czxx = –czyy. The absence of any detectable bulk
SHG signal in a Sin–Sout geometry [section S5
of (36)] imposes the additional constraint czxx =
–2cxxz, reducing the number of independent tensor elements to just one. For a Sin–Pout geometry,
this tensor structure produces a bulk nonlinear
polarization of the form c T2usinðfÞ, which taken
alone would lead to a RA-SHG pattern that is
even in f. However, a coherent superposition of
the c T2usinðfÞ, cEucosðfÞ, and cScosð3fÞ terms,
illustrated in the right column of Fig. 3, generates
a RA-SHG pattern that breaks f→−f symmetry.
To quantitatively assess the validity of this
model, we performed fits to the RA-SHG data
in all four polarization geometries simultaneously.
We fixed cS at its fitted T > Tc value (Fig. 3A)
because it does not measurably change across
Tc [section S5 of (36)], leaving the two complex
numbers cEu and c T2u as the only free parameters.
This model provides an excellent and unique fit
to the data (Fig. 3), providing further evidence of
coupled T2u and Eu order parameters below Tc.
As the temperature is further cooled to just sev-
eral kelvin below Tc, cEu becomes dominant, and
a pronounced transformation of the RA-SHG
pattern toward a jcEucosðfÞj2 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 nat-
urally explains the absence of any detectable
T2u distortion by structure-sensitive probes
Distinguishing a genuine electronic nematic
phase transition from a simple ferrodistortive
transition is a well-known experimental chal-
lenge (11) because the electronic and structural
order parameters are typically coupled and have
a concurrent temperature onset, as is the case for
Cd2Re2O7. The task of disentangling primary from
secondary order parameters can be approached
by studying the critical exponents b of the order
parameter temperature scaling law j1 − T=Tcjb.
SHG is particularly well-suited for this because
cijk is linearly proportional to order parameters
that are parity-odd [section S7 of (36)]. To obtain
the temperature dependence of the Eu and T2u
order parameters, we acquired RA-SHG patterns
over a series of finely spaced temperatures below
Tc and fit them to the model previously described.
Because cosð3fÞ, cosðfÞ, and sinðfÞ are orthogonal
functions, the fitted values of cEu and c T2u at any
given temperature can be determined uniquely.
Furthermore, because the Eu and T2u tensors
have no elements in common, every bulk SHG re-
sponse channel cijk couples to only one of the two
order parameters. The temperature dependence
of jc T2uj, 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 b ≈ 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 re-
sponse (Fig. 3C) and can no longer be reliably ex-
tracted from the data.
The temperature dependence of jcEu j, which
is proportional to the Eu order parameter, also
exhibits an onset at Tc ≈ 201 K (Fig. 4B), demonstrating that it is coupled to the T2u order
parameter. It has a linear temperature dependence
(b ≈ 1), extending over a wide temperature range
below Tc. This behavior is contrary to that expected
of a primary order parameter because critical
fluctuations may reduce b from its mean-field value
but can never increase it. Instead, the Eu structural
distortion must be a secondary order parameter.
Fig. 2. Spatially resolved optical SHG anisotropy measurements. (A) Schematic
of the RA-SHG setup. A circularly polarized laser beam (red) with center wave-
length at 800 nm is sent through a linear polarizer (to select either Pin or Sin
polarization) and onto a phase mask. The +1 diffracted order is isolated and
redirected parallel to the optical axis by a collimating lens. After passing through
a dichroic mirror, the beam is focused by an objective lens onto the stationary
sample to a spot size of ~30 mm. The reflected SHG beam (blue) is recollimated
by the objective, passes through a second linear polarizer (to select either Pout or
Sout polarization), and is deflected by the dichroic mirror onto a 2D electron-
multiplying charge-coupled device (EMCCD) camera. The polarizers and phase
mask rotate rapidly (black arrows), causing the SHG beam to trace out a circle
on the camera as the scattering plane angle f 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.