# Virial Theorems and Virial Stresses of Micropolar Media

Martin Ostoja-Starzewski

###### Abstract

A generalization of virial theorems and virial stresses to micropolar
continuum mechanics is explored. The linear momentum balance in dyadic product
with translation leads to (i) the first virial theorem of micropolar continuum
mechanics involving the infinite-time limit of the kinetic translational
energy and (ii) a classical formula for computing the virial force-stress
known in molecular dynamics. The angular momentum balance in dyadic product
with rotation leads to (i) the second virial theorem of micropolar continuum
mechanics involving the infinite-time limit of the kinetic rotational energy
and (ii) a virial couple-stress along with a formula for its computation. The
latter stress is also uncovered in the dyadic product of linear momentum
balance with rotation. The virial force-stress and virial couple-stress
contain, respectively, the Reynolds force-stress and turbulent couple-stress.

Department of Mechanical Science & Engineering
also Institute for Condensed Matter Theory and Beckman Institute
University of Illinois at Urbana-Champaign
Urbana, IL 61801, U.S.A.
e-mail<EMAIL_ADDRESS>
##  1 Introduction

This communication reports a generalization of the virial theorem and virial
stress from classical to micropolar continuum mechanics. The study is set in
the spatial (Eulerian) description [Eringen, 2001; Łukaszewicz, 1999] in
$\mathbb{E}^{d}$, $d=2$ or $3$. For reference, the kinematics of such a
continuum is described through a field with two degrees of freedom
$\left(u_{i},\varphi_{i}\right)$ and their rates:

$\text{velocity: \ }\dot{u}_{i}\text{; \ \ \ \ \ microrotation: \
}\dot{\varphi}_{i}.$ (1)

The overdot stands for the material time derivative. Standard index as well as
symbolic tensor notations are used.

The local form of linear momentum balance reads

$\rho\dot{v}_{i}=\rho f_{i}+\tau_{ki},_{k},$ (2)

where $\rho$ is the mass density, $f_{i}$ is the body force per unit mass, and
$\tau_{ji}$ is the (generally non-symmetric) Cauchy force-stress. The latter
is related to the force-traction on any surface element specified by the outer
normal $n_{k}$ through $t_{i}^{\left(\mathbf{n}\right)}=\tau_{ji}n_{j}$.

The local form of angular momentum balance, after accounting for (2), reads

$\rho J\frac{D\dot{\varphi}_{i}}{Dt}=\rho g_{i}+\mu_{ki},_{k}+T_{i}\text{ \ \
where \ \ }T_{i}=\epsilon_{ijk}\tau_{jk},$ (3)

where $J$ is the (micro)inertia tensor of a particle, $g_{i}$ is the body
torque per unit mass, and $\mu_{ki}$ is the couple-stress, which is related to
the surface moment-traction $m_{i}^{\left(\mathbf{n}\right)}$ through
$m_{i}^{\left(\mathbf{n}\right)}=\mu_{ki}n_{k}$. If $T_{i}=0$, $\tau_{jk}$
becomes symmetric, in which case we write $\sigma_{jk}$ instead. In (2) and
henceforth, we work with a microisotropic medium, i.e. where
$J_{ij}=J\delta_{ij}$.

In general, we consider the material properties to be constant, while the
displacement, rotation, force-stress, and couple stress, as well as their
derivatives, to be tensor-valued random fields [Malyarenko and Ostoja-
Starzewski, 2019]. Denoting by $F(\mathbf{x},t,\omega)$ a realization of any
such random field, it is clearly seen to be parametrized by the (spatial)
position $\mathbf{x}$ and time $t$, where $\omega$ is a realization drawn from
a probability space $\left(\Omega,\digamma,\mathsf{P}\right)$. Thus, there are
three basic types of averaging:

(i) Space (volume) averaging of volume integrals:

$\overline{F}_{V}(\mathbf{x},t,\omega):=\frac{1}{V}\int_{\mathcal{B}}F(\mathbf{x}^{\prime},t,\omega)dV,$
(4)

where $V=|\mathcal{B}|$ is the volume of domain $\mathcal{B}$ having a
bounding surface $\partial\mathcal{B}$. In the language of stochastic
processes, this is local averaging transforming the original random field $F$
into a new, smoother random field $\overline{F}_{V}$. In general,
$\overline{F}_{V}$ is a function of $\mathbf{x}$ but, if spatial ergodicity
(i.e., equivalence of spatial averaging with time averaging and/or ensemble
averaging) is assumed, this dependence vanishes. This situation corresponds to
passing from a statistical to a representative volume element (RVE), with the
RVE size heavily dependent on the random microstructural properties involved
[Ostoja-Starzewski et al., 2016]. In the following, to simplify the notation,
we will simply write $\overline{F}$.

(ii) Time averaging, wherein fluctuations observed on a fine time scale get
smoothed (smeared) on a coarse time scale;

$F_{T}(\mathbf{x},t,\omega):=\frac{1}{T}\int_{0}^{T}F(\mathbf{x},t^{\prime},\omega)dV.$
(5)

In the language of stochastic processes, this is local averaging in the time
domain, transforming the original random field $F$ into a new, smoother-in-
time random field $F_{T}$. In general, $F_{T}$ is still a random function of
$t$ but, if temporal ergodicity (i.e., equivalence with space and/or ensemble
averaging) is assumed, this dependence vanishes. The latter situation
typically implies taking the $T\rightarrow\infty$ limit.

(iii) Ensemble (statistical) averaging.

While in either case, one obtains a coarser description than the original one,
in the present study only space averaging over arbitrary volumes and time
averaging over infinitely long times is considered. The latter operation,
called infinite-time averaging, is indicated by

$\left\\{F(\mathbf{x},t,\omega)\right\\}_{\infty}=\lim_{T\rightarrow\infty}\left\\{F(\mathbf{x},t,\omega)\right\\}_{T}.$
(6)

The reason for considering such a limit as well as the spatial averaging
resides in our focus on the exploration of the virial theorem(s) and virial
stress(es) in micropolar media. The ensuing study aims to generalize the
deterministic exposition of those topics in the setting of classical media
[Podio-Guidugli, 2019], whereby we follow a similar line of reasoning. The
statistical exposition in the vein of that paper is not discussed here.

##  2 Consequences of linear momentum balance

First, we take a dyadic product of a test function $\phi_{j}$ with (2) and
integrate over an arbitrary (and simply-connected) domain $\mathcal{B}$ to get

$\int_{\mathcal{B}}\left(\phi_{j}\rho
f_{i}+\phi_{j}\tau_{ki,k}\right)dV=\int_{\mathcal{B}}\phi_{j}\rho\dot{v}_{i}dV.$
(7)

We now consider two choices for the test function.

###  2.1 Translation as test function

Taking translation as the test function ($\phi_{j}=x_{j}$) in (7), we perform
these steps

$\begin{array}[]{c}\int_{\mathcal{B}}\left(x_{j}\rho
f_{i}+x_{j}\tau_{ki,k}\right)dV=\int_{\mathcal{B}}\left[\rho\left(x_{j}v_{i}\right)^{\cdot}-\rho\left(\dot{x}_{j}v_{i}\right)\right]dV\\\
\int_{\mathcal{B}}x_{j}\rho
f_{i}dV+\int_{\mathcal{B}}\left[\left(x_{j}\tau_{ki}\right),_{k}-x_{j},_{k}\tau_{ki}\right]dV=\frac{d}{dt}\left(\int_{\mathcal{B}}x_{j}\rho
v_{i}dV\right)-\int_{\mathcal{B}}\left(\dot{x}_{j}\rho v_{i}\right)dV\\\
\int_{\mathcal{B}}x_{j}\rho
f_{i}dV+\int_{\partial\mathcal{B}}x_{j}\tau_{ki}n_{k}dS-\int_{\mathcal{B}}\tau_{ji}dV=\frac{d}{dt}\left(\int_{\mathcal{B}}x_{j}\rho
v_{i}dV\right)-V\overline{\rho v_{j}v_{i}}\\\ \int_{\mathcal{B}}x_{j}\rho
f_{i}dV+\int_{\partial\mathcal{B}}x_{j}\tau_{ki}n_{k}dS-\int_{\mathcal{B}}\tau_{ji}dV=\frac{d}{dt}\left(\int_{\mathcal{B}}x_{j}\rho
v_{i}dV\right)-V\overline{\rho v_{j}v_{i}}.\end{array}$ (8)

Hence, we can write

$\int_{\mathcal{B}}\rho
x_{j}f_{i}dV+\int_{\partial\mathcal{B}}x_{j}t_{i}^{\left(\mathbf{n}\right)}dS-V\tau_{ji}^{\ast}=\frac{d}{dt}\left(\int_{\mathcal{B}}\rho
x_{j}v_{i}dV\right),$ (9)

where

$\tau_{ji}^{\ast}:=\overline{\tau_{ji}}-\overline{\rho v_{j}v_{i}}$ (10)

is the virial stress. The term $\overline{\rho v_{j}v_{i}}$ contains
$\overline{\rho v_{j}^{\prime}v_{i}^{\prime}}$, as can be seen from the
Reynolds decomposition ($v_{j}=\overline{v_{j}}+v_{j}^{\prime}$ with
$\overline{v_{j}^{\prime}}=0$). Depending on a convention, $\overline{\rho
v_{j}v_{i}}$ or $\overline{\rho v_{j}^{\prime}v_{i}^{\prime}}$ (with a plus or
minus sign) is called the Reynolds stress. From the standpoint of turbulent
flows, $\tau_{ji}^{\ast}$ is the modified stress in the sense that the linear
momentum, angular momentum, and energy balances are unchanged in form upon
volume averaging [Ostoja-Starzewski, 2021].

Going from continuum to discrete systems such as molecular dynamics, (10) is
consistent with the very well-known prescription for computation of the
instantaneous volume-averaged virial (generally non-symmetric) force-stress

$\tau_{ji}^{\ast}:=\frac{1}{V}\sum_{k\in\mathcal{B}}\left[\frac{1}{2}\sum_{l\in\mathcal{B}}\left(x_{i}^{\left(l\right)}-x_{i}^{\left(k\right)}\right)f_{j}^{\left(kl\right)}-m^{\left(k\right)}\left(v_{j}^{\left(k\right)}-\overline{v_{j}}\right)\left(v_{i}^{\left(k\right)}-\overline{v_{i}}\right)\right],$
(11)

where

$k$ and $l$ are particles in the domain $\mathcal{B}$,

$m^{\left(k\right)}$ is the mass of particle $k$,

$v_{j}^{\left(k\right)}$ is the $j$-th component of the velocity of particle
$k$,

$\overline{v_{j}}$ is the $j$-th component of the average velocity of
particles in the volume,

$x_{i}^{\left(k\right)}$ is the $i$-th component of the position of particle
$k$, and

$f_{j}^{\left(kl\right)}$ is the $j$-th component of the force applied on
particle $k$ by particle $l$.

No symmetry of the Cauchy stress is required in the above derivation. Also,
(10) does not imply the symmetry of $\tau_{ji}^{\ast}$ unless all the body
torques and couple-stresses are zero: this is seen by applying (3) on the
lengthscale $L=\sqrt[d]{V}$ ($d=2$ or $3$) on which the space averaging is
carried out. This would imply writing $\rho
J_{ij}\frac{D\dot{\varphi}_{j}}{Dt}=\rho
g_{i}+\mu_{ki,k}^{\ast}+\epsilon_{ijk}\tau_{jk}^{\ast}$, where
$\mu_{ki}^{\ast}$ is given by (26) below. In essence, the formula (2.5) alone
is not sufficient to judge whether this force-stress will be symmetric or not.

Taking the trace of (9) we get:

$\int_{\mathcal{B}}\rho
x_{i}f_{i}dV+\int_{\partial\mathcal{B}}x_{i}t_{i}^{\left(\mathbf{n}\right)}dS-\int_{\mathcal{B}}\tau_{ii}dV=\dot{W}_{u}\left(\mathcal{B}\right)-2K_{u}\left(\mathcal{B}\right),$
(12)

where

$W_{u}\left(\mathcal{B}\right):=V\text{ }\overline{\rho x_{i}v_{i}},\text{ \ \
}K_{u}\left(\mathcal{B}\right):=\frac{V}{2}\overline{\rho v_{i}v_{i}},$ (13)

with the latter being the kinetic translational energy.

Performing the infinite-time averaging (6), with the argument given in [Podio-
Guidugli, 2019], the above becomes

$\left\\{\int_{\mathcal{B}}\rho
x_{i}f_{i}dV+\int_{\partial\mathcal{B}}x_{i}t_{i}^{\left(\mathbf{n}\right)}dS-\int_{\mathcal{B}}\tau_{ii}dV\right\\}_{\infty}=-2\left\\{K_{u}\left(\mathcal{B}\right)\right\\}_{\infty},$
(14)

which, in view of Section 3 below, is called the first virial theorem of
micropolar continuum mechanics. The arguments accompanying this infinite time
limit include the assumption of kinematic uniform or periodic boundary
conditions (Costanzo et al., 2005).

Taking the dual vector of (9) we find two relations to be satisfied by the
skew-symmetric part of the Cauchy stress tensor and, essentially, linking all
the quantities appearing in (2) and (3):

$\begin{array}[]{c}\int_{\mathcal{B}}\left(\rho J\ddot{\varphi}_{k}-\rho
g_{k}-\mu_{ik},_{i}\right)dV=\int_{\mathcal{B}}T_{k}dV\\\
=\int_{\mathcal{B}}\rho\epsilon_{kji}x_{j}f_{i}dV+\int_{\partial\mathcal{B}}\epsilon_{kji}x_{j}t_{i}^{\left(\mathbf{n}\right)}dS-\int_{\mathcal{B}}\rho\epsilon_{kji}x_{j}\dot{v}_{i}dV.\end{array}$
(15)

In the special case of micropolar effects in (2)-(3) being absent,
$W_{\varphi}\left(\mathcal{B}\right)=0$ and
$K_{\varphi}\left(\mathcal{B}\right)=0$ in (22), so that (14) simplifies to
the virial theorem of classical continuum mechanics

$\left\\{\int_{\mathcal{B}}\rho
x_{i}f_{i}dV+\int_{\partial\mathcal{B}}x_{i}t_{i}^{\left(\mathbf{n}\right)}dS-\int_{\mathcal{B}}\sigma_{ii}dV\right\\}_{\infty}=-2\left\\{K\left(\mathcal{B}\right)\right\\}_{\infty}.$
(16)

Note: there are various ways for couple-stress effects to arise, typically
involving some internal material structure, and these include:

\- presence of moment interactions in granular media or suspensions, e.g.
[Cowin, 1974; Eringen, 2001; Vardoulakis, 2019];

\- presence of defects, phase boundaries, e.g. [Chen et al., 2006;
Rigelesaiyin et al. 2018];

\- homogenization of random media, e.g. [Trovalusci et al., 2015];

\- fractal media in which the fractal structure (homogenized in the vein of
dimensional regularization) is anisotropic [Li & Ostoja-Starzewski, 2011].

The symmetric part of (9) was discussed in [Podio-Guidugli, 2019].

###  2.2 Rotation as test function

Taking rotation as the test function ($\phi_{j}=x_{j}$) in (7), we obtain

$\int_{\mathcal{B}}\varphi_{j}\rho
f_{i}dV+\int_{\partial\mathcal{B}}\varphi_{j}t_{i}^{\left(\mathbf{n}\right)}dS-\int_{\mathcal{B}}\varphi_{j},_{k}\tau_{ki}dV+\int_{\mathcal{B}}\rho\text{
}\dot{\varphi}_{j}v_{i}dV=\frac{d}{dt}\left(\int_{\mathcal{B}}\rho\varphi_{j}v_{i}dV\right)$
(17)

Multiplying this equation by $J$ yields

$\int_{\mathcal{B}}\rho
J\varphi_{j}f_{i}dV+\int_{\partial\mathcal{B}}J\varphi_{j}t_{i}^{\left(\mathbf{n}\right)}dS-\int_{\mathcal{B}}J\varphi_{j},_{k}\tau_{ki}dV-\left(-\int_{\mathcal{B}}\rho
J\dot{\varphi}_{j}v_{i}dV\right)=\frac{d}{dt}\left(\int_{\mathcal{B}}\rho
J\varphi_{j}v_{i}dV\right).$ (18)

The new term in parenthesis on the left-hand side, $-V\overline{\rho
J\dot{\varphi}_{j}v_{i}}$, will independently arise in Section 3.2.

##  3 Consequences of angular momentum balance

Consider a dyadic product of test function $\phi_{j}$ with the vector (3) and
integrate over $\mathcal{B}$ to get

$\int_{\mathcal{B}}\phi_{j}\left(\rho
g_{i}+\mu_{ki},_{k}+\epsilon_{ink}\tau_{nk}\right)dV=\int_{\mathcal{B}}\phi_{j}\rho
J\ddot{\varphi}_{i}dV.$ (19)

Again, there are two basic options for choosing the test function.

###  3.1 Rotation as test function

On account of the Green-Gauss Theorem and taking rotation as the test function
($\phi_{j}=\varphi_{j}$), we find

$\begin{array}[]{c}\int_{\mathcal{B}}\varphi_{j}\rho
g_{i}dV+\int_{\partial\mathcal{B}}\varphi_{j}m_{i}^{\left(\mathbf{n}\right)}dS-\int_{\mathcal{B}}\varphi_{j},_{k}\mu_{ki}dV+\int_{\mathcal{B}}\varphi_{j}\epsilon_{ink}\tau_{nk}dV\\\
=\frac{d}{dt}\left(\int_{\mathcal{B}}\rho
J\varphi_{j}\dot{\varphi}_{i}dV\right)-\int_{\mathcal{B}}\rho
J\dot{\varphi}_{j}\dot{\varphi}_{i}dV.\end{array}$ (20)

Taking the trace of (20) we obtain (with $T_{i}$ defined in (3)2)

$\int_{\mathcal{B}}\varphi_{i}\rho
g_{i}dV+\int_{\partial\mathcal{B}}\varphi_{i}m_{i}^{\left(\mathbf{n}\right)}dS-\int_{\mathcal{B}}\varphi_{i},_{k}\mu_{ki}dV+\int_{\mathcal{B}}\varphi_{i}T_{i}dV=\dot{W}_{\varphi}\left(\mathcal{B}\right)-2K_{\varphi}\left(\mathcal{B}\right),$
(21)

where

$W_{\varphi}\left(\mathcal{B}\right):=V\text{ }\overline{\rho
J\varphi_{j}\dot{\varphi}_{i}},\text{ \ \
}K_{\varphi}\left(\mathcal{B}\right):=\frac{V}{2}\overline{\rho
J\dot{\varphi}_{i}\dot{\varphi}_{i}},$ (22)

with the latter being the kinetic rotational energy; compare with (13).

Under the infinite-time averaging (6), we find

$\left\\{\int_{\mathcal{B}}\rho\varphi_{i}g_{i}dV+\int_{\partial\mathcal{B}}\varphi_{i}m_{i}^{\left(\mathbf{n}\right)}dS-\int_{\mathcal{B}}\varphi_{i},_{k}\mu_{ki}dV+\int_{\mathcal{B}}\varphi_{i}T_{i}dV\right\\}_{\infty}=-2\left\\{K_{\varphi}\left(\mathcal{B}\right)\right\\}_{\infty},$
(23)

which we call the second virial theorem of micropolar continuum mechanics.

###  3.2 Translation as test function

Considering $\phi_{j}=x_{j}$ in (19) we get:

$\begin{array}[]{c}\int_{\mathcal{B}}x_{j}\rho
g_{i}dV+\int_{\partial\mathcal{B}}x_{j}m_{i}^{\left(\mathbf{n}\right)}dS-\int_{\mathcal{B}}\mu_{ji}dV+\int_{\mathcal{B}}x_{j}\epsilon_{ink}\tau_{nk}dV=\\\
\frac{d}{dt}\left(\int_{\mathcal{B}}\rho
Jx_{j}\dot{\varphi}_{i}dV\right)-\int_{\mathcal{B}}\rho
J\dot{x}_{j}\dot{\varphi}_{i}dV,\end{array}$ (24)

which can be written as

$\int_{\mathcal{B}}x_{j}\rho
g_{i}dV+\int_{\partial\mathcal{B}}x_{j}m_{i}^{\left(\mathbf{n}\right)}dS-V\mu_{ji}^{\ast}+\int_{\mathcal{B}}x_{j}T_{i}dV=\frac{d}{dt}\left(\int_{\mathcal{B}}\rho
Jx_{j}\dot{\varphi}_{i}dV\right),$ (25)

where

$\mu_{ji}^{\ast}:=\overline{\mu_{ji}}-\overline{\rho J\dot{\varphi}_{j}v_{i}}$
(26)

is the virial couple-stress. As can be seen from the Reynolds decomposition of
$\varphi_{j}$ and $v_{i}$, the term $-\overline{\rho J\dot{\varphi}_{j}v_{i}}$
contains

$S_{ji}:=-\overline{\rho J\dot{\varphi}_{j}^{\prime}v_{i}^{\prime}}.$ (27)

which was called the turbulent couple-stress in [Ostoja–Starzewski, 2021]. No
microisotropy restriction was assumed there, so the formula for $S_{ji}$ was a
bit more general. This couple-stress may be thought of as an analogue of the
Reynolds stress $-\overline{\rho v_{j}^{\prime}v_{i}^{\prime}}$ brought up in
Section 2.1.

Interestingly, $-\overline{\rho J\dot{\varphi}_{j}v_{i}}$ appeared via a
completely different route in (18). As discussed in [Ostoja–Starzewski, 2021],
from the standpoint of turbulent flows, $\mu_{ji}^{\ast}$ is the modified
stress in the sense that, if it replaces $\mu_{ji}$ along with the heat flux
and internal energy density being also replaced by modified expressions, then
the angular momentum and energy balance equations are unchanged in form upon
volume averaging.

Again, going from continuum to discrete systems such as the molecular/particle
dynamics, (26) gives a prescription for computation of the instantaneous
volume-averaged virial couple-stress

$\mu_{ji}^{\ast}:=\frac{1}{V}\sum_{k\in\mathcal{B}}\left[\frac{1}{2}\sum_{l\in\mathcal{B}}\left(x_{i}^{\left(l\right)}-x_{i}^{\left(k\right)}\right)m_{j}^{\left(kl\right)}-J^{\left(k\right)}\left(\dot{\varphi}_{j}^{\left(k\right)}-\overline{\dot{\varphi}_{j}}\right)\left(v_{i}^{\left(k\right)}-\overline{v_{i}}\right)\right],$
(28)

where

$k$ and $l$ are particles in the domain $\mathcal{B}$,

$J^{\left(k\right)}$ is the mass moment of inertia of particle $k$,

$v_{j}^{\left(k\right)}$ is the $j$-th component of the velocity of particle
$k$,

$\overline{v_{j}}$ is the $j$-th component of the average velocity of
particles in the volume,

$\dot{\varphi}_{i}^{\left(k\right)}$ is the $i$-th component of the velocity
of particle $k$,

$\overline{\dot{\varphi}_{i}}$ is the $j$-th component of the average velocity
of particles in the volume,

$x_{i}^{\left(k\right)}$ is the $i$-th component of the position of particle
$k$, and

$m_{j}^{\left(kl\right)}$ is the $j$-th component of the moment applied on
particle $k$ by particle $l$.

Note the analogy between the formulas (28) and (11).

##  4 Conclusions

This communication generalizes the virial theorem and virial stress to
micropolar continuum mechanics. The study hinges on exploring the dyadic
products of linear and angular momentum balances with translational and
rotational test functions. In generic terms, the results are summarized as
follows.

  * •

The linear momentum balance with translation leads to:

(i) A virial theorem known in classical continuum theories involving the
infinite-time limit of the kinetic translational energy. It is called the
first virial theorem of micropolar continuum mechanics.

(ii) A classical formula for computing the virial force-stress known in
molecular dynamics.

  * •

The angular momentum balance with rotation leads to a new virial theorem
involving the infinite-time limit of the kinetic rotational energy. It is
called the second virial theorem of micropolar continuum mechanics.

  * •

The angular momentum balance with translation leads to a virial couple-stress
along with a formula for its computation. The same stress arises independently
in the linear momentum balance with rotation.

  * •

The virial couple-stress contains the turbulent couple-stress, just like the
virial force-stress contains the Reynolds stress.

  * •

The virial force-stress and couple-stress formulas do not require ‘infinitely’
large volumes (on the RVE level) and can be used to assess tensor-valued
random fields on mesoscale levels as in [Ostoja-Starzewski and Laudani [2020].

These results have applicability in computational studies as well as a
stepping-stone to micropolar extensions of various complex phenomena such as
presented in, say, [Ganghoffer, 2015; Chen and Diaz, 2016, 2018].

Acknowledgment

Comments by two anonymous reviewers are gratefully acknowledged.

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# The Superconductivity of Sr2RuO4 Under c-Axis Uniaxial Stress

Fabian Jerzembeck  Max Planck Institute for Chemical Physics of Solids,
Nöthnitzer Str 40, 01187 Dresden, Germany  Henrik S. Røising  Nordita, KTH
Royal Institute of Technology and Stockholm University, Hannes Alfvéns väg 12,
SE-106 91 Stockholm, Sweden  Alexander Steppke  Max Planck Institute for
Chemical Physics of Solids, Nöthnitzer Str 40, 01187 Dresden, Germany  Helge
Rosner  Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Str
40, 01187 Dresden, Germany  Dmitry A. Sokolov  Max Planck Institute for
Chemical Physics of Solids, Nöthnitzer Str 40, 01187 Dresden, Germany  Naoki
Kikugawa  National Institute for Materials Science, Tsukuba 305-0003, Japan
Thomas Scaffidi  Department of Physics, University of Toronto, Toronto,
Ontario, M5S 1A7, Canada  Steven H. Simon  Rudolf Peierls Center for
Theoretical Physics, Oxford OX1 3PU, United Kingdom  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, United Kingdom
Clifford W. Hicks  School of Physics and Astronomy, University of Birmingham,
Birmingham B15 2TT, United Kingdom  Max Planck Institute for Chemical Physics
of Solids, Nöthnitzer Str 40, 01187 Dresden, Germany

###### Abstract

We compress the unconventional superconductor Sr2RuO4 along its $c$ axis,
achieving a maximum stress of 3.2 GPa, a record uniaxial stress for single-
crystal Sr2RuO4. The energy of the Ru $4d$ $xz$ and $yz$ bands is raised
relative to the $xy$ band, which causes the largest Fermi surface sheet to
expand. We present evidence that 3.2 GPa is about halfway to an electron-to-
hole Fermi surface topological transition of this sheet. In line with
qualitative expectations for increasing density of states, the upper critical
field increases. However, $T_{\text{c}}$ decreases. We present calculations of
$T_{\text{c}}$ in the limit of weak interactions that indicate that decreasing
$T_{\text{c}}$ could be obtained for order parameters that have nodes along
the $\Gamma$-$X$ and $\Gamma$-$Y$ lines, but this conclusion is opposite to
that reached from measurements under in-plane uniaxial stress. The data here
therefore provide an important new constraint on theories of the
superconductivity of Sr2RuO4, and the demonstration of a method to weaken the
pairing interaction in Sr2RuO4 may provide a clue as to what the pairing
interaction is.

##  I Introduction

Sr2RuO4 is a famous exemplar of unconventional superconductivity, due to the
quality of the available samples and the precision of knowledge about its
normal state, and because the origin of its superconductivity remains
unexplained in spite of strenuous effort Maeno  _et al._ (1994); Mackenzie and
Maeno (2003); Maeno  _et al._ (2012); Mackenzie  _et al._ (2017). No proposed
order parameter is able straightforwardly to account for all experimental
results. The greatest conundrum is posed by evidence that the order parameter
combines even parity with time reversal symmetry breaking, with evidence for
even parity provided by a drop in the spin susceptibility below $T_{\text{c}}$
Pustogow  _et al._ (2019); Ishida  _et al._ (2020); Chronister  _et al._
(2021); Petsch  _et al._ (2020) and Pauli-like limiting of the in-plane upper
critical field Yonezawa  _et al._ (2013); Kittaka  _et al._ (2014), and
evidence for time reversal symmetry breaking by muon spin rotation Luke  _et
al._ (1998); Grinenko  _et al._ (2021a) and Kerr rotation Xia  _et al._
(2006). Further discussion is given in Refs. Mackenzie  _et al._ (2017);
Kivelson  _et al._ (2020). The difficulty in combining even parity and time
reversal symmetry breaking is that, without fine tuning, the implied order
parameter is $d_{xz}\pm id_{yz}$ Grinenko  _et al._ (2021b). The horizontal
line node at $k_{z}=0$ implies, under conventional understanding, interlayer
pairing, which is unlikely because the electronic structure of Sr2RuO4 is
highly two-dimensional Bergemann  _et al._ (2003); Ohmichi  _et al._ (2000).

This puzzle has led to substantial theoretical activity. Two recent proposals
are $s\pm id$ Rømer  _et al._ (2019, 2021) and $d\pm ig$ Kivelson  _et al._
(2020); Wagner  _et al._ (2021) order parameters, which require tuning to
obtain $T_{\text{TRSB}}\approx T_{\text{c}}$ (where $T_{\text{TRSB}}$ is the
time reversal symmetry breaking temperature) on the tetragonal lattice of
Sr2RuO4, but avoid horizontal line nodes. A mixed-parity state Scaffidi (2020)
and superconductivity that breaks time reversal symmetry only in the vicinity
of extended defects Willa  _et al._ (2021) have been proposed to account for
the absence of a resolvable heat capacity anomaly at $T_{\text{TRSB}}$ Li  _et
al._ (2021). Interorbital pairing through Hund’s coupling is also under
discussion Suh  _et al._ (2020); Clepkens  _et al._ (2021); Gingras  _et al._
(2019); Ramires and Sigrist (2019); this mechanism could yield $d_{xz}\pm
id_{yz}$ order without interlayer pairing. Thermal conductivity and
quasiparticle interference data, on the other hand, have been interpreted as
evidence for a single-component, $d_{xy}$ or $d_{x^{2}-y^{2}}$ gap Sharma  _et
al._ (2020); Hassinger  _et al._ (2017).

The electronic structure of Sr2RuO4 is tunable through in-plane uniaxial
stress, providing a further avenue for probing its superconductivity. Under
uniaxial stress along the $[100]$ lattice direction the largest Fermi surface
sheet (the $\gamma$ sheet— see Fig. 1) transitions from an electron-like to an
open topology Sunko  _et al._ (2019). $T_{\text{c}}$ increases from 1.5 K in
unstressed Sr2RuO4 to 3.5 K at this transition, while the $c$-axis upper
critical field $H_{\text{c2}}$ increases by a factor of twenty Steppke  _et
al._ (2017). This strong enhancement in $H_{\text{c2}}$ is further evidence
for even parity, because it is difficult to obtain without a gap that is large
at the point in $k$-space where this Lifshitz transition occurs, the $Y$
point. In a two-dimensional picture this point is parity-invariant, so the gap
of odd-parity order parameters must vanish there.

These effects of in-plane stress suggest that $T_{\text{c}}$ could rise even
further under strong compression along the $c$ axis. $c$-axis compression
raises the energy of the $xz$ and $yz$ bands relative to the $xy$ band, and
expands the $\gamma$ sheet. This sheet is expected eventually to undergo a
transition from an electron-like to a hole-like geometry. Because this
transition occurs at two points in $k$-space [the $X$ and $Y$ points— see Fig.
1(c-d)], the spike in the Fermi-level density of states (DOS) is expected to
be stronger than for in-plane stress, and $H_{\text{c2}}$ and $T_{\text{c}}$
might therefore increase correspondingly further. Alternatively, there could
be a transition into a magnetically ordered state. The electron-to-hole
Lifshitz transition has been studied theoretically Hsu  _et al._ (2016); Liu
_et al._ (2018); Facio  _et al._ (2018); František  _et al._ (2019); Rømer
_et al._ (2019), and Ref. Liu  _et al._ (2018) predicts a transition to spin
density wave order. In cubic BaRuO3, proximity of the Fermi level to a Van
Hove singularity in the DOS induces ferromagnetism Jin  _et al._ (2008); Han
_et al._ (2016). The electron-to-hole Lifshitz transition has been achieved in
Sr2RuO4 through epitaxial strain Burganov  _et al._ (2016), and substitution
of La for Sr Kikugawa  _et al._ (2004a, b); Shen  _et al._ (2007). However, in
both cases the superconductivity was suppressed by disorder.

We demonstrate here elastic compression by 3.2 GPa, a record for bulk Sr2RuO4,
and present evidence that this is roughly halfway to the electron-to-hole
Lifshitz transition. $H_{\text{c2}}$ increases, following general expectations
for increasing Fermi-level DOS. However, in striking contrast to the effect of
in-plane uniaxial stress, $T_{\text{c}}$ decreases, which provides a major new
constraint on theories of the superconductivity of Sr2RuO4.

##  II Electronic structure calculations

We start with density functional theory calculations of the band structure of
Sr2RuO4 under $c$-axis compression. Unstrained lattice parameters were taken
from the $T=15$ K data of Ref. Chmaissem  _et al._ (1998). Longitudinal strain
$\varepsilon_{zz}$ is taken as the independent variable, and
$\varepsilon_{xx}$ and $\varepsilon_{yy}$ are set following the low-
temperature Poisson’s ratio from Ref. Ghosh  _et al._ (2021), which is $0.223$
for stress along the $c$ axis. Calculations were performed as described in
Ref. Steppke  _et al._ (2017). We note in particular that spin-orbit coupling
was treated nonperturbatively by solving the four-component Kohn-Sham-Dirac
equation Eschrig  _et al._ (2004), the calculation was done in the local
density approximation, and, due to proximity of a Van Hove singularity to the
Fermi level, calculations were carried out on a fine-scale mesh in $k$ space.
The apical oxygen position was relaxed.

Results are shown in Fig. 1. The electron-to-hole Lifshitz transition is
predicted to occur at $\varepsilon_{zz}=-0.025$ (where negative values denote
compression). Low-temperature ultrasound data give a $c$-axis Young’s modulus
of 219 GPa Ghosh  _et al._ (2021), so this corresponds to stress
$\sigma_{zz}\approx-5.5$ GPa. Because meV-level energy shifts can
substantially alter the distance to the Lifshitz transition, there is
considerable uncertainty in these values Barber  _et al._ (2019a). The
transition occurs approximately at the $X$ and $Y$ points of the Brillouin
zone of the RuO2 sheet, indicated in panel (c). We note also that while
$k_{z}$ warping increases on all the Fermi sheets, as expected for $c$-axis
compression, the $\beta$ sheet has the strongest $k_{z}$ warping both at
$\varepsilon_{zz}=0$ and at the Lifshitz transition.

Figure 1: (a) DFT calculation of the Fermi-level DOS against energy for a
series of strains $\varepsilon_{zz}$. (b) Calculated Fermi-level DOS against
$\varepsilon_{zz}$. (c) and (d) Calculated Fermi surfaces of Sr2RuO4 at,
respectively, $\varepsilon_{zz}=0$ and $-0.025$. The sheets are projected onto
the $k_{x}$-$k_{y}$ plane, so the width of the lines shows the magnitude of
warping along the $k_{z}$ axis. The first Brillouin zone of the RuO2 sheet is
indicated by the green dashed line, and the $X$ and $Y$ points of this zone
are also indicated.

##  III Experimental methods

Sr2RuO4 samples were grown using a floating-zone method Mao  _et al._ (2000);
Bobowski  _et al._ (2019). Four samples were measured. All were taken from the
same original rod, and from a portion where sharp superconducting transitions
were observed in field sweeps at $T\approx 0.22$ K, indicating a low density
of Ru inclusions Maeno  _et al._ (1998). All had $T_{\text{c}}$ between 1.45
and 1.50 K.

Figure 2: (a) Schematic of the sample configuration for samples 2 and 3. The
sample ends are embedded in epoxy. (b) Schematic of the sample carrier
employed for samples $2$ and $3$. The cap blocks incorporate slots into which
the sample fits. The sample is compressed by bringing parts A and B into
contact. (c) $T_{\text{c}}$ versus applied displacement for samples 2 and 3.
When the mechanical contact between carrier parts A and B opens, the force on
the sample falls to approximately zero, and $T_{\text{c}}$ stops changing.

Uniaxial stress was applied using piezoelectric-driven apparatus Hicks  _et
al._ (2014); Barber  _et al._ (2019b). There are technical challenges in
applying $c$-axis stress. Inadvertent in-plane strain must be minimized: we
will show that $T_{\text{c}}$ decreases by $0.13$ K under a $c$-axis stress of
$\sigma_{zz}=-3.0$ GPa, while an increase of $T_{\text{c}}$ by $0.13$ K is
obtained from an in-plane uniaxial stress of only $0.2$ GPa Steppke  _et al._
(2017). Applied $c$-axis pressure could generate in-plane stress through
bending and/or sample inhomogeneity. In a previous experiment Kittaka  _et
al._ (2010), $c$-axis compression raised $T_{\text{c}}$ and broadened the
transition. However, the stress was applied at room temperature, where the
elastic limit of Sr2RuO4 is low Barber  _et al._ (2019b), so these effects may
have been a consequence of in-plane strain due to defects introduced by the
applied stress.

To obtain high stress homogeneity, samples should be long along the stress
axis. But Sr2RuO4 cleaves easily along the $ab$ plane, so it is a challenge to
prepare samples elongated along the $c$ axis. Here, we employed a plasma
focused ion beam (PFIB), in which material is milled using a beam of Xe ions.
Sample $1$ was prepared with a uniform cross section, and a large enough
stress, $\sigma_{zz}=-0.84$ GPa, was achieved to observe a clear change in
$T_{\text{c}}$. To go further, the other samples were all sculpted into
dumbell shapes, as shown in Fig. 2(a). The wide ends provide large surfaces
for coupling force into the sample. For measurement of $T_{\text{c}}$ in the
neck portion, two concentric coils of a few turns each were wound around the
neck. Samples 1 and 4 also had electrical contacts, for measurement of the
$c$-axis resistivity $\rho_{zz}$. All samples were mounted using Stycast
$2850$ epoxy; the epoxy constitutes a conformal layer that ensures even
application of stress Hicks  _et al._ (2014).

Samples 2–4 were mounted into two-part sample carriers; that for samples 2 and
3 is diagrammed in Fig. 2(b). The purpose was to protect samples from
inadvertent application of tensile stress. Samples are mounted across a gap
between a fixed and a moving portion of part B of the carrier, and can be
compressed, but not tensioned, by bringing part A into contact with part B. In
Fig. 2 (c), we show $T_{\text{c}}$ of samples $2$ and $3$ versus applied
displacement, and the point where parts A and B come into contact and
$T_{\text{c}}$ starts changing is clearly visible. For sample 2 the point of
contact is rounded on the scale of a few microns, due to roughness and/or
misalignment of the contact faces, and in all figures below we exclude data
points that we estimate to be affected by this rounding.

For samples 1–3 a stress cell was used that had a sensor only of the
displacement applied to the sample. Conversion from applied displacement to
sample strain involves large uncertainties, so sample 4 was measured using a
cell that also had a sensor of the applied force Barber  _et al._ (2019b); the
stress is the force divided by cross-sectional area. To present data from all
four samples together, we apply displacement-to-stress conversions for samples
1–3 that bring the rate of change of $T_{\text{c}}$ over the stress range
$-0.92$ to $-0.20$ GPa into agreement with that observed in sample 4. In other
words, we impose on our data an assumption that $dT_{\text{c}}/d\sigma_{zz}$
for all four samples is the same over this stress range, which is reasonable
because their $T_{\text{c}}$’s are so similar. The zero-stress $T_{\text{c}}$
could be measured accurately for samples 2–4 by bringing carrier parts A and B
out of contact, and for sample 1 by deliberately fracturing the sample under
tension.

##  IV Experimental results

We begin by showing resistivity data, in Fig. 3. The plotted resistivities are
corrected for the expected stress-induced change in sample geometry (reduced
length and increased width) under an assumption that stress and strain remain
proportional, and using the low-temperature elastic moduli reported in Ref.
Ghosh  _et al._ (2021). At zero stress the resistivity of sample 4 shows a
sharp transition into the superconducting state at $1.55$ K. This sharpness,
and the fact that it only slightly exceeds the transition temperature seen in
susceptibility, indicate high sample quality. With compression, $T_{\text{c}}$
decreases. The normal-state resistivity also decreases, following the general
expectations that $c$-axis compression should increase $k_{z}$ dispersion.

We find elastoresistivities $(1/\rho_{zz})d\rho_{zz}/d\varepsilon_{zz}$,
obtained with a linear fit over the range $-0.5<\sigma_{zz}<0$ GPa, of 37 and
32 for samples 1 and 4, respectively. Sample 4 was compressed to $-1.7$ GPa,
and its resistivity does not show any major deviation from linearity over this
range. There is some scatter in the data at large compression, which may be a
consequence of cracking in the electrical contacts— we show below that the
sample deformation was almost certainly elastic.

Figure 3: Main panel: $c$-axis resistivity $\rho_{zz}$ versus stress
$\sigma_{zz}$ at 1.9 K, normalized by its $\sigma_{zz}=0$ value. Note that the
stress scale of sample $1$ is adjusted so that $dT_{c}/d\sigma_{zz}$ as
measured through the Meissner effect matches that from sample 4 over the range
$-0.92<\sigma_{zz}<-0.2$ GPa. At $\sigma_{zz}=0$, $\rho_{zz}(1.9~{}\text{K})$
of samples 1 and 4 is $0.278$ and $0.228$ m$\Omega$-cm, respectively. Inset:
$\rho_{zz}$ versus temperature for sample 4 at 0, $-0.37$, $-0.85$, $-1.25$,
and $-1.64$ GPa. Figure 4: (a–c) Mutual inductance $M$ of the sense coils
versus temperature for samples 2–4, at various applied stresses $\sigma_{zz}$.
The lines indicate the selected thresholds for determination of
$T_{\text{c}}$. For sample $2$, under large $|\sigma_{zz}|$ a high-temperature
tail appears on the transitions, which we attribute to in-plane stress, and a
lower threshold is chosen to avoid this tail. (d) $T_{\text{c}}$ versus stress
for all the samples. The stress scales for samples 1–3 are scaled to bring
$dT_{\text{c}}/d\sigma_{zz}$ into agreement with that of sample 4 over the
stress range labeled “fitted range,” $-0.92$ to $-0.20$ GPa.

In Fig. 4 we show the dependence of $T_{\text{c}}$ measured through
susceptibility on $c$-axis stress, for all four samples. Panels (a–c) show the
actual transitions — the mutual inductance $M$ of the sense coils versus
temperature — for samples 2–4. To check that sample deformation remained
elastic, we repeatedly cycled the stress to confirm that the form of the
$M(T)$ curves remained unchanged; see the Appendix for examples. For samples 3
and 4, the transition remained narrow as stress was applied, indicating high
stress homogeneity. For sample 2, there was a tail on the high-temperature
side of the transition, that was stronger at higher compressions. We attribute
it to in-plane strain, possibly originating in the fact that sample 2 was not
as well aligned as samples 3 and 4. A similar, though weaker, tail is also
visible for sample 3.

For sample 3, some non-elastic compression of the epoxy was observed. The data
shown in Fig. 4 are therefore those taken as the stress was released, after
the epoxy had been maximally compressed. The full data set is shown in
Appendix Fig. 8.

Panel (d) shows $T_{\text{c}}$ versus stress for all the samples.
$T_{\text{c}}$ is taken as the temperature where $M$ crosses a threshold. For
samples 1, 3, and 4, we select a threshold at $\approx$50% of the height of
the transition, and for sample 2, 20%, in order to minimize the influence from
the high-temperature tail. $T_{\text{c}}$ is seen to decrease almost linearly
out to $\sigma_{zz}\approx-1.8$ GPa. For sample 4 (to which, as described
above, the other samples are referenced), $dT_{\text{c}}/d\sigma_{zz}$ in the
limit $\sigma_{zz}\rightarrow 0$ is $76\pm 5$ mK/GPa. The error is 6%: we
estimate a 5% error on the calibration of the force sensor of the cell, and a
3% error on the cross-sectional area of the sample ($155\times 106$ $\mu$m2).

At $\sigma_{zz}\lesssim-1.8$ GPa, the stress dependence of $T_{\text{c}}$
flattens markedly. In sample 3, $T_{\text{c}}$ is seen to resume its decrease
for $\sigma_{zz}<-3$ GPa. We show in Appendix Fig. 8 that both the flattening
and this further decrease reproduce when the stress is cycled, which, in
combination with the narrowness of the transitions, shows that this behavior
is intrinsic, and not an artefact of any drifts or non-elastic deformation in
the system.

Figure 5: (a–b) Susceptibility versus applied field $\parallel c$ for samples
$2$ and $3$, at fixed temperatures of $300$ mK and $310$ mK. For sample $3$,
the raw data are plotted. For sample $2$, the signal magnitude shifted from
run to run, so data are normalized by the readings at $0$ and $100$ mT. This
shift was probably due to motion of the coils against the sample; when they
were fixed more securely to collect the data in Fig. 4(a), the problem
disappeared. The horizontal lines indicate the thresholds for determination of
$H_{\text{c2}}$. (c) $H_{\text{c2}}(T\approx 0.3~{}\text{K})$ versus stress
for samples 2 and 3. (d) $H_{\text{c2}}/T_{\text{c}}^{2}$, normalized by its
zero-stress value, versus stress for sample $3$.

We noted in the Introduction the possibility of stress-driven transition into
magnetic order. However, we observe no spikes in the sense coil mutual
inductance at $T>T_{\text{c}}$ that could indicate transition into a
ferromagnetic state. The coils would not have had enough sensitivity to detect
transition into an antiferromagnetic state. The small stress dependence that
is visible at $T>T_{\text{c}}$ is likely to be an artefact of a shift in coil
geometry as stress is applied.

Fig. 5 shows measurements of the $c$-axis upper critical field. $M(H)$ for
samples 2 and 3 at constant temperature $T\approx 0.3$ K is shown in panels
(a) and (b). In Fig. 5(c), we plot $H_{\text{c2}}$ versus stress, taking
$H_{\text{c2}}$ as the fields at which $M$ crosses the thresholds indicated in
panels (a–b). $H_{c2}$ is seen to increase as stress is applied, as generally
expected when the density of states increases. The increase is faster for
sample $2$ than sample $3$, which may be an artefact of the tail on the
transition for sample 2.

For an isotropic system,
$H_{\text{c2}}\propto(T_{\text{c}}/v_{\text{F}})^{2}$, where $v_{\text{F}}$ is
the Fermi velocity, and so in panel (d) we plot
$H_{\text{c}}/T_{\text{c}}^{2}$ normalized by its zero-stress value for sample
3. It increases by $\approx$40% by $\sigma_{zz}=-3.0$ GPa, which, if the gap
structure does not change drastically, suggests an increase in the Fermi-level
DOS of $\approx$20%.

Another feature visible in the $M(H)$ traces of Fig. 5(a–b) is a peak effect —
a local maximum in the susceptibility just below $H_{\text{c2}}$. It occurs
when there is a range of temperature below $T_{\text{c}}$ where vortex motion
is uncorrelated, allowing individual vortices to find deeper pinning sites
Yamazaki  _et al._ (2002). The peak is suppressed by $c$-axis compression, and
it is suppressed downward rather than by being smeared horizontally along the
$H$ axis, meaning that it is not an artefact of a spread of $H_{\text{c2}}$
due to strain inhomogeneity. It could indicate stronger pinning, due to the
reduction in the coherence length.

##  V Weak-coupling calculations

We supplement these results with weak-coupling calculations for repulsive
Hubbard models, as developed in Refs. Kohn and Luttinger (1965); Baranov and
Kagan (1992); Kagan and Chubukov (1989); Chubukov and Lu (1992); Baranov  _et
al._ (1992); Fukazawa and Yamada (2002); Hlubina (1999); Raghu  _et al._
(2010a, b); Scaffidi  _et al._ (2014). We employ three-dimensional Fermi
surfaces, because the two-dimensional form of the Fermi surfaces of Sr2RuO4
masks substantial variation of orbital content along $k_{z}$ Veenstra  _et
al._ (2014), that could cause substantial gap variation along $k_{z}$ Røising
_et al._ (2019). $c$-axis compression will modify this $k_{z}$ structure, so
the effect on superconductivity could be substantial.

For the band structure we employ a tight-binding model for the three bands
that cross the Fermi level, which takes the form

$H_{0}=\sum_{\bm{k},s}\bm{\psi}^{\dagger}_{s}(\bm{k})\mathcal{H}_{s}(\bm{k})\bm{\psi}_{s}(\bm{k}).$
(1)

$\bm{\psi}_{s}(\bm{k})=\left[c_{xz,s}(\bm{k}),c_{yz,s}(\bm{k}),c_{xy,\bar{s}}(\bm{k})\right]^{T}$,
and $\mathcal{H}_{s}(\bm{k})$ incorporates spin-orbit coupling, inter-orbital
and intra-orbital terms as extracted from the DFT calculations. The complete
set of tight-binding parameters retained here is given in the Appendix. In
Fig. 6(a), we show the tight-binding Fermi surfaces at $\varepsilon_{zz}=0$
and $-0.02$. In Fig. 6(b), we show the orbital weight on the $\gamma$ sheet at
$k_{z}=0$. Because the $\gamma$ sheet expands under $c$-axis compression, the
orbital mixing around its avoided crossings with the $\beta$ sheet is reduced,
and it becomes more dominated by $xy$ orbital weight.

Figure 6: (a) Cuts through the tight-binding Fermi surfaces employed in our
weak-coupling calculations, at $\varepsilon_{zz}=0$ and $-0.02$. The surfaces
at $\varepsilon_{zz}=-0.02$ are colored by orbital content. (b) Orbital
weights on the $\gamma$ sheet at $k_{z}=0$ in this model. (c) Eigenvalues as a
function of $\varepsilon_{zz}$ for $J/U=0.15$. In the legend, $f_{i}$ is any
function that transforms as $\sin k_{i}$; $d_{0}(\mathbf{k})$ is the gap
function for the even-parity irreducible representations, and
$\mathbf{d}(\mathbf{k})$ the $d$-vector for $E_{u}$. In the $A_{2g}$ channel,
there is a transition in the gap structure between $\varepsilon_{zz}=0$ and
$-0.0075$. (d) $H_{\text{c2}}(T\rightarrow 0)/T_{\text{c}}^{2}$ versus
$\varepsilon_{zz}$ in each channel. (e) Gap structure in the $B_{\text{2g}}$
channel.

To $H_{0}$ we add on-site Coulomb terms projected onto the $t_{2g}$ orbitals
Dagotto  _et al._ (2001) (Appendix Eq. 7) and study the solutions to the
linearized gap equation in the weak-coupling limit $U/t\ll 1$, where $U$ is
the intraorbital Coulomb repulsion and $t$ is the leading tight-binding term.
We take the interorbital on-site Coulomb repulsion to be $U^{\prime}=U-2J$,
where $J$ is the Hund’s coupling, and the pair-hopping Hund’s interaction
$J^{\prime}$ to be equal to the spin-exchange Hund’s interaction $J$. Under
these assumptions, the remaining free parameter is $J/U$. We take $J/U=0.15$,
which is close to the value $J/U=0.17$ found in Refs. Mravlje  _et al._
(2011); Tamai  _et al._ (2019). The linearized gap equation reads

$\sum_{\nu}\int_{S_{\nu}}\frac{\mathrm{d}\bm{k}_{\nu}}{\lvert
S_{\nu}\rvert}\bar{\Gamma}(\bm{k}_{\mu},\bm{k}_{\nu})\varphi(\bm{k}_{\nu})=\lambda\varphi(\bm{k}_{\mu}),$
(2)

where $\mu$ and $\nu$ are band indices, $\lvert S_{\nu}\rvert$ is the area of
Fermi surface sheet $\nu$, and $\bar{\Gamma}$ is the two-particle interaction
vertex calculated consistently to order $\mathcal{O}(U^{2}/t^{2})$. Solutions
to Eq. (2) with $\lambda<0$ signal the onset of superconductivity, at the
critical temperature $T_{c}\sim W\exp(-1/\lvert\lambda\rvert)$, where $W$ is
the bandwidth.

In a pseudo-spin basis each eigenvector $\varphi$ belongs to one of the ten
irreducible representations of the crystal point group $D_{4h}$ Sigrist and
Ueda (1991); Raghu  _et al._ (2010a). We calculate the leading eigenvalues in
four even-parity channels, $B_{1g}$, $B_{2g}$, $A_{1g}$, and $A_{2g}$ — see
the legend of Fig. 6(c–d). The $E_{g}$ channel — $d_{xz}\pm id_{yz}$ — has
been found to be strongly disfavored in weak-coupling calculations Røising
_et al._ (2019), and so is not considered here. We calculate only one odd-
parity channel, $E_{u}$. The splitting among the odd-parity channels has been
found to be small in comparison with that between odd- and even-parity states
for reasonable values of $J/U$ and spin-orbit coupling Wang  _et al._ (2020).

The leading eigenvalues in each channel as a function of $\varepsilon_{zz}$
are shown in Fig. 6(c). Although, as in Ref. Røising  _et al._ (2019), odd-
parity order is found to be favored, calculations in the random phase
approximation at similar $J/U$ tend to favor even-parity order Rømer  _et al._
(2019); Wang  _et al._ (2020). A tendency towards odd-parity order appears to
be a feature of calculations in the weak-coupling limit.

The weak-coupling results show a dichotomy in the strain dependence of
$T_{\text{c}}$: $T_{\text{c}}$ in the channels that have symmetry-imposed
nodes at the $X$ and $Y$ points ($E_{u}$, $A_{2g}$, and $B_{2g}$) decreases
with initial $c$-axis compression. These nodes coincide with the regions of
highest local density of states, and this result is an indication that order
parameters in these channels are less able to take advantage of the increase
in density of states induced by $c$-axis compression. We note, however, that
under stronger compression $T_{\text{c}}$ increases modestly in all the
channels.

We also calculate $H_{\text{c2}}$, following the procedure described in Ref.
Steppke  _et al._ (2017). Results are shown in Fig. 6(d). $H_{\text{c2}}$ in
the $A_{2g}$ channel increases very substantially, due to a transition in the
gap structure to one dominated by weight on the $\gamma$ sheet, where the
average Fermi velocity is smallest. Setting aside this feature in the results,
a dichotomy between the even-parity channels with ($A_{2g}$ and $B_{2g}$) and
without ($A_{1g}$ and $B_{1g}$) symmetry-imposed nodes along the $\Gamma$-$X$
and $\Gamma$-$Y$ lines is apparent only at the strongest compressions:
$H_{\text{c2}}/T_{\text{c}}^{2}$ in the $A_{2g}$ and $B_{2g}$ channels
decreases between $\varepsilon_{zz}=-0.015$ and $-0.020$, while that in the
$A_{1g}$ and $B_{1g}$ channels increases. In Fig. 6(e), we show the calculated
gap structure in the $B_{2g}$ channel; that for the other even-parity channels
is shown in Appendix Fig. 9. It can be seen that
$H_{\text{c2}}/T_{\text{c}}^{2}$ tracks the gap weight on the $\gamma$ sheet,
which also increases then decreases with compression. In general, the
increasing density of states on the $\gamma$ sheet under $c$-axis compression
favors a shift in the superconducting gap towards this sheet, and this shift
is seen to occur for all the even-parity channels. The increase in
$H_{\text{c2}}$ in the $B_{1g}$ channel is small because it starts off heavily
weighted towards the $\gamma$ sheet. In the $A_{2g}$ and $B_{2g}$ channels,
the gap changes sign upon reflection across the zone boundary of the RuO2
sheet, and so as the $\gamma$ sheets in adjacent zones approach each other the
gap is expected to be suppressed in the $\gamma$ sheet. It appears that this
suppression only becomes substantial for $|\varepsilon_{zz}|$ larger than
$\sim 0.015$.

##  VI Discussion

We have shown that the response of the superconductivity of Sr2RuO4 to
compression along the $c$ axis contrasts sharply with that along the $a$ axis:
although the Fermi-level DOS increases in both cases, $T_{\text{c}}$ decreases
under $c$-axis compression. We begin the Discussion with an estimate of how
far along the way to the electron-to-hole Lifshitz transition we reached; from
the absence of any sharp feature in $T_{\text{c}}$ or $H_{\text{c2}}$, it is
almost certain that we did not reach this transition. $\sigma_{zz}=-3.2$ GPa
corresponds to strain $\varepsilon_{zz}\approx-0.014$, about 60% of the
calculated Lifshitz strain, $-0.025$.

We have noted however that there is considerable uncertainty in the calculated
value, and so we look also at measured physical quantities. As noted above,
$H_{\text{c2}}$ data suggest an increase in the Fermi-level DOS of around 20%
between $\sigma_{zz}=0$ and $-3.0$ GPa. In the DFT calculation, the increase
in the DOS reaches 20% at $\varepsilon_{zz}\approx-0.017$, around 70% of the
way to the Lifshitz transition. The $c$-axis resistivity also allows an
estimate. $c$-axis conductivity is proportional to the square of the amplitude
of $k_{z}$ warping. We observe $\rho_{zz}$ to fall by 14–17% between
$\sigma_{zz}=0$ and $-1.0$ GPa, implying an increase in warping amplitude of
8–10%. The $\beta$ sheet is the most strongly warped, and in the DFT
calculations, its warping — its area projected onto the $k_{x}$-$k_{y}$ plane
— is 85% larger at the Lifshitz strain than at zero strain. Therefore, if
warping increases linearly with $c$-axis compression the Lifshitz transition
is expected to occur between $-8$ and $-11$ GPa.

Overall, $\sigma_{zz}=-3.2$ GPa appears to be roughly halfway to the Lifshitz
transition, which makes the contrast with in-plane stress very stark. Halfway
to the electron-to-open Lifshitz transition under in-plane stress,
$T_{\text{c}}$ is 0.5 K higher than in unstressed Sr2RuO4 Steppke  _et al._
(2017), but halfway to the electron-to-hole transition under $c$-axis stress,
where the rise in Fermi-level DOS should be even stronger, $T_{\text{c}}$ is
0.15 K lower.

We now resolve the stress dependence of $T_{\text{c}}$ into components. By
comparing the effect of hydrostatic compression (which also suppresses
$T_{\text{c}}$) with $c$-axis compression, we obtain the coefficients $\alpha$
and $\beta$ in the expression

$T_{\text{c}}=T_{\text{c,0}}+\alpha\times\frac{\Delta
V}{V}+\beta\times\left(\varepsilon_{zz}-\frac{\varepsilon_{xx}+\varepsilon_{yy}}{2}\right),$

where $\Delta V/V=\varepsilon_{xx}+\varepsilon_{yy}+\varepsilon_{zz}$ is the
fractional volume change of the unit cell, and
$\varepsilon_{zz}-(\varepsilon_{xx}+\varepsilon_{yy})/2$ is a volume-
preserving tetragonal distortion. Refs. Forsythe  _et al._ (2002); Grinenko
_et al._ (2021b); Svitelskiy  _et al._ (2008) report
$dT_{\text{c}}/d\sigma_{\text{hydro}}=0.22\pm 0.02$, $0.24\pm 0.02$, and
$0.21\pm 0.03$ K/GPa; we take $dT_{\text{c}}/d\sigma_{\text{hydro}}=0.23\pm
0.01$ K/GPa. Employing the low-temperature elastic moduli from Ref. Ghosh  _et
al._ (2021) to convert stress to strain, we find $\alpha=34.8\pm 1.6$ K and
$\beta=-2.2\pm 1.2$ K 111Under hydrostatic stress,
$\sigma_{zz}=(396~{}\text{GPa})\times\varepsilon_{zz}$ and
$\varepsilon_{xx}=0.814\varepsilon_{zz}$. Under $c$-axis uniaxial stress,
$\sigma_{zz}=(219~{}\text{GPa})\times\varepsilon_{zz}$ and
$\varepsilon_{xx}=-0.223\varepsilon_{zz}$.. The small value of $\beta$ means
that a volume-preserving reduction in the lattice parameter ratio $c/a$ would
have little effect on $T_{\text{c}}$: the increase in density of states by
approaching the electron-to-hole Lifshitz transition is balanced by weakening
of the pairing interaction. This demonstration, of a specific method to weaken
the pairing interaction in clean Sr2RuO4, might be a vital clue on what that
pairing interaction is.

For the remainder of the Discussion, we consider implications of our results
for the superconducting order parameter of Sr2RuO4. We note that the weak-
coupling renormalization group study of Ref. Hsu  _et al._ (2016) and
functional renormalization group study of Ref. Liu  _et al._ (2018) both
predict a rapid increase in $T_{\text{c}}$ with approach to the electron-to-
hole Lifshitz transition, even for odd-parity order parameters: the overall
increase in the density of states was found to overcome the suppression of the
gap around the $X$ and $Y$ points. These results heighten the mystery of the
observed decrease in $T_{\text{c}}$.

Our weak-coupling results suggest that, in $H_{\text{c2}}$ data at least, the
distinction between order parameters with and without nodes along the
$\Gamma$-$X$ and $\Gamma$-$Y$ lines becomes sharpest very close to the
electron-to-hole Lifshitz transition, and motivate extending measurements to
larger compressions. Nevertheless, at lower compressions it is the $A_{2g}$
and $B_{2g}$ channels that give the best match to observations. Due to
differences between the actual and calculated electronic structures the
$\varepsilon_{zz}=0$ point in the calculations should not be considered too
literally as equivalent to $\varepsilon_{zz}=0$ in reality, and so the key
point is that it is only in the $A_{2g}$ and $B_{2g}$ channels that
$T_{\text{c}}$ is found to decrease and $H_{\text{c2}}/T_{\text{c}}^{2}$ to
increase over some range of strain. However, $A_{2g}$ and $B_{2g}$ order
parameters do not appear to be consistent with the results of in-plane stress
experiments, in which the strong increase in $H_{\text{c2}}$ as the electron-
to-open Lifshitz transition is approached indicates that there are not nodes
along the $\Gamma$-$X$ and $\Gamma$-$Y$ lines.

Figure 7: (a–c) Photographs of samples 2, 3, and 4. The graphics at the lower
right of each panel are schematic cross sections: the end tabs of samples 2
and 3 were epoxied into slots, while sample 4 was sandwiched between two
surfaces. (d) A photograph of the sample carrier for sample 4. Figure 8: (a)
$T_{\text{c}}$ of sample 3 versus applied displacement, with the data points
colored by the order they were taken. The sequence of data points with the
thick borders, during which $|\sigma_{zz}|$ was monotonically decreased, are
those shown in Fig. 4(d). (b) Sense coil mutual inductance $M(T)$ at points A
and B in the left-hand panel. (c) Force versus displacement of sample 4. (d)
$M(T)$ at points A, B, and C in the left-hand panel.

We comment on the possibility of interorbital pairing Suh  _et al._ (2020);
Clepkens  _et al._ (2021); Gingras  _et al._ (2019); Ramires and Sigrist
(2019). Interorbital pairing depends on mixing orbital weight within Fermi
surfaces through spin-orbit coupling, because the superconducting energy scale
by itself is far too small to mix bands. The proximity of the $\gamma$ and
$\beta$ sheets, and the resulting mixing of $xy$ and $xz$/$yz$ orbital weight
over substantial sections of Fermi surface, is crucial to these models
Clepkens  _et al._ (2021), and the reduction in this mixing that occurs under
$c$-axis compression [illustrated in Fig. 6(b)] could therefore weaken the
pairing interaction and suppress $T_{\text{c}}$. This hypothesis would need to
be tested against that fact that $T_{\text{c}}$ increases under $a$-axis
compression, which also alters the $\gamma$-$\beta$ sheet separation.

In summary, we have demonstrated that it is technically feasible to apply very
large compressive stress along the $c$ axis of Sr2RuO4. This is important,
because $c$-axis stress has its effect primarily through shifts in orbital
occupation, and large stress is required to get a substantial effect. We find
that $T_{\text{c}}$ decreases even though the Fermi-level DOS increases, which
is in stark contrast to the effects of in-plane uniaxial compression.
Approaching the Lifshitz transition at either the $X$ or $Y$ point, through
in-plane stress, dramatically strengthens the superconductivity, while
approaching both suppresses superconductivity. This contrast presents a strong
constraint on any proposed theories of the superconductivity in this material.
Certain order parameter symmetries that have previously been considered
possibilities may not satisfy this constraint, while others not previously
considered strong contenders may need to be reexamined.

Acknowledgements. We thank Aline Ramires and Carsten Timm for helpful
discussions, Markus König for training on the focused ion beam, and Felix
Flicker for help with development and running of the code. F.J., A.P.M., and
C.W.H. acknowledge the financial support of the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) - TRR 288 - 422213477
(project A10). H.S.R. and S.H.S. acknowledge the financial support of the
Engineering and Physical Sciences Research Council (UK). H.S.R. acknowledges
support from the Aker Scholarship. T.S. acknowledges the support of the
Natural Sciences and Engineering Research Council of Canada (NSERC), in
particular the Discovery Grant [RGPIN-2020-05842], the Accelerator Supplement
[RGPAS-2020-00060], and the Discovery Launch Supplement [DGECR-2020-00222].
This research was enabled in part by support provided by Compute Ontario
(www.computeontario.ca) and Compute Canada (www.computecanada.ca). 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). Raw data shown in this
article are available at website to be determined.

##  VII Appendix

###  VII.1 Additional data

Photographs of samples 2–4 are shown in Fig. 7(a–c). The carrier for sample 4,
which has a different design to those used for samples 2 and 3, is shown in
Fig. 7(d). Where electrical contacts were made, Du-Pont 6838 silver paste
annealed at $450^{\circ}$ for typically 30 minutes was used. This is longer
than usual, in order to penetrate a thin insulating layer deposited during the
ion beam milling.

In Fig. 8(a) the complete set of measurements of $T_{\text{c}}$ of sample 3,
plotted against applied displacement, are shown. Data points are colored by
the order in which they were collected. It is apparent that over time the data
drifted leftward: stronger compression was needed to reach the same
$T_{\text{c}}$. This shows that the epoxy underwent non-elastic compression.
However, the qualitative form of the curve — initial decrease in
$T_{\text{c}}$, then a flattening, and then further decrease at the largest
achieved compression — reproduced over multiple stress cycles. In the right-
hand panel the transition at the beginning and end of this set of measurements
is shown. Its width and form are unchanged, which shows that the sample
deformation was elastic; plastic deformation has previously been found to
broaden the superconducting transition Taniguchi  _et al._ (2015). We
attribute the small apparent shift in $T_{\text{c}}$ to an artefact of
inadvertent mechanical contact between the stress cell and inner vacuum can of
the cryostat.

Figure 9: Gap structures at $J/U=0.15$ for, from top to bottom, the $A_{1g}$,
$A_{2g}$, and $B_{1g}$ channels, at the indicated strains. For each channel,
the top, middle, and bottom rows show the gap on the $\alpha$, $\beta$, and
$\gamma$ sheets, respectively.

Similar data on sample 4 are shown in Fig. 8(c–d). Because there was a force
sensor for this sample, we plot force against displacement. Very substantial
non-elastic deformation of the epoxy is apparent, but the shape of the
transition before and after application of large stress is again unchanged:
the sample deformation remained elastic. Over regions where the sample and
epoxy deformed elastically, the combined spring constant was 1.45 N/$\mu$m.
The spring constant of the flexures in the carrier, on the other hand, is
calculated to be $\sim$0.03 N/$\mu$m, meaning that almost all of the applied
force was transferred to the sample.

In Fig. 9 the calculated gap structures in the $A_{1g}$, $A_{2g}$, and
$B_{1g}$ channels are shown; the $B_{2g}$ gap structures are shown in Fig.
6(e). $c$-axis compression favors large gaps on the $\gamma$ sheet in all
channels; however, at the largest compression reached, gap weight in the
$A_{2g}$ channel, but not in the $A_{1g}$ and $B_{1g}$ channels, shifts back
away from the $\gamma$ sheet.

###  VII.2 Details of the weak-coupling calculation

The tight-binding Hamiltonian from Eq. (1) takes the form

$\mathcal{H}_{s}(\bm{k})=\begin{pmatrix}\varepsilon_{AA}(\bm{k})&\varepsilon_{AB}(\bm{k})-is\eta_{1}&+i\eta_{2}\\\
\varepsilon_{BA}(\bm{k})+is\eta_{1}&\varepsilon_{BB}(\bm{k})&-s\eta_{2}\\\
-i\eta_{2}&-s\eta_{2}&\varepsilon_{CC}(\bm{k})\end{pmatrix},$ (3)

where we used the Ru orbital shorthand notation $A=xz$, $B=yz$, $C=xy$, and
where $\bar{s}=-s$ ($s$ being spin). In Eq. (3) the energies
$\varepsilon_{AB}(\bm{k})$ account for intra-orbital ($A=B$) and inter-orbital
($A\neq B$) hopping, and $\eta_{1}$, $\eta_{2}$ parametrize the spin-orbit
coupling. We define
$\varepsilon_{AA}(\bm{k})=\varepsilon_{\mathrm{1D}}(k_{x},k_{y},k_{z})$,
$\varepsilon_{BB}(\bm{k})=\varepsilon_{\mathrm{1D}}(k_{y},k_{x},k_{z})$, and
$\varepsilon_{CC}(\bm{k})=\varepsilon_{\mathrm{2D}}(k_{x},k_{y},k_{z})$, and
we retain the following terms in the matrix elements:

$\displaystyle\varepsilon_{\mathrm{1D}}(k_{\parallel},k_{\perp},k_{z})$
$\displaystyle=-\mu_{\mathrm{1D}}-2t_{1}\cos(k_{\parallel})-2t_{2}\cos(k_{\perp})-4t_{3}\cos(k_{\parallel})\cos(k_{\perp})$
$\displaystyle\hskip
10.0pt-8t_{4}\cos(k_{\parallel}/2)\cos(k_{\perp}/2)\cos(k_{z}/2)-2t_{5}\cos(2k_{\parallel})-4t_{6}\cos(2k_{\parallel})\cos(k_{\perp})-2t_{7}\cos(3k_{\parallel}),$
(4) $\displaystyle\varepsilon_{\mathrm{2D}}(\bm{k})$
$\displaystyle=-\mu_{\text{2D}}-2\bar{t}_{1}\left[\cos(k_{x})+\cos(k_{y})\right]-2\bar{t}_{2}\left[\cos(2k_{x})+\cos(2k_{y})\right]-4\bar{t}_{3}\cos(k_{x})\cos(k_{y})$
$\displaystyle\hskip
10.0pt-4\bar{t}_{4}\left[\cos(2k_{x})\cos(k_{y})+\cos(2k_{y})\cos(k_{x})\right]-4\bar{t}_{5}\cos(2k_{x})\cos(2k_{y})$
(5) $\displaystyle\hskip
10.0pt-4\bar{t}_{6}\left[\cos(3k_{x})\cos(k_{y})+\cos(3k_{y})\cos(k_{x})\right]-2\bar{t}_{7}\left[\cos(3k_{x})+\cos(3k_{y})\right]$
$\displaystyle\hskip
10.0pt-8\bar{t}_{8}\cos(k_{z}/2)\cos(k_{x}/2)\cos(k_{y}/2),$
$\displaystyle\varepsilon_{AB}(\bm{k})$
$\displaystyle=-8\tilde{t}\sin(k_{x}/2)\sin(k_{y}/2)\cos(k_{z}/2).$ (6)

Here the first Brillouin zone is defined as
$\text{BZ}=[-\pi,\pi]^{2}\times[-2\pi,2\pi]$. For the four values of $c$-axis
compression $\varepsilon_{zz}=0,~{}-0.0075,~{}-0.015,~{}-0.020$ we extract the
entire set of parameters from DFT calculations consistent with Fig. 1; see
Table LABEL:tab:Parameters.

Table 1: Tight-binding parameters (in units of meV) used for Eqs. (3), (4),
(5), and (6), yielding Fig. 6(a).

$\lvert\varepsilon_{zz}\rvert$ |  $\mu_{\text{1D}}$ |  $t_{1}$ |  $t_{2}$ |  $t_{3}$ |  $t_{4}$ |  $t_{5}$ |  $t_{6}$ |  $t_{7}$ |  $\mu_{\text{2D}}$ |  $\bar{t}_{1}$ |  $\bar{t}_{2}$ |  $\bar{t}_{3}$ |  $\bar{t}_{4}$ |  $\bar{t}_{5}$ |  $\bar{t}_{6}$ |  $\bar{t}_{7}$ |  $\bar{t}_{8}$ |  $\tilde{t}$ |  $\eta_{1}$ |  $\eta_{2}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$0.00$ |  $316$ |  $296$ |  $53$ |  $-16$ |  $17$ |  $-57$ |  $-15$ |  $-12$ |  $433$ |  $370$ |  $-6$ |  $123$ |  $20$ |  $14$ |  $3$ |  $3$ |  $-2$ |  $9$ |  $-51$ |  $-51$
$0.0075$ |  $294$ |  $284$ |  $54$ |  $-16$ |  $18$ |  $-56$ |  $-15$ |  $-10$ |  $437$ |  $369$ |  $-5$ |  $122$ |  $20$ |  $14$ |  $3$ |  $3$ |  $-3$ |  $10$ |  $-51$ |  $-51$
$0.015$ |  $273$ |  $271$ |  $55$ |  $-17$ |  $19$ |  $-55$ |  $-15$ |  $-9$ |  $441$ |  $368$ |  $-5$ |  $121$ |  $20$ |  $13$ |  $3$ |  $3$ |  $-3$ |  $10$ |  $-51$ |  $-51$
$0.020$ |  $259$ |  $264$ |  $56$ |  $-18$ |  $20$ |  $-55$ |  $-15$ |  $-8$ |  $443$ |  $367$ |  $-4$ |  $120$ |  $20$ |  $13$ |  $3$ |  $3$ |  $-3$ |  $11$ |  $-52$ |  $-51$

For the interactions we use the (on-site) Hubbard–Kanamori Hamiltonian

$H_{I}=\frac{U}{2}\sum_{i,a,s\neq
s^{\prime}}n_{ias}n_{ias^{\prime}}+\frac{U^{\prime}}{2}\sum_{i,a\neq
b,s,s^{\prime}}n_{ias}n_{ibs^{\prime}}+\frac{J}{2}\sum_{i,a\neq
b,s,s^{\prime}}c_{ias}^{\dagger}c_{ibs^{\prime}}^{\dagger}c^{\phantom{\dagger}}_{ias^{\prime}}c^{\phantom{\dagger}}_{ibs}+\frac{J^{\prime}}{2}\sum_{i,a\neq
b,s\neq
s^{\prime}}c_{ias}^{\dagger}c_{ias^{\prime}}^{\dagger}c^{\phantom{\dagger}}_{ibs^{\prime}}c^{\phantom{\dagger}}_{ibs},$
(7)

where $i$ is site, $a$ is orbital, and $n_{ias}=c_{ias}^{\dagger}c_{ias}$ is
the density operator. We further assume that $U^{\prime}=U-2J$ and
$J^{\prime}=J$ Dagotto  _et al._ (2001). In the weak-coupling limit this
leaves $J/U$ as a single parameter fully characterizing the interactions.

Figure 10: Second-order diagrams taken into account in $\Gamma$ in (a) the
even-parity channel, and (b) the odd-parity channel. The vertical arrows
denote pseudo-spin, and the dashed lines contain all the terms of Eq. (7). The
approach is asymptomatically exact in the weak-coupling limit, $U/t\to 0$.

In the linearized gap equation (2) the (dimensionless) two-particle
interaction vertex $\bar{\Gamma}$ is defined as Raghu  _et al._ (2010a)

$\bar{\Gamma}(\bm{k}_{\mu},\bm{k}_{\nu})=\sqrt{\frac{\rho_{\mu}\bar{v}_{\mu}}{v_{\mu}(\bm{k}_{\mu})}}\Gamma(\bm{k}_{\mu},\bm{k}_{\nu})\sqrt{\frac{\rho_{\nu}\bar{v}_{\nu}}{v_{\nu}(\bm{k}_{\nu})}\vphantom{\frac{\rho_{\mu}\bar{v}_{\mu}}{v_{\mu}(\bm{k}_{\mu})}}},$
(8)

where $\rho_{\mu}=\lvert S_{\mu}\rvert/[\bar{v}_{\mu}(2\pi)^{3}]$ is the
density of states, and
$1/\bar{v}_{\mu}=\int_{S_{\mu}}\mathrm{d}\bm{k}/\left(\lvert S_{\mu}\rvert
v_{\mu}(\bm{k})\right)$. Here, $\Gamma$ is the irreducible two-particle
interaction vertex which to leading order retains the diagrams shown in Fig.
10.

An eigenfunction $\varphi$ of Eq. (2) corresponding to a negative eigenvalue
$\lambda$ yields the superconducting order parameter

$\Delta(\bm{k}_{\mu})\sim\sqrt{\frac{v_{\mu}(\bm{k}_{\mu})}{\bar{v}_{\mu}\rho_{\mu}}}\varphi(\bm{k}_{\mu}).$
(9)

In the chosen pseudo-spin basis each eigenvector $\varphi$ belongs to one of
the ten irreducible representations of the crystal point group $D_{4h}$
Sigrist and Ueda (1991); Raghu  _et al._ (2010a).

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