## 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 URu_{2}Si_{2} 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

## Abstract

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 URu_{2}Si_{2} 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 URu_{2}Si_{2}. It displays a nonmagnetic second-order phase transition into an electronically ordered state at K, and then becomes superconducting below 1.5 K (*3*, *4*). Despite numerous theoretical proposals to explain the properties below in the past 30 years (*5*–*10*), 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 (*12*–*18*). 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.

URu_{2}Si_{2} crystallizes in a body-centered tetragonal structure belonging to the point group (space group no. 139 *I*4*/mmm*, Fig. 1A). The uniqueness of URu_{2}Si_{2} 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 and trivalent (*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.

In the dominant atomic configuration, the orbital angular momentum and spin of the two quasi-localized U-5f electrons add up to total momentum 4, having ninefold degeneracy (*6*, *22*). In the crystal environment of URu_{2}Si_{2}, these states split into seven energy levels denoted by irreducible representations of the group: five singlet states and two doublet states . Each irreducible representation possesses distinct symmetry properties under operations such as reflection, inversion, and rotation. For example, the states are invariant under all symmetry operations of the group, whereas the 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 excitations, whereas these excitations can be probed by Raman spectroscopy (*23*–*28*).

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 , , , , and in the 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 channel, where the most significant temperature dependence was observed. The Raman susceptibility above can be described within a low-energy minimal model suggested in (*6*) (illustrated in Fig. 1B) that contains two singlet states of and symmetries, split by an energy , and a conduction band of predominantly symmetry. In the following, we denote the singlet states of and symmetries by |〉 and ; the conduction band is labeled .

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 (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 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 (left axis) and the *c*-axis static magnetic susceptibility (right axis), showing that the responses are proportional to each other at temperatures above . This proportionality can be understood by noting that both susceptibilities probe -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 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 symmetry. The mode can be interpreted as a meV resonance between the |〉 and quasi-localized states, which can only appear in the channel of the group. A weaker intensity is also observed at the same energy in and geometries commonly containing the excitations of the symmetry, and a much weaker intensity is barely seen within the experimental uncertainty in RL geometry.

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, . For example, the mode intensity “leakage” from the into the channel implies that the irreducible representation and of the point group merge into the representation of the lower group . 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 -like interaction operator |〉 mixes the |〉 and states, leading to two new local states

with being the interaction strength (*6*). A pair of such states cannot be transformed into one another by any remaining 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, or , 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 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 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 ; 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 excitations. The sharp resonance is explained by excitation from the ground state, in which a chirality density wave staggers and , to the excited collective state (*22*).

A local order parameter of primary symmetry, breaking vertical and diagonal reflections, with a subdominant component, breaking fourfold rotational symmetry, can be expressed in terms of the composite hexadecapole local order parameter of the form (*6*, *22*) (3)where , 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 *A*_{2}* _{g}* 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

## References and Notes

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**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.