To place this interpretation on firmer theoretical
grounds, we used a phenomenological Landau free-energy analysis. A system with an odd-parity primary
order parameter Yu and a secondary Eu order parameter FEu is described by the generic Landau
free-energy expansion F ¼ F0− 1 − T Tc
ðag Y2 g þ
au Y2 uÞ þ bF2 Eu − g Yg Yu FEuþ higher-order terms,
where ag, au, b, and g are temperature-independent
parameters. To realize a linear coupling between
Yu and FEu, an additional even-parity primary
order parameter Yg that transforms like the
product YuFEu must be introduced. By construction, minimization of the free energy gives
Yuº Ygºj1−T=Tcj1=2 and FEuº Yu Ygºj1− T=Tcj,
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 represen-
tations of Yu and Yg 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 elec-
trons in Re t2g levels may realize a correlated
spin-triplet state with T1g symmetry [section S8
of (36)], and nuclear quadrupole resonance mea-
surements have in fact detected a moderate fer-
romagnetic 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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298 21 APRIL 2017 • VOL 356 ISSUE 6335 sciencemag.org SCIENCE
Fig. 4. Critical exponents of the T2u and Eu order parameters. Temperature dependence of (A) jc T2u j
and (B) jcEuj normalized to the fixed surface contribution jcSj. Solid lines show least-squares fits to the
scaling law j1 − T=Tcjb. Specified uncertainties in the fit parameters are 1 SD. The critical exponent of the
T2u order parameter is consistent with the mean-field prediction b = 1/2, and the critical exponent of the Eu
order parameter is consistent with a linear temperature dependence (b = 1). The inset in (B) shows that the
linearity of the Eu secondary structural order persists over a wide temperature range below Tc.
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 cS = 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.