# Electronic nematic order in the normal state of strontium ruthenate

Ryan Russell  Materials Department, University of California, Santa Barbara,
California 93106, USA  Hari P. Nair  Department of Materials Science and
Engineering, Cornell University, Ithaca, New York 14853, USA  Kyle M. Shen
Physics Department, Cornell University, Ithaca, New York 14853, USA  Kavli
Institute at Cornell for Nanoscale Science, Ithaca, New York 14853, USA
Darrell G. Schlom  Department of Materials Science and Engineering, Cornell
University, Ithaca, New York 14853, USA  Kavli Institute at Cornell for
Nanoscale Science, Ithaca, New York 14853, USA  Leibniz-Institut für
Kristallzüchtung, 12489 Berlin, Germany  John W. Harter<EMAIL_ADDRESS>Materials Department, University of California, Santa Barbara, California
93106, USA

###### Abstract

Despite significant achievements in characterizing the properties of Sr2RuO4
over the last three decades, the precise nature of its electronic ground state
is still unresolved. In this work, we provide a missing piece of the puzzle by
uncovering evidence of electronic nematic order in the normal state of
Sr2RuO4, revealed by ultrafast time-resolved optical dichroism measurements of
uniaxially strained thin films. This nematic order, whose domains are aligned
by the strain, spontaneously breaks the four-fold rotational symmetry of the
crystal. The temperature dependence of the dichroism resembles an Ising-like
order parameter, and optical pumping induces a coherent oscillation of its
amplitude mode. A microscopic model of intra-unit-cell nematic order is
presented, highlighting the importance of Coulomb repulsion between
neighboring oxygen $p$-orbitals. The existence of electronic nematic order in
the normal state of Sr2RuO4 may have consequences for the form and mechanism
of superconductivity in this material.

The study of Sr2RuO4 entered a new era after the recent observation of a
reduction in the Knight shift at the onset of superconductivity [1], which
overturned the long-favored chiral $p$-wave triplet pairing scenario.
Researchers are presently faced with an array of sometimes contradictory
results, and it is clear that a fundamental piece of the Sr2RuO4 puzzle is
still missing [2]. A complete understanding of the superconducting phase in
any material depends upon a detailed knowledge of the normal state electronic
structure out of which it emerges. It has long been believed that the normal
state of Sr2RuO4 is a conventional quasi-two-dimensional Fermi liquid with
moderate correlations [2, 3]. Evidence of any deviation from conventional
Fermi liquid behavior would have significant consequences for both theoretical
proposals and the interpretation of experiments. In this work, we present such
evidence. Motivated by recent transverse resistivity experiments indicating
possible electronic nematicity [4], we use optical dichroism measurements of
high-quality epitaxially strained thin films to uncover evidence of electronic
nematic order in the normal state of Sr2RuO4.

Figure 1:  Static and transient optical measurements of Sr2RuO4. (a)
Crystallographic directions in the film and substrate. For Sr2RuO4 on (110)
NdGaO3, the $a$-axis corresponds to [1$\bar{1}$0] NdGaO3 and the $b$-axis
corresponds to [001] NdGaO3. (b) Static reflectivity anisotropy $R_{a}/R_{b}$
versus temperature for films grown on NdGaO3 (SRO/NGO) and LSAT (SRO/LSAT). A
fit to an Ising model of nematic order is shown for the NdGaO3 sample. Also
shown is the temperature-dependent lattice constant ratio $a/b$ for films
strained to NdGaO3 from Ref. 14. (c) Illustration of the pump-probe transient
reflectivity experimental setup. The pump and probe beams remain fixed in a
cross-polarized configuration while the sample is rotated about its surface
normal. $\theta=0^{\circ}$ when the probe is polarized along the $a$-axis (as
shown) and $\theta=90^{\circ}$ when polarized along the $b$-axis. (d) Raw
transient reflectivity curves for several values of $\theta$ between
$0^{\circ}$ and $90^{\circ}$ at 12 K. A clear anisotropy is observed in the
picosecond-scale transient optical response, with a peak at $0^{\circ}$ and a
flat step at $90^{\circ}$. (e) Polar plot of static reflectivity versus
$\theta$, showing approximate isotropy. (f) Polar plot of $\Delta R/R$
integrated from 0 to 2 ps [gray shaded region in (c)] versus $\theta$, showing
pronounced anisotropy.

Nematic order spontaneously breaks rotational symmetry while preserving
translational symmetry. Nematicity in unconventional superconductors is far
from rare, occurring in both the iron-based superconductors and the cuprates
[5, 6, 7]. A nematic transition in these systems, however, is usually
accompanied by a structural or magnetic transition. Such a transition is
absent in Sr2RuO4. In a tetragonal crystal, nematicity results in a lowering
of symmetry to an orthorhombic unit cell where the $a$ and $b$ directions
become nonequivalent. The spontaneous nature of the symmetry breaking
typically leads to the formation of opposing domains related by 90∘ rotations,
whose average over long length and time scales retains the parent tetragonal
symmetry. Thus, directly detecting electronic nematic symmetry breaking with
macroscopic probes generally requires stabilizing the formation of one type of
domain over the other, akin to structural detwinning. Uniaxial strain is an
ideal external conjugate field by which one can achieve this effect. In the
iron-based superconductors, for example, uniaxial strain has been used
successfully to study nematic order with optical, transport, and photoemission
experiments [8, 9, 10, 11, 12]. Here, we take advantage of uniaxial epitaxial
strain to enable macroscopic transient optical reflectivity measurements of
electronic nematic order in Sr2RuO4.

We study high-quality thin films of (001)-oriented Sr2RuO4 grown by molecular-
beam epitaxy, as described in Ref. 13. The films are coherently strained by
growing them on (110) NdGaO3 single-crystal substrates [14], guaranteeing a
clean, uniform strain field over the entire sample area. Electrical transport
measurements confirm the films are superconducting, with a critical
temperature of $T_{c}\approx 1.4$ K (see Supplemental Materials [15]). NdGaO3
induces a small uniaxial ($B_{1g}$) strain of $a/b-1\approx 0.3\%$ at low
temperatures, where $a$ and $b$ are the in-plane lattice constants of the
coherently strained Sr2RuO4 film. Specifically, $a$ and $b$ correspond to the
[1$\bar{1}$0] and [001] spacings, respectively, of the (110) NdGaO3 substrate
in the $Pbnm$ setting [14], as illustrated in Fig. 1(a). At room temperature,
the uniaxial strain is reduced to $\sim$0.2$\%$. Strain has been used
extensively in recent years to study the Lifshitz transition in Sr2RuO4 and to
enhance the superconducting critical temperature [16, 17, 18, 22, 19, 20, 21].
We emphasize, however, that the magnitude of uniaxial strain that we employ is
not sufficient to induce a Lifshitz transition (estimated in Ref. 22 to be
$\varepsilon_{\mathrm{xx}}-\varepsilon_{\mathrm{yy}}\approx 0.7$%), and
moreover the epitaxial strain is biaxially compressive ($-0.1$% in the $a$
direction and $-0.4$% in the $b$ direction at low temperatures) relative to
bulk Sr2RuO4 [23, 24], which is known to move the Fermi level away from the
van Hove singularity rather than towards it [17]. Thus, the effects that we
measure are due to a relatively weak uniaxial perturbation of the electronic
structure and are not expected to be driven by density of states enhancements
or changes in band topology.

To determine whether signatures of electronic nematic order are evident in the
optical response of Sr2RuO4 at our probe wavelength of 800 nm, we first
examine the static reflectivity anisotropy, which we define as the ratio of
optical reflectivity with electric field polarized along the $a$-axis to that
along the $b$-axis of the film. Figure 1(b) shows the measured reflectivity
anisotropy as a function of temperature, where a striking dichroism is
apparent. We find a decrease in reflectivity along the (long) $a$-axis
relative to the (short) $b$-axis, which becomes more pronounced at lower
temperatures. At the lowest temperatures, we measure a maximum reflectivity
anisotropy of $1.7\%$, more than five times larger than the uniaxial strain
imposed by the substrate ($0.3\%$). Similar behavior has been observed in
detwinned FeSe, where the optical reflectivity anisotropy was taken as a proxy
for the nematic order parameter [12]. To eliminate the possibility that the
dichroism is merely due to structural orthorhombicity, we include in the
figure the temperature-dependent lattice constant ratio $a/b$ for films
strained to NdGaO3. Under the hypothesis that the reflectivity anisotropy is
entirely due to simple $a\neq b$ structural effects, we would expect a linear
proportionality with the substrate orthorhombicity as it changes with
temperature (see Supplemental Materials [15]). This is not what we observe.
Instead, the dichroism grows by an order of magnitude upon cooling from room
temperature to low temperature, while the substrate orthorhombicity only
increases by $30\%$. As a control, we also measure the reflectivity anisotropy
of a Sr2RuO4 thin film grown on (001) (LaAlO3)0.29(SrTa1/2Al1/2O3)0.71 (LSAT),
which is tetragonal and induces no uniaxial strain ($a=b$) [25]. In this case,
we detect no appreciable dichroism. Our static optical anisotropy measurements
suggest that the electronic structure of Sr2RuO4 possesses either spontaneous
nematic order (scenario I) or a large nematic susceptibility (scenario II). In
scenario I, an electronic instability results in the formation of microscopic
nematic domains fluctuating in space and time, with one orientation favored by
the uniaxial strain over the other. This imbalance induces a net macroscopic
anisotropy in the optical response of the electrons. In scenario II, the weak
uniaxial strain drives a strong electronic nematic response via a large
nematic susceptibility. Without the uniaxial strain, however, the electronic
structure would retain tetragonal symmetry even at low temperatures. Both
scenarios are consistent with an absence of dichroism in films grown on LSAT.

Figure 2:  Temperature dependence of transient reflectivity measurements. (a)
$\Delta R/R$ measurements for selected temperatures between 12 K and 300 K for
polarization along the $a$-axis (upper left) and the $b$-axis (upper right),
as well as their difference (lower left) and their sum (lower right). (b)
Temperature dependence of the average value of $\Delta(R_{a}+R_{b})/R$ between
1.5 and 2 ps, representing the isotropic component of the dynamical response
of the material. The response monotonically decreases with increasing
temperature. The gray line is a fit to the sample temperature change, as
described in the text. (c) Temperature dependence of the average value of
$\Delta(R_{a}-R_{b})/R$ between 0 and 2 ps, representing the nematic component
of the dynamical response of the material. An unusual non-monotonic response
is observed. The gray line is a guide to the eye. (d) Two-dimensional
depiction of the joint time delay and temperature dependence of
$\Delta(R_{a}-R_{b})/R$. (e) Corresponding depiction generated by fitting to
the Ising model of nematic order discussed in the text.

Now that signatures of electronic nematic order in the optical dichroism have
been established, we next turn to measuring the ultrafast response of the
order. As we will show, this reveals much larger signatures of nematicity than
the static reflectivity and enables the extraction of dynamical information
about the order. We use a standard pump-probe transient reflectivity technique
to measure changes in the dichroism induced by an ultrafast ($\sim$50 fs) pump
pulse of wavelength 760 nm and fluence $\sim$25 $\mu$J/cm2. To eliminate any
possible anisotropy due to the polarization of the pump, we keep the pump and
probe pulses fixed in a cross-polarized configuration and rotate the sample
about its surface normal, as illustrated in Fig. 1(c). Like the static
reflectivity, we uncover a pronounced anisotropy of the transient optical
response. Figure 1(d) shows raw transients of the relative change in
reflectivity $\Delta R/R$ versus pump-probe time delay at 12 K for several
angles between the $a$-axis (0∘) and the $b$-axis (90∘). A clear breaking of
tetragonal symmetry is apparent: for optical polarization parallel to the
$a$-axis, we observe a sharp peak in $\Delta R/R$ that decays over a $\sim$1
ps timescale, whereas for polarization parallel to the $b$-axis, the peak
vanishes and only a flat step remains. This nematic ($C_{4}\rightarrow C_{2}$)
rotational symmetry breaking can be seen more clearly in a polar plot of the
area under the transient reflectivity curve between 0 and 2 ps, as shown in
Fig. 1(f). Here, no symmetrization has been performed; independent
reflectivity transients were collected over a full 360∘ angular range. While
the static reflectivity shows a modest 1.7% anisotropy, the transient
reflectivity exhibits a giant anisotropy exceeding 100%. Ultrafast optical
anisotropies of this magnitude are routinely observed in the nematic phase of
iron-based superconductors [26, 27, 28, 29, 30]. Sub-picosecond timescales are
dominated by low-energy electronic degrees of freedom, affirming that the
dichroism we observe is a result of electronic nematic order rather than
structural orthorhombicity.

A saturation at low temperatures is evident in the static dichroism shown in
Fig. 1(b). This behavior resembles an Ising-like order parameter. To
quantitatively analyze the data, we therefore develop a heuristic mean-field
Ising model of the nematic order in Sr2RuO4. This model can describe both
scenarios discussed above. Within the model, we define an order parameter
$\phi$ that takes the values $+1$ or $-1$, corresponding to a nematic director
oriented along the $a$\- or $b$-axis, respectively. The Hamiltonian is given
by

${\mathcal{H}=-\frac{1}{2}\sum_{i,j}U(\textbf{r}_{i}-\textbf{r}_{j})\phi_{i}\phi_{j}-F\sum_{i}\phi_{i},}$

where $i$ and $j$ label unit cells, $U(\textbf{r})$ is the interaction energy,
related to the stiffness of the nematic order and the size of domains, and $F$
is the uniaxial strain field, which favors $+1$ domains over $-1$ domains.
Within mean-field theory, the average order parameter
$\left\langle\phi\right\rangle$ satisfies the self-consistency equation

${\left\langle\phi\right\rangle=\tanh\left(\frac{U\left\langle\phi\right\rangle+F}{k_{B}T}\right),}$

where $U=\sum_{i}U(\textbf{r}_{i})/2=k_{B}T_{c}$ defines the critical
temperature of the nematic phase transition in the absence of the external
strain field $F$. Henceforth, we drop the angle brackets and call the Ising
nematic order parameter $\phi$. By assuming the measured optical anisotropy is
proportional to the order parameter ($R_{a}/R_{b}-1\propto\phi$), we can fit
the data to this functional form. Such a least-squares fit, included in Fig.
1(b), yields the fitting parameters $U=53\pm 8$ K and $F=53\pm 7$ K.
Reasonably good agreement with the data is achieved despite the relative
simplicity of our model. Taking the model at face value, we predict a critical
temperature of $T_{c}\approx 50$ K, where electronic nematic order would
spontaneously condense following scenario I described above. In the measured
sample, however, this transition occurs gradually over a wide temperature
range due to the large magnitude of the symmetry-breaking uniaxial strain
field (i.e., $F\approx U$). Scenario II, where nematic order does not occur
without strain, would require $U\leq 0$, which is not consistent with our
data. Based on the success of this simple Ising nematic model, we will
continue to use it to extract quantitative information about our data.

Turning now to the temperature dependence of the nematicity, Fig. 2(a) shows
transient reflectivity measurements at selected temperatures between 12 K and
300 K for polarization along the $a$-axis ($R_{a}$) and $b$-axis ($R_{b}$). To
isolate temperature-dependent changes in the anisotropy from other effects, we
also show the difference in $\Delta R/R$ between the $a$\- and $b$-axis
($R_{a}-R_{b}$) as well as the sum of the two ($R_{a}+R_{b}$). From this
process, we see that the sharp peak in $\Delta R/R$ is entirely due to the
nematicity of the material, while the flat step after time-zero represents the
isotropic component of the transient optical response. Moreover, while the
isotropic response monotonically decreases with increasing temperature, as
shown in Fig. 2(b), we uncover an unusual non-monotonic temperature dependence
of the nematic component of the response, displayed in Fig. 2(c). The
isotropic transient response is likely caused by an increase in temperature
${\Delta T}$ after absorption of the pump. Under the assumption that each pump
pulse deposits a fixed amount of energy $\Delta E$ into the electronic
subsystem (with heat capacity $C=\gamma T$), one can show that ${\Delta
T}_{\mathrm{max}}(T)=\sqrt{T^{2}+(2\Delta E/\gamma)}-T$. Using the value of
$\gamma$ from Ref. 31 and $\Delta E$ calculated from our pump fluence, we find
$(2\Delta E/\gamma)\approx(80~{}\textrm{K})^{2}$ (see Supplemental Materials
[15]). A function of the form $g(T)=\alpha+\beta{\Delta T}_{\mathrm{max}}(T)$
with only two free parameters, $\alpha$ and $\beta$, fits the isotropic
component extremely well, as shown by the gray line in Fig. 2(b). We therefore
use this fact to scale the data at each temperature to match $g(T)$ and
correct for laser fluctuations during the duration of the experiment, where,
for example, the two data points near $\sim$35 K are otherwise anomalously
low. This normalization procedure is used for the remainder of our data
analysis, including Fig. 2(c).

Figure 3:  Ising model fit of optical anisotropy. (a) Measured
$\Delta(R_{a}-R_{b})/R$ versus temperature for selected time delays after the
pump pulse. Curves are offset by $10^{-4}$ for clarity. Gray lines show the
global fit to the Ising model of electronic nematic order, as described in the
text. (b) Confidence regions for the Ising parameters $U$ and $F$ computed
through an $F$-test. $U>0$ with greater than 99% confidence, implying the
existence of spontaneous nematic order. The inset shows the order parameter
$\phi(T)$ together with the critical temperature $T_{c}$ for spontaneous
nematic order when $F=0$. (c) Electronic temperature increase versus time
delay as a fraction of the calculated maximum increase. Also shown is a fit to
the solution of the heat equation $df/dt=k(f_{0}-f)$ (purple curve) with
background electronic temperature $f_{0}$ (gray curve). (d) Relative
suppression of the nematic order versus time delay, showing a coherent
oscillation of the order parameter amplitude. Also shown is a fit to a damped
harmonic oscillator (purple curve) with a pump-induced displacement of the
equilibrium amplitude (gray curve).

Figure 1(b) shows that the onset of nematic order reduces the optical
reflectivity along the $a$-axis. After the pump pulse, this reflectivity
increases. It follows that the effect of the pump is to suppress the nematic
order, likely through a transient increase in temperature. To model this, we
assume that the dichroism is proportional to the nematic order parameter
$\phi$. The measured change in dichroism is therefore proportional to the
change in $\phi$:

${\Delta(R_{a}-R_{b})/R\propto(1+\delta)\phi(T+\Delta T)-\phi(T),}$

where $\phi(T)$ is the equilibrium value of the order parameter at the
temperature $T$ of the measurement, $\Delta T$ is the increase in electronic
temperature after absorption of the pump energy, and $\delta<0$ represents a
possible non-thermal suppression of the order parameter amplitude by the pump
pulse. The previously developed Ising model provides a concrete expression for
$\phi(T)$. Due to equilibration, the temperature increase $\Delta T$ is not
necessarily instantaneous. Instead, we allow the temperature of the electrons
participating in the nematic order to rise to some time-dependent fraction
$0\leq f(t)\leq 1$ of ${\Delta T}_{\mathrm{max}}$ calculated from the pump
fluence as discussed above [$\Delta T(t)=f(t){\Delta T}_{\mathrm{max}}$]. In
doing so, we assume a temperature-independent electron equilibration rate.
With this model, we fit the transient optical data at all time delays and all
temperatures simultaneously. The fit includes three global parameters ($U$,
$F$, and a fixed scaling factor) and two time-dependent parameters
[$\delta(t)$ and $f(t)$]. The results of the fit, shown in Fig. 3(a), are
excellent ($R^{2}=0.994$).

A number of salient features of the data are uncovered by the fitting
analysis: (1) The unusual non-monotonic temperature dependence of the
dichroism [Fig. 2(c)] is a result of the flattening of the nematic order
parameter at low temperatures ($d\phi/dT\rightarrow 0$ as $T\rightarrow 0$),
which reduces the effect of pump heating. (2) Uncertainties in the fit
parameters $U$ and $F$ are correlated, which is a result of their coupled
influence on the functional form of $\phi(T)$. The $F$-test-derived confidence
regions displayed in Fig. 3(c), however, show that $U>0$ with high confidence.
Within the applicability of our Ising model, this implies that the nematic
order we observe is spontaneous (scenario I). (3) While we extract a value of
$F$ that is similar to that of the static reflectivity measurement (46 K
versus 53 K), the value of $U$ in the transient optical measurements is
substantially smaller (23 K versus 53 K). We conjecture that repeatedly
exciting the sample with high-fluence pump and probe pulses does not allow the
sample to completely return to equilibrium, causing a net suppression of the
order parameter. This subsequently results in a renormalization of the average
interaction energy $U$. (4) Figure 3(c) shows that the nematic electron
temperature rises to $\sim$50% of the maximum calculated increase ${\Delta
T}_{\mathrm{max}}$ over a timescale of $\sim$0.5 ps. This suggests that the
energy of the pump is first absorbed by background electrons during the
lifetime of the pulse ($\sim$50 fs) and then is quickly transferred to the
nematic electrons via thermal equilibration mediated by electron–electron
interactions. At a much slower rate dictated by electron–phonon interactions,
thermal equilibration with the lattice commences, causing a gradual cooling.
To make this picture more quantitative, we fit $f(t)$ to the solution of the
heat equation $df/dt=k(f_{0}-f)$, where $f_{0}(t)=e^{-t/\tau}$ represents the
background electronic temperature, with lattice equilibration timescale
$\tau$. We find excellent agreement with the data for $\tau=0.82$ ps. (5) The
non-thermal suppression of the order parameter amplitude after the pump pulse
[Fig. 3(d)] reveals a coherent oscillation. This behavior can be understood
within the context of a simple damped harmonic oscillator model:
$d^{2}\delta/dt^{2}+2\gamma d\delta/dt+\omega_{0}^{2}(\delta-\delta_{0})=0$,
where $\gamma$ is the damping rate, $\omega_{0}$ is the natural frequency of
oscillation, and $\delta_{0}$ is the pump-induced displacement of the
equilibrium order parameter amplitude. We find that a solution to this
equation fits the data well if $\delta_{0}(t)=\alpha e^{-t/\tau}+\beta$,
representing a sharp impulsive suppression at $t=0$ followed by a rapid
recovery ($\tau=0.09$ ps) to a displaced equilibrium amplitude $\beta=-5.0$%
relative to the initial value, as shown by the gray curve in Fig. 3(d). We
determine a frequency of $\omega_{0}/2\pi=0.89$ THz for the amplitude mode of
the nematic order.

Figure 4:  Microscopic model of nematic order in Sr2RuO4. (a) Unit cell of the
RuO2 plane showing the three-orbital basis (Ru $d_{xy}$ and O $p_{x}$ and
$p_{y}$) of the Emery model. Hopping integrals (solid lines) and interaction
parameters (dashed lines) are defined following Ref. 34. (b) Resulting tight-
binding band structure, with parameters taken from Ref. 35. (c)
Interaction–filling phase diagram showing the critical value of $V_{pp}$
necessary to produce a nematic ground state ($\eta\neq 0$). (d)
Temperature–filling phase diagram for different values of $V_{pp}$. Panels (c)
and (d) were calculated for $U_{p}=4.4$ eV [35].

The foregoing optical experiments suggest that Sr2RuO4 supports a high-
temperature Ising-like nematic order parameter with an order-disorder phase
transition below $\sim$50 K. To put forth a possible microscopic origin for
this order, we turn to the Emery model, first used to describe hole doping of
CuO2 planes in cuprate superconductors away from half filling [32]. Here, we
use the model to study intra-unit-cell nematic order in RuO2 planes with
fillings adjacent to the van Hove singularity of the $\gamma$ band [33]. Our
mean-field analysis closely follows Ref. 34 (see Supplemental Materials [15]).
Figure 4(a) shows the hopping and interaction parameters defined within the
model, and Fig. 4(b) shows the resulting band structure. Of central importance
are the Coulomb repulsion between neighboring oxygen $p$-orbitals ($V_{pp}$),
which favors a nematic charge order $\eta=n_{x}^{p}-n_{y}^{p}$ with different
electron densities on the $p_{x}$ and $p_{y}$ orbitals, and the van Hove
singularity of the $\gamma$ band at the $X$ point, which leads to a diverging
density of states that supports electronic instabilities by reducing the
energy cost to redistribute orbital density. Figure 4(c) shows the mean-field
ground state phase diagram as a function of $V_{pp}$ and filling. A nematic
phase is stabilized for sufficiently large values of $V_{pp}$ that depend on
the proximity to the van Hove filling. Bulk Sr2RuO4 has
$n-n_{\mathrm{vH}}\approx-0.2$ [33], which requires an interaction strength
$V_{pp}>1.7$ eV to stabilize nematic order. The full temperature–filling phase
diagram is displayed in Fig. 4(d), where it is seen that such order, if it
exists, condenses well above room temperature. The model shows that Sr2RuO4 is
exceptionally close to a nematic instability, and offers a plausible
microscopic picture of an Ising-like ($\eta=\pm\eta_{0}$) nematic charge
order. We conjecture that at high temperatures this nematicity is spatially
disordered at the nanoscale, but shows an order-disorder phase transition to a
globally nematic state at lower temperatures.

In conclusion, we have presented static and ultrafast time-resolved optical
dichroism measurements of epitaxially strained thin films of Sr2RuO4 that
strongly support the existence of electronic nematic order. By fitting our
data to a simple Ising model, we conjecture that this order emerges
spontaneously at low temperatures through an order-disorder transition, rather
than through a large nematic susceptibility driven by strain. These results
corroborate the angle-resolved transverse resistivity measurements that first
uncovered electronic nematicity in Sr2RuO4 [4]. Electrical transport and
optical reflectivity are disparate probes operating at different energy
scales. In both cases, however, clear signatures of two-fold electronic
anisotropy are evident. Furthermore, while transport can be sensitive to non-
local effects such as percolation and boundary scattering, especially in a
material with ostensible microscopic nematic domains, optical reflectivity is
a local, bulk-sensitive probe. This difference may explain the minor
discrepancies between experiments, such as the absence of nematic order in
tetragonal Sr2RuO4/LSAT and the alignment of the nematic director with respect
to the crystallographic directions. Our findings also offer a new perspective
on the Fermi liquid crossover at $T_{\mathrm{FL}}\sim 40$ K observed by Hall
transport [36, 37, 38], optical spectroscopy [39], and nuclear magnetic
resonance [40, 41]. We speculate that the crossover is directly related to
nematicity, which appears to develop at the same temperature. Indeed, such a
Fermi liquid crossover has been theoretically shown to emerge in the vicinity
of an Ising nematic quantum critical point [42, 43, 44]. Nematic order also
offers an explanation for the checkerboard charge order observed at the
surface of Sr2RuO4 by scanning tunneling microscopy [45]. Rather than surface
nematic order emerging as a secondary effect, we find it likely that the
checkerboard charge order is the secondary response, emerging from a
combination of bulk nematicity and surface octahedral rotations. An important
open question is to what extent electronic nematic order in the normal state
of Sr2RuO4 influences the likelihood of different forms of superconductivity
at lower temperatures. Future experimental and theoretical work is necessary
to address this principal question.

## Acknowledgments

We thank Dr. Ludi Miao for performing electrical resistivity measurements.
This work was primarily supported by the U.S. Department of Energy, Office of
Basic Energy Sciences under Award No. DE-SC0019414. Thin film synthesis was
funded in part by the Gordon and Betty Moore Foundation’s EPiQS Initiative
through Grant Nos. GBMF3850 and GBMF9073 to Cornell University. Substrate
preparation was, in part, facilitated by the Cornell NanoScale Facility, a
member of the National Nanotechnology Coordinated Infrastructure (NNCI), which
is supported by the National Science Foundation (Grant No. NNCI-2025233).
Additional support was provided by AFOSR FA9550-21-1-0168.

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