# Probing momentum-dependent scattering in uniaxially stressed Sr2RuO4 through
the Hall effect

Po-Ya Yang  Max Planck Institute for Chemical Physics of Solids, Nöthnitzer
Str 40, 01187 Dresden, Germany  Hilary M. L. Noad  Max Planck Institute for
Chemical Physics of Solids, Nöthnitzer Str 40, 01187 Dresden, Germany  Mark E.
Barber  Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Str
40, 01187 Dresden, Germany  Department of Applied Physics, Stanford
University, Stanford, CA 94305, USA  Geballe Laboratory for Advanced
Materials, Stanford, CA 94305, USA  Naoki Kikugawa  National Institute for
Materials Science, Tsukuba 305-0003, Japan  Dmitry Sokolov  Max Planck
Institute for Chemical Physics of Solids, Nöthnitzer Str 40, 01187 Dresden,
Germany  Andrew P. Mackenzie  Max Planck Institute for Chemical Physics of
Solids, Nöthnitzer Str 40, 01187 Dresden, Germany  Scottish Universities
Physics Alliance, School of Physics and Astronomy, University of St. Andrews,
St. Andrews KY16 9SS, U.K.  Clifford W. Hicks  School of Physics and
Astronomy, University of Birmingham, Birmingham B15 2TT, U.K.  Max Planck
Institute for Chemical Physics of Solids, Nöthnitzer Str 40, 01187 Dresden,
Germany

###### Abstract

Under in-plane uniaxial stress, the largest Fermi surface sheet of the
correlated metal Sr2RuO4 undergoes a Lifshitz transition from an electron-like
to an open geometry. We investigate the effects of this transition on
transport through measurement of the longitudinal resistivity $\rho_{xx}$ and
the Hall coefficient $R_{\text{H}}$. At temperatures where scattering is
dominated by electron-electron scattering, $R_{\text{H}}$ becomes more
negative across the Lifshitz transition, opposite to expectations from the
change in Fermi surface topology. We show that this change in $R_{\text{H}}$
is explainable only if scattering changes throughout the Brillouin zone, not
just at the point in $k$-space where the Lifshitz transition occurs. In a
model of orbital-dependent scattering, the electron-electron scattering rate
on sections of Fermi surface with $xy$ orbital weight decreases dramatically.
On the other hand, at temperatures where defect scattering dominates
$\rho_{xx}$ and $R_{\text{H}}$ are essentially constant across the Lifshitz
transition.

Introduction. The Hall coefficient $R_{\text{H}}$ of multi-band metals is a
challenging quantity to analyze. In the low-field limit, it is determined by
an integral of the mean free path around the Fermi surfaces [1]. It can
therefore be used to probe the momentum dependence of scattering in metals
with simple Fermi surfaces [2, 3, 4]. However, as complexity increases models
become badly underconstrained. The case of the correlated metal and
superconductor Sr2RuO4 [5] highlights the challenge. With 1% substitution of
La onto the Sr site, the low-temperature Hall coefficient changes from
electron-like to hole-like, even though the Fermi surfaces are almost
unchanged [6, 7].

The advent of strain tuning might make $R_{\text{H}}$ a more broadly useful
measurement quantity. By correlating changes in $R_{\text{H}}$ with specific
strain-driven changes in electronic structure, models of scattering can be
tested with greater rigor. Crucially, if the deformation is elastic the defect
landscape is not altered.

Here, we study Sr2RuO4, which has become an important test case for
understanding of correlated electron materials [8, 9, 10]. It is a highly two-
dimensional metal, in which correlations renormalize but do not destroy Landau
quasiparticles [11, 12]. Under uniaxial stress $\sigma$ applied along the
$[100]$ lattice direction, the largest Fermi surface sheet [the $\gamma$
sheet— see Fig. 1(a)] undergoes a Lifshitz transition from an electron-like to
an open geometry [13]. This transition is illustrated in Fig. 1(b). It has
been reported to occur at $\sigma=\sigma_{\text{L}}=-0.71\pm 0.08$ GPa [14],
where negative values denote compression. Under the common assumption that the
Hall effect is determined by the topology of the Fermi surfaces, this
transition should make $R_{\text{H}}$ more positive. If scattering time rather
than mean free path is isotropic, the quantitative change in $R_{\text{H}}$
might be small, but is still expected to be positive [15, 16].

Figure 1: (a) Model Fermi surfaces of Sr2RuO4, coloured by orbital weight,
from Ref. [17]. (b) Fermi surfaces calculated with a simple tight-binding
model, presented later in this Letter, under uniaxial stresses along $[100]$
of $0$, $-0.75$, and $-1.1$ GPa. (c,d) Photographs of samples 1 and 2.

We report here that in fact the Lifshitz transition makes $R_{\text{H}}$ more
negative, opposite to the expectation from the change in Fermi surface
topology. Furthermore, the magnitude of the change implies changes in
scattering throughout the Brillouin zone: going through the Lifshitz
transition appears to make Sr2RuO4 a considerably less-strongly correlated
metal. Sr2RuO4 therefore provides an important example of how electronic
correlations emerge in metals, and how correlations may potentially be
“switched on and off.”

Methods. Measurements were performed using a piezoelectric-driven uniaxial
stress device similar to that described in Ref. [18], that incorporates
sensors of both the applied displacement and applied force. As in Ref. [19],
samples were mounted onto carriers, which were then mounted onto the cell.
These carriers comprise two parts, and the sample is compressed when they are
brought together. This mechanism allows in situ calibration of the zero-force
reading of the force sensor.

Two samples, photographed in Figs. 1(c,d), were studied. Sample 1 is a beam of
constant cross-section with electrical contacts attached by hand, while for
sample 2 a plasma focused ion beam was used to mill contacts from the sample
itself. The residual resistivities are 0.4 and 0.09 $\mu\Omega$-cm,
respectively. Due to its lower residual resistivity, most of the data
presented here are from sample 2, though some data from sample 1 are included
to demonstrate reproducibility. For both, the Hall voltage $V_{\text{H}}$ at
field $B$ is taken as $\frac{1}{2}[V_{13}(B)-V_{13}(-B)]$ or
$\frac{1}{2}[V_{24}(B)-V_{24}(-B)]$, where $V_{ij}\equiv V_{i}-V_{j}$, $V_{i}$
is the voltage in contact $i$, and contact numbering is shown in Figs. 1(c,d).
This procedure cancels contributions to $V_{H}$ from contact misalignment. The
Hall coefficient $R_{\text{H}}$ is then obtained as
$R_{\text{H}}=V_{\text{H}}t/(IB)$, where $I$ is the applied current and $t$
the sample thickness.

Figure 2: (a) $[\rho_{xx}(B=0)/\Delta\rho_{xx}]^{0.5}$, where
$\Delta\rho_{xx}$ is the change in $\rho_{xx}$ between 0 and 1.5 T, against
$T^{2}$ for sample 2 at various stresses. The dashed lines are linear fits.
(b) $\rho_{xx}(T)$ of sample 2 at $\sigma=\sigma_{\text{L}}=-0.75$ GPa. The
black line is an average of data at $+1.5$ and $-1.5$ T, with the
magnetoresistance subtracted as described in the text. (c) The difference
between various fitting models and the black line in panel (b).

Longitudinal resistivity. We look first at the temperature dependence of the
longitudinal resistivity $\rho_{xx}$ at the Lifshitz transition, testing our
data against three hypothesized temperature dependences. (1) $\rho_{xx}\propto
T^{2}\log T$. Boltzmann transport theory applied to a Fermi liquid tuned to an
electron-to-hole Lifshitz transition yields a $T^{2}\log T$ form [20, 21]. To
obtain this result, it is necessary to take into account the fact that, in the
absence of impurity scattering, only umklapp scattering generates resistivity.
Ref. [22] finds that $\rho_{xx}\propto T^{2}\log T$ also for an electron-to-
open transition, as occurs here, but only if there are other bands present.
(2) $\rho_{xx}\propto T^{3/2}$. The non-umklapp scattering rate, which affects
e.g. the thermal but not the electrical resistivity, is expected to scale as
$T^{3/2}$ [22]. Also, $\rho_{xx}\propto T^{3/2}$ is expected for extended
saddle points [20]. (3) $\rho_{xx}\propto T^{5/3}$. This form has been
observed near ferromagnetic quantum critical points [23, 24, 25]. It is
understood theoretically as a result of loss of quasiparticle coherence due to
the quantum criticality [24, 25, 26].

Figure 3: (a) $V_{\text{H}}t/I$, where $t$ is sample thickness and $I$ the
applied current, versus $B$ for sample 2 at various temperatures. The dotted
lines are extrapolations of linear fits over $0<B<0.35$ T. Equivalent data for
Sample 1 are shown in [27]. (b) $R_{H}(T)$ of both samples at zero stress, and
$B$ up to $\pm 0.5$ T for sample 1 and $\pm 0.35$ T for sample 2. Data from
Ref. [28] are also shown. Figure 4: (a) $R_{\text{H}}(\sigma)$ of sample 2 at
various temperatures, measured at $B=\pm 0.35$ T. Data that are clearly
affected by superconductivity are excluded. The expected
$R_{\text{H}}(\sigma)$ for isotropic mean free path and $T\rightarrow 0$,
calculated using a simple tight-binding model, is also shown. (b)
$R_{\text{H}}(\sigma)$ of sample 2 at $B=\pm 1.5$ T, a strong enough field to
fully suppress the superconductivity. (c) $\rho_{xx}(\sigma)$ of sample 2 at
$B=0$ and $0.35$ T. (d) $\rho_{xx}(\sigma)$ of sample 2 at $B=1.5$ T. (e)
$R_{\text{H}}(\sigma)$ of sample 1 at $B=\pm 1$ T. (f) The phase boundary of
the magnetic order, taken as the stresses where $d^{2}R_{\text{H}}/dT^{2}$ in
panel (c) is maximum. The error bars are the FWHM of the peaks in
$d^{2}R_{\text{H}}/dT^{2}$. Also shown is the phase boundary found in
elastocaloric effect data [29], with the Lifshitz stress normalized to $-0.75$
GPa.

The superconductivity of Sr2RuO4 is strongly enhanced at the Lifshitz
transition [30]. To measure $\rho_{xx}$ at low temperatures, we suppress it
with a 1.5 T field applied along the $c$ axis. We apply a magnetoresistance
model to subtract an estimate for $\Delta\rho_{xx}$, the change in $\rho_{xx}$
under the 1.5 T field, yielding an estimate for the zero-field resistivity
that would be obtained without superconductivity. For isotropic metals in weak
magnetic fields,

$\Delta\rho_{xx}\propto\omega_{\text{c}}^{2}\tau^{2}\times\rho_{xx}(B=0),$ (1)

where the cyclotron frequency $\omega_{\text{c}}=eB/m^{*}$ and $\tau$ is the
relaxation time. $\tau^{-1}=\alpha+\beta T^{2}$ is expected in the Fermi
liquid regime, and in unstressed Sr2RuO4 it has been shown that
$(\rho_{xx}/\Delta\rho_{xx})^{1/2}$ can be fitted by $\alpha+\beta T^{2}$ up
to at least 80 K [31], even though $\rho_{xx}$ follows a $T^{2}$ form only up
to $\sim 30$ K [32, 31]. In Fig. 2(a) it is shown that a model
$(\rho_{xx}/\Delta\rho_{xx})^{1/2}=\alpha+\beta T^{2}$ fits the observed
magnetoresistance at each stress that we tested. We use this model to
extrapolate $\Delta\rho_{xx}$ into the superconducting regime where it cannot
be directly measured.

Fig. 2(b) shows $\rho_{xx}(T)$ and $\rho_{xx}(T)-\Delta\rho_{xx}(T)$ of sample
2 at $\sigma=\sigma_{L}$. Fig. 2(c) shows the differences between
$\rho_{xx}-\Delta\rho_{xx}$ and the three hypothesised fitting models.
$\rho_{xx}=\rho_{0}+AT^{3/2}$ clearly does not fit the data well, while
$\rho_{xx}=\rho_{0}+AT^{5/3}$ and $\rho_{xx}=\rho_{0}+AT^{2}\log(T_{0}/T)$
work equally well. For the latter, we obtain $T_{0}=95$ K. In Ref. [27], we
show that an alternative magnetoresistance model,
$(\rho_{xx}/\Delta\rho_{xx})^{1/2}=\alpha+\beta T^{3/2}$, can also be applied
at $\sigma=\sigma_{\text{L}}$, and does not alter the conclusion that
$\rho_{xx}=\rho_{0}+AT^{2}\log(T_{0}/T)$ is a better fit to the data than
$\rho_{xx}=\rho_{0}+AT^{3/2}$. In contrast, in earlier work it was found that
the $T^{2}\log T$ and $T^{3/2}$ forms worked equally well at
$\sigma=\sigma_{\text{L}}$, over a fitting range of 4–40 K [33]. We
hypothesize that this temperature range was too high to accurately capture the
low-temperature behavior.

Although the $T^{5/3}$ and $T^{2}\log T$ models work equally well, the
hypothesis under which $T^{5/3}$ is obtained theoretically, ferromagnetic
quantum criticality, appears not to apply here. NMR data on Sr2RuO4 show that
the Stoner factor of Sr2RuO4 is enhanced by $\sim$30% at the Lifshitz
transition [34], and that there is no dramatic change in quasiparticle weight
[35]. Tuning to the Lifshitz transition strengthens ferromagnetic
fluctuations, but apparently not to point that they become critical.

Hall effect. We now look at the Hall effect. To investigate appropriate
measurement fields, in Fig. 3(a) we show $V_{\text{H}}$ versus $B$ for sample
2. For temperatures below $\sim 10$ K, the low-field regime in which
$V_{\text{H}}\propto B$ extends only up to $\sim 0.3$ T. Fig. 3(b) shows
$R_{\text{H}}(T)$ of both samples in the low-field limit. $R_{\text{H}}$ is
hole-like at 50 K, and electron-like below $\sim 20$ K, as reported previously
[36, 28].

Figs. 4(a,c) show $R_{\text{H}}(\sigma)$ and $\rho_{xx}(\sigma)$ of sample 2
at various temperatures and at $\pm 0.35$ T, which is close to the low-field
limit. Figs. 4(b,d) show the equivalent data at $\pm 1.5$ T, which fully
suppresses the superconductivity. Fig. 4(e) shows data on $R_{\text{H}}$ from
sample 1 at 1 T. We identify the Lifshitz stress $\sigma_{\text{L}}$ as the
peak in $\rho_{xx}$ at 5 K, yielding $\sigma_{\text{L}}=-0.75\pm 0.08$ GPa for
sample 2. For sample 1, $\rho_{xx}$ was not measured, but comparison of
$R_{\text{H}}(\sigma)$ with that of sample 2 yields
$\sigma_{\text{L}}=-0.82\pm 0.08$ GPa. For temperatures of $\sim 2$ K and
above, the data shown in Figs. 4(a, b, e) reveal that $R_{\text{H}}$ decreases
as samples are compressed through $\sigma=\sigma_{\text{L}}$, that is, moving
from right to left in the graphs. Below $\sim 5$ K, this decrease is somewhat
less steep for sample 1 than sample 2, possibly due to the stronger defect
scattering, but the difference is not large. Fig. 4(a) also shows the expected
$R_{\text{H}}(\sigma)$ for an isotropic mean free path, calculated using a
tight-binding model that will be presented, and the contrast with the data is
clear.

In the 1.5 T data in Figs. 4(b,d) it can be seen that below $\sim 1$ K, both
$R_{\text{H}}$ and $\rho_{xx}$ become essentially constant across the Lifshitz
transition. In other words, even in the $T\rightarrow 0$ limit, $R_{\text{H}}$
does not respond as expected to the change in Fermi surface topology.

Before discussing the Hall data further, we comment briefly on stress-induced
magnetic order. This order is probably an incommensurate spin density wave
[37], and has a clear effect on $R_{\text{H}}$ at 1.5 T. The stress range
where the magnetic order condenses is indicated in Fig. 4(b). However, at
$0.35$ T the magnetic order has no clear effect on $R_{\text{H}}(\sigma)$ [see
Fig. 4(a)], and at 1 T $R_{\text{H}}(\sigma)$ shows some response to the order
but less than at 1.5 T [see Fig. 4(e)]. This variation is not because the
magnetic phase boundary has strong field dependence: in Fig. 4(f), we show
that the magnetic phase boundary identified in the 1.5 T data matches well
that identified from elastocaloric effect measurements in zero field [29].
Rather, there must be a strong field dependence in the effect of the magnetic
phase on transport. We leave this as a topic for future investigation.

Discussion. We focus first on $R_{\text{H}}$ at 5–10 K, where the downturn in
$R_{\text{H}}$ at $\sigma<\sigma_{\text{L}}$ is large and quite sharp.
$\rho_{xx}$ peaks at $\sigma=\sigma_{\text{L}}$ due to enhanced electron-
electron scattering [21, 38], and the fact that both the downturn in
$R_{\text{H}}$ and the peak in $\rho_{xx}$ fade over the same temperature
range shows that the downturn in $R_{\text{H}}$ is also associated with
electron-electron scattering. In this section, we show that the magnitude of
the change in $R_{\text{H}}$ implies changes in scattering throughout the
Brillouin zone, and that a model of orbital differentiation can account for
the change.

Following Ref. [17], in this orbital differentiation model the band-dependent
electron-electron scattering rate $\eta_{\nu}(\mathbf{k})$, where
$\nu=\alpha$, $\beta$, or $\gamma$, is given by:

$\eta_{\nu}(\mathbf{k})=\sum_{m}\left|\left\langle\chi_{m}(\mathbf{k})|\psi_{\nu}(\mathbf{k})\right\rangle\right|^{2}\eta_{m}.$
(2)

$\alpha$, $\beta$, and $\gamma$ label the Fermi surfaces— see Fig. 1(a).
$\left|\left\langle\chi_{m}(\mathbf{k})|\psi_{\nu}(\mathbf{k})\right\rangle\right|^{2}$
is the weight of orbital $m$ in band $\nu$ at momentum $\mathbf{k}$, and
$\eta_{m}$ is an orbital-dependent scattering rate. Scattering is taken to be
a local property, so $\eta_{m}$ has no $\mathbf{k}$ dependence. This model is
supported by photoemission and Raman scattering data [39, 40], and applies
when Hund’s interactions control electronic correlations, as appears to be the
case in Sr2RuO4 [9, 10].

We calculate the orbital weights using the simplest possible tight-binding
model that reproduces the Fermi surface topology of Sr2RuO4 [41, 42, 43, 27].
Our model Fermi surfaces are shown in Fig. 1(b). Near-neighbour hopping
integrals are taken to vary linearly with strain, with a scaling constant that
is set so that the Lifshitz transition occurs at $\sigma_{\text{L}}=-0.75$
GPa. The Hall conductivity $\sigma_{xy}$ is calculated using the Ong
construction [1]:

$\sigma_{xy}=\frac{e^{3}}{2\pi^{2}\hbar^{2}}\bm{B}\cdot\int_{FS}\frac{d\mathbf{l}_{\text{mfp}}(\mathbf{k})\times\mathbf{l}_{\text{mfp}}(\mathbf{k})}{2},$
(3)

where $\mathbf{l}_{\text{mfp}}(\mathbf{k})$ is the mean free path at point
$\mathbf{k}$. Stated in words, Eq. 3 states that the Hall conductivity is
determined by the curvature of the sections of Fermi surface with longer mean
free path. To obtain $R_{\text{H}}$, the longitudinal conductivities
$\sigma_{xx}$ and $\sigma_{yy}$ are needed:
$R_{\text{H}}\approx(1/B)\sigma_{xy}/(\sigma_{xx}\sigma_{yy})$. $\sigma_{xx}$
is taken to be:

$\sigma_{xx}=\frac{e^{2}}{2\hbar\pi^{2}}\int_{FS}ds\>l_{\text{mfp}}(\mathbf{k})\left(\mathbf{l}_{\text{mfp}}\cdot\mathbf{\hat{x}}\right)^{2}.$
(4)

Eq. 4 neglects the distinction between umklapp and non-umklapp processes; it
is an approximate model. We also neglect impurity and phonon scattering, so
$l_{\text{mfp}}(\mathbf{k})=v_{\text{F}}(\mathbf{k})/\eta_{\nu}(\mathbf{k})$,
where $v_{\text{F}}(\mathbf{k})$ is the Fermi velocity.

Figure 5: (a) $R_{\text{H}}$ calculated within the orbital-dependent
scattering model described in the text, as a function of the ratio of $xy$ and
$xz$, $yz$ scattering rates, $\eta_{xy}/\eta_{xz,yz}$. The calculation employs
a simple tight-binding model of the Fermi surfaces of Sr2RuO4. (b)
$\eta_{xy}/\eta_{xz,yz}$ versus stress at 5 and 10 K, determined from
$R_{\text{H}}$ of sample 2 at $\pm 0.35$ T and the calculation results shown
in panel (a).

For simplicity, we assume $\eta_{xz}=\eta_{yz}=\eta_{xz,yz}$, even at nonzero
stress where this equality is not symmetry-protected. $R_{\text{H}}$
calculated as a function of the ratio $\eta_{xy}/\eta_{xz,yz}$ and at three
different stresses is shown in Fig. 5(a). Our results at $\sigma=0$ match well
the more accurate model of Ref. [17]. $R_{\text{H}}>0$ for large
$\eta_{xy}/\eta_{xz,yz}$ because both the $\beta$ and $\gamma$ sheets have
substantial $xy$ orbital weight. A high $xy$ scattering rate suppresses the
contribution to the Hall conductivity from these sheets, leaving that from the
$\alpha$ sheet, which is hole-like.

We use the results shown in Fig. 5(a) to convert $R_{\text{H}}$ measured at
$\pm 0.35$ T and at 5 and 10 K, temperatures where electron-electron
scattering dominates [44], to $\eta_{xy}/\eta_{xz,yz}$. Results are shown in
Fig. 5(b). At $\sigma=0$, we find $\eta_{xy}/\eta_{xz,yz}>1$. This is expected
because in unstressed Sr2RuO4 it is the $xy$ band that is most strongly
renormalized, and the electron-electron scattering rate is correlated with the
degree of renomalization [45]. Dynamical mean-field theory calculations
indicate that this stronger renormalization is due to proximity to the
Lifshitz transition [9, 8], which occurs in the $xy$ band. It is therefore
reasonable to hypothesize that tuning even closer to this transition would
cause a further increase in $\eta_{xy}/\eta_{xz,yz}$. Such an increase is
indeed visible in Fig. 5(b), but it is small. The much more prominent feature
is a steep fall in $\eta_{xy}/\eta_{xz,yz}$ for $\sigma<\sigma_{\text{L}}$.
Consistent with this interpretation that scattering in the $xy$ band falls
steeply for $\sigma<\sigma_{\text{L}}$, the magnetoresistance becomes much
stronger for $\sigma<\sigma_{\text{L}}$: in Fig. 4(c), a noticeable difference
appears between $\rho_{xx}(T=5\;\text{K})$ at 0 and 0.35 T.

Even if this specific model is called into question, our data show that
scattering must change throughout the Brillouin zone. Our tight-binding model
yields $R_{\text{H}}=-1.74\times 10^{-10}$ m3/C for unstressed Sr2RuO4 when
$l_{\text{mfp}}$ is isotropic and equal on the $\beta$ and $\gamma$ sheets but
zero on the $\alpha$ sheet. The fact that $R_{\text{H}}$ falls to values in
this range when $\sigma<\sigma_{\text{L}}$ means that $l_{\text{mfp}}$ must
become long on essentially all convex portions of the $\beta$ and $\gamma$
sheets.

We briefly discuss the $T\rightarrow 0$ limit, where impurity scattering
dominates. In this limit it is frequently assumed that $l_{\text{mfp}}$ is set
by defect spacing and is therefore isotropic. However, we have already noted
that with isotropic $l_{\text{mfp}}$ $R_{\text{H}}$ would jump upward at the
Lifshitz transition, due to the change in Fermi surface topology. The fact
that $R_{\text{H}}$ remains essentially constant across
$\sigma=\sigma_{\text{L}}$ indicates that $l_{\text{mfp}}$ must be very short
in the neck region of the $\gamma$ sheet. The observation here that
$\rho_{xx}\propto T^{2}\log T$ at $\sigma=\sigma_{\text{L}}$ indicates
standard quasiparticle dispersion right up to the saddle point, and we
therefore hypothesize that small-angle scattering from charge disorder is
stronger in these regions. Similar anisotropy is seen in overdoped cuprate
superconductors, where $l_{\text{mfp}}$ is shorter along the $(0,\pi)$ and
$(\pi,0)$ directions (the anti-nodal directions) [3, 4, 46], and enhanced
small-angle scattering due to charge disorder is thought to play a role [3,
47].

In summary, going through the Lifshitz transition drives large changes in
electron-electron scattering in Sr2RuO4. In a model of orbital
differentiation, the $xy$ band transitions over a small strain range from
being the most- to being the least-strongly corrleated band. This result
provide important information on the origin of electronic correlations in
metals, and should be tested with other techniques.

##  I APPENDIX

Sample carrier. The sample carrier used here is illustrated in Fig. S1(a). The
two parts are respectively bolted to the uniaxial stress cell, which controls
the distance between them. Bringing the two parts into contact compresses the
sample. The set of four flexures in the sample carrier, which protect the
sample from inadvertent transverse and twisting forces, are calculated to have
a combined spring constant of 0.14 N/$\mu$m. This spring constant is used to
calculate the force in the flexures when the sample gets compressed, and is
subtracted from the force reading from the cell to obtain the force applied to
the sample.

Sample 2. As described in the main text, for sample 2 a plasma focused ion
beam was used to mill voltage contacts from the sample itself, ensuring highly
accurate placement. In Fig. S1(b), a scanning electron micrograph of sample 2
is shown after mounting onto the sample carrier, but before the upper plates
were installed. The electrical contacts at the end of the sample are visible
in this SEM. In Fig. S1(c), a photograph taken after the upper plates were
installed is shown. The upper plate is held in place with Stycast 2850.

Figure 6: (a) Illustration of the sample carrier used here. This carrier gets
mounted to a uniaxial stress cell similar to that described in Ref. [18]. (b)
Scanning electron micrograph of sample 2 before installation of the upper
plates. (c) Photograph of sample 2 after installation of the upper plates.

The longitudinal resistivity at $\sigma=\sigma_{\text{L}}$. In the main text,
we used a relationship $[\rho_{xx}(B=0)/\Delta\rho_{xx}]^{0.5}=\alpha+\beta
T^{2}$ to subtract the magnetoresistance and estimate the $\rho_{xx}(T)$ that
would be obtained at $B=0$ without superconductivity. We show in Fig. S2 that
the form $[\rho_{xx}(B=0)/\Delta\rho_{xx}]^{0.5}=\alpha+\beta T^{1.5}$ can
also be fit to the data at $\sigma=\sigma_{\text{L}}$. Potentially, it is a
better model because non-umklapp scattering is expected to be proportional to
$T^{1.5}$ at $\sigma=\sigma_{\text{L}}$. Results when this model is used to
subtract the magnetoresistivity are also shown in Fig. S2. The conclusion is
the same as with the $T^{2}$ magnetoresistance model shown in the main text:
$\rho_{xx}=\rho_{0}+AT^{3/2}$ is a poor fit to the data, while
$\rho_{0}+AT^{2}\log(T_{0}/T)$ and $\rho_{0}+AT^{5/3}$ work equally well.

Figure 7: An alternative magnetoresistance model. (a)
$(\rho_{xx}(B=0)/\Delta\rho_{xx})^{0.5}$, where $\Delta\rho_{xx}$ is the
change in $\rho_{xx}$ between 0 and 1.5 T, against $T^{3/2}$ for sample 2 at
$\sigma=\sigma_{\text{L}}$. (b) $\rho_{xx}(T)$ of sample 2 at
$\sigma=\sigma_{\text{L}}$. The black line is an average of the data at $+1.5$
and $-1.5$ T with the magnetoresistance subtracted as described in the
supplemental text. (c) Difference between three fitting models and the data
with the magnetoresistance subtracted.

$V_{\text{H}}(B)$ for sample 1. In Fig. S3 we show the Hall voltage
$V_{\text{H}}(B)$ for sample 1, at various temperatures. Due to its higher
defect scattering, the low-field range where $V_{\text{H}}\propto B$ extends
slightly further than for sample 2.

Figure 8: Hall voltage $V_{\text{H}}$ versus field $B$ for sample 1 at various
temperatures.

Complete data set at $B=\pm 0.35$ T. In Fig. 4 in the main text, we excluded
from panels (a) and (c), which show data at $B=\pm 0.35$ T, data that we
estimated to be affected by the superconductivity of Sr2RuO4. In Fig. S4(a–b),
we show the complete data set.

In Fig. S4(c) we also show an example of unsymmetrized data at 0.35 T: the
voltage between contacts 1 and 3 divided by the applied current. We noted in
the main text that the stress-induced magnetic order has a strong effect on
$R_{\text{H}}$ at 1.5 T, but no discernible effect on $R_{\text{H}}$ at 0.35
T. Here, it can be seen that there are anomalies in the unsymmetrized data at
0.35 T. In Fig. 4(c) in the main text, there are also no clear anomalies in
$\rho_{xx}(\sigma)$ when the magnetic order onsets, so the appearance of
anomalies in the unsymmetrized data suggests that the magnetic order affects
$\rho_{yy}$ and/or $\rho_{zz}$ more strongly than $R_{\text{H}}$. $\rho_{yy}$
and $\rho_{zz}$ are expected to enter the unsymmetrized Hall data at some
level due to misalignment of the sample axes with respect to the crystal axes;
without particular effort, misalignment of $\sim 1^{\circ}$ is typical. The
main point of showing this data is to make clear that the magnetic order is
present at 0.35 T, as at 1.5 T.

Figure 9: (a) $R_{\text{H}}(\sigma)$ of sample 2, measured at $B=\pm 0.35$ T.
This is a copy of Fig. 4(a) from the main text, but including the data that
are affected by the superconductivity. (b) $\rho_{xx}(\sigma)$ of sample 2 at
$B=+0.35$ T— Fig. 4(b) from the main text, with the data affected by the
superconductivity included. (c) Unsymmetrized data at $T=5$ K and $\pm 0.35$
T: $V_{13}$, the voltage difference between contacts $1$ and $3$, divided by
the current applied between contacts $5$ and $6$. The onset of the magnetic
order is taken as the stress where $d^{2}R_{\text{H}}/d\sigma^{2}$ at 1.5 T is
maximum; see Fig. 4(d).

Tight-binding model. Our tight-binding model is:

$\displaystyle H$ $\displaystyle=$
$\displaystyle\psi^{\dagger}_{s}(\mathbf{k})\left(\begin{matrix}E_{xz}(\mathbf{k})&-isg&ig\\\
isg&E_{yz}(\mathbf{k})&-sg\\\
-ig&-sg&E_{xy}(\mathbf{k})\end{matrix}\right)\psi_{s}(\mathbf{k}),$
$\displaystyle E_{xz}(\mathbf{k})$ $\displaystyle=$
$\displaystyle-\mu-2t_{1}\cos k_{x}-2t_{2}\cos k_{y},$ $\displaystyle
E_{xy}(\mathbf{k})$ $\displaystyle=$ $\displaystyle-\mu-2t_{3}(\cos k_{x}+\cos
k_{y})-4t_{4}\cos k_{x}\cos k_{y}-$ $\displaystyle 2t_{5}(\cos 2k_{x}+\cos
2k_{y}).$

$\psi_{s}(\mathbf{k})=[c_{\mathbf{k},{xz},s},c_{\mathbf{k},{yz},s},c_{\mathbf{k},{xy},-s}]^{T}$,
$s=\pm 1$ for spin, and
$\\{t_{1},t_{2},t_{3},t_{4},t_{5},\mu,g\\}=\\{88,9,80,40,5,109,35\\}$ meV.
This model was derived as a fit to ARPES data in Ref. [41], and was also
applied in in Ref. [42, 43]. These parameters yield cyclotron effective masses
for the $\alpha$, $\beta$, and $\gamma$ sheets of 5.4, 4.8, and $16.7m_{e}$,
respectively.

To apply uniaxial stress, $x$-oriented hopping integrals are scaled as
$t\rightarrow t\times(1-\alpha\varepsilon_{xx})$, and $y$-oriented hopping
integrals as $t\rightarrow t\times(1+\alpha\nu_{xy}\varepsilon_{xx})$.
$\varepsilon_{xx}$ is the longitudinal strain, $\nu_{xy}=0.508$ is the
Poisson’s ratio [48], and $\alpha$ is a constant that places the Lifshitz
transition at $\sigma_{\text{L}}=-0.75$ GPa. The diagonal hopping integral,
$t_{4}$, is scaled as $t_{4}\rightarrow
t_{4}\times[1-\frac{\alpha}{2}(1-\nu_{xy})\varepsilon_{xx}]$. The Fermi
surfaces resulting from this tight-binding model are shown in Fig. 1 in the
main text. In Figs. S5(a–c) we show these Fermi surfaces again, but populated
with arrows whose lengths are proportional to the mean free path. For this
illustration, we take $\eta_{xy}=\eta_{xz,yz}$, so $l_{\text{mfp}}$ is
proportional to the Fermi velocity. The Ong construction is illustrated in
Figs. S5(d–f). The curves shown in these panels are the paths traced when each
mean free path vector $\mathbf{l_{\text{mfp}}}$ is shifted to the origin. Ong
showed that the Hall conductivity $\sigma_{xy}$ is proportional to the sum of
the areas enclosed.

Figure 10: (a–c) Fermi surfaces calculated using the tight-binding model
described in the text. The arrow length is proportional to the Fermi velocity.
(d–f) l curves under an assumption of isotropic scattering time, that is,
$\eta_{xy}=\eta_{xz,yz}$. In this case,
$\mathbf{l}_{\text{mfp}}\propto\mathbf{v}_{\text{F}}$.

###### Acknowledgements.

Acknowledgements. P.Y. thanks K. Shirer and M. König for assistance with the
plasma focused ion beam. We thank A. Georges, M. Zingl, and J. Mravlje for a
critical read of the manuscript. We additionally thank E. Berg, N. Hussey, and
J. Schmalian for useful comments. We acknowledge the financial support of the
Max Planck Society. A.P.M. and C.W.H. acknowledge the financial support of the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - TRR 288 -
422213477 (project A10). N.K. is supported by a KAKENHI Grants-in-Aids for
Scientific Research (Grant Nos. 17H06136, 18K04715, and 21H01033), and Core-
to-Core Program (No. JPJSCCA20170002) from the Japan Society for the Promotion
of Science (JSPS) and by a JST-Mirai Program (Grant No. JPMJMI18A3). H. M. L.
N. acknowledges support from the Alexander von Humboldt Foundation through a
Research Fellowship for Postdoctoral Researchers. Research in Dresden benefits
from the environment provided by the DFG Cluster of Excellence ct.qmat (EXC
2147, project ID 390858940). Raw data for this publication are available at
url to be determined.

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