Strong electron interactions can drive metallic systems toward a variety of well-known symmetry-broken phases, but the instabilities of correlated metals with strong spin-orbit coupling have only recently begun to be explored. We uncovered a multipolar nematic phase of matter in the metallic pyrochlore Cd2Re2O7 using spatially resolved second-harmonic optical anisotropy measurements. Like previously discovered electronic nematic phases, this multipolar phase spontaneously breaks rotational symmetry while preserving translational invariance. However, it has the distinguishing property of being odd under spatial inversion, which is allowed only in the presence of spin-orbit coupling. By examining the critical behavior of the multipolar nematic order parameter, we show that it drives the thermal phase transition near 200 kelvin in Cd2Re2O7 and induces a parity-breaking lattice distortion as a secondary order.
In the presence of strong Coulomb interactions, the fluid of mobile electrons in a metal can spontaneously break the point group symmetries of the underlying crystal lattice, realizing the quantum analog of a nematic liquid crystal (1). Like their classical counterparts, quantum nematic phases generally preserve spatial inversion symmetry and are therefore anisotropic but centrosymmetric fluids. Experimental evidence of such nematic order was first detected in a two-dimensional (2D) GaAs/AlGaAs quantum well interface on the basis of a pronounced resistivity anisotropy between the two principal directions of the underlying square lattice (2, 3). Subsequently, similar behavior has been reported in a number of quasi-2D square lattice compounds, including Sr3Ru2O7 (4), URu2Si2 (5), and several families of both copper- (6, 7) and iron-based (8–11) high-temperature superconductors, suggesting possible connections between even-parity nematic fluctuations and unconventional s- and d-wave Cooper pairing (12).
Extending earlier work on Fermi liquid instabilities in the p-wave spin interaction channel (13), it has recently been predicted that correlated metals with strong spin-orbit coupling may realize a fundamentally new class of electronic nematic phases with spontaneously broken spatial inversion symmetry (14), including a quantum analog of the unusual NT nematic phase discussed in the context of classical bent-core liquid crystals (15). Theoretical models have shown that parity-breaking nematic fluctuations can induce odd-parity p- or f-wave Cooper pairing and thus provide a route to topological superconductivity (16, 17). In addition, because inversion symmetry-breaking necessarily lifts the spin degeneracy of bulk energy bands in a spin-orbit coupled system, odd-parity nematic order offers a potential mechanism for generating topologically protected Weyl and nodal-line semimetals and for designing highly tunable charge-to-spin current conversion technologies for spintronics applications.
, where σi are the Pauli matrices and kj is the crystal momentum (14, 18). In a cubic material, for example, this order parameter can have either Eu or T2u symmetry. An example of a spin-polarized Fermi surface distortion induced by T2u multipolar nematic order is shown in Fig. 1A.
Fig. 1 Illustration of electronic and structural order parameters in Cd2Re2O7.
) mirror plane, and arrows show the displacement directions of the O(1) atoms.
axis (Fig. 1, B and C). This phase transition has traditionally been attributed to the freezing of a soft phonon mode with Eu symmetry, dominated by the displacement of O(1) atoms (19, 24, 25). However, the observation of extremely small changes in lattice parameters (21, 22), an anomalous temperature dependence of superlattice Bragg peaks (20), and a dramatic reduction in the electronic density of states across Tc (26–29) calls this interpretation into question and raises the possibility that a hitherto undetected electronic order is driving the transition.
onto a circular locus of points on a stationary 2D detector (Fig. 2A), the experiment can be carried out at very high rotational frequencies (~4 Hz), which greatly enhances the sensitivity to small changes in symmetry by averaging over laser fluctuations.
Fig. 2 Spatially resolved optical SHG anisotropy measurements.
changes. (B) Optical micrograph of the (111) surface of a Cd2Re2O7 single crystal. (C) Wide-field SHG image of a striation-free region measured at T = 210 K and T = 150 K. All three types of tetragonal domains [section S4 of (36)] are visible in the low-temperature image. (D) Enlarged SHG image of the region over which scanning RA-SHG was performed. Pin‒Pout RA-SHG patterns for two opposite parity domains are shown, and an approximate domain boundary is drawn.
) arising from interference between the surface and bulk terms. This in turn produces distinct RA-SHG patterns (Fig. 2D).
tensor element. The resulting RA-SHG intensity pattern, which is proportional to the squared magnitude of the nonlinear polarization, exactly reproduces the high-temperature data and reaffirms its surface origin.
Fig. 3 Detection of Eu and T2u symmetry breaking by RA-SHG.
Raw RA-SHG images acquired with a Sin‒Pout polarization geometry (left column) at temperatures of (A) 204.5 K, (B) 199.7 K, and (C) 196.6 K. Concentric white circles show the radial integration region used to generate the RA patterns (middle column). Numbers on the outer boundaries of the polar plots indicate the intensity scale in units where χS = 1. The RA patterns are fit to the squared magnitude of a sum of surface, bulk Eu, and bulk T2u polarization terms, as described in the text. Fits are overlaid on the RA patterns (black curves), and each component is illustrated in the right column, where solid petals denote a positive sign and white petals denote a negative sign for the polarization.
. This is incompatible with the data and suggests that an additional inversion-symmetry–breaking order parameter emerges together with the Eu structural order parameter below Tc.
form takes place (Fig. 3C), obscuring the T2u order parameter. The relative faintness of the T2u signal is consistent with a nematic instability that predominantly affects states only near the Fermi level and naturally explains the absence of any detectable T2u distortion by structure-sensitive probes (19–22).
, which is proportional to the T2u order parameter, was extracted from such fits (Fig. 4A). An onset temperature of Tc ≈ 201 K and a critical exponent of β ≈ 1/2 are obtained from a least-squares fit to the scaling law, which is consistent with the mean-field prediction for a primary order parameter. At temperatures below ~198 K, the T2u response is overwhelmed by the Eu response (Fig. 3C) and can no longer be reliably extracted from the data.
Fig. 4 Critical exponents of the T2u and Eu order parameters.
. Specified uncertainties in the fit parameters are 1 SD. The critical exponent of the T2u order parameter is consistent with the mean-field prediction β = 1/2, and the critical exponent of the Eu order parameter is consistent with a linear temperature dependence (β = 1). The inset in (B) shows that the linearity of the Eu secondary structural order persists over a wide temperature range below Tc.
, which exactly reproduces our experimental results. By performing a general symmetry-based analysis of the Landau expansion [section S8 of (36)], it is possible to constrain the irreducible representations of Ψu and Ψg to T2u and T1g, respectively. This serves as a strong self-consistency check of our RA-SHG data analysis. The T1g order parameter uncovered by this analysis preserves inversion symmetry and is therefore not detectable with SHG. It is possible that interactions among electrons in Re t2g levels may realize a correlated spin-triplet state with T1g symmetry [section S8 of (36)], and nuclear quadrupole resonance measurements have in fact detected a moderate ferromagnetic enhancement below Tc (37). Further study, however, is required to firmly establish a microscopic origin of the T1g order.
Our data and analysis reveals the existence of a T2u electronic order in Cd2Re2O7 that drives the 200 K phase transition and induces the Eu lattice distortion as a secondary order parameter. The assignment of the T2u order to a multipolar nematic phase is supported by previous experiments that show only weak Eu structural distortions (20–22) accompanied by large electronic anomalies across Tc (26–29), the absence of any charge or magnetic order below Tc (37), and full agreement with theoretical prediction (14). More generally, our results establish a distinct class of odd-parity multipolar electronic nematic phases in spin-orbit–coupled correlated metals and demonstrates an experimental strategy for uncovering further realizations of such order. Carefully examining the competing phases in the vicinity of odd-parity nematic order, including the superconducting phase below ~1 K in Cd2Re2O7 (38–40), may prove fruitful for uncovering other unconventional phases of matter.
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Acknowledgments: We thank J. P. Eisenstein, L. Fu, T. Hsieh, P. A. Lee, A. de la Torre, and L. Zhao for useful discussions. RA-SHG experiments were supported by the U.S. Department of Energy under grant DE-SC0010533. Instrumentation for the RA-SHG setup was partially supported by a U.S. Army Research Office Defense University Research Instrumentation Program award under grant W911NF-13-1-0293 and the Alfred P. Sloan Foundation under grant FG-BR2014-027. D.H. also acknowledges funding from the Institute for Quantum Information and Matter, a NSF Physics Frontiers Center (PHY-1125565) with support of the Gordon and Betty Moore Foundation through grant GBMF1250. J.-Q.Y. and D.G.M. were supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. Z.Y.Z. acknowledges the Center for Emergent Materials and NSF Materials Research Science and Engineering Center under grant DMR-1420451. D.H. and D. H. Torchinsky are inventors on U.S. patent application #14/705,831 submitted by the California Institute of Technology, which covers a spectrometer apparatus for the study of the crystallographic and electronic symmetries of crystals and methods of using said apparatus. The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.