Chirality density wave of the 'hidden order' phase in URu$_2$Si$_2$
A second-order phase transition is associated with emergence of an “order parameter” and a spontaneous symmetry breaking. For the heavy fermion superconductor URuSi, the symmetry of the order parameter associated with its ordered phase below 17.5 K has remained ambiguous despite 30 years of research, and hence is called “hidden order” (HO). Here we use polarization resolved Raman spectroscopy to specify the symmetry of the low energy excitations above and below the HO transition. These excitations involve transitions between interacting heavy uranium 5f orbitals, responsible for the broken symmetry in the HO phase. From the symmetry analysis of the collective mode, 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.
Electrons occupying 5f orbitals often possess dual characters in solids, partly itinerant and partly localized, which leads to a rich variety of self-organization at low temperature, such as magnetism, superconductivity, or even more exotic states. These ordered states are in general 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 encodes 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.
One particularly interesting example among this class of materials is the uranium-based inter-metallic compound URuSi. It displays a non-magnetic second-order phase transition into an electronically ordered state at K, which becomes superconducting below 1.5 K[3, 4]. Despite numerous theoretical proposals to explain the properties below in the past 30 years [5, 6, 7, 8, 9, 10], the symmetry and microscopic mechanism for the order parameter remains ambiguous, hence the term “hidden order” (HO). In this ordered state, an energy gap in both spin and charge response have been reported [12, 13, 14, 15, 16, 17, 18]. In addition, an in-gap collective excitation at a commensurate wave vector has been observed in neutron scattering experiments[16, 17, 18]. Recently, four-fold rotational symmetry breaking under an in-plane magnetic field and a lattice distortion along the crystallographic a-axis has been reported in high quality small crystals. However, the available experimental works can not yet conclusively determine the symmetry of the order parameter in the HO phase.
URuSi crystallizes in a body-centered tetragonal structure belonging to the point group (space group No. 139 , Fig 1A). The uniqueness of URuSi is rooted in the coexistence of the broad conduction bands, comprised 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 . When the temperature is lowered below approximately 70 K, the hybridization with the conduction band allows a small fraction of a U-5f electron to participate in formation of a narrow quasiparticle band at the Fermi level, while the rest of the electron remains better described as localized on the uranium site.
In the dominant atomic configuration[7, 22], the orbital angular momentums and spins of the two quasi-localized U-5f electrons add up to total momentum 4, having nine-fold degeneracy. In the crystal environment of URuSi, these states split into seven energy levels denoted by irreducible representations of the group: 5 singlet states and 2 doublet states . Each irreducible representation possess distinct symmetry properties under operations such as reflection, inversion, and rotation. For example, the states are invariant under all symmetry operations of the group (Fig. 1A), but the state changes sign under all diagonal and vertical reflections, and thereby possesses 8 nodes (Fig. 1A). Most of the measurable physical quantities, such as density-density and stress tensors, or one particle response functions, are symmetric under exchange of x- and y-axis in tetragonal crystal structure and therefore do not probe the excitations. In contrast, these excitations are accessible to Raman spectroscopy [23, 24, 25, 26].
Raman scattering is an inelastic process which promotes excitations of controlled symmetry (Fig. 1A) defined by the scattering geometries, namely polarizations of the incident and scattered light (blue and red arrows in Fig. 5). It enables to separate the spectra of excitations into single symmetry representation, such as , , , , and in the group (Fig. 5), and thereby classify the symmetry of the collective excitations. The temperature evolution of these excitations across a phase transition provides an unambiguous identification of the broken symmetries. Unlike most other symmetry sensitive probes requiring external perturbations, such as magnetic, electric or strain fields, the photon field used by Raman probe is weak. Thus, Raman spectroscopy presents an ideal tool to study the broken symmetries across phase transitions without introducing external symmetry breaking perturbations.
We employ linearly and circularly polarized light to acquire the temperature evolution of the Raman response functions in all symmetry channels. In figure 2 we plot the Raman response in the channel, where the most significant temperature dependence was observed. The Raman response in the paramagnetic state can be described within a low energy minimal model (illustrated in Fig. 1A, B) that contains two singlet states of and symmetries, split by , and a predominantly symmetry conduction band. In the following, we denote the singlet states of and symmetries by and , as suggested in Ref. , and conduction band labeled by .
At high temperatures, the Raman response exhibits quasielastic scattering, with maximum decreasing from 5 meV at room temperature to 2 meV at low temperature (Fig. 2). We interpret these excitations as transitions from the state into conduction band . Below 50 K, a new maximum around 1 meV develops. This feature resembles a Fano-type interference, where a resonance interacts with the electronic continuum. Here, we interpret the two interacting excitations as the quasielastic scattering (blue lines in Fig. 2), and an overdamped resonance between and states (green lines in Fig. 2). After turning on the hybridization between and , some of the Raman spectral weight is redistributed to lower energy, whereby the spectra acquires the observed feature. Such hybridization tracks the formation of the heavy fermion states in URuSi.
Figure 3 displays a comparison between the static Raman susceptibility (left axis) and the c-axis static magnetic susceptibility (right axis), showing that in the paramagnetic phase the responses are proportional to each other. This proportionality can be understood by noting that both susceptibilities probe -like excitations, which are dominated by transitions from to conduction band , hence in the minimal model of figure 1B, they are proportional to each other. The extreme anisotropy of the magnetic susceptibility (Fig. 3) also follows from this minimal model .
Below 18 K, the Raman response in the channel (Fig. 2) shows the suppression of low energy spectral weight below 6 meV and the emergence of a sharp in-gap mode at 1.6 meV. Figure 4 shows the detailed development of these features. The temperature dependence of the gap qualitatively follows the gap function expected from a mean-field BCS model (pink line in Fig. 4).
Having established the Raman response of symmetry and its correspondence with the magnetic susceptibility in the paramagnetic state, we now present our main results describing the symmetry breaking in the HO state. Figure 5 shows the Raman response in all six proper 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 (Fig. 1B). A weaker intensity is also observed at the same energy in XX and XX geometries commonly containing the excitations of the symmetry, and a much weaker intensity is barely seen within the experimental uncertainty in RL geometry.. The in-gap mode intensity in the channel is about four times weaker than in the channel.
The observation of this intensity “leakage” into forbidden scattering geometries implies the lowering of symmetry in the HO phase, allowing some of the irreducible representations of point group to mix. 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 breaking of the local vertical and diagonal reflection symmetries at the uranium sites in the HO phase. Similarly, the tiny intensity leakage into the RL scattering geometry measure the strength of orthorhombic distortion due to broken four-fold rotational symmetry.
When the reflection symmetries are broken, an -like interaction operator mixes the and states leading to two new local states: and , with being the interaction strength. 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 left-handed or the right-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 the commensurate crystal momentum, but only in the HO state [30, 17, 18]. The Raman measurement proves that this resonance is a long-wavelength excitation of character. The appearance of the same resonance in the neutron scattering at different wavelength, corresponding to the c-axis lattice constant, requires HO to be a staggered alternating electronic order in 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 tetragonal lattice, hence it is hidden to all probes but scattering in symmetry. We reveal this hidden order to be a chirality density wave depicted in figure 1D.
The chirality density wave doubles the translational periodicity of the paramagnetic phase, hence it folds the electronic Brillouin zone, as recently observed by angle-resolved photoemission spectroscopy. It also gives rise to an energy gap, as previously observed in optics[12, 13, 14] and tunneling experiments[15, 32], and shown in figure 4 to originate in expelling the continuum of excitations. The sharp (0.3 meV) resonance is explained by excitation from the ground state, which posses chirality density wave staggering and , to the excited state depicted in figure 1D, which staggers and .
A local order parameter of primary symmetry, breaking vertical/diagonal reflections, with subdominant component, breaking four-fold rotational symmetry, can be expressed in terms of the composite hexadecapole local order parameter of the form:
where , are in-plane angular momentum operators[7, 22]. A spatial order alternating the sign of this hexadecapole for neighboring basal planes is the chirality density wave (see Fig. 1D) that consistently explains the HO phenomena as it is observed by Raman and neutron scattering [16, 30, 17, 18], magnetic torque, X-ray diffraction, and other data[12, 13, 14, 31, 11]. Our finding is a new example of exotic electronic ordering, emerging from strong interaction among f electrons, which should be a more generic phenomenon relevant to other intermetallic compounds.
Acknowledgments We thank J. Buhot, P. Chandra, P. Coleman, G. Kotliar, M.-A. Méasson, D.K. Morr, L. Pascut, A. Sacuto and J. Thompson for discussions. G.B. and V.K.T. acknowledge support from the US 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 Career DMR-1405303. W.-L. Z. acknowledges support by ICAM (NSF-IMI grant DMR-0844115). Work at Los Alamos National Laboratory was performed under the auspices of the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering.
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