Chirality density wave of the “hidden order” phase in URu2Si2

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Science  20 Mar 2015:
Vol. 347, Issue 6228, pp. 1339-1342
DOI: 10.1126/science.1259729

Uncovering the symmetry of a hidden order

Cooling matter generally makes it more ordered and may induce dramatic transitions: Think of water becoming ice. With increased order comes loss of symmetry; water in its liquid form will look the same however you rotate it, whereas ice will not. Kung et al. studied the symmetry properties of a mysteriously ordered phase of the material URu2Si2 that appears at 17.5 K. They shone laser light on the crystal and studied the shifts in the frequency of the light. The electron orbitals of the uranium had a handedness to them that alternated between the atomic layers.

Science, this issue p. 1339


A second-order phase transition in a physical system is associated with the emergence of an “order parameter” and a spontaneous symmetry breaking. The heavy fermion superconductor URu2Si2 has a “hidden order” (HO) phase below the temperature of 17.5 kelvin; the symmetry of the associated order parameter has remained ambiguous. Here we use polarization-resolved Raman spectroscopy to specify the symmetry of the low-energy excitations above and below the HO transition. We determine that the HO parameter breaks local vertical and diagonal reflection symmetries at the uranium sites, resulting in crystal field states with distinct chiral properties, which order to a commensurate chirality density wave ground state.

In solids, electrons occupying 5f orbitals often have a partly itinerant and partly localized character, which leads to a rich variety of low-temperature phases, such as magnetism and superconductivity (1). Generally, these ordered states are characterized by the symmetry they break, and an order parameter may be constructed to describe the state with reduced symmetry. In a solid, the order parameter reflects the microscopic interactions among electrons that lead to the phase transition. In materials containing f-electrons, exchange interactions of the lanthanide or actinide magnetic moments typically generate long-range antiferromagnetic or ferromagnetic order at low temperatures, but multipolar ordering such as quadrupolar, octupolar, and hexadecapolar is also possible (2).

One particularly interesting example is the uranium-based intermetallic compound URu2Si2. It displays a nonmagnetic second-order phase transition into an electronically ordered state at Embedded Image K, and then becomes superconducting below 1.5 K (3, 4). Despite numerous theoretical proposals to explain the properties below Embedded Image in the past 30 years (510), the symmetry and microscopic mechanism for the order parameter remain ambiguous, hence the term “hidden order” (HO) (11). In this ordered state, an energy gap in both the spin and the charge response has been reported (1218). In addition, an in-gap collective excitation at a commensurate wave vector has been observed in neutron scattering experiments (13, 14, 16). Recently, fourfold rotational symmetry breaking under an in-plane magnetic field (19) and a lattice distortion along the crystallographic a axis (20) have been reported in high-quality small crystals. However, the available experimental works cannot yet conclusively determine the symmetry of the order parameter in the HO phase.

URu2Si2 crystallizes in a body-centered tetragonal structure belonging to the Embedded Image point group (space group no. 139 I4/mmm, Fig. 1A). The uniqueness of URu2Si2 is rooted in the coexistence of the broad conduction bands, composed mostly of Si-p and Ru-d electronic states, and more localized U-5f orbitals, which are in a mixed-valent configuration between tetravalent Embedded Image and trivalent Embedded Image (21). When the temperature is lowered below ~70 K, the hybridization with the conduction band allows a small fraction of each U-5f electron to participate in formation of a narrow quasiparticle band at the Fermi level, whereas the rest of the electron remains better described as localized on the uranium site.

Fig. 1 Schematics of the local symmetry of the quasi-localized states.

(A) The crystal structure of URu2Si2 above Embedded Image, belonging to the Embedded Image point group. Presented in three dimensions and xy-plane cut are illustrations showing the symmetry of the Embedded Image state |Embedded Image〉 and Embedded Image state Embedded Image, where the positive (negative) amplitude is denoted by red (blue) color. The Embedded Image state is symmetric with respect to the vertical (Embedded Image) and diagonal (Embedded Image) reflections, whereas the Embedded Image state is antisymmetric with respect to these reflections. (B) Schematic of the band structure of a low-energy minimal model. The green dashed line denotes the conduction band Embedded Image; the red and black dashed lines denote crystal field states of the U-5f electrons: the ground state |Embedded Image〉 and the first excited state Embedded Image (22). Blue and red arrows denote the incident and scattered light in a Raman process, respectively. Embedded Image eV is the incoming photon energy (energy levels not to scale), Embedded Image is the hybridization strength between Embedded Image and Embedded Image; Embedded Image and Embedded Image are the resonance energies for |Embedded ImageEmbedded ImageEmbedded Image and |Embedded ImageEmbedded Image excitations, respectively. (C) The crystal structure of URu2Si2 in the HO phase, and illustrations showing the symmetry of the chiral states Embedded Image and Embedded Image, and the excited state Embedded Image. The left- and right-handed states, denoted by red and blue atoms, respectively, are staggered in the lattice. Embedded Image and Embedded Image denotes the two nonequivalent uranium sites in the HO phase. (D) Schematics of the chirality density wave in the HO phase. The uranium sites Embedded Image and Embedded Image are occupied by Embedded Image and Embedded Image states, respectively.

In the dominant atomic configuration, the orbital angular momentum and spin of the two quasi-localized U-5f electrons add up to total momentum 4Embedded Image, having ninefold degeneracy (6, 22). In the crystal environment of URu2Si2, these states split into seven energy levels denoted by irreducible representations of the Embedded Image group: five singlet states Embedded Image and two doublet states Embedded Image. Each irreducible representation possesses distinct symmetry properties under operations such as reflection, inversion, and rotation. For example, the Embedded Image states are invariant under all symmetry operations of the Embedded Image group, whereas the Embedded Image state changes sign under all diagonal and vertical reflections, and thereby has eight nodes (Fig. 1A). Most of the physical observables, such as density-density and stress tensors, or one-particle spectral functions, are symmetric under exchange of x and y axis in tetragonal structure and therefore are impervious to the Embedded Image excitations, whereas these Embedded Image excitations can be probed by Raman spectroscopy (2328).

Raman scattering is an inelastic process that promotes excitations of controlled symmetry defined by the scattering geometries, namely, polarizations of the incident and scattered light (22, 23). Polarization-resolved Raman spectroscopy enables separation of the spectra of excitations into distinct symmetry representations, such as Embedded Image, Embedded Image, Embedded Image, Embedded Image, and Embedded Image in the Embedded Image group, thereby classifying the symmetry of the collective excitations (22, 26). The temperature evolution of these excitations across a phase transition provides an unambiguous identification of the broken symmetries; furthermore, the photon field used by the Raman probe is weak, which avoids introducing external symmetry-breaking perturbations.

We use linearly and circularly polarized light to acquire the temperature evolution of the Raman response functions in all symmetry channels. In Fig. 2, we plot the Raman susceptibility in the Embedded Image channel, where the most significant temperature dependence was observed. The Raman susceptibility above Embedded Image can be described within a low-energy minimal model suggested in (6) (illustrated in Fig. 1B) that contains two singlet states of Embedded Image and Embedded Image symmetries, split by an energy Embedded Image, and a conduction band of predominantly Embedded Image symmetry. In the following, we denote the singlet states of Embedded Image and Embedded Image symmetries by |Embedded Image〉 and Embedded Image; the conduction band is labeled Embedded Image.

Fig. 2 Temperature dependence of the Embedded Image Raman susceptibility.

(A) The Embedded Image Raman response function decomposed from the spectra measured in the Embedded Image, and Embedded Imagescattering geometries (22). The solid lines are a guide to the eye, illustrating the narrowing of the Drude function (25): Embedded Image, where Embedded Image is the Drude scattering rate (indicated by the arrows), which decreases on cooling. Below 70 K, the Raman response deviates from the Drude function. Below Embedded Image, the Raman response shows spectral weight suppression below 6 meV and the appearance of an in-gap mode at 1.6 meV (7 and 13 K). (B) Temperature dependence of the static Raman susceptibility in Embedded Image channel Embedded Image (red dots), and the static magnetic susceptibility along c and a axis from (3) are plotted as blue squares and black circles, respectively. Embedded Image is marked by the dashed line. (C) Temperature dependence of the low-frequency Raman response in the XY scattering geometry, dominantly composed of Embedded Image excitations. A gaplike suppression develops on cooling, and an in-gap mode at 1.6 meV (black dashed line) emerges below Embedded Image. The full width at half-maximum of the mode decreases on cooling from Embedded Image meV at 13 K to Embedded Image meV at 7 K. The white line shows the temperature dependence of the BCS gap function.

At high temperatures, the Raman spectra exhibit a Drude-like line shape, which in (25) was attributed to quasi-elastic scattering. The maximum in the Raman response function decreases from 5 meV at room temperature to 1 meV just above Embedded Image (Fig. 2A). Below 70 K, the line shape deviates slightly from the Drude function, tracking the formation of the heavy fermion states by the hybridization of the itinerant conduction band and the U-5f states. Below 17.5 K, the Embedded Image Raman response function shows suppression of low-energy spectral weight resembling the temperature dependence of the Bardeen-Cooper-Schrieffer (BCS) gap function, and the emergence of a sharp in-gap mode at 1.6 meV (Fig. 2, A and C).

Figure 2B displays a comparison between the static Raman susceptibility Embedded Image (left axis) and the c-axis static magnetic susceptibility Embedded Image (right axis), showing that the responses are proportional to each other at temperatures above Embedded Image. This proportionality can be understood by noting that both susceptibilities probe Embedded Image-like excitations, as given by the minimal model of Fig. 1B. The extreme anisotropy of the magnetic susceptibility (Fig. 2B) also follows from this minimal model (22).

Having established the Raman response of Embedded Image symmetry and its correspondence with the magnetic susceptibility, we now present our main results describing the symmetry breaking in the HO state. Figure 3 shows the Raman response in six scattering geometries at 7 K. The intense in-gap mode is observed in all scattering geometries containing Embedded Image symmetry. The mode can be interpreted as a Embedded Image meV resonance between the |Embedded Image〉 and Embedded Image quasi-localized states, which can only appear in the Embedded Image channel of the Embedded Image group. A weaker intensity is also observed at the same energy in Embedded Image and Embedded Image geometries commonly containing the excitations of the Embedded Image symmetry, and a much weaker intensity is barely seen within the experimental uncertainty in RL geometry.

Fig. 3 The Raman response function in six scattering geometries at 7 K.

The arrows in each panel show the linear or circular polarizations for incident (blue) and scattered (red) light. The six scattering geometries are denoted as Embedded Image, and Embedded Image, with Embedded Image being the direction vector for incident light polarization and Embedded Image being the scattered light polarization. Embedded Image, Embedded Image are aligned along crystallographic axes; Embedded Image, Embedded Image are at Embedded Image to the a axes; Embedded Image and Embedded Image are right and left circularly polarized light, respectively (22). The irreducible representations for each scattering geometry are shown within the Embedded Image point group. The data are shown in black circles, where the error bars show 1 SD. The red solid lines are fits of the in-gap mode to a Lorentzian, and the fitted intensity using the method of maximum likelihood is noted in each panel. By decomposition, the in-gap mode intensity in each symmetry channels areEmbedded Image, Embedded Image, Embedded Image, and Embedded Image. The full width at half-maximum of the in-gap mode is about 0.3 meV at 7 K (instrumental resolution of 0.17 meV is shown in the Embedded Image panel).

The observation of this intensity “leakage” into forbidden scattering geometries marks the lowering of symmetry in the HO phase, indicating the reduction in the number of irreducible representations of the parent point group, Embedded Image. For example, the Embedded Image mode intensity “leakage” from the Embedded Image into the Embedded Image channel implies that the irreducible representation Embedded Image and Embedded Image of the Embedded Image point group merge into the Embedded Image representation of the lower group Embedded Image. This signifies the removal of the local vertical and diagonal reflection symmetry operators at the uranium sites in the HO phase. Similarly, the tiny intensity leakage into the RL scattering geometry measures the strength of orthorhombic distortion caused by broken fourfold rotational symmetry.

When the reflection symmetries are broken, an Embedded Image-like interaction operator Embedded Image |Embedded Image〉 mixes the |Embedded Image〉 and Embedded Image states, leading to two new local states

Embedded Image

with Embedded Image being the interaction strength (6). A pair of such states cannot be transformed into one another by any remaining Embedded Image group operators: a property known as chirality (or handedness). The choice of either the right-handed or the left-handed state on a given uranium site, Embedded Image or Embedded Image, defines the local chirality in the HO phase (Fig. 1C). Notice that these two degenerate states both preserve the time-reversal symmetry, carry no spin, and contain the same charge, but differ only in handedness.

The same 1.6-meV sharp resonance has also been observed by inelastic neutron scattering at momentum commensurate with the reciprocal lattice vector, but only in the HO state (14, 16, 29). The Raman measurement proves that this resonance is a long-wavelength excitation of Embedded Image character. The appearance of the same resonance in neutron scattering at a different wavelength, corresponding to the c-axis lattice constant, requires HO to be a staggered alternating electronic order along the c direction. Such order with alternating left- and right-handed states at the uranium sites for neighboring basal planes has no modulation of charge or spin and does not couple to the tetragonal lattice; hence, it is hidden to all probes but the scattering of Embedded Image symmetry. We reveal this hidden order to be a chirality density wave depicted in Fig. 1D.

The chirality density wave doubles the translational periodicity of the phase above Embedded Image; hence, it folds the electronic Brillouin zone, as recently observed by angle-resolved photoemission spectroscopy (30). It also gives rise to an energy gap, as previously observed in optics (12, 17, 18) and tunneling experiments (15, 31) and shown in Fig. 2C to originate in expelling the continuum of Embedded Image excitations. The sharp resonance is explained by excitation from the ground state, in which a chirality density wave staggers Embedded Image and Embedded Image, to the excited collective state (22).

A local order parameter of primary Embedded Image symmetry, breaking vertical and diagonal reflections, with a subdominant Embedded Image component, breaking fourfold rotational symmetry, can be expressed in terms of the composite hexadecapole local order parameter of the form (6, 22)Embedded Image (3)where Embedded Image, Embedded Image are in-plane angular momentum operators and the overline stands for symmetrization. A spatial order alternating the sign of this hexadecapole for neighboring basal planes is the chirality density wave (Fig. 1D) that consistently explains the HO phenomena as it is observed by Raman and neutron scattering (13, 14, 16, 29), magnetic torque (19), x-ray diffraction (20), and other data (11, 12, 17, 18, 30). Our finding is an example of exotic electronic ordering emerging from strong interaction among f electrons, which should be a more generic phenomenon relevant to other intermetallic compounds.

Note added in proof: While this paper was being reviewed, J. Buhot et al. (32) reproduced the A2g symmetry in-gap mode in a Raman experiment with 561-nm laser excitation and showed that the mode does not split in up to 10 T magnetic field.

Supplementary Materials

Material and Methods

Figs. S1 to S4

References (3346)

References and Notes

  1. See supplementary materials on Science Online.
  2. Acknowledgments: We thank J. Buhot, P. Chandra, P. Coleman, G. Kotliar, M.-A. Méasson, D. K. Morr, L. Pascut, A. Sacuto, J. Thompson, and V. M. Yakovenko for discussions. G.B. and V.K.T. acknowledge support from the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-SC0005463. H.-H.K. acknowledges support from the National Science Foundation under Award NSF DMR-1104884. K.H. acknowledges support by NSF DMR-1405303. W.-L.Z. acknowledges support by the Institute for Complex Adaptive Matter (NSF-IMI grant DMR-0844115). Work at Los Alamos National Laboratory was performed under the auspices of the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering.
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