstprinciples calculation is able to give a comprehensive explanation to the problem of HO, which has been a quartercentury mystery. Why has the HO been hidden for a long time? The reason is that in conventional experimental techniques, such as resonant Xray and neutron measurements, extremely highresolution measurements should be required for the direct detection of the highrank multipole order parameter. We also point out that the present rank5 order induces a very tiny but finite inplane dipole moment belonging to the same symmetry E^{−}, which is roughly estimated as 10^{−2}−10^{−3}μ_{B}(Supplementary Section SVI). The detection of such a tiny moment remains a future issue. The itinerant multipole ordering with nematicity revealed in the present study is a new type of electron ordering, which is expected to be ubiquitously present in strongly correlated electron systems when spin and orbital degrees of freedom are entangled.
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Acknowledgements
We thank K. Ueda, K. Haule, G. Kotliar, M. Sigrist and T. M. Rice for helpful discussions and suggestions. This work was supported by a GrantinAid for the Global COE programme ‘The Next Generation of Physics, Spun from Universality and Emergence’, a GrantinAid for Scientific Research on Innovative Areas ‘Heavy Electrons’ (20102002, 20102006) from MEXT, and KAKENHI from JSPS.
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H.I. and R.A. developed a methodology for the DFT+RPA and beyond. MT.S. analysed the fermiology in some ordered states. T.T. provided grouptheoretical arguments of multipoles. H.I., T.S. and Y.M. wrote the text. All authors contributed to critical discussion of the physical interpretation of the results.
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Ikeda, H., Suzuki, MT., Arita, R. et al. Emergent rank5 nematic order in URu_{2}Si_{2}. Nature Phys 8, 528–533 (2012). https://doi.org/10.1038/nphys2330
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