††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Strain control of a bandwidth-driven spin reorientation in Ca3Ru2O7

C. D. Dashwood<EMAIL_ADDRESS>A. H. Walker
<EMAIL_ADDRESS>London Centre for Nanotechnology and Department of
Physics and Astronomy, University College London, London, WC1E 6BT, United
Kingdom  M. P. Kwasigroch  Department of Mathematics, University College
London, London, WC1H 0AY, United Kingdom  Trinity College, Cambridge, CB2 1TQ,
United Kingdom  L. S. I. Veiga  London Centre for Nanotechnology and
Department of Physics and Astronomy, University College London, London, WC1E
6BT, United Kingdom  Diamond Light Source, Harwell Science and Innovation
Campus, Didcot, Oxfordshire, OX11 0DE, United Kingdom  Q. Faure  London Centre
for Nanotechnology and Department of Physics and Astronomy, University College
London, London, WC1E 6BT, United Kingdom  Laboratoire León Brillouin, CEA,
CNRS, Université Paris-Saclay, CEA-Saclay, 91191 Gif-sur-Yvette, France  J. G.
Vale  London Centre for Nanotechnology and Department of Physics and
Astronomy, University College London, London, WC1E 6BT, United Kingdom  D. G.
Porter  Diamond Light Source, Harwell Science and Innovation Campus, Didcot,
Oxfordshire, OX11 0DE, United Kingdom  P. Manuel  D. D. Khalyavin  F. Orlandi
ISIS Neutron and Muon Source, STFC Rutherford Appleton Laboratory, Didcot,
Oxfordshire, OX11 0QX, United Kingdom  C. V. Colin  Université Grenoble Alpes,
CNRS, Institut Néel, 38000 Grenoble, France  O. Fabelo  Institut Laue-
Langevin, 71 Avenue des Martyrs, CS 20156, 38042 Grenoble, France  F. Krüger
London Centre for Nanotechnology and Department of Physics and Astronomy,
University College London, London, WC1E 6BT, United Kingdom  ISIS Neutron and
Muon Source, STFC Rutherford Appleton Laboratory, Didcot, Oxfordshire, OX11
0QX, United Kingdom  R. S. Perry  London Centre for Nanotechnology and
Department of Physics and Astronomy, University College London, London, WC1E
6BT, United Kingdom  R. D. Johnson  Department of Physics and Astronomy,
University College London, London, WC1E 6BT, United Kingdom  A. G. Green  D.
F. McMorrow  London Centre for Nanotechnology and Department of Physics and
Astronomy, University College London, London, WC1E 6BT, United Kingdom

###### Abstract

The layered-ruthenate family of materials possess an intricate interplay of
structural, electronic and magnetic degrees of freedom that yields a plethora
of delicately balanced ground states. This is exemplified by Ca3Ru2O7, which
hosts a coupled transition in which the lattice parameters jump, the Fermi
surface partially gaps and the spins undergo a $None$ in-plane reorientation.
Here, we show how the transition is driven by a lattice strain that tunes the
electronic bandwidth. We apply uniaxial stress to single crystals of Ca3Ru2O7,
using neutron and resonant x-ray scattering to simultaneously probe the
structural and magnetic responses. These measurements demonstrate that the
transition can be driven by externally induced strain, stimulating the
development of a theoretical model in which an internal strain is generated
self-consistently to lower the electronic energy. We understand the strain to
act by modifying tilts and rotations of the RuO6 octahedra, which directly
influences the nearest-neighbour hopping. Our results offer a blueprint for
uncovering the driving force behind coupled phase transitions, as well as a
route to controlling them.

## Introduction

The coupling between structural and electronic degrees of freedom in quantum
materials generates a variety of ground states and drives transitions between
them. Textbook examples include the Peierls transition in 1D materials, in
which a periodic lattice deformation leads to a metal-insulator transition and
the formation of a charge-density wave Peierls (1955); Fröhlich (1954).
Similarly, the cooperative Jahn-Teller effect describes the spontaneous
distortion of a crystalline lattice to lower the electronic degeneracy and
give rise to orbital ordering Kanamori (1960). Recently, there has been
considerable interest in twisted bilayer materials, which host a spectrum of
electronic phases—from Mott insulators Cao  _et al._ (2018a) to unconventional
superconductors Cao  _et al._ (2018b)—as the bandwidth is tuned by the twist
angle. The key role of the lattice in many quantum materials offers a powerful
set of control parameters with which to tune their phases, but also presents a
considerable challenge in developing a comprehensive understanding of the
interactions that give rise to these phases.

In this context, the application of stress has arisen as a powerful method to
tune the electronic properties of quantum materials, including superconducting
Hicks  _et al._ (2014); Steppke  _et al._ (2017), charge/spin-density wave Kim
_et al._ (2018, 2021); Wang  _et al._ (2022); Choi  _et al._ (2022); Simutis
_et al._ (2022), nematic Malinowski  _et al._ (2020); Bartlett  _et al._
(2021); Worasaran  _et al._ (2021); Sanchez  _et al._ (2021) and topological
Nicholson  _et al._ (2021); Lin  _et al._ (2021) phases. Here, we use uniaxial
stress to drive a coupled spin reorientation and Fermi surface reconstruction
in Ca3Ru2O7, uncovering the central role of the lattice and facilitating a
microscopic understanding of the transition.

Fig. 1: Neutron scattering under stress. a Schematic of the strain cell used
in the neutron scattering experiment. The bridges of the cell and sample
plates are shown in grey, and the sample is shown with a blue-to-yellow
colourmap to indicate the strain gradient induced along its length. Neutrons
are scattered in transmission through the central strained region of the
sample. b Magnetic structure of Ca3Ru2O7 in the AFMa phase Bohnenbuck  _et
al._ (2008). c Integrated intensity of the commensurate $(0,0,1)$ and
incommensurate $(\pm\delta,0,1)$ peaks as a function of applied strain,
$\Delta L/L$. The background signal comes from scattering from the strain
cell, and errors are standard deviations. The insets show detector images at
zero and maximum compressive/tensile applied strain. d Magnetic structure of
Ca3Ru2O7 in the AFMb phase Bohnenbuck  _et al._ (2008).

Ca3Ru2O7 is a bilayer member of the Ruddlesden-Popper ruthenates,
An+1RunO3n+1, across which varying structural distortions lead to a diversity
of ground states. In the monolayer compounds these range from
superconductivity in Sr2RuO4 Maeno  _et al._ (1994); Hicks  _et al._ (2014) to
a Mott insulating state in Ca2RuO4 Nakatsuji and Maeno (2000); Riccò  _et al._
(2018), while bilayer Sr3Ru2O7 displays an electronic nematic phase with spin-
density wave order Yamase and Katanin (2007); Berridge  _et al._ (2009);
Lester  _et al._ (2015); Lindquist  _et al._ (2022). Ca3Ru2O7 crystallises in
the $Bb2_{1}m$ space group ($a\approx$
$5.3\text{\,}\mathrm{\SIUnitSymbolAngstrom}$, $b\approx$
$5.5\text{\,}\mathrm{\SIUnitSymbolAngstrom}$, $c\approx$
$19.5\text{\,}\mathrm{\SIUnitSymbolAngstrom}$), in which a combination of
octahedral tilts around $\mathbf{b}$ ($X_{3}^{-}$, $a^{-}a^{-}c^{0}$ in Glazer
notation) and rotations around $\mathbf{c}$ ($X_{2}^{+}$, $a^{0}a^{0}c^{+}$)
combine to unlock polar lattice displacements Lei  _et al._ (2018). It
undergoes a coupled structural, electronic and magnetic transition, where the
lack of inversion symmetry causes a magnetic cycloid to form and mediate a
spin-reorientation transition (SRT). Below $T_{N}\approx$
$60\text{\,}\mathrm{K}$, the spins align along $\mathbf{a}$, coupled
ferromagnetically within the bilayers and antiferromangetically between them,
in the AFMa phase (see Fig. 1b) Bohnenbuck  _et al._ (2008). On cooling
through $49\text{\,}\mathrm{K}$, an incommensurate cycloid (ICC) develops with
$\mathbf{q}=(\delta,0,1)$ ($\delta\approx 0.023$) and the spins rotating in
the $\mathbf{a}$–$\mathbf{b}$ plane Dashwood  _et al._ (2020); Faure  _et al._
(2023). The envelope of the cycloid evolves from elongated along $\mathbf{a}$,
to circular, to elongated along $\mathbf{b}$. The SRT concludes at
$46\text{\,}\mathrm{K}$ with the cycloid collapsing into the collinear AFMb
phase, where the spins are globally rotated by $None$ from AFMa (Fig. 1d). The
SRT is accompanied by a rapid change in the lattice parameters Yoshida  _et
al._ (2005) and a partial gapping of the Fermi surface Baumberger  _et al._
(2006); Xing  _et al._ (2018); Horio  _et al._ (2021); Marković  _et al._
(2020).

Although there have been various proposals for the mechanism behind the SRT,
including recent work that attributes it to the energy gain associated with a
Rashba-based hybridisation of bands at the Fermi level Marković  _et al._
(2020), these proposals have generally neglected the structural degrees of
freedom. We use piezoelectric cells to apply continuously tuneable uniaxial
stresses to Ca3Ru2O7 single crystals. A combination of neutron and resonant
x-ray scattering enables us to directly probe the magnetic structure and offer
insight into the response of the lattice to applied stress. Our measurements
reveal that the SRT can be fully driven by strain at fixed temperature, and
allow the construction of temperature-strain phase diagrams for orthogonal in-
plane stresses. This motivates the development of a minimal theoretical model
in which strain tunes the electronic hopping. As well as reproducing our
strain data, a self-consistent solution of the model (with realistic values
for the electronic parameters) reveals that the transition with temperature is
driven by an internal strain that arises to lower the electronic energy. We
interpret this strain as acting via the tilts and rotations of the RuO6
octahedra, as corroborated by temperature-dependent diffraction measurements.
Our results therefore offer a promising mechanism to drive electronic and
magnetic phase transitions in correlated perovskite materials.

## Results

### Driving the spin reorientation with strain

As a well-established bulk probe of magnetic order, neutron scattering is a
natural choice for our experiments. The large sample size generally needed is,
however, at odds with the inverse scaling of achievable strain on sample
length – an issue that is compounded by the increased background scattering
from the strain cell. To overcome these issues, we used the latest generation
of piezoelectric strain cells from Razorbill Instruments Raz , which can apply
large strains to samples measured in a transmission scattering geometry with
minimal background from the cell. Combined with the high count-rate of the
WISH instrument at the ISIS Neutron and Muon Source Chapon  _et al._ (2011),
this enabled us to achieve sufficient signal at strains of up to $0.5\%$. A
schematic of the strain cell is shown in Fig. 1a, with a Ca3Ru2O7 sample
spanning a distance $L$ between the sample plates. Applying a voltage across
the piezoelectric stacks changes the distance between the plates by $\Delta
L$, producing an applied strain $\Delta L/L$ along the $a$-axis of the sample.
Further experimental details can be found in the Methods.

Figure 1c shows the result of a strain sweep at a fixed temperature in the
centre of the ICC phase. We tracked the intensities of the commensurate
$(0,0,1)$ magnetic peak, which arises from both the AFMa and AFMb structures,
and satellite $(\pm\delta,0,1)$ peaks that arise from the mediating cycloidal
phase. At $\Delta L/L=0$, we see intense satellite peaks and a small remnant
commensurate peak due to temperature and strain gradients in the sample. Under
both tensile ($\Delta L/L>0$) and compressive ($\Delta L/L<0$) strain the
satellite peaks are suppressed and the commensurate peak is simultaneously
enhanced. The commensurate intensity is lower under compression, consistent
with entering the AFMa phase, while the higher commensurate intensity under
tension is consistent with the AFMb phase Dashwood  _et al._ (2020). This
assignment is confirmed by strain dependences at temperatures just above and
below the SRT (see the Supplementary Information). The slight change in
incommensurate intensity at low strains is consistent with the variation of
moment size through the ICC phase. Our neutron data therefore provide strong
evidence that we can continuously drive the cycloid-mediated SRT with strain
at fixed temperature.

Despite this, our neutron setup has a number of drawbacks. The transitions
between the magnetic phases can be seen to occur over a finite width of around
$0.3\%$ due to the neutron beam illuminating a large region of the sample
across which there is a significant strain gradient (see Fig. 1a). Further,
despite the large displacements that can be generated by the cell, the maximum
strains of $|\Delta L/L|\approx 0.5\%$ are not large enough to fully enter the
collinear phases. Finally, and most significantly, the scattering geometry
necessitated by the strain cell blocks access to any nuclear Bragg peaks with
a finite component along the stress direction. This precludes a determination
of the lattice parameters, and therefore the true strain generated in the
sample, forcing us to rely on the displacement of the sample plates as a
measure of the applied strain. As we will see in the following section, the
fact that the epoxy is significantly softer than the sample leads to only a
fraction of the applied strain being transmitted to the sample.

### Response of the lattice to applied stress

Fig. 2: X-ray scattering under stress. a Schematic of the strain cell used for
x-ray scattering. The principles of operation are the same as the neutron
setup, except that the sample is mounted on top of raised sample plates to
give a large sphere of access for the incident and scattered beams. b
$2\theta$ scans of the structural $(1,0,7)$ Bragg peak at various applied
strains. c True strain, $\Delta b/b$, as a function of applied strain, $\Delta
L/L$, at a range of temperatures through the SRT. d Temperature dependences of
the Poisson ratios determined from the relative strains along orthogonal axes.
All errors are standard deviations.

A full characterisation of the response of the lattice to applied stress is
crucial to understanding the magnetic response, as highlighted by previous
contradictory reports of the electronic and magnetic changes under uniaxial
pressure in Ca3Ru2O7 Cao  _et al._ (2003); Yoshida  _et al._ (2008). To
achieve this, we turned to synchrotron x-ray scattering at beamline I16 of the
Diamond Light Source. Size limitations of the closed-cycle cryostat required
use of a smaller strain cell than the neutron measurements. The lower stresses
are mitigated, however, by the high flux and small focus of the x-ray beam,
allowing the use of smaller samples to achieve applied strains over $2\%$. To
maximise beam access we mounted the samples on top of raised sample plates, as
shown in Fig. 2a (further details can be found in the Methods). Two samples
were measured, one with stress applied along $\mathbf{a}$ and the other along
$\mathbf{b}$.

We tracked the positions of multiple structural Bragg peaks while applying
compressive stresses. Figure 2b shows representative $2\theta$ scans of the
$(1,0,7)$ peak for stress applied along $\mathbf{b}$. The shifting of the peak
arises from the changing lattice parameters, from which we can determine the
true strain, $\Delta b/b$. This is shown in Fig. 2c, where we see a linear,
temperature-insensitive dependence on $\Delta L/L$, with around 40% of the
applied strain transmitted to the sample. We were unable to apply tensile
strains due to the asymmetric sample mounting, which also induced a slight
bending of the sample under compression (see the Supplementary Information).
The bending did not induce significant strain gradients through the probed
region, however, as evidenced by the minimal broadening of the peaks in Fig.
2b.

Fig. 3:  Temperature-strain phase diagrams. a, b Experimental phase diagrams
for stress applied along the $a$\- and $b$-axes respectively. The points with
errors (standard deviations) are transition temperatures extracted from the
strain measurements, and the squares indicate the zero-strain transition
temperatures from ref. Dashwood  _et al._ (2020). The dashed lines are guides
to the eye. The insets indicate the tensile strains induced along the
orthogonal directions according to the Poisson ratios. c Theoretical phase
diagram under applied strain $\varepsilon_{\text{app}}$, calculated by self-
consistently minimising the free energy in Eq. (2) using the parameter values
$U=8$, $\lambda=0.5$, $\theta=$15\text{\,}\mathrm{\SIUnitSymbolDegree}$$,
$\mu=8.5$, $\kappa=400$, $\nu=12$, $t_{0}=1.12$ and $\varepsilon_{0}=0.066$.

Our x-ray measurements also enable us to quantify the strains along the other
crystalline axes, $\varepsilon_{x}=\Delta x/x$, that arise from the uniaxial
stresses applied by the cell, and thereby calculate the Poisson ratios
$\nu_{xy}=-\varepsilon_{y}/\varepsilon_{x}$. The results are plotted as a
function of temperature in Fig. 2d. The anisotropy of the in-plane
($\nu_{ab}\approx\nu_{ba}\approx 0.5$) and out-of-plane
($\nu_{ac}\approx\nu_{bc}\approx 0.2$) Poisson ratios is as expected for a
layered material like Ca3Ru2O7.

We can also use x-ray scattering to probe the magnetic phases by exploiting
the resonant scattering enhancement on tuning the incident x-ray energy to the
Ru $L_{2}$ edge ($2.967\text{\,}\mathrm{keV}$). Alongside the positions of the
structural peaks, we monitored the intensities of the $(\delta,0,5)$ and
$(0,0,5)$ magnetic peaks at each strain value, the latter at two orthogonal
azimuthal angles to differentiate between the AFMa and AFMb phases Dashwood
_et al._ (2020). Repeating these strain sweeps at a range of temperatures
enabled us to construct the phase diagrams shown in Fig. 3a, b for stress
applied along $\mathbf{a}$ and $\mathbf{b}$ respectively (further details of
how the phase diagrams were constructed can be found in the Supplementary
Information).

The phase diagrams show the transition temperatures varying linearly with
compressive strain along both $\mathbf{a}$ and $\mathbf{b}$. The data quality
is worse for the sample with stress applied along $\mathbf{a}$, most likely
due to the poorer crystalline quality of the sample causing a heterogeneous
distribution of strain (see the Supplementary Information). Despite this, we
can still determine a rate of change of the transition temperatures of
approximately $40\text{\,}\mathrm{K}$ per percent strain along $\mathbf{a}$.
The transition temperatures change at a slower rate for the sample stressed
along $\mathbf{b}$, at around $7\text{\,}\mathrm{K}$ per percent strain along
$\mathbf{b}$. Although we could not access the tensile sides of the phase
diagrams in our x-ray experiments, our neutron results suggest that the trends
should continue unchanged. It is interesting to note that the phase boundaries
move in the same direction in both phase diagrams, despite the opposite in-
plane deformations of the lattice (as depicted in the insets of Fig. 3a, b).

### A strain-coupled electronic model

To understand the role of strain in the SRT, we developed a phenomenological
model in which strain tunes the electronic hopping between nearest-neighbour
Ru orbitals. In the spirit of developing a minimal model that captures the
physics under study, we included only the Ru $d_{xz}$ and $d_{yz}$ orbitals in
a perovskite monolayer. Such a two-band model is consistent with ARPES studies
that show that the $d_{xy}$ orbital does not contribute to the Fermi surface
Horio  _et al._ (2021). We consider a Hamiltonian
$\hat{H}=\hat{H}_{t}+\hat{H}_{U}+\hat{H}_{\lambda}-\mu\hat{N}$. $\hat{H}_{t}$
is a tight-binding Hamiltonian fitted to ARPES data Horio  _et al._ (2021)
(see the Methods).
$\hat{H}_{U}=U\sum_{i\tau\alpha}\hat{n}_{i\tau\alpha\uparrow}\hat{n}_{i\tau\alpha\downarrow}$
is the on-site intra-orbital Hubbard interaction, where the lattice site is
labelled $i$, the orbital is labelled $\alpha\in\\{xz,yz\\}$, and
$\tau\in\\{A,B\\}$ is a sublattice index that is introduced to allow for the
staggered octahedral tilting. The interaction is treated in the Hartree-Fock
mean-field approximation, and all other local electron-electron interactions
are neglected for simplicity. The Hamiltonian is then a function of site-
averaged charge, $\rho_{\tau\alpha}$, and magnetisation,
$\mathbf{M}_{\tau\alpha}$, fields for each orbital and sublattice.
$\hat{H}_{\lambda}$ is the spin-orbit interaction Mohapatra and Singh (2020),

$\hat{H}_{\lambda}=-\frac{i\lambda}{2}\sum_{i,\tau}\left(\underline{c}_{i\tau,xz}^{\dagger}\sigma_{z}^{\tau}\underline{c}_{i\tau,yz}+\text{h.c.}\right),$
(1)

where
$\underline{c}_{i\tau\alpha}^{\dagger}=(c_{i\tau\alpha\uparrow}^{\dagger},c_{i\tau\alpha\downarrow}^{\dagger})$.
This is an on-site term that couples the two orbitals. Octahedral tilting
affects the spin-orbit term by rotating the spin quantisation axes on the
sublattice sites by $\pm\theta$ about the $b$-axis, as
$\sigma_{z}^{A(B)}=\cos(\theta)\sigma_{z}\pm\sin(\theta)(\sigma_{x}-\sigma_{y})/\sqrt{2}$.
The octahedral rotation (as well as the polar distortion that leads to Rashba-
like spin-orbit effects) does not enter the Hamiltonian explicitly, although
it does influence the hopping as described in the next section.

Strain enters the model via a phenomenological field, $\varepsilon$, that
couples to the nearest-neighbour hopping, $t$, of the tight-binding model (see
the Methods for further details). We take a minimal coupling of
$t=t_{0}+\nu\varepsilon$, where $t_{0}$ and $\nu$ are parameters, which is
valid when the strain magnitude is small. The coupling of strain to further-
neighbour hopping is neglected. We also introduce a strain cost set by a
parameter, $\kappa$, in the free energy

$F(\mathbf{M},\rho,\varepsilon,\varepsilon_{\text{app}})=-T\sum_{n,\mathbf{k}}\ln\left(1+e^{-(\epsilon_{n\mathbf{k}}-\mu)/T}\right)\\\
+U\sum_{\tau,\alpha}\left(\mathbf{M}_{\tau\alpha}^{2}-\rho_{\tau\alpha}^{2}\right)+\frac{1}{2}\kappa\left(\varepsilon_{0}+\varepsilon-\varepsilon_{\text{app}}\right)^{2},$
(2)

where $\epsilon_{n\mathbf{k}}$ are the eigenvalues of the electronic
Hamiltonian, $\mu$ is the chemical potential, and $\varepsilon_{0}$ and
$\varepsilon_{\text{app}}$ are parameters. The last term is the contribution
of the non-electronic degrees of freedom to quadratic order. Within this
parameterisation, $\varepsilon$ is the strain measured relative to the
unstrained system which we define, via our choice of value of
$\varepsilon_{0}$ and without loss of generality, as the system at the Néel
temperature. This is a natural reference point for studying the magnetic phase
diagram. As well as the internal strain $\varepsilon$, we include an
externally applied strain $\varepsilon_{\text{app}}$ that moves the minimum of
the strain cost function away from zero. The electronic state is determined at
a particular temperature and externally applied strain by minimising the free
energy with respect to the electronic degrees of freedom,
$\mathbf{M}_{\tau\alpha}$ and $\rho_{\tau\alpha}$, and the strain field,
$\varepsilon$, in a self-consistent manner.

In the absence of strain, $\kappa=\nu=0$, the solution of this model exhibits
easy $a$\- and easy $b$-axis ferromagnetism that corresponds to the AFMa and
AFMb phases in the full structure. This results from the interplay of the on-
site interaction and spin-orbit coupling, which generate the magnetic
anisotropy, and the octahedral tilting that breaks its symmetry in the
$\mathbf{a}$–$\mathbf{b}$ plane. The anisotropy is strongly dependent on the
nearest-neighbour hopping, and an $\mathbf{a}\rightarrow\mathbf{b}$
reorientation of the ferromagnetic order is achieved by increasing $t$ and
therefore the electronic bandwidth. Although our minimal model cannot
reproduce the cycloid that mediates the SRT, we would expect this to appear
naturally when the full bilayer is considered, due to a uniform
Dzyaloshinskii-Moriya interaction that becomes dominant close to the
reorientation transition Dashwood  _et al._ (2020).

Fig. 4:  Self-consistent internal strain. Strain field, $\varepsilon$, as a
function of temperature, with the magnetic phase transitions indicated by
vertical dashed lines. The corresponding change in the effective nearest-
neighbour hopping parameter, $t$, is measured on the right-hand axis. The
zoomed region shows a small discontinuity in $\varepsilon$ across the SRT of
magnitude $|\Delta\varepsilon|\sim 0.005\%$. The insets show Fermi surfaces
calculated in the two magnetic phases. The parameter values are the same as in
Fig. 3c. The calculated value of the hopping parameter at the transition
corresponds to $U/t\sim 6.6$, which is consistent with values of $t$ Horio
_et al._ (2021) and $U$ Puggioni  _et al._ (2020) previously fitted to the
observed electronic structure of Ca3Ru2O7.

With a finite $\kappa$ and $\nu$, we find an internal strain being generated
self-consistently even with zero applied strain, $\varepsilon_{\text{app}}=0$,
as shown in Fig. 4. We see $\varepsilon$ increase as the temperature is
reduced through the AFMa phase, lowering the energy of the electronic system
at the expense of an elastic energy cost. When the hopping reaches a threshold
value the SRT is triggered. This transition is first order and results in a
small positive jump in the strain (see zoomed region in Fig. 4) as well as a
jump in the magnetisation. We also capture some of the changes in the Fermi
surface that are seen in ARPES studies. We find a continuous Fermi surface
reconstruction (insets in Fig. 4) that is not directly driven by the
transition, but instead by the increase in magnetisation (and Stoner gap) as
temperature is reduced. We emphasise that in this self-consistent calculation,
the SRT and Fermi surface reconstruction are achieved as a function of
temperature without varying the band filling. The model does not capture the
partial gapping of the Fermi surface that is also seen by ARPES Baumberger
_et al._ (2006); Horio  _et al._ (2021).

To make contact with our strain experiments, we also solve the model with a
negative applied strain, $\varepsilon_{\text{app}}<0$. This reduces the self-
consistent strain, $\varepsilon$, and thus the nearest-neighbour hopping,
resulting in a reduction in the critical temperature. As shown in Fig. 3c, the
linear suppression of the transition temperature matches the experimental
phase diagrams for compressive stress applied along $\mathbf{a}$ and
$\mathbf{b}$ in Fig. 3a, b. As well as reproducing the main sequence of
magnetic phases with temperature, our minimal model is therefore able to
capture the qualitative features of our strain measurements.

### Microscopic understanding of the transition

Fig. 5: Octahedral tilt and rotation modes. a Schematic of the tilt mode,
consisting of staggered tilts of the RuO6 octahedra around the $b$-axis. b
Schematic of the rotation mode, consisting of staggered rotations of the
octahedra around the $c$-axis. c Temperature dependence of the lattice
parameters of Ca3Ru2O7 from x-ray scattering, normalised to the values at
$40.4\text{\,}\mathrm{K}$. d Temperature dependence of the octahedral rotation
and tilt amplitudes determined from a symmetry analysis of single-crystal
neutron diffraction data, normalised to the values at $44\text{\,}\mathrm{K}$.
Errors are standard deviations.

Our theoretical model is able to reproduce the phenomenology of the SRT by
introducing a strain field that couples to the nearest-neighbour hopping. For
a full understanding of the mechanism behind the transition, however, we need
to connect this strain field to microscopic distortions of the Ca3Ru2O7
lattice. Given their strong influence on the ground states across the
Ruddlesden-Popper series of ruthenates, the octahedral tilt and rotation
degrees of freedom (depicted in Fig. 5a, b) are the most obvious candidates.

To be able to identify the tilt and rotation modes with the strain field of
the model, we must first show that they directly influence the nearest-
neighbour hopping. This can be achieved with a heuristic model of the overlap
between neighbouring Ru $d$ orbitals, assuming perfectly rigid RuO6 octahedra.
Considering the tilt and rotation modes separately (valid for small
distortions), we associate them with angles $\theta$ and $\phi$ respectively.
In a similar manner to the two-centre method of Slater and Koster Slater and
Koster (1954), the orbitals on a distorted Ru–O–Ru bond are expanded in a
basis of orbitals quantised with respect to an undistorted bond axis (further
details can be found in the Methods). We find that the intra-orbital hopping
is equal along both nearest-neighbour bonds, with angle-dependent hopping
parameters

$\displaystyle t(\theta)$
$\displaystyle=\frac{1}{4}\left[\cos(2\sqrt{2}\theta)+4\cos(\sqrt{2}\theta)-1\right]\
(dpd\pi)$ (3) $\displaystyle+\ \cos^{3}(\sqrt{2}\theta)\ (dd\pi)\ +\
\frac{3}{4}\sin^{2}(\sqrt{2}\theta)\ (dd\sigma)$ $\displaystyle+\
\frac{1}{4}\sin^{2}(\sqrt{2}\theta)\cos(2\sqrt{2}\theta)\ (dd\delta)$
$\displaystyle t(\phi)$ $\displaystyle=(dpd\pi)+\cos^{2}(\phi)\
(dd\pi)+\sin^{2}(\phi)\ (dd\delta)$

where $(ll^{\prime}m)$ are Slater-Koster integrals due to direct hopping
between the $d$ orbitals, and $(dpd\pi)$ is the $\pi$-bonding integral due to
indirect hopping via an oxygen $p$ orbital on an undistorted bond. We see that
increasing the rotation reduces the contribution from the direct $\pi$-bonding
while allowing a weaker $\delta$-bond to develop, therefore reducing the
effective hopping. The effect of tilt on the hopping depends on the precise
hierarchy of bonding integrals, but we expect from the behaviour of the
(Sr1-xCax)3Ru2O7 series that increasing tilt will also reduce the hopping Peng
_et al._ (2010); Rivero  _et al._ (2017, 2018).

The importance of the octahedral tilts and rotations in the transition is
further supported by (zero-stress) measurements of the Ca3Ru2O7 structure as a
function of temperature. Figure 5c shows the lattice parameters determined by
resonant x-ray scattering. The in-plane lattice parameters are seen to
increase and $c$ to decrease on cooling through the SRT, all by around 0.05%.
The lack of significant changes either side of the SRT points to the low
phonon populations at these temperatures, and therefore the importance of the
electronic coupling in the structural changes. We also directly determined the
symmetry-adapted tilt ($X_{3}^{-}$) and rotation ($X_{2}^{+}$) mode amplitudes
from neutron diffraction measurements, the results of which are shown in Fig.
5d. On cooling through the SRT, the tilts increase by around 0.6% while the
rotations decrease by around 1%. These changes are an order of magnitude
larger than the changes in bond lengths Yoshida  _et al._ (2005), supporting
our earlier assumption that the octahedra remain rigid. Based on our orbital-
overlap argument above, we would expect the rotations to increase the hopping
on cooling, slightly offset by the increase in tilt. This is in agreement with
our model, in which the self-consistent strain and thus the hopping increase
with decreasing temperature. We therefore have a clear picture of how the
transition proceeds with temperature: an internal strain, corresponding to
changes in the octahedral tilts and rotations, arises through feedback with
the electronic system to modulate the hopping and thereby alter the magnetic
anisotropy.

Finally, we can apply our understanding of the effects of the tilts and
rotations to our strain experiments. While the precise changes in tilt and
rotation angle under applied stress are unknown, we can make some general
deductions from our knowledge of the Poisson ratios (ignoring strain along
$\mathbf{c}$ which is likely to be mostly accommodated within the CaO rock-
salt layers). When compressing along $\mathbf{a}$, the dominant effect is
likely to be an increase in the tilt angle to accomodate the increased
orthorhombicity, together with a smaller reduction in the rotation angle to
account for the accompanying expansion along $\mathbf{b}$. This leads to a
reduction in hopping (offset by the reduced rotation) which suppresses the
transition temperature in agreement with Fig. 3a. By contrast, the dominant
effect when compressing along $\mathbf{b}$ is likely to be an increase in the
rotation angle. This alone would cause an equal contraction along
$\mathbf{a}$, such that a significant decrease in the tilt is needed to allow
an expansion along $\mathbf{a}$ according to the Poisson ratio. This could
explain the slope of the transition temperature in Fig. 3b if the net effect
is still a reduction in the hopping due to the increased rotation, but one
that is largely offset by the reduction in tilt.

## Discussion

In this work, we have demonstrated that the spin orientation in Ca3Ru2O7 can
be controlled with applied stress. By isolating the lattice degrees of
freedom, our measurements have provided a unique window into the role of
strain in a coupled structural, electronic and magnetic phase transition.
Neutron scattering gave us access to the bulk magnetic phases, while resonant
x-ray scattering enabled us to probe in detail how strain is transmitted to
the lattice, and thereby construct temperature-strain phase diagrams. These
experiments motivated a theoretical model in which a strain field couples to
the electronic hopping, and drives a spin reorientation when it reaches a
critical value.

Our combination of experiment and theory has therefore uncovered a mechanism
whereby strain, either externally induced or thermally generated through
feedback with the electronic system, drives a global reorientation of the
spins and concomitant reconstruction of the Fermi surface in Ca3Ru2O7. Given
the ubiquity of the octahedral structural unit, we expect this mechanism to be
relevant to other transition-metal oxides. For the Ruddlesden-Popper
ruthenates in particular, octahedral tilts and rotations have a profound
effect on the ground state, whether that is a Mott insulator or unconventional
superconductor. Strain then offers the possibility of continuously tuning
between these phases without the disorder introduced by chemical doping, and
possibly uncovering phases not seen in the ambient structures.

## Methods

### Samples

Single crystals of Ca3Ru2O7 were grown using the floating zone method and
characterised by wavelength-dispersive x-ray spectrometry, resistivity and
magnetization measurements, and x-ray and neutron diffraction Dashwood  _et
al._ (2020); Faure  _et al._ (2023). The strain axis was aligned with Laue
diffraction. A polarised-light microscope was used to identify orthorhombic
twin domains, and single-domain pieces were then cut out using a wire saw. The
single-domain nature of the samples was checked during the scattering
measurements. The samples were found to naturally cleave during cutting,
producing long, thin bars with clean $(0,0,1)$ faces for scattering (and
precluding measurements with stress applied along $\mathbf{c}$). The
dimensions of the sample used for neutron scattering were $L\times W\times
T\approx 1\times 0.3\times$ $0.05\text{\,}\mathrm{mm}$, for x-ray scattering
with stress along $\mathbf{a}$ were $0.3\times 0.1\times$
$0.04\text{\,}\mathrm{mm}$, and for x-ray scattering with stress along
$\mathbf{b}$ were $0.2\times 0.1\times$ $0.03\text{\,}\mathrm{mm}$.

### Neutron scattering under stress

Neutron scattering measurements under stress were performed at the WISH
instrument of the ISIS Neutron and Muon Source Chapon  _et al._ (2011). We
used the CS200T strain cell from Razorbill Instruments Raz . The cell
incorporates a $None$ access cone allowing transmission measurements with
minimal background signal, and a capacitive displacement sensor using which we
can calculate the applied strain. The cell was mounted on a custom cryostat
stick with feedthroughs for the cables to power the cell and measure the
capacitance of the displacement sensor. Voltage was applied to the cell with
an RP100 power supply from Razorbill Instruments, and the capacitance was
measured with a Keysight E4980AL LCR meter, both of which were integrated into
the instrument control software to enable automated control and data
acquisition. A temperature sensor was thermally contacted with the body of the
cell to give accurate readings of the sample temperature. Cadmium shielding
was used to reduce background scattering from the bridges of the cell and from
unstrained regions of the sample near the sample plates, while still allowing
full beam access in the horizontal plane. The sample was sandwiched between
two pairs of sample plates and secured with Stycast 2850FT epoxy, such that
$\mathbf{a}$ lies along the stress direction and vertical detector banks. The
sample was cooled without any applied stress, and held at fixed temperatures
during the strain sweeps. Data analysis was performed with Mantid Arnold  _et
al._ (2014). All data are normalised to the cumulative current and a beam
monitor. The intensities were obtained by diffraction focussing a small area
on the detector around the peak, and then integrating the resulting time-of-
flight spectra.

### X-ray scattering under stress

X-ray scattering measurements were performed at beamline I16 of the Diamond
Light Source, with the incident energy tuned to the Ru $L_{2}$ edge
($2.967\text{\,}\mathrm{keV}$) by performing 4 bounces on the monochromator
and using an extended helium-filled beam pipe. Here, we used the CS100 strain
cell from Razorbill Instruments Raz , which is smaller than the CS200T and
necessitates a reflection scattering geometry. A custom mount was designed to
hold the cell in the closed-cycle cryocooler, with feedthroughs added for the
power and capacitance cables. As at WISH, the RP100 power supply and E4980AL
LCR meter were interfaced with the beamline software to enable remote control
and scripting of the measurements. A Cernox temperature sensor was thermally
contacted with the body of the cell. The sample was mounted to the raised
sample plates with Stycast 2850FT epoxy, with either $\mathbf{a}$ or
$\mathbf{b}$ along the stress direction. No top-plates were used in order to
maximise beam access, but the asymmetric mounting resulted in stress being
transmitted mostly through the lower surface of the sample causing it to bend
(see the Supplementary Information). This, along with the difference in
elastic moduli of the sample ($\sim$$100\text{\,}\mathrm{GPa}$) and epoxy
($\sim$$4\text{\,}\mathrm{GPa}$), leads to the true strain being significantly
lower than the applied strain. The diffractometer was operated with a vertical
scattering geometry in fixed-azimuth mode, and the scattered intensity was
measured with an in-vacuum Pilatus 100K area detector in ultrahigh gain mode.
All temperature changes were conducted with zero applied stress, and the
temperature was kept constant during strain measurements. Data analysis was
carried out using the Py16 program Porter (2020), and reciprocal-space maps
were reconstructed from rocking scans using the MillerSpaceMapper software
Basham  _et al._ (2015). Intensities were obtained by summing over a region-
of-interest on the area detector and fitting the resulting rocking curves with
pseudo-Voigt profiles plus a constant background.

### Neutron diffraction

The neutron diffraction data shown in Fig. 5d were taken on the four-circle
diffractometer D9 at the Institut Laue Langevin. A wavelength of
$0.836\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ was obtained from a Cu(220)
monochromator giving access to 800 reflections (up to $h=10$, $k=10$, $l=15$).
The symmetry mode analysis was performed using the Amplimodes software
Orobengoa  _et al._ (2009) from the Bilbao Crystallographic Server,
considering the high-symmetry parent phase $I4/mmm$ (139) and the low symmetry
phase $Bb2_{1}m$ (36). The mode amplitudes were directly refined at each
temperature by the least-squares method using Fullprof Rodríguez-Carvajal
(1993).

### Strain-coupled electronic model

We apply the tight-binding model for Ca3Ru2O7 that has been fitted to ARPES
data Horio  _et al._ (2021). The tight-binding Hamiltonian in a crystal
momentum basis is

$\displaystyle\hat{H}_{t}(\mathbf{k})=\sum_{\tau}\Big{[}$
$\displaystyle\epsilon_{x}(\mathbf{k})\
\underline{c}_{xz,\tau}^{\dagger}(\mathbf{k})\ \sigma_{0}\
\underline{c}_{xz,\bar{\tau}}(\mathbf{k})$ (4) $\displaystyle+$
$\displaystyle\epsilon_{y}(\mathbf{k})\
\underline{c}_{yz,\tau}^{\dagger}(\mathbf{k})\ \sigma_{0}\
\underline{c}_{yz,\bar{\tau}}(\mathbf{k})\Big{]}$
$\displaystyle+\sum_{\alpha,\tau}\Big{[}$
$\displaystyle\epsilon_{ab}(\mathbf{k})\
\underline{c}_{\alpha,\tau}^{\dagger}(\mathbf{k})\ \sigma_{0}\
\underline{c}_{\alpha,\tau}(\mathbf{k})$ $\displaystyle+$
$\displaystyle\epsilon_{ab}^{\prime}(\mathbf{k})\
\underline{c}_{\alpha,\tau}^{\dagger}(\mathbf{k})\ \sigma_{0}\
\underline{c}_{\bar{\alpha},\tau}(\mathbf{k})\Big{]}$

where
$\underline{c}_{\alpha\tau}^{\dagger}(\mathbf{k})=(c_{\alpha\tau\uparrow}^{\dagger}(\mathbf{k}),c_{\alpha\tau\downarrow}^{\dagger}(\mathbf{k}))$,
$\sigma_{0}$ is the $2\times 2$ identity matrix in spin space,
$\alpha\in\\{xz,yz\\}$ and $\tau\in\\{A,B\\}$. Bars are used to denote flipped
orbital and sublattice indices, i.e. $\bar{xz}=yz$, $\bar{A}=B$, and vice
versa. In the tetragonal crystal momentum basis, $\mathbf{k}=(k_{x},k_{y})$,
the dispersions are

$\begin{split}\epsilon_{x}(\mathbf{k})&=-2t\cos(k_{x}a)\\\
\epsilon_{y}(\mathbf{k})&=-2t\cos(k_{y}a)\\\
\epsilon_{ab}(\mathbf{k})&=-2t_{a}\cos\left((k_{x}-k_{y})a\right)-2t_{b}\cos\left((k_{x}+k_{y})a\right)\\\
\epsilon_{ab}^{\prime}(\mathbf{k})&=-2t_{a}^{\prime}\cos\left((k_{x}-k_{y})a\right)-2t_{b}^{\prime}\cos\left((k_{x}+k_{y})a\right)\\\
\end{split}$ (5)

where $k_{x,y}\in[-\frac{\pi}{a},\frac{\pi}{a}]$ with lattice constant $a$.
$\epsilon_{x(y)}$ describes the nearest-neighbour hopping between $xz$($yz$)
orbitals along the $x$($y$) direction, and $\epsilon_{ab}$ and
$\epsilon_{ab}^{\prime}$ describe the intra- and inter-orbital next-nearest-
neighbour hopping along the $a$ and $b$ orthorhombic directions respectively.
We use $t_{a}=0.04\tilde{t}$, $t_{b}=0.16\tilde{t}$ and
$t_{a}^{\prime}=-t_{b}^{\prime}=0.079\tilde{t}$, with $\tilde{t}=1.20$. The
coefficients were chosen by fitting the Fermi surfaces in both magnetic phases
to those measured using ARPES Horio  _et al._ (2021). Only the intra-orbital
nearest-neighbour hopping parameter couples to the strain field, as
$t=t_{0}+\nu\varepsilon$. Strictly, the octahedral tilting and rotation also
induce inter-orbital nearest-neighbour hopping between the $d_{xz}$ and
$d_{yz}$ orbitals Mohapatra and Singh (2020). This hopping mode is neglected
in ref. Horio  _et al._ (2021), and we also neglect it as a sub-leading effect
of the lattice distortions.

In our minimal model, we consider only the on-site intra-orbital repulsion,
$\hat{H}_{U}=U\sum_{i\tau\alpha}\hat{n}_{i\tau\alpha\uparrow}\hat{n}_{i\tau\alpha\downarrow}$,
and neglect the inter-orbital repulsion and Hund’s coupling for simplicity.
$\hat{H}_{U}$ is treated in the Hartree-Fock mean-field approximation by
decoupling in the charge, $\hat{\rho}_{i\tau\alpha}=\frac{1}{2}\
\underline{c}_{i\tau\alpha}^{\dagger}\ \sigma_{0}\
\underline{c}_{i\tau\alpha}$, and spin,
$\hat{\mathbf{S}}_{i\tau\alpha}=\frac{1}{2}\
\underline{c}_{i\tau\alpha}^{\dagger}\ \boldsymbol{\sigma}\
\underline{c}_{i\tau\alpha}$, channels, where
$\boldsymbol{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ is a vector of Pauli
matrices. Defining $\rho_{\tau\alpha}$ and $\mathbf{M}_{\tau\alpha}$ as the
site-averaged charge and spin densities, the interaction decouples to

$\displaystyle\hat{H}_{U}^{\text{MF}}$
$\displaystyle=U\sum_{i,\tau,\alpha}\underline{c}_{i\tau\alpha}^{\dagger}\left(\rho_{\tau\alpha}\sigma_{0}-\mathbf{M}_{\tau\alpha}\cdot\boldsymbol{\sigma}\right)\underline{c}_{i\tau\alpha}$
(6)
$\displaystyle+U\sum_{\tau,\alpha}\left(\mathbf{M}_{\tau\alpha}^{2}-\rho_{\tau\alpha}^{2}\right).$

The system is described in the mean-field and at fixed chemical potential,
$\mu$, by the Hamiltonian,
$\hat{H}^{\text{MF}}=\hat{H}_{t}+\hat{H}_{\lambda}+\hat{H}_{U}^{\text{MF}}-\mu\hat{N}=\sum_{n\mathbf{k}}\left(\epsilon_{n\mathbf{k}}-\mu\right)\hat{n}_{n\mathbf{k}}+U\sum_{\tau\alpha}\left(\mathbf{M}_{\tau\alpha}^{2}-\rho_{\tau\alpha}^{2}\right)$.
The free energy is evaluated via the partition function to give Eq. (2) (in
the absence of coupling to strain, $\nu=\kappa=0$) in terms of the energy
eigenvalues, $\epsilon_{n\mathbf{k}}$, in a band and crystal momentum basis.
The strain field is then introduced into the model as described in the
Results.

The values of strain parameters ($\nu$, $\kappa$) are chosen such that the
calculated strain is of the same order of magnitude as seen in experiment (see
Figs. 4 and 5c), and the magnitude of strain-hopping coupling, $\nu$, is
comparable to a value calculated from first principles for a related material
Li  _et al._ (2022). It is worth noting that the solution of this model is
invariant under the scaling of the strain parameters as $\nu\rightarrow x\nu$,
$\kappa\rightarrow x^{2}\kappa$, $\varepsilon_{0}\rightarrow\varepsilon_{0}/x$
and $\varepsilon_{\text{app}}\rightarrow\varepsilon_{\text{app}}/x$. This
scales the self-consistent strain solution as
$\varepsilon\rightarrow\varepsilon/x$ but leaves the electronic solution
unchanged.

### Effective hopping on a distorted lattice

Fig. 6: Rotation- and tilt-dependence of hopping. a, b Dependence of the in-
plane nearest-neighbour hopping, $t$, between Ru $d_{xz}$ orbitals along
$\mathbf{x}$ on the octahedral rotation angle, $\phi$, and octahedral tilt
angle, $\theta$, respectively. The orbitals on the distorted Ru–O–Ru bond are
shown in blue on the left side of each equation. Direct hopping is expressed
in terms of Slater-Koster integrals, $(ll^{\prime}m)$, illustrated in red on
the right side of each equation. The indirect hopping, due to $\pi$-bonding
with oxygen $p$ orbitals, is expressed in terms of $(dpd\pi)$, the $d$–$p$–$d$
$\pi$-bonding integral on a straight Ru–O–Ru bond (also shown in red). All
oxygen $p$ orbitals are considered but only the $p_{z}$ orbital is shown for
clarity. The coefficients in (b) are
$A(\theta)=\frac{1}{4}\left[\cos(2\sqrt{2}\theta)+4\cos(\sqrt{2}\theta)-1\right]$,
$B(\theta)=\cos^{3}(\sqrt{2}\theta)$,
$C(\theta)=\frac{1}{4}\sin^{2}(\sqrt{2}\theta)\cos(2\sqrt{2}\theta)$ and
$D(\theta)=\frac{3}{4}\sin^{2}(\sqrt{2}\theta)$. Identical results are
obtained for the hopping between $d_{yz}$ orbitals along $\mathbf{y}$.

We calculate the effective nearest-neighbour hopping between ruthenium $d$
orbitals in the presence of octahedral tilting and rotational distortions
(separately), considering the direct orbital overlap and the indirect overlap
via oxygen $p$ orbitals.

The hopping is first defined in the absence of lattice distortions, i.e. for a
straight Ru–O–Ru bond. In the case of hopping due to direct $d$ orbital
overlap, we apply the Slater-Koster two-centre approximation,

$t_{lm,l^{\prime}m^{\prime}}^{\text{Ru--
Ru}}=\langle\psi_{lm}^{\text{Ru}}(\mathbf{r})|\hat{H}^{\text{Ru--
Ru}}|\psi_{l^{\prime}m^{\prime}}^{\text{Ru}}(\mathbf{r}+R\hat{\mathbf{x}})\rangle$
(7)

where $\psi_{lm}^{\text{Ru}}$ is an atomic orbital centred on a ruthenium ion
whose angular momentum quantisation axis is dictated by the crystal field, and
$\hat{H}^{\text{Ru--Ru}}$ is the sum of the kinetic energy operator and ionic
potentials on the bond. In the case of indirect hopping, the effective hopping
is calculated as a second order process in which an electron or hole hops
between ruthenium $d$ orbitals via an oxygen $p$ orbital. The hopping matrix
element for a straight Ru–O–Ru bond is

$t_{lm,l^{\prime}m^{\prime}}^{\text{Ru--O--
Ru}}=-\sum_{l^{\prime},m^{\prime}}\frac{\langle\psi_{lm}^{\text{Ru}}(\mathbf{r})|\hat{H}^{\text{Ru
--
O}}|\psi_{l^{\prime}m^{\prime}}^{\text{O}}(\mathbf{r}+\frac{R}{2}\hat{\mathbf{x}})\rangle\langle\psi_{l^{\prime}m^{\prime}}^{\text{O}}(\mathbf{r}+\frac{R}{2}\hat{\mathbf{x}})|\hat{H}^{\text{O
--
Ru}}|\psi_{lm}^{\text{Ru}}(\mathbf{r}+R\hat{\mathbf{x}})\rangle}{E_{lm}-E_{l^{\prime}m^{\prime}}^{\prime}}$
(8)

where $E_{lm}$ and $E_{l^{\prime}m^{\prime}}^{\prime}$ are the total energies
of the system when the electron/hole occupies a ruthenium orbital,
$\psi_{lm}^{\text{Ru}}$, or an oxygen orbital,
$\psi_{l^{\prime}m^{\prime}}^{\text{O}}$, respectively. The numerator is a
product of matrix elements for Ru–O and O–Ru hopping processes. We make a key
assumption that the lattice distortions are a weak perturbation on the oxygen
$p$ orbitals, which remain degenerate. Every indirect hopping process then has
the same intermediate energy, $E^{\prime}$, and the effect of lattice
distortions is only to modify the two-centre integrals in the numerator.

The lattice distortions are introduced by rotating all orbitals in space
accordingly. Since the orbital orientation is dictated most strongly by the
local (octahedral) crystal field, we assume that the ruthenium $d$ orbitals
are rotated/tilted along with the octahedra. Note that we decompose the
(small) tilt about $\mathbf{b}$ into tilts of $\theta/\sqrt{2}$ about the
$x$\- and $y$-axes. We also assume that the change in Ru–Ru distance due to
the distortions has a sub-leading effect on the hopping and as such is
neglected.

The rotated orbitals are then expanded in a basis of atomic orbitals that are
quantised with respect an appropriate bond axis. In the direct case this is
the Ru–Ru bond. The effective hopping can then be written in terms of matrix
elements for the hopping of an electron between the basis orbitals, which are
the Slater-Koster integrals

$(ll^{\prime}m)\delta_{mm^{\prime}}=\langle\phi_{lm}^{\text{Ru}}(\mathbf{r})|\hat{H}^{\text{Ru
--
Ru}}|\phi_{l^{\prime}m^{\prime}}^{\text{Ru}}(\mathbf{r}+R\hat{\mathbf{x}})\rangle$
(9)

where $\phi_{lm}$ is an orbital that is quantised with respect to the bond
axis. As is convention, we use the labels $l,l^{\prime}\in\\{s,p,d\\}$ and
$m\in\\{\sigma,\pi,\delta\\}$ on the left-hand side. In the indirect case we
expand in bases of orbitals that are quantised with respect to the Ru–O and
O–Ru bonds. We find that the effective hopping can be written as a function of
the hopping along a straight bond,

$(ll^{\prime}lm)\delta_{mm^{\prime}}=-\frac{\langle\phi_{lm}^{\text{Ru}}(\mathbf{r})|\hat{H}^{\text{Ru
--
O}}|\phi_{l^{\prime}m^{\prime}}^{\text{O}}(\mathbf{r}+R\hat{\mathbf{x}})\rangle\langle\phi_{l^{\prime}m^{\prime}}^{\text{O}}(\mathbf{r}+R\hat{\mathbf{x}})|\hat{H}^{\text{O
--
Ru}}|\phi_{lm}^{\text{Ru}}(\mathbf{r}+2R\hat{\mathbf{x}})\rangle}{E_{lm}-E_{l^{\prime}m^{\prime}}^{\prime}}.$
(10)

On the LHS we have introduced a similar notation for the indirect hopping
matrix element to that of the Slater-Koster integrals. The numerator is a
product of two Slater-Koster integrals.

By summing the direct and indirect hoppings in each case, we find the
effective nearest-neighbour hoppings, Eq. (3). We find that increasing the
rotation angle reduces the contribution from the direct $\pi$-bonding while
allowing a weaker $\delta$-bond to develop, therefore reducing the effective
hopping. Increasing the tilt angle reduces the contribution from both
$\pi$-bonding modes, while allowing $\sigma$\- and $\delta$-bonding modes to
develop. Since the indirect $\pi$-bonding, $(dpd\pi)$, likely dominates, we
expect tilting to cause a net reduction in the effective hopping.

## Data availability

Raw data from the neutron scattering measurements under stress are available
at https://doi.org/10.5286/ISIS.E.RB1920210. Raw data from the neutron
diffraction measurements are available at https://doi.org/10.5291/ILL-
DATA.EASY-951. Source data used to plot the figures are available at
https://doi.org/10.6084/m9.figshare.23508354.

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## Acknowledgements

We thank Richard Thorogate for assistance with the resistivity measurements,
Daniel Nye and Gavin Stenning for assistance with the powder x-ray diffraction
and Laue alignment in the Materials Characterisation Laboratory at the ISIS
Neutron and Muon Source, Mike Matthews for technical support at I16, and Jacob
Simms, Katherine Mordecai, Jon Bones and David Keymer for technical support at
WISH. C.D.D. was supported by the Engineering and Physical Sciences Research
Council (EPSRC) Centre for Doctoral Training in the Advanced Characterisation
of Materials under Grant No. EP/L015277/1. A.H.W. was supported by the EPSRC
under Grant No. EP/N509577/1. D.D.K. was supported by the EPSRC under Grant
No. EP/W00562X/1. Work at UCL was supported by the EPSRC under Grants No.
EP/W005786/1, EP/N027671/1, EP/P013449/1 and EP/N509577/1. Experiments at the
ISIS Neutron and Muon Source were supported by beamtime allocation RB1920210
from the Science and Technology Facilities Council. We acknowledge the Diamond
Light Source for time on beamline I16 under proposals MM23580 and MM25554. We
thank Institut Laue Langevin for access to the neutron diffractometer D9 under
proposal EASY-951.

## Author Contributions

R.S.P. and C.D.D. grew and prepared the single crystal samples respectively.
C.D.D., L.S.I.V., Q.F., R.S.P., R.D.J., P.M., D.D.K. and F.O. performed the
neutron scattering measurements under stress, and C.D.D., P.M., D.D.K. and
F.O. analysed the data. C.D.D., L.S.I.V., Q.F., R.S.P., R.D.J. and D.G.P.
performed the x-ray scattering measurements under stress, and C.D.D. and
D.G.P. analysed the data. Q.F., C.V.C. and O.F. performed the neutron
diffraction measurements and all analysed the data. A.H.W., M.P.K., F.K. and
A.G.G. developed the theoretical model. C.D.D. and A.H.W. wrote the paper,
with input from all authors. A.G.G. and D.F.M. oversaw the study.

## Competing Interests

The authors declare no competing interests.