# Constraints on the superconducting state of Sr2RuO4 from elastocaloric
measurements

Grgur Palle<EMAIL_ADDRESS>Institute for Theory of Condensed Matter,
Karlsruhe Institute of Technology, Karlsruhe, Germany  Clifford Hicks  School
of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom
Max Planck Institute for Chemical Physics of Solids, Dresden, Germany  Roser
Valentí  Institute for Theoretical Physics, Goethe-University Frankfurt,
Frankfurt am Main, Germany  Zhenhai Hu  Max Planck Institute for Chemical
Physics of Solids, Dresden, Germany  You-Sheng Li  Max Planck Institute for
Chemical Physics of Solids, Dresden, Germany  Andreas Rost  Scottish
Universities Physics Alliance, School of Physics and Astronomy, University of
St Andrews, St Andrews, United Kingdom  Michael Nicklas  Max Planck Institute
for Chemical Physics of Solids, Dresden, Germany  Andrew P. Mackenzie  Max
Planck Institute for Chemical Physics of Solids, Dresden, Germany  Scottish
Universities Physics Alliance, School of Physics and Astronomy, University of
St Andrews, St Andrews, United Kingdom  Jörg Schmalian  Institute for Theory
of Condensed Matter, Karlsruhe Institute of Technology, Karlsruhe, Germany
Institute for Quantum Materials and Technologies, Karlsruhe Institute of
Technology, Karlsruhe, Germany

###### Abstract

Strontium ruthenate Sr2RuO4 is an unconventional superconductor whose pairing
symmetry has not been fully clarified, despite more than two decades of
intensive research. Recent NMR Knight shift experiments have rekindled the
Sr2RuO4 pairing debate by giving strong evidence against all odd-parity
pairing states, including chiral $p$-wave pairing that was for a long time the
leading pairing candidate. Here, we exclude additional pairing states by
analyzing recent elastocaloric measurements [YS. Li et al., Nature 607,
276–280 (2022)]. To be able to explain the elastocaloric experiment, we find
that unconventional even-parity pairings must include either large
$d_{x^{2}-y^{2}}$-wave or large $\\{d_{xz}\mid d_{yz}\\}$-wave admixtures,
where the latter possibility arises because of the body-centered point group
symmetry. These $\\{d_{xz}\mid d_{yz}\\}$-wave admixtures take the form of
distinctively body-centered-periodic harmonics that have horizontal line
nodes. Hence $g_{xy(x^{2}-y^{2})}$-wave and $d_{xy}$-wave pairings are
excluded as possible dominant even pairing states.

##  I Introduction

The nature of the superconductivity of strontium ruthenate (SRO) remains
elusive. In the three decades following its discovery [1], an impressive array
of experiments have been performed with high precision and on exceedingly pure
samples [2, 3, 4, 5, 6]. Yet the most straightforward interpretations of the
various experimental results are regularly at odds with one another. Although
many proposals [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21] have
been made on how the assortment of experimental results might be reconciled,
no consensus has formed around which proposal is the correct one. Before
presenting our results, in the next six paragraphs we review what is currently
known about the pairing state. This literature review is not essential to our
argument and can be skipped.

The superconductivity (SC) of SRO is unconventional. This has been established
early on by the absence of a Hebel-Slichter peak [22, *Hebel1959] in the NMR
relaxation rate $1/T_{1}$ [24, 25, 26], and by the large suppression of the SC
transition temperature $T_{c}$ by non-magnetic impurities [27,
*Mackenzie1998-E, 29, 30, 31] that saturates the Abrikosov-Gor’kov bound [32,
33]. Subsequent experiments have only further confirmed the unconventional
character of SRO’s SC.

The pairing of SRO is more likely to be even than not. Recent34 NMR Knight
shift [35, 36, 37] and polarized neutron scattering [38] experiments strongly
favor singlet pairing, as do numerous studies [6, 39] indicating that the in-
plane critical field $B_{c2\parallel ab}$ is Pauli limited [40]. Although the
observation of $\pi$ phase shifts [41] and half-quantum vortices [42, 43, 44]
is at tension with even-parity SC, possible explanations do exist [10, 45,
46]. Reconciling an $80\text{\,}\mathrm{\char 37\relax}$ drop in the in-plane
Knight shift [37] with triplet pairing, or a strained critical field
anisotropy $B_{c2\parallel ab}/B_{c2\parallel c}\sim 3$ [47] far below the SC
anisotropy $\xi_{ab}/\xi_{c}\sim 60$ [48, 49] without Pauli limiting [6], is
significantly more challenging, but perhaps possible [50, 51].

11footnotetext:  The heating caused by NMR pulses [35, 36] has rendered early
NMR Knight shift experiments [134], nicely summarized in Figure 14 of Ref.
[26], invalid. The NMR pulse heat-up effect acts on a time scale much shorter
than $T_{1}$ and has not invalidated the early NMR relaxation rate studies
[35]. An early polarized neutron scattering study [135] has been superseded by
a new one [38] with better statistics, carried out at a smaller magnetic
field. 22footnotetext:  The evidence for a Pauli-limited $B_{c2\parallel ab}$
is threefold: (i) the SC-normal state transition is first-order below
$0.5T_{c}$, as seen in the hysteresis [136, 137, 49] and jumps in the specific
heat [138, 137], thermal conductivity [138], magnetocaloric effect [136], ac
magnetic susceptibility [139], magnetization [49], and Knight shift [37]; (ii)
the measured intrinsic SC anisotropy $\xi_{ab}/\xi_{c}\sim 60$ [48, 49]
exceeds the critical field anisotropy $B_{c2\parallel ab}/B_{c2\parallel
c}\sim 20$ [140] by a factor of $3$ in the unstrained case, and by a factor of
$20$ under $\langle 100\rangle$ uniaxial pressure that maximally enhances
$T_{c}$ [47], whereas for orbitally limited $B_{c2\parallel ab}$ the two
ratios would be comparable; and (iii) $B_{c2\parallel
ab}\propto\Delta/\mu_{B}\propto T_{c}$ under small uniaxial strain [109], as
expected for Pauli limiting.

The evidence for time-reversal symmetry breaking (TRSB) is mixed. Zero-field
muon spin relaxation (ZF-$\mu$SR) [52, 53, 54, 55, 56] and polar Kerr effect
[57, 58] experiments indicate TRSB at a $T_{\text{TRSB}}$ at or very near
$T_{c}$, yet the current response of micron-sized Josephson junctions
[[][Note:contrarytowhattheysay,
theinversionsymmetry$I_c^+(H)=-I_c^-(-H)$thatbecomesrestoredforsmalljunctionsispreciselytime-
reversalsymmetry.]Saitoh2015, 60] exhibits time-reversal invariance. Under
$\langle 100\rangle$ uniaxial pressure, ZF-$\mu$SR [55] observes a large
splitting between $T_{\text{TRSB}}$ and $T_{c}$,61 yet no signatures of a TRSB
phase transition below $T_{c}$ have been found in heat capacity [62] or
elastocaloric [63] measurements. Under disorder and hydrostatic pressure, no
splitting between SC and TRSB is observed in ZF-$\mu$SR [56]. Preliminary
ZF-$\mu$SR measurements point towards splitting of SC and TRSB under $\langle
110\rangle$ uniaxial stress [64]. In the presence of TRSB, spontaneous
magnetization and currents are generically expected to appear around domain
walls, edges, and defects, yet scanning SQUID and Hall probe microscopy [65,
66, 67, 68, 69, 70, 71, 72] has failed to find any evidence for them.
Josephson junction experiments [59, 73, 74, 75] show signs of SC domains in
their interference patters, switching behavior, and size dependence of their
transport properties, but the domains themselves need not be chiral.

33footnotetext:  In one sample [55], $T_{\text{TRSB}}$ and $T_{c}$ split even
without any external pressure.

The coupling of SC to strain is partially known from measurements of elastic
constants. The main obstacle to making these measurements conclusive is the
fact that strain inhomogeneities, such as stacking faults or lattice
dislocations, mix elastic waves of different symmetry.76 That said, according
to elastic constant measurements, the SC order appears to couple quadratically
to $\varepsilon_{xx}-\varepsilon_{yy}\in B_{1g}$ strain and possibly linearly
to $\varepsilon_{xy}\in B_{2g}$ strain. (Irreducible representations (irreps)
of SRO are summarized in Table 1.) The evidence for the former is the
quadratic dependence of $T_{c}$ on $\varepsilon_{xx}-\varepsilon_{yy}$,
whether measured globally [77, 47, 78] or locally [79], and the absence of a
jump at $T_{c}$ in the shear elastic modulus
$C_{B_{1g}}=\tfrac{1}{2}(C_{11}-C_{12})$ [80, 81, 82]. The evidence for the
latter is a jump at $T_{c}$ in the shear elastic constant $C_{66}\in B_{2g}$
[83, 81, 82], as measured by ultrasound. However, the magnitude of this jump
varies by a factor of $50$ between the two experimental groups [81, 82] and
direct measurements of $T_{c}$ under $[110]$ strain show linear dependence
without any splitting and whose magnitude can be fully accounted without any
linear coupling to $\varepsilon_{xy}$ [84]. This raises the possibility that
the observed jump in $C_{66}$ is due to lattice defect effects that, however,
need to be channel selective so as to not generate a jump in $C_{B_{1g}}$. One
such proposal [9] is that a subleading pairing channel activates near
dislocations; the product of the leading and subleading pairing irreps then
determines which elastic modulus experiences a jump. No jump has been observed
for the elastic modulus $C_{44}\in E_{g}$ [80, 82], indicating that the
coupling to $E_{g}$ strain is quadratic. Large jumps in the $A_{1g}$
components of the viscosity tensor have recently been discovered at $T_{c}$
[85].

44footnotetext:  As pointed out in [9], dislocations give contributions to elastic constants that are on the order of $1\text{\,}\mathrm{\char 37\relax}$, which is two orders of magnitude larger than the (larger of the two sets of) measured jumps of the elastic constants at $T_{c}$ [82].  Table 1: Examples of functions transforming according to the irreps of the point group $D_{4h}$ of SRO. $D_{4h}$ is generated by fourfold rotations around $z$, twofold rotations around $x$ and $y$, twofold rotations around the diagonals $x\pm y$, and parity. It has five even ($A_{1g}$, $A_{2g}$, $B_{1g}$, $B_{2g}$, $E_{g}$) and five odd ($A_{1u}$, $A_{2u}$, $B_{1u}$, $B_{2u}$, $E_{u}$) irreps, of which $E_{g}$ and $E_{u}$ are 2D. $A_{1g}$ | $A_{2g}$ | $B_{1g}$ | $B_{2g}$ | $E_{g}$
---|---|---|---|---
$1$, $x^{2}+y^{2}$, $z^{2}$ | $xy(x^{2}-y^{2})$ | $x^{2}-y^{2}$ | $xy$ | $\\{yz\mid-xz\\}$
$A_{1u}$ | $A_{2u}$ | $B_{1u}$ | $B_{2u}$ | $E_{u}$
$xyz(x^{2}-y^{2})$ | $z$ | $xyz$ | $(x^{2}-y^{2})z$ | $\\{x\mid y\\}$

The preponderance of evidence points towards line nodes. The expected
dependence on temperature is found in the heat capacity [86, 87, 88],
ultrasound attenuation rate [80, 89], NMR relaxation rate [25], and London
penetration depth [90]. In weak in-plane fields, the heat capacity [91, 88]
and Knight shift [37] obey Volovik scaling ($\propto\sqrt{B/B_{c2}}$) expected
of line nodes [92]. The in-plane thermal conductivity [93, 94] exhibits
universal transport, which is a type of transport found only in nodal SC [95,
96, 97, 98]. Finally, STM spectroscopy [99, 100] shows a $V$-shaped
conductance minimum.101 The only evidence to the contrary is an STM/S study
[102] that scanned micron-sized grains ($\sim 10\,\xi_{ab}$) situated on top
of SC aluminium and found an implausibly large SC gap $\Delta$ of
$3.5\text{\,}\mathrm{K}$. Given that so many studies [86, 87, 88, 80, 89, 25,
90, 91] found nodal behavior, in some cases down to as low as
$$0.04\text{\,}\mathrm{K}$\approx T_{c}/30$, any fully gapped SC must have
extraordinarily deep minima.

55footnotetext:  One should keep in mind that STM mostly probes the
$\alpha,\beta$ bands because of their $d_{xz},d_{yz}$ orbital characters which
make their overlaps with the tip (along $z$) large.

The location and orientation of the line node(s) is not settled. Heat capacity
[88] and in-plane thermal conductivity [103, 104] both display a fourfold
anisotropy in their dependence on the in-plane $\bm{B}$ orientation.105 Since
these anisotropies are small ($\sim$1\text{\,}\mathrm{\char 37\relax}$$), they
can be explained by both horizontal and vertical nodes. That the heat capacity
anisotropy has the same sign down to $T_{c}/20$ appears to exclude
$d_{xy}$-wave pairing [88], and maybe other pairings too. The universal heat
transport along $c$ has been found finite with $2\sigma$ significance [94],
indicating that nodal quasi-particles have a finite $c$-axis velocity. If
true, this result is strong evidence against symmetry-enforced horizontal line
nodes. A resonance at transfer energy $\approx 2\Delta$ and momentum with a
finite $z$ component was reported below $T_{c}$ in the inelastic neutron
scattering intensity [106], suggesting horizontal line nodes, but was not
reproduced in subsequent measurements [107]. In the Fourier transform of the
real-space STM tunneling conductance [100], peaks were found at nesting
vectors expected of $d_{x^{2}-y^{2}}$-wave SC. However, the peaks are not
clearly resolved because of noise and compatibility with other pairings was
not investigated.

66footnotetext:  As pointed out in [88], little useful information can be
extracted from the out-of-plane field-angle anisotropy.

Compelling evidence on SRO’s gap structure has recently emerged from
measurements performed under uniaxial pressure. When $\langle 100\rangle$
uniaxial pressure is applied on SRO, its SC is drastically enhanced [77, 108,
47, 78, 109], with $T_{c}$ increasing from $1.5\text{\,}\mathrm{K}$ to a
maximal $3.5\text{\,}\mathrm{K}$ before decaying again. The most likely cause
of this enhancement is the Lifshitz transition that occurs at
$\varepsilon_{xx}=$-0.44\text{\,}\mathrm{\char
37\relax}$\equiv\varepsilon_{\text{VH}}$ strain [47, 78, 110] and is
accompanied by an increase in the density of states (DOS). The DOS peaks at
$\varepsilon_{\text{VH}}$, as does the normal-state entropy [63]. In the SC
state, however, the entropy becomes a _minimum_ at $\varepsilon_{\text{VH}}$,
as directly measured by the elastocaloric effect [63]. As we later explain,
this is only possible if SRO’s SC does not have vertical line nodes at the Van
Hove lines that induce the DOS peak at $\varepsilon_{\text{VH}}$. This is a
severe constraint on possible pairing states, one whose implications we
explore in this article. The final piece of the argument is that these
properties of strained SRO carry over to the unstrained SC state, which is
supported by the absence of any signatures of a bulk SC state change at finite
strain in the heat capacity [62], elastocaloric effect [63], or NMR Knight
shift [35, 37].

The main result of this work is that, among even pairings, only $s$-wave
($A_{1g}$), $d_{x^{2}-y^{2}}$-wave ($B_{1g}$), and body-centered periodic
$\\{d_{xz}\mid d_{yz}\\}$-wave ($E_{g}$) pairings gap the Van Hove lines. Thus
the SC state must include admixtures from at least one of these three pairings
to be consistent with the elastocaloric experiment. The logic of our argument
does not put any constraints on the subleading channels. For instance, almost
degenerate states like $d_{x^{2}-y^{2}}+\mathrm{i}\,g_{xy(x^{2}-y^{2})}$ [7]
or $s^{\prime}+\mathrm{i}\,d_{xy}$ [14, 15] are consistent with a dominant
$d_{x^{2}-y^{2}}$-wave or $s$-wave state, respectively. Among odd-parity
pairings, all irreps can gap the Van Hove lines. However, $A_{2u}$ and
$B_{2u}$ pairings must be made of body-centered periodic wavefunctions, and
for the rest we find non-trivial constraints on the orientations of their
Balian-Werthamer $\bm{d}$-vectors [111].

The paper is organized as follows. In Sec. II we review some basic properties
of SRO. After that, in Sec. III, we explain what has been measured in the
elastocaloric experiment [63] and why these measurements forbid vertical line
nodes at the Van Hove lines. The precise location of the Van Hove lines is the
subject of Sec. IV. Because of its multiband nature, SRO supports a richer set
of pairing states than single-band SC [112, 113, *Kaba2019-E, 115], which is
briefly discussed at the beginning of Sec. V and at length in Appendix C.
Section V contains the main results of our work: how the momentum and spin-
orbit parts of the SC gap behave near the Van Hove lines and which SC states
are excluded by the elastocaloric measurements. Table 6b is our main result.
In the last section, we discuss our results.

##  II Crystal and electronic structure

SRO is a layered perovskite with a body-centered tetragonal lattice
($a=$3.86\text{\,}\mathrm{\text{\r{A}}}$$,
$c=$12.7\text{\,}\mathrm{\text{\r{A}}}$$), space group $I4/mmm$, and point
group $D_{4h}$ [2, 116]. The character table of $D_{4h}$ is given in Table 2.

SRO has three conduction bands, conventionally referred to as $\alpha$,
$\beta$, and $\gamma$, with cylindrical Fermi sheets [2, 116]. They are
depicted in Figure 1. These bands primarily derive from the $t_{2g}$ orbital
manifold of the Ru atoms, which is made of $d_{yz}$, $d_{zx}$, and $d_{xy}$
orbitals [2, 116, 20]. To a first approximation, due to the high anisotropy,
$d_{yz}$ and $d_{zx}$ have 1D tight-binding dispersions:

$\displaystyle\epsilon_{yz}(\bm{k})$ $\displaystyle=-\mu-2t\cos ak_{2},$ (1)
$\displaystyle\epsilon_{zx}(\bm{k})$ $\displaystyle=-\mu-2t\cos ak_{1},$ (2)

whereas $d_{xy}$ has a 2D tight-binding dispersion:

$\displaystyle\begin{aligned} \epsilon_{xy}(\bm{k})&=-\mu-2t(\cos ak_{1}+\cos
ak_{2})\\\ &\qquad-4t^{\prime}\cos ak_{1}\cos ak_{2},\end{aligned}$ (3)

where $(\mu,t,t^{\prime})\approx(0.35,0.3,0.1)\,$\mathrm{eV}$$ [117, 16].
After introducing interorbital mixing and spin-orbit coupling,
$\epsilon_{yz}(\bm{k})$ and $\epsilon_{zx}(\bm{k})$ hybridize into the
quasi-1D $\alpha$ and $\beta$ bands, whereas $\epsilon_{xy}(\bm{k})$
hybridizes into the quasi-2D $\gamma$ band [Figure 1]. Interlayer hopping adds
warping along $k_{3}$. Below $25\text{\,}\mathrm{K}$, SRO is a quasi-2D Fermi
liquid. Its quasi-particles are strongly renormalized by electronic
correlations [2, 118]. In the clean limit, SRO develops SC below
$1.5\text{\,}\mathrm{K}$ [2].

Table 2: The character table of the point group $D_{4h}$ [119]. Irreps are divided into even ($g$) and odd ($u$) ones. $C_{4}$ are rotations by $\pm\pi/2$ around the $z$ axis. $C_{2}$, $C_{2}^{\prime}$, and $C_{2}^{\prime\prime}$ are rotations by $\pi$ around the $z$ axis, $x$ or $y$ axes, and diagonals $x\pm y$, respectively. $P$ is parity. $S_{4}$, $\Sigma_{h}$, $\Sigma_{v}^{\prime}$, $\Sigma_{d}^{\prime\prime}$ are compositions of $C_{4}$, $C_{2}$, $C_{2}^{\prime}$, $C_{2}^{\prime\prime}$ with $P$, respectively. $D_{4h}$ | $E$ | $2C_{4}$ | $C_{2}$ | $2C_{2}^{\prime}$ | $2C_{2}^{\prime\prime}$ | $P$ | $2S_{4}$ | $\Sigma_{h}$ | $2\Sigma_{v}^{\prime}$ | $2\Sigma_{d}^{\prime\prime}$
---|---|---|---|---|---|---|---|---|---|---
$A_{1g}$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$
$A_{2g}$ | $1$ | $1$ | $1$ | $-1$ | $-1$ | $1$ | $1$ | $1$ | $-1$ | $-1$
$B_{1g}$ | $1$ | $-1$ | $1$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $1$ | $-1$
$B_{2g}$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $1$ | $-1$ | $1$ | $-1$ | $1$
$E_{g}$ | $2$ | $0$ | $-2$ | $0$ | $0$ | $2$ | $0$ | $-2$ | $0$ | $0$
$A_{1u}$ | $1$ | $1$ | $1$ | $1$ | $1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$
$A_{2u}$ | $1$ | $1$ | $1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ | $1$ | $1$
$B_{1u}$ | $1$ | $-1$ | $1$ | $1$ | $-1$ | $-1$ | $1$ | $-1$ | $-1$ | $1$
$B_{2u}$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $-1$
$E_{u}$ | $2$ | $0$ | $-2$ | $0$ | $0$ | $-2$ | $0$ | $2$ | $0$ | $0$
Figure 1: The Fermi surfaces of SRO. The solid black lines are the $k_{3}=0$
cross-sections of the cylindrical $\alpha$, $\beta$, and $\gamma$ Fermi sheets
of unstrained SRO, as determined by our tight-binding model [Appendix B]. The
dashed red line is the $\gamma$ band of SRO under Van Hove unaxial strain
$\varepsilon_{100}=$-0.44\text{\,}\mathrm{\char
37\relax}$\equiv\varepsilon_{\text{VH}}$ [110, 63]. At this strain, the
$\gamma$ band opens at the Van Hove lines
$\left(0,\pm\frac{\pi}{a},k_{3}\right)$, here denoted with red dots.

Below $25\text{\,}\mathrm{K}$, SRO is well-described by a tight-binding model
based on the $t_{2g}$ orbitals of ruthenium [117, 16, 120, 121]. Within it,
the hopping amplitudes $t_{\bm{\delta}}$ between neighboring lattice sites are
significantly constrained by the symmetries of SRO. In a body-centered
lattice, hopping amplitudes along the half-diagonal
$\bm{\delta}=\frac{1}{2}\left(a\bm{\hat{e}}_{1}+a\bm{\hat{e}}_{2}+c\bm{\hat{e}}_{3}\right)$,
as well as many other $\bm{\delta}$, are additionally possible. However, all
such characteristically body-centered hoppings necessarily connect different
layers and are thus suppressed by SRO’s anisotropy. For the purpose of making
estimates, throughout this paper we employ the normal-state model of Ref.
[117], the details of which are provided in Appendix B.

##  III Implications of elastocaloric measurements

The elastocaloric effect describes the change in the temperature that
accompanies an adiabatic change in the strain $\varepsilon_{\alpha\beta}$. By
measuring it, one may determine the dependence of the entropy $S$ on strain.
This is made possible by the thermodynamic identity:

$\left.\frac{\partial
T}{\partial\varepsilon_{\alpha\beta}}\right|_{S}=-\frac{T}{C_{\varepsilon}(T)}\left.\frac{\partial
S}{\partial\varepsilon_{\alpha\beta}}\right|_{T},$ (4)

where $C_{\varepsilon}(T)=T\left(\partial S/\partial T\right)_{\varepsilon}$
is the heat capacity at constant strain. Recently, important progress has been
made in the experimental techniques for measuring the elastocaloric effect and
in their analysis for correlated electron systems [122, 123, 124].

The elastocaloric effect has been measured last year for strain applied along
the $[100]$ direction [63]. Numerical analysis of this dense data set [125]
enables the separation of the contribution from $C_{\varepsilon}$ and the
reconstruction of the dependence of the entropy on strain; see Figure 2.

77footnotetext:  The elastocaloric data of Ref. [63] are available at
https://doi.org/10.17630/6a4a06c6-38d3-464f-88d1-df8d2dbf1e75.

As clearly seen in the figure, the normal-state entropy has a maximum at the
Van Hove strain $\varepsilon_{100}=$-0.44\text{\,}\mathrm{\char
37\relax}$\equiv\varepsilon_{\text{VH}}$. As we enter the SC state, however,
this maximum becomes a _minimum_ as a function of strain. To understand this
behavior, let us recall that the entropy of a Fermi liquid is given by [126,
127]:

$\displaystyle
S=V\frac{\pi^{2}}{3}k_{B}^{2}T\int\mathop{}\\!\mathrm{d}{E}\,g(E)\operatorname{\updelta}_{T}(E),$
(5)
$\displaystyle\operatorname{\updelta}_{T}(E)=\frac{3}{\pi^{2}k_{B}T}\big{[}-f\log
f-(1-f)\log(1-f)\big{]},$ (6)

where $V$ is the volume, $g(E)$ the DOS, $E$ is relative to the chemical
potential, and $f=1/(\mathrm{e}^{E/k_{B}T}+1)$.
$\operatorname{\updelta}_{T}(E)\to\operatorname{\updelta}(E)$ as $T\to 0$ so
$S\sim Tg(0)$. This formula applies to both the normal and the SC state. Thus
to understand the entropy, we need to study the DOS near the Fermi level
$E=0$.

In the normal state, at Van Hove strain the $\gamma$ band experiences a
Lifshitz transition in which its cylindrical Fermi surface opens at the Van
Hove lines $\bm{k}_{\text{VH}}\approx\left(0,\pm\frac{\pi}{a},k_{3}\right)$
along the $k_{2}$-direction [47, 78, 110]. This is shown in Figure 1. Because
of the particularly weak $k_{3}$-dispersion of the $\gamma$ band at
$\bm{k}_{\text{VH}}$ ($\sim$1\text{\,}\mathrm{K}$$), the Van Hove lines
contribute a pronounced peak in the DOS that is only rounded on an energy
scale of about one kelvin [63]. It is this peak in the DOS that explains the
observed normal-state entropy maximum.

To gain a qualitative understanding of what sort of pairings can induce an
entropy minimum at $\varepsilon_{\text{VH}}$ strain, it is sufficient to
consider the $\gamma$ band near the Van Hove lines. This is justified by the
fact that the $\gamma$ band contributes $60\text{\,}\mathrm{\char 37\relax}$
of the total DOS [Appendix B] and is solely responsible for the normal-state
peak in the entropy. For the moment, we shall also neglect the
$k_{3}$-dispersion.

Figure 2: The entropy $S$ as a function of strain $\varepsilon_{100}$ at
constant temperatures $T$ ranging from $2.5\text{\,}\mathrm{K}$ (blue) to
$4.0\text{\,}\mathrm{K}$ (red) in $0.1\text{\,}\mathrm{K}$ increments. The
entropies at different temperatures are naturally offset from each other by
their temperature dependence. The entropy has been reconstructed from
elastocaloric measurements [63] using Eq. (4). At Van Hove strain
$\varepsilon_{100}=$-0.44\text{\,}\mathrm{\char
37\relax}$\equiv\varepsilon_{\text{VH}}$, $T_{c}$ attains its maximal value of
$3.5\text{\,}\mathrm{K}$. Above (below) $3.5\text{\,}\mathrm{K}$, the entropy
has a maximum (minimum) at $\varepsilon_{\text{VH}}$ strain.

The DOS of a band in 2D with a dispersion $\epsilon_{\bm{k}}$ and SC gap
$\Delta_{\bm{k}}$ is given by:

$g_{\text{sc}}(E)=2\int\frac{\mathop{}\\!\mathrm{d}{k_{1}}\mathop{}\\!\mathrm{d}{k_{2}}}{(2\pi)^{2}}\operatorname{\updelta}\\!\big{(}E-\xi_{\bm{k}}\big{)},$
(7)

where the $2$ is due to spin and
$\xi_{\bm{k}}=\sqrt{\epsilon_{\bm{k}}^{2}+\left\lvert{\Delta_{\bm{k}}}\right\rvert^{2}}$
is the Bogoliubov quasi-particle dispersion. It is often easier to calculate
the integrated DOS

$\mathcal{N}_{\text{sc}}(E)=\int_{0}^{E}\mathop{}\\!\mathrm{d}{E^{\prime}}\,g_{\text{sc}}(E^{\prime})=2\int_{\xi_{\bm{k}}\leq
E}\frac{\mathop{}\\!\mathrm{d}{k_{1}}\mathop{}\\!\mathrm{d}{k_{2}}}{(2\pi)^{2}}$
(8)

and then differentiate it to get $g_{\text{sc}}(E)$. Near the Van Hove point
$\left(0,\pi\right)$, the dispersion of the $\gamma$ band is approximately
given by [see Appendix B or Eq. (18)]:

$\epsilon_{\bm{k}}=\frac{1}{2m_{1}}k_{1}^{2}-\frac{1}{2m_{2}}k_{2}^{2}=\frac{1}{m_{*}}q_{+}q_{-},$
(9)

where $m_{*}=\sqrt{m_{1}m_{2}}=1/$3200\text{\,}\mathrm{K}$$,
$r=\sqrt[4]{m_{2}/m_{1}}=0.59$, and $q_{\pm}=\frac{1}{\sqrt{2}}\left(rk_{1}\pm
k_{2}/r\right)$. Since this expression for $\epsilon_{\bm{k}}$ only applies
near the Van Hove point, we impose a momentum cutoff
$\left\lvert{q_{\pm}}\right\rvert\leq\Lambda$.

In the normal state (NS), $\Delta_{\bm{k}}=0$ and the DOS at the Van Hove
strain equals:

$g_{\text{sc}}^{\text{NS}}(E)=\frac{8m_{*}}{(2\pi)^{2}}\log\frac{\Lambda^{2}}{m_{*}E}.$
(10)

This diverges logarithmically as $E\to 0$. As we move away from
$\varepsilon_{100}=\varepsilon_{\text{VH}}$, the logarithmic divergence is
moved away from the Fermi level $E=0$, explaining the normal-state entropy
maximum.

If we fully gap (FG) the saddle point, $\Delta_{\bm{k}}=\Delta_{0}$, then the
DOS vanishes up to $\Delta_{0}$,
$g_{\text{sc}}^{\text{FG}}(E\leq\Delta_{0})=0$, and diverges above it
according to ($E>\Delta_{0}$):

$g_{\text{sc}}^{\text{FG}}(E)=\frac{8m_{*}}{(2\pi)^{2}}\frac{E}{\sqrt{E^{2}-\Delta_{0}^{2}}}\log\frac{\Lambda^{2}}{m_{*}\sqrt{E^{2}-\Delta_{0}^{2}}}.$
(11)

Since $\operatorname{\updelta}_{T}(E)$ in Eq. (5) has a width $\sim k_{B}T$,
for sufficiently large $\Delta_{0}/k_{B}T$ the normal-state entropy maximum
can be suppressed so strongly that it becomes a minimum as a function of
strain. Hence fully gapping the Van Hove lines reproduces the features of
Figure 2. Note that a constant gap does not necessarily mean an $s$-wave
state, but merely that the gap is finite in the vicinity of the Van Hove
point. For instance, $d_{x^{2}-y^{2}}$-wave pairing is finite at the Van Hove
point $\left(0,\pi\right)$ and approximately constant around it. Our analysis
focuses only on the behavior of the pairing gap near the saddle point of the
dispersion.

Can pairings with nodal lines at the Van Hove lines also reproduce the SC
entropy minimum? To answer this question, let us calculate the DOS for a
vertical and horizontal line node. For vertical line nodes (VLN), there are
two cases to distinguish: when $\Delta_{\bm{k}}$ is linear and when
$\Delta_{\bm{k}}$ is quadratic in $\bm{k}$.

In the linear case, we may always write the gap as:

$\Delta_{\bm{k}}=\Delta_{0}\left(q_{+}\cos\varphi+q_{-}\sin\varphi\right)/\Lambda=\Delta_{0}(p_{1}/\Lambda).$
(12)

In the limit of small $E$, the inequality $\xi_{\bm{k}}\leq E$ that determines
$\mathcal{N}_{\text{sc}}(E)$ simplifies to

$\frac{\Delta_{0}^{2}}{\Lambda^{2}}p_{1}^{2}+\frac{\sin^{2}(2\varphi)}{4m_{*}^{2}}p_{2}^{4}\leq
E^{2},$ (13)

where $p_{2}=q_{-}\cos\varphi-q_{+}\sin\varphi$. The area enclosed by this
inequality equals
$\pi^{\prime}\left\lvert{p_{1,\text{max}}}\right\rvert_{E}\left\lvert{p_{2,\text{max}}}\right\rvert_{E}$,
where
$\pi^{\prime}=4\int_{0}^{1}\mathop{}\\!\mathrm{d}{x}\sqrt{1-x^{4}}\approx
3.496$, and therefore for small $E$:

$g_{\text{sc}}^{\text{VLN}}(E\to
0)=\frac{3\pi^{\prime}}{(2\pi)^{2}}\frac{\Lambda}{\Delta_{0}}\sqrt{\frac{2m_{*}E}{\left\lvert{\sin
2\varphi}\right\rvert}}.$ (14)

This $g_{\text{sc}}^{\text{VLN}}\propto\sqrt{E}$ behavior persists up to the
point where $g_{\text{sc}}^{\text{VLN}}(E_{w})\approx
g_{\text{sc}}^{\text{NS}}(E_{w})$. By solving this equation with
$\Delta_{0}\sim$3\text{\,}\mathrm{K}$$ (the $T_{c}$ at
$\varepsilon_{xx}=\varepsilon_{\text{VH}}$) and $\Lambda\sim 0.5$, one obtains
$E_{w}\sim$0.2\text{\,}\mathrm{K}$$.128 Exceptionally, when $\varphi=0$ or
$\pi/2$, one finds a constant DOS up to $\Delta_{0}$:

$g_{\text{sc}}^{\text{VLN}^{\prime}}(E\leq\Delta_{0})=\frac{8m_{*}}{(2\pi)^{2}}\operatorname{arcsinh}\frac{\Lambda^{2}}{m_{*}\Delta_{0}}.$
(15)

Thus if a single line node cuts through the Van Hove point, the DOS
generically vanishes like $\sqrt{E}$ in a very narrow range
$E\lesssim$0.2\text{\,}\mathrm{K}$$. If this line node is fine-tuned to
coincide with the lines $q_{+}=0$ or $q_{-}=0$, then the DOS becomes finite
and large.

99footnotetext:  The solution of $\sqrt{x}=\frac{1}{2}\delta\log(1/x)$ is
$x=\delta^{2}W^{2}(1/\delta)$, where $W(x)$ is the Lambert $W$-function. In
our case $x=m_{*}E_{w}/\Lambda^{2}$ and
$\delta=(8\sqrt{2}/3\pi^{\prime})(m_{*}\Delta_{0}/\Lambda^{2})$.

The second case is when $\Delta_{\bm{k}}$ is quadratic in $\bm{k}$. Quadratic
$\Delta_{\bm{k}}$ may correspond to a line node with a quadratic orthogonal
dispersion, a pair of line nodes that intersect at $\bm{k}=\bm{0}$, or a point
node, depending on the eigenvalues of the Hessian. The inequality
$\xi_{\bm{k}}\leq E$ is in this case invariant under the scaling
$\bm{k}\mapsto\sqrt{\alpha}\bm{k}$, $E\mapsto\alpha E$. Hence
$\mathcal{N}_{\text{sc}}(E)$ is linear in $E$ for small $E$, yielding a finite
$g_{\text{sc}}^{\text{VLN}^{\prime\prime}}(E=0)$ and no opening of a gap.
Exceptionally, when we have two SC line nodes that coincide with the Van Hove
strain Fermi surfaces $q_{\pm}=0$, the SC gap equals
$\Delta_{\bm{k}}=\Delta_{0}(q_{+}q_{-}/\Lambda^{2})$, from which we see that
$g_{\text{sc}}^{\text{VLN}^{\prime\prime\prime}}$ retains the normal-state
logarithmic singularity, albeit with a renormalized
$1/m_{*}\mapsto\sqrt{1/m_{*}^{2}+\Delta_{0}^{2}/\Lambda^{4}}$.

Figure 3: The Van Hove line contributions to the DOS $g_{\text{sc}}(E)$ for
the four possible types of gaps considered in the text: normal state (NS), SC
with a fully gapped (FG) Van Hove line, SC with a vertical line node (VLN) on
the Van Hove line, and SC with a horizontal line node (HLN) crossing the Van
Hove line. These correspond to Eqs. (10), (11), (14), and (16), respectively.
The VLN case (with $\varphi=\pi/4$) was calculated numerically. The parameter
values $m_{*}^{-1}=$3200\text{\,}\mathrm{K}$$,
$\Delta_{0}=$3\text{\,}\mathrm{K}$$, and $\Lambda=0.5$ were used in all four
cases. Note that the Fermi energy ($E=0$) is tuned precisely to the saddle
point, so this depicts the DOS at the Van Hove strain
$\varepsilon_{100}=$-0.44\text{\,}\mathrm{\char 37\relax}$$.

Lastly, there’s the possibility of a horizontal line node (HLN) crossing the
vertical Van Hove line $\left(0,\pi,k_{3}\right)$. For a schematic
$\Delta_{\bm{k}}=\Delta_{0}(k_{3}/\pi)$, the 3D DOS can be calculated by
averaging Eq. (11):

$\displaystyle g_{\text{sc}}^{\text{HLN}}(E)$
$\displaystyle=\int_{-\pi}^{\pi}\frac{\mathop{}\\!\mathrm{d}{k_{3}}}{2\pi}\left.g_{\text{sc}}^{\text{FG}}(E)\right|_{\Delta_{0}\to\Delta_{0}\left\lvert{k_{3}}\right\rvert/\pi}$
(16)
$\displaystyle=\frac{4m_{*}}{(2\pi)^{2}}\frac{E}{\Delta_{0}}\left[\pi\log\frac{2\Lambda^{2}}{m_{*}E}-\zeta(E)\right],$

where $\zeta(E\leq\Delta_{0})=0$ and for $E>\Delta_{0}$:

$\displaystyle\zeta(E)$ $\displaystyle=(\pi-2\arccos
x)\log\frac{\Lambda^{2}}{m_{*}E}+2\arcsin x\log(2x)$
$\displaystyle\quad-2\arctan\frac{x}{\sqrt{1-x^{2}}}\log
x+\operatorname{Cl}_{2}(\phi),$ (17)

where $x=\sqrt{1-\Delta_{0}^{2}/E^{2}}$, $\phi=\arccos(1-2x^{2})$, and
$\operatorname{Cl}_{2}(\phi)=\sum_{k=1}^{\infty}\sin(k\phi)/k^{2}$ is the
Clausen function. $g_{\text{sc}}^{\text{HLN}}$ is thus roughly linear in $E$
up to $\Delta_{0}$.

The dependence of the DOS $g_{\text{sc}}(E)$ for different realizations of the
SC gap $\Delta_{\bm{k}}$ near the saddle point is summarized in Figure 3.

Now we come back to the question of whether line nodes at the Van Hove lines
are consistent with an entropy minimum. To clarify this issue, we need to take
into account the $k_{3}$-dispersion, the energy integral in Eq. (5), and the
DOS contributions of the other bands.

The $k_{3}$-dispersion of the $\gamma$ band smears all characteristically 2D
features of the DOS by the scale of its energy variation
$\mathop{}\\!\delta{\epsilon_{\text{VH}}}\sim$2\text{\,}\mathrm{K}$$ [Eq.
(18)]. The normal-state logarithmic singularity becomes a peak. The
$g_{\text{sc}}^{\text{VLN}}\propto\sqrt{E}$ ascent is cut off to give a finite
zero-energy DOS that is because of
$E_{w}/\mathop{}\\!\delta{\epsilon_{\text{VH}}}\ll 1$ of the same magnitude as
the normal-state DOS. Finally, the HLN DOS attains a finite zero-energy DOS
that is at most a factor of three or so smaller than the normal-state DOS
(since $\mathop{}\\!\delta{\epsilon_{\text{VH}}}/\Delta_{0}\sim 1$). The
$\operatorname{\updelta}_{T}(E)$ factor in Eq. (5) leads to a temperature
smearing that has a similar effect: the “effective DOS” that enters the
entropy is not $g_{\text{sc}}(0)$, but $g_{\text{sc}}(E)$ averaged over $E\sim
k_{B}T$. All in all, because of these smearing effects, vertical line nodes at
the Van Hove lines $\left(0,\pm\pi,k_{3}\right)$ do not suppress the entropy
contribution coming from the Van Hove lines, whereas horizontal line nodes can
indeed suppress it.

Because of the strain-dependence of $T_{c}$, the SC gap becomes
$\varepsilon_{100}$-dependent at constant $T$, peaking at Van Hove strain. A
strong enough gapping of the $\alpha$ and $\beta$ bands could then, in
principle, suppress the entropy more than the Van Hove singularities enhance
it, resulting in a minimum. To exclude this scenario, we have calculated the
entropy for when the $\alpha$, $\beta$, and $80\text{\,}\mathrm{\char
37\relax}$ of the $\gamma$ band have $\Delta_{\bm{k}}=\Delta_{0}$, and the
remaining $20\text{\,}\mathrm{\char 37\relax}$ of the $\gamma$ band that
includes the Van Hove lines has $\Delta_{\bm{k}}=0$.129 The result of this
calculation is that a minimum as a function of strain does develop, but the
drop in the entropy is $20\text{\,}\mathrm{\char 37\relax}$ too small when
compared to experiment at $2.5\text{\,}\mathrm{K}$. Thus even in this worst-
case scenario, where line nodes that are known [86, 87, 88, 80, 89, 25, 90,
91] to be present in the system are neglected, the Van Hove lines must be
gapped in some way to agree with experiment.

1010footnotetext:  For the total DOS and gap we have assumed the form:
$\displaystyle g_{\text{sc}}^{\text{tot}}(E)$
$\displaystyle=g_{\text{VH}}+\Theta(E-\Delta_{0})\frac{E}{\sqrt{E^{2}-\Delta_{0}^{2}}}g_{\text{rest}},$
$\displaystyle\Delta_{0}$
$\displaystyle=1.76\,k_{B}T_{c}\tanh\\!\bigg{(}1.76\sqrt{\frac{T_{c}}{T}-1}\bigg{)},$
where $g_{\text{VH}}$ is the normal-state DOS coming from the parts of the
$\gamma$ sheet that are close to the Van Hove lines and $g_{\text{rest}}$ is
the remaining normal-state DOS. Both
$g_{\text{VH}}(\varepsilon_{100})+g_{\text{rest}}(\varepsilon_{100})\propto\left.S(\varepsilon_{100},T)/T\right|_{T>T_{c}}$
and $T_{c}(\varepsilon_{100})$ are known experimentally. Only the ratio
$g_{\text{VH}}/g_{\text{rest}}$ needs to be calculated, which we have done
using the tight-bind model of Appendix B. One finds that
$g_{\text{rest}}(\varepsilon_{100})$ is roughly strain-independent, as
expected. The entropy is given by Eq. (5).

The final conclusion that follows from all of these considerations is that the
Van Hove lines
$\bm{k}_{\text{VH}}\approx\left(0,\pm\frac{\pi}{a},k_{3}\right)$ must be
either fully gapped or can at most have a horizontal line node crossing them.
Hence, we may exclude vertical line nodes at $\bm{k}_{\text{VH}}$ near Van
Hove strain [63]. That the heat capacity jump is maximal at the Van Hove
strain [62] also supports this conclusion. Vertical line nodes away from the
Van Hove lines are still possible.

To draw conclusions for the unstrained tetragonal system from measurements
performed at uniaxial strain
$\varepsilon_{100}\approx\varepsilon_{\text{VH}}$, we rely on the assumption
that the pairing states of the strained and unstrained system are
adiabatically connected. Measurements of the highly-sensitive elastocaloric
effect [63] and heat capacity [62] show no hints of a transition between two
different bulk SC states under $[100]$ strain. By contrast, the onset of spin-
density waves, previously found through muon spin relaxation [55], is clearly
visible in the elastocaloric data of Ref. [63]. So the elastocaloric effect is
able to identify a variety of phase transitions.

We may thus exclude all SC states of the unstrained system that are
adiabatically connected to SC states of the $\varepsilon_{xx}$ strained system
which have a vertical line node at
$\bm{k}_{\text{VH}}\approx\left(0,\pm\frac{\pi}{a},k_{3}\right)$. Given that
$\varepsilon_{xx}$ strain preserves all the symmetry operations that map the
Van Hove lines to themselves, as we shall see in Sec. V, we may conclude that
there are no vertical line nodes at either
$\left(\pm\frac{\pi}{a},0,k_{3}\right)$ nor
$\left(0,\pm\frac{\pi}{a},k_{3}\right)$ in the unstrained tetragonal system.
Intuitively, this means that SRO’s SC takes full advantage of the enhanced DOS
induced by the Van Hove lines. Indeed, the drastic enhancement of $T_{c}$ and
$B_{c2}$ under uniaxial pressure [77, 108, 47, 78, 109] were suggestive of
this conclusion long ago, but only with the recent elastocaloric measurements
of Ref. [63] could more conclusive statements be made.

##  IV Location of the Van Hove lines

Here we establish that the Van Hove lines are adequately approximated with
$\left(\pm\frac{\pi}{a},0,k_{3}\right)$ and
$\left(0,\pm\frac{\pi}{a},k_{3}\right)$. For a simple-tetragonal lattice, the
Van Hove lines are lines of high symmetry. However, they are not located
precisely on the boundary of the body-centered first Brillouin zone relevant
here, which could in principle allow for large deviations away from
$\left(\pm\frac{\pi}{a},0,k_{3}\right)$ and
$\left(0,\pm\frac{\pi}{a},k_{3}\right)$. As we shall see, the high anisotropy
of SRO makes these deviations negligible, justifying the subsequent analysis.

Van Hove points are points in momentum space where the gradient of the band
energy $\epsilon_{\bm{k}}$ vanishes. In 3D, the solutions of
$\bm{\nabla}\epsilon_{\bm{k}}=\bm{0}$ are generically isolated points.
However, quasi-2D dispersions may yield Van Hove _lines_ , that is, lines on
which a number of Van Hove points are situated of similar energy. The quality
of the emergent Van Hove lines is quantified by how well-aligned the Van Hove
points are to a line and by how close the energies of the Van Hove points are.

Consider the Van Hove line $\left(0,\frac{\pi}{a},k_{3}\right)$. Then for any
two
$\bm{k}=\left(\mathop{}\\!\delta{k_{1}},\frac{\pi}{a}+\mathop{}\\!\delta{k_{2}},k_{3}\right)$
and $\bm{k}^{\prime}=R(g)\bm{k}$ related by a symmetry operation $g\in
D_{4h}$, $\epsilon_{\bm{k}}=\epsilon_{\bm{k}^{\prime}+\bm{K}}$ for any
reciprocal lattice vector $\bm{K}$. Applying this to parity gives
$\bm{\nabla}\epsilon_{\bm{k}}=\bm{0}$ at the mid-points of the Brillouin zone
faces, which for body-centered tetragonal SRO are
$\left(0,\frac{\pi}{a},\pm\frac{\pi}{c}\right)$. These are the first two Van
Hove points. The positions of the other two Van Hove points are restricted by
symmetry to be at
$\left(0,\frac{\pi}{a}+\mathop{}\\!\delta{k_{\text{VH},2}},0\right)$ and
$\left(0,\frac{\pi}{a}-\mathop{}\\!\delta{k_{\text{VH},2}},\pm\frac{2\pi}{c}\right)$.
Reflection across the $k_{1}=0$ plane implies
$\partial_{k_{1}}\epsilon_{\bm{k}}=0$ in the $k_{1}=0$ plane and reflection
across the $k_{3}=0$ plane implies $\partial_{k_{3}}\epsilon_{\bm{k}}=0$ in
the planes $k_{3}=0,\pm\frac{2\pi}{c}$. If the system were simple tetragonal-
periodic, then reflection across the $k_{2}=0$ plane would imply
$\partial_{k_{2}}\epsilon_{\bm{k}}=0$ in the $k_{2}=\pm\frac{\pi}{a}$ planes,
making $\mathop{}\\!\delta{k_{\text{VH},2}}=0$. Because of the smallness of
the characteristically body-centered hopping in SRO, which is always between
layers, $\mathop{}\\!\delta{k_{\text{VH},2}}$ is very close to zero.

From the tight-binding model [Appendix B], we may extract the following
simplified expression for the dispersion of the $\gamma$ band near the Van
Hove line $\left(0,\frac{\pi}{a},k_{3}\right)$:

$\displaystyle\begin{aligned}
\epsilon_{\bm{k}}&=\mu_{\text{VH}}+\frac{a^{2}}{2m_{1}}k_{1}^{2}-\frac{a^{2}}{2m_{2}}\left(k_{2}-\frac{\pi}{a}\right)^{2}\\\
&-\mathop{}\\!\delta{\epsilon_{\text{VH}}}\cos
ck_{3}+\frac{a^{2}}{m_{2}}\mathop{}\\!\delta{k_{\text{VH},2}}\left(k_{2}-\frac{\pi}{a}\right)\cos\frac{ck_{3}}{2}.\end{aligned}$
(18)

Its form follows from symmetry; only the lowest powers in $k_{1},k_{2}$ and
lowest harmonics in $k_{3}$ were retained. Here
$\mu_{\text{VH}}=$54\text{\,}\mathrm{meV}$$,
$\mathop{}\\!\delta{\epsilon_{\text{VH}}}=$2.4\text{\,}\mathrm{K}$$,
$\mathop{}\\!\delta{k_{\text{VH},2}}=0.013/a$,
$m_{1}^{-1}=$1100\text{\,}\mathrm{K}$$, and
$m_{2}^{-1}=$9300\text{\,}\mathrm{K}$$. While this dispersion was derived from
a model of unstrained SRO, it offers a good understanding of the effects of
the $k_{3}$-dispersion on the Van Hove line. The deviation of the Van Hove
points from the $(\frac{\pi}{a},0,k_{3})$-line is characterized by
$\mathop{}\\!\delta{k_{\text{VH},2}}\ll\frac{2\pi}{a}$, which is a factor of
$500$ smaller than the width of the Brillouin zone. Furthermore, the
difference in the $\gamma$ band energies of the Van Hove points is given by
$\mathop{}\\!\delta{\epsilon_{\text{VH}}}$ which is on the order of a few
kelvins. We may thus conclude that the four Van Hove points, illustrated in
Figure 4, together constitute a Van Hove line
$\left(0,\frac{\pi}{a},k_{3}\right)$ to a high degree of accuracy. The same is
true for the Van Hove lines $\left(0,-\frac{\pi}{a},k_{3}\right)$ and
$\left(\pm\frac{\pi}{a},0,k_{3}\right)$.

##  V Behavior on the Van Hove lines

To see which SC states are excluded by the fact that vertical line nodes on
the Van Hove lines are incompatible with the elastocaloric effect data, we
first need to see which SC states are possible. This is significant because
the multiband nature of SRO allows for a richer set of possibilities than
usual. Since this has already been analyzed [112, 113, *Kaba2019-E, 115], here
we only briefly discuss how the multiband case differs from the singleband
one, delegating the details of the categorization of all possible SC states to
Appendix C.

To describe SRO’s SC, we employ an effective model based on the $t_{2g}$
orbitals of Ru [Appendix B]. Within it, SC is described by a gap matrix
$\Delta_{\alpha\beta}(\bm{k})$ which is characterized by its momentum
dependence and spin-orbit structure. It is the possibility of a non-trivial
orbital structure that sets multiband systems apart from singleband ones.
Thus, for instance, when dealing with even pairings, we cannot simply assume a
spin singlet that transforms trivially ($A_{1g}$) under all symmetry
operations and equate the irrep of the momentum wavefunction with the irrep of
the total gap matrix. The irrep of the gap matrix is determined by the
_product_ of the irreps of its momentum and spin-orbit parts. Within the
effective model, there are spin-orbit matrices belonging to all the possible
irreps of $D_{4h}$ for both even and odd pairings. The details of how
$\Delta_{\alpha\beta}(\bm{k})$ are constructed by combining pairing
wavefunctions $d(\bm{k})$ with spin-orbit matrices $\Gamma$ can be found in
Appendix C.

Figure 4: The body-centered tetragonal Brillouin zone of SRO (a), its
$k_{1}=0$ cross-section (b), and the region around the
$\left(0,\frac{\pi}{a},k_{3}\right)$ Van Hove line (c). Shaded in blue is the
simple tetragonal Brillouin zone. The red crosses are the
$\left(0,\frac{\pi}{a},\pm\frac{\pi}{c}\right)$ Van Hove points. The blue dots
are the $\left(0,\frac{\pi}{a}+\mathop{}\\!\delta{k_{\text{VH},2}},0\right)$
and
$\left(0,\frac{\pi}{a}-\mathop{}\\!\delta{k_{\text{VH},2}},\pm\frac{2\pi}{c}\right)$
Van Hove points. Together they constitute the Van Hove line
$\left(0,\frac{\pi}{a},k_{3}\right)$, drawn here with a dashed red line. The
displacement length $\mathop{}\\!\delta{k_{\text{VH},2}}\approx 0.013/a$ is
designated in (c).

Now we analyze which SC states of the $\varepsilon_{xx}$ strained system gap
the Van Hove lines sufficiently strongly to be able to explain the
elastocaloric experiment [63]. Viable unstrained SC states must be
adiabatically connected to these states. As we shall see, in the arguments of
this section the key symmetry operations are those that map the Van Hove lines
$\bm{k}_{\text{VH}}=\left(0,\pm\frac{\pi}{a},k_{3}\right)$ to themselves. As
it turns out, although $\varepsilon_{xx}$ strain reduces the point group from
$D_{4h}$ to $D_{2h}$ [Table 3], the symmetries that map the Van Hove lines to
themselves are the same for both $D_{4h}$ and $D_{2h}$. Hence we may do the
whole analysis either with or without $\varepsilon_{xx}$ strain. We have opted
for the latter. Using Table 4, one may translate all the results for irreps of
$D_{4h}$ of this section into results for irreps of $D_{2h}$. Table 4 also
specifies which irreps of $D_{2h}$ are adiabatically connected to which irreps
of $D_{4h}$, which brings us back to the initial $D_{4h}$ irreps.

Table 3: The character table of the point group $D_{2h}$ [119]. Irreps are divided into even ($g$) and odd ($u$) ones. Primes have been added on the irreps to distinguish them from $D_{4h}$ irreps. $C_{2}^{x}$, $C_{2}^{y}$, and $C_{2}^{z}$ are rotations by $\pi$ around the $x$, $y$, and $z$ axes, respectively. $P$ is parity. $\Sigma_{x}$, $\Sigma_{y}$, $\Sigma_{z}$ are compositions of $C_{2}^{x}$, $C_{2}^{y}$, and $C_{2}^{z}$ with $P$, respectively. $D_{2h}$ | $E$ | $C_{2}^{z}$ | $C_{2}^{y}$ | $C_{2}^{x}$ | $P$ | $\Sigma_{z}$ | $\Sigma_{y}$ | $\Sigma_{x}$
---|---|---|---|---|---|---|---|---
$A_{1g}^{\prime}$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$
$B_{1g}^{\prime}$ | $1$ | $1$ | $-1$ | $-1$ | $1$ | $1$ | $-1$ | $-1$
$B_{2g}^{\prime}$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $-1$
$B_{3g}^{\prime}$ | $1$ | $-1$ | $-1$ | $1$ | $1$ | $-1$ | $-1$ | $1$
$A_{1u}^{\prime}$ | $1$ | $1$ | $1$ | $1$ | $-1$ | $-1$ | $-1$ | $-1$
$B_{1u}^{\prime}$ | $1$ | $1$ | $-1$ | $-1$ | $-1$ | $-1$ | $1$ | $1$
$B_{2u}^{\prime}$ | $1$ | $-1$ | $1$ | $-1$ | $-1$ | $1$ | $-1$ | $1$
$B_{3u}^{\prime}$ | $1$ | $-1$ | $-1$ | $1$ | $-1$ | $1$ | $1$ | $-1$
Table 4: Reduction of the $D_{4h}$ irreps (top) to $D_{2h}$ irreps (bottom)
that occurs under $\varepsilon_{xx}$ uniaxial strain. Parity stays the same so
we have suppressed the $g/u$ subscripts. $\\{x\mid y\\}$ transforms according
to the $\rho^{(E)}(g)$ of Eq. (55), Appendix C.

Let us consider the Van Hove line
$\bm{k}_{\text{VH}}=\left(0,\frac{\pi}{a},k_{3}\right)$. For a SC gap matrix
$\Delta_{a}(\bm{k})$ to be able to gap the $\gamma$ band at
$\bm{k}_{\text{VH}}$, both its pairing wavefunction $d_{a}(\bm{k})$ and the
projection of its spin-orbit matrix $\Gamma_{a}$ onto the $\gamma$ band must
be finite there.

The only point group symmetries $g\in D_{4h}$ that constrain
$d_{a}\\!\left(\bm{k}_{\text{VH}}\right)$ or the band projections of
$\Gamma_{a}$ are those that map the $\left(0,\frac{\pi}{a},k_{3}\right)$ line
to itself, modulo body-centered reciprocal lattice vectors. One readily find
that these are

$\displaystyle\begin{aligned} \Sigma_{x}^{\prime}\colon k_{3}&\mapsto
k_{3},\\\ \Sigma_{y}^{\prime},C_{2z}\colon k_{3}&\mapsto
k_{3}+\frac{2\pi}{c},\\\ \Sigma_{h},C_{2y}^{\prime}\colon k_{3}&\mapsto-
k_{3},\\\ C_{2x}^{\prime}\colon k_{3}&\mapsto-
k_{3}+\frac{2\pi}{c}.\end{aligned}$ (19)

Here, $C_{2x}^{\prime}$, $C_{2y}^{\prime}$, $C_{2z}$ are rotations by $\pi$
around $x$, $y$, and $z$, respectively, and
$\Sigma_{x}^{\prime}=PC_{2x}^{\prime}$,
$\Sigma_{y}^{\prime}=PC_{2y}^{\prime}$, $\Sigma_{h}=PC_{2z}$ are reflections.
Given that $C_{2z}=\Sigma_{x}^{\prime}\Sigma_{y}^{\prime}$ and
$C_{2y}^{\prime}=\Sigma_{x}^{\prime}\Sigma_{h}$, we may focus solely on the
reflections and $C_{2x}^{\prime}$. Their matrices are listed in Table 5. The
strongest constraints follow from $\Sigma_{x}^{\prime}$ because it maps
$k_{3}\mapsto k_{3}$. In the simple tetragonal limit, $k_{3}\cong
k_{3}+\frac{2\pi}{c}$ so $\bm{k}_{\text{VH}}$ are on the Brillouin zone
boundary and $\Sigma_{y}^{\prime},C_{2z}$ give strong constraints too.

Table 5: The $\rho_{ab}^{(\lambda)}(g)$ of non-trivial irreps $\lambda$ of $D_{4h}$. $g$ | $A_{2g}$ | $B_{1g}$ | $B_{2g}$ | $E_{g}$ | $A_{1u}$ | $A_{2u}$ | $B_{1u}$ | $B_{2u}$ | $E_{u}$
---|---|---|---|---|---|---|---|---|---
$\Sigma_{x}^{\prime}$ | $-1$ | $1$ | $-1$ | $\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$ | $-1$ | $1$ | $-1$ | $1$ | $\begin{pmatrix}-1&0\\\ 0&1\end{pmatrix}$
$\Sigma_{y}^{\prime}$ | $-1$ | $1$ | $-1$ | $\begin{pmatrix}-1&0\\\ 0&1\end{pmatrix}$ | $-1$ | $1$ | $-1$ | $1$ | $\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$
$\Sigma_{h}$ | $1$ | $1$ | $1$ | $\begin{pmatrix}-1&0\\\ 0&-1\end{pmatrix}$ | $-1$ | $-1$ | $-1$ | $-1$ | $\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}$
$C_{2x}^{\prime}$ | $-1$ | $1$ | $-1$ | $\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$ | $1$ | $-1$ | $1$ | $-1$ | $\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$

Consider one of the four $g$ from Table 5 and a $k_{3}$ that $g$ maps to
itself, modulo $\frac{4\pi}{c}$. Periodicity and the symmetry transformation
rule of pairing wavefunctions [Eq. (56), Appendix C] then give the constraint:

$\displaystyle d_{a}\\!\left(0,\tfrac{\pi}{a},k_{3}\right)$
$\displaystyle=\sum_{b=1}^{\dim\lambda}\rho_{ab}^{(\lambda)}(g)d_{b}\\!\left(0,\tfrac{\pi}{a},k_{3}\right).$
(20)

By analyzing it, we find the following symmetry-enforced behavior of
$d_{a}\\!\left(0,\frac{\pi}{a},k_{3}\right)$, depending on its irrep and
$k_{3}$:

  * •

$d$ belonging to $A_{2g}$, $B_{2g}$, $A_{1u}$, and $B_{1u}$ vanish for all
$k_{3}$.

  * •

For $\\{d_{1}\mid d_{2}\\}\in E_{g}$, $d_{2}$ vanishes for all $k_{3}$,
whereas $d_{1}$ vanishes only at $k_{3}=0,\pm\frac{2\pi}{c}$.

  * •

For $\\{d_{1}\mid d_{2}\\}\in E_{u}$, $d_{1}$ vanishes for all $k_{3}$,
whereas $d_{2}$ vanishes only at $k_{3}=\pm\frac{\pi}{c}$.

  * •

For those $\\{d_{1}\mid d_{2}\\}\in E_{g/u}$ that are periodic under simple
tetragonal translations [$k_{3}\cong k_{3}+\frac{2\pi}{c}$], both components
vanish for all $k_{3}$.

  * •

$d$ from irreps $A_{2u}$ and $B_{2u}$ vanish only at $k_{3}=0$,
$\pm\frac{\pi}{c}$, and $\pm\frac{2\pi}{c}$, but are otherwise unconstrained.

  * •

$d$ from $A_{1g}$ and $B_{1g}$ are completely unconstrained for all $k_{3}$.

To proceed, we consider the pairing of the band eigenstates of the problem and
focus on intraband pairing. To find it, we need to project $\Gamma_{a}$ onto
the bands. Call
$V_{\bm{k}}=\left(v_{\bm{k}\uparrow},v_{\bm{k}\downarrow}\right)$ the Kramers-
degenerate eigenvectors of the $\gamma$ band,
$H_{\bm{k}}V_{\bm{k}}=\epsilon_{\bm{k}}V_{\bm{k}}$. The projection is then
given by:

$\displaystyle P_{\bm{k}a}$
$\displaystyle=V_{\bm{k}}^{{\dagger}}\Gamma_{a}V_{-\bm{k}}^{*}=\sum\nolimits_{\mu}P_{a}^{\mu}(\bm{k})\upsigma_{\mu}(\mathrm{i}\upsigma_{y}),$
(21)

where the Pauli matrices act in pseudospin space. Since all three $t_{2g}$
orbitals are even, we may locally choose a gauge in which
$V_{-\bm{k}}=V_{\bm{k}}$ so that
$P_{\bm{k}a}^{\intercal}=sP_{-\bm{k}a}=sP_{\bm{k}a}$, where
$\Gamma_{a}^{\intercal}=s\Gamma_{a}$. $\mu=0$ for antisymmetric $\Gamma_{a}$
($s=-1$), whereas $\mu\in\\{x,y,z\\}$ for symmetric $\Gamma_{a}$ ($s=+1$).

Whenever a $g\in D_{4h}$ maps a $\bm{k}$ to itself modulo periodicity, its
symmetry transformation matrix $U(g)=M(g)\otimes S(g)$ [Appendix B, Table 7]
commutes with the normal-state Hamiltonian $H_{\bm{k}}$:

$\displaystyle U^{{\dagger}}(g)H_{\bm{k}}U(g)$
$\displaystyle=H_{R(g^{-1})\bm{k}}=H_{\bm{k}}.$ (22)

This means that the interband parts of $U(g)$ vanish. As for the intraband
part, we may always choose a basis for the Kramers degenerate subspace such
that it takes a spin-like form:

$\displaystyle V_{\bm{k}}^{{\dagger}}U(g)V_{\bm{k}}$ $\displaystyle=S(g).$
(23)

The symmetry transformation rule of spin-orbit matrices [Eq. (57), Appendix C]
now gives the constraint:

$\displaystyle S^{{\dagger}}(g)P_{\bm{k}a}S^{*}(g)$
$\displaystyle=\sum_{b=1}^{\dim\lambda}\rho_{ab}^{(\lambda)}(g)P_{\bm{k}b}.$
(24)

For $\bm{k}$ on the Van Hove line $\left(0,\frac{\pi}{a},k_{3}\right)$, the
$g$ from Table 5 constrain certain $P_{a}^{\mu}(\bm{k})$ to vanish, depending
on the (anti-)symmetry, irrep, and $k_{3}$. The (anti-)symmetry
$\Gamma_{a}^{\intercal}=s\Gamma_{a}$ we shall denote with an irrep superscript
$s=\pm$. Thus, for instance, $\Gamma\in A_{1g}^{-}$ are antisymmetric under
transposition, whereas $\Gamma\in B_{1g}^{+}$ are symmetric under
transposition. The symmetry-enforced behavior of
$P_{a}^{\mu}\\!\left(0,\frac{\pi}{a},k_{3}\right)$ we may summarize as
follows:

  * •

$\Gamma$ belonging to $A_{2g}^{-}$ and $B_{2g}^{-}$ have $P^{0}=0$ for all
$k_{3}$.

  * •

$\\{\Gamma_{1}\mid\Gamma_{2}\\}\in E_{g}^{-}$ have $P_{2}^{0}=0$ for all
$k_{3}$, whereas $P_{1}^{0}=0$ only at $k_{3}=0,\pm\frac{2\pi}{c}$.

  * •

$\Gamma\in A_{1g}^{+},B_{1g}^{+}$ have $P^{y}=P^{z}=0$ for all $k_{3}$, and
$P^{x}=0$ only at $k_{3}=0,\pm\frac{2\pi}{c}$.

  * •

$\Gamma\in A_{2g}^{+},B_{2g}^{+}$ have $P^{x}=0$ for all $k_{3}$, and
$P^{y}=0$ only at $k_{3}=0,\pm\frac{2\pi}{c}$. $P^{z}$ is unconstrained.

  * •

$\\{\Gamma_{1}\mid\Gamma_{2}\\}\in E_{g}^{+}$ have
$P_{1}^{y}=P_{1}^{z}=P_{2}^{x}=0$ for all $k_{3}$, and $P_{2}^{z}=0$ only at
$k_{3}=0,\pm\frac{2\pi}{c}$. The remaining $P_{1}^{x}$ and $P_{2}^{y}$ are
unconstrained.

  * •

The $P^{0}$ of $\Gamma$ from $A_{1g}^{-}$ and $B_{1g}^{-}$ are completely
unconstrained for all $k_{3}$.

In the limit of vanishing body-centered tetragonal hopping, the following
$P_{a}^{\mu}$ vanish in addition:

  * •

For $\\{\Gamma_{1}\mid\Gamma_{2}\\}\in E_{g}^{-}$, $P_{1}^{0}$ vanishes for
all $k_{3}$ so both $P_{a}^{0}$ are zero.

  * •

For $\Gamma\in A_{1g}^{+},B_{1g}^{+}$, $P^{\mu}$ completely vanish for all
$k_{3}$.

  * •

For $\Gamma\in A_{2g}^{+},B_{2g}^{+}$, $P^{y}=0$ for all $k_{3}$, but $P^{z}$
is still unconstrained.

  * •

For $\\{\Gamma_{1}\mid\Gamma_{2}\\}\in E_{g}^{+}$, $P_{2}^{z}=0$ for all
$k_{3}$, but $P_{1}^{x}$ and $P_{2}^{y}$ are still unconstrained.

Owning to the fact that all characteristically body-centered hopping is
necessarily between layers and that these hoppings are very small in SRO
because of its high anisotropy, the vanishing $P_{a}^{\mu}$ listed above are
very small for SRO, although not precisely zero. Using the tight-binding model
of Ref. [117], described in Appendix B, we have quantified their smallness:
the vanishing $P_{a}^{\mu}$ listed above are by a factor of $50$ or more
smaller than the largest possible $P_{a}^{\mu}\sim 1$, where all $\Gamma_{a}$
have been normalized to $\operatorname{tr}\Gamma_{a}^{{\dagger}}\Gamma_{a}=1$
for a fair comparison.

Unlike the above anisotropy argument, arguments based on the $d_{xy}$ orbital
character of the $\gamma$ band do not suppress any irreps, but only inform us
on which $\Gamma_{a}$ from within a given irrep have large $P_{a}^{\mu}$.

Finally, we synthesize the results found for $d_{a}$ and $\Gamma_{a}$. This is
done by going through the multiplication table of irreps [Table 11 in Appendix
C] and seeing which entries yield a $\Delta_{a}(\bm{k})$ with a finite
$\gamma$ band projection. The results are summarized in Table 6b. Table 6b is
the main result of this paper. As mentioned, SRO’s anisotropy suppresses the
blue entries of the table by two orders of magnitude. This means that a
$\Delta$ with a maximal value $\sim k_{B}T_{c}$ is way too small on the Van
Hove lines to explain the observed entropy quenching [63]. Hence the blue
entries of Table 6b are excluded as possible leading SC states as well.

From Table 6b we see that, among even pairings, only $A_{1g}$, $B_{1g}$, and
$E_{g}$ irreps have pairings that do not have symmetry-enforced vertical line
nodes on the Van Hove lines. Thus even pairings must have admixtures from one
of these three irreps to be able to explain the elastocaloric experiment of
Ref. [63]. It is worth noting that within these three irreps, pairings with
symmetry-enforced vertical line nodes on $\bm{k}_{\text{VH}}$ do exist, like
for instance $\Delta(\bm{k})=\Lambda_{1}(\mathrm{i}\upsigma_{y})\sin
ak_{1}\sin ak_{2}\in B_{2g}^{-}\otimes B_{2g}=A_{1g}$ [$\Lambda_{1}$ is given
in Appendix A]. So Table 6b also yields non-trivial information on the spin-
orbit and momentum structure of these Van Hove line-gapping admixtures. Three
representatives of such even-parity $\bm{k}_{\text{VH}}$-gapping SC states are
plotted in Figure 5.

Table 6: Even-parity (a) and odd-parity (b) pairings that do not have a
vertical line node at $\left(0,\frac{\pi}{a},k_{3}\right)$, constructed by
combining pairing wavefunctions $d_{a}(\bm{k})$ with spin-orbit matrices
$\Gamma_{a}$ according to the multiplication table of $D_{4h}$ irreps [Table
11, Appendix C]. A ‘$+$’ superscript on a spin-orbit matrix irrep means that
the matrices are symmetric ($\Gamma^{\intercal}=+\Gamma$), whereas a ‘$-$’
superscript indicates antisymmetry under transposition. A zero component of
$E_{g/u}$ means that it vanishes on $\left(0,\frac{\pi}{a},k_{3}\right)$.
Highlighted red are those $d_{a}$ that must be periodic under body-centered
translations, but not under simple tetragonal translations, to be finite on
$\left(0,\frac{\pi}{a},k_{3}\right)$. For examples, see Table 9 from Appendix
C. Such $d_{a}$ have horizontal line nodes at $k_{3}=0,\pm\frac{2\pi}{c}$.
Highlighted blue are those $\Gamma_{a}$ whose projections onto the $\gamma$
band are suppressed by two orders of magnitude because of the weakness of
body-centered interlayer hopping. Such $\Gamma_{a}$ are unable to account for
the elastocaloric experiment, but are listed for the sake of completeness.

$\otimes$ | $A_{1g}\\{d\\}$ | $B_{1g}\\{d\\}$ | $E_{g}\\{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}d_{1}}\mid 0\\}$
---|---|---|---
$A_{1g}^{-}\\{\Gamma\\}$ | $A_{1g}\\{\Gamma d\\}$ | $B_{1g}\\{\Gamma d\\}$ | $E_{g}\\{\Gamma{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}d_{1}}\mid 0\\}$
$B_{1g}^{-}\\{\Gamma\\}$ | $B_{1g}\\{\Gamma d\\}$ | $A_{1g}\\{\Gamma d\\}$ | $E_{g}\\{\Gamma{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}d_{1}}\mid 0\\}$
$E_{g}^{-}\begin{Bmatrix}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Gamma_{1}}\\\ 0\end{Bmatrix}$ | $E_{g}\begin{Bmatrix}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Gamma_{1}}d\\\ 0\end{Bmatrix}$ | $E_{g}\begin{Bmatrix}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Gamma_{1}}d\\\ 0\end{Bmatrix}$ | $\begin{matrix}A_{1g}\\{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Gamma_{1}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}d_{1}}+0\\}\\\ \oplus B_{1g}\\{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Gamma_{1}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}d_{1}}-0\\}\end{matrix}$
(a) Even pairings that are finite on $\left(0,\frac{\pi}{a},k_{3}\right)$.

$\otimes$ | $A_{2u}\\{d\\}$ | $B_{2u}\\{d\\}$ | $E_{u}\\{0\mid{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}d_{2}}\\}$
---|---|---|---
$A_{1g}^{+}\\{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Gamma}\\}$ | $A_{2u}\\{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Gamma}d\\}$ | $B_{2u}\\{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Gamma}d\\}$ | $E_{u}\\{0\mid{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Gamma}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}d_{2}}\\}$
$A_{2g}^{+}\\{\Gamma\\}$ | $A_{1u}\\{\Gamma d\\}$ | $B_{1u}\\{\Gamma d\\}$ | $E_{u}\\{\Gamma{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}d_{2}}\mid 0\\}$
$B_{1g}^{+}\\{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Gamma}\\}$ | $B_{2u}\\{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Gamma}d\\}$ | $A_{2u}\\{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Gamma}d\\}$ | $E_{u}\\{0\mid-{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Gamma}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}d_{2}}\\}$
$B_{2g}^{+}\\{\Gamma\\}$ | $B_{1u}\\{\Gamma d\\}$ | $A_{1u}\\{\Gamma d\\}$ | $E_{u}\\{\Gamma{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}d_{2}}\mid 0\\}$
$E_{g}^{+}\begin{Bmatrix}\Gamma_{1}\\\ \Gamma_{2}\end{Bmatrix}$ | $E_{u}\begin{Bmatrix}\Gamma_{2}d\\\ -\Gamma_{1}d\end{Bmatrix}$ | $E_{u}\begin{Bmatrix}\Gamma_{2}d\\\ \Gamma_{1}d\end{Bmatrix}$ | $\begin{matrix}A_{1u}\\{0+\Gamma_{2}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}d_{2}}\\}\\\ \oplus A_{2u}\\{\Gamma_{1}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}d_{2}}-0\\}\\\ \oplus B_{1u}\\{0-\Gamma_{2}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}d_{2}}\\}\\\ \oplus B_{2u}\\{\Gamma_{1}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}d_{2}}+0\\}\end{matrix}$
(b) Odd pairings that are finite on $\left(0,\frac{\pi}{a},k_{3}\right)$.

One such piece of information is that $E_{g}$ pairing must be made of
wavefunctions $d_{a}$ that are body-centered periodic, but not simple
tetragonal periodic. The lowest order such $\\{d_{xz}\mid d_{yz}\\}\in E_{g}$
is:

$\left\\{\sin\frac{ak_{1}}{2}\cos\frac{ak_{2}}{2}\sin\frac{ck_{3}}{2}\middle|\cos\frac{ak_{1}}{2}\sin\frac{ak_{2}}{2}\sin\frac{ck_{3}}{2}\right\\}.$
(25)

It is this pairing state, only allowed because of the body-centered tetragonal
structure of SRO, that opens a gap at the Van Hove line and that we cannot
exclude based on the elastocaloric data. In Ref. [16] it was shown that such a
pairing state can be stabilized by a strongly momentum-dependent spin-orbit
coupling. A better understanding of the origin of such momentum dependence
might help elucidate whether this state is a viable option for SRO’s SC. In
distinction, the $E_{g}$ pairing state

$\\{\sin ak_{1}\sin ck_{3}\mid\sin ak_{2}\sin ck_{3}\\},$ (26)

which would be the only allowed one for simple-tetragonal lattices, cannot be
the only pairing state as it does not open a gap on the Van Hove line. An
important difference between these two types of states [(25) vs. (26)] is that
the former always have horizontal line nodes at $k_{3}=0,\pm\frac{2\pi}{c}$.

(a) $\Delta_{\bm{k}}=\Lambda_{4}(\mathrm{i}\upsigma_{y})\in A_{1g}$

(b)
$\Delta_{\bm{k}}=\Lambda_{4}(\mathrm{i}\upsigma_{y})\left(\cos\kappa_{1}-\cos\kappa_{2}\right)\in
B_{1g}$

(c)
$\Delta_{\bm{k}}=\Lambda_{4}(\mathrm{i}\upsigma_{y})\left(\sin\frac{\kappa_{1}}{2}\cos\frac{\kappa_{2}}{2}\pm\mathrm{i}\cos\frac{\kappa_{1}}{2}\sin\frac{\kappa_{2}}{2}\right)\sin\frac{\kappa_{3}}{2}\in
E_{g}$

Figure 5: Projections onto the Fermi sheets of three Van Hove line-gapping SC
states $\Delta_{\bm{k}}$ from Table 6b, belonging to irreps $A_{1g}$ (a),
$B_{1g}$ (b), and chiral $E_{g}$ (c), respectively.
$\kappa_{1}=ak_{1}\in[-\pi,\pi]$, $\kappa_{2}=ak_{2}\in[-\pi,\pi]$, and
$\kappa_{3}=ck_{3}\in[-2\pi,2\pi]$. In the $\gamma$ sheet plots, the Van Hove
lines $\left(\pm\frac{\pi}{a},0,k_{3}\right)$ and
$\left(0,\pm\frac{\pi}{a},k_{3}\right)$ are highlighted red. Because of the
$\Lambda_{4}=\operatorname{diag}(0,0,\sqrt{2})$ orbital structure we have
chosen, the projections are only large where the bands have $d_{xy}$ orbital
character, which is on the $\gamma$ sheet and where the $\gamma$ and $\beta$
sheets nest. More plots can be found in Appendix D.

Among odd pairings, all irreps have pairings without symmetry-enforced
vertical line nodes on $\bm{k}_{\text{VH}}$. However, the orientations of the
Balian-Werthamer $\bm{d}$-vectors [111] are non-trivially restricted and the
non-suppressed $A_{2u}$ and $B_{2u}$ pairings are necessarily made of
characteristically body-centered periodic $d_{a}$.

In multiband systems with spin-orbit coupling, a $\bm{d}$-vector is associated
with each band in its pseudospin (Kramers) space. It is defined through:

$\displaystyle V_{\bm{k}n}^{{\dagger}}\Delta(\bm{k})V_{-\bm{k}n}^{*}$
$\displaystyle=\bm{d}_{\bm{k}n}\bm{\cdot}\bm{\upsigma}(\mathrm{i}\upsigma_{y}),$
(27)

where $V_{\bm{k}n}=\left(v_{\bm{k}n\uparrow},v_{\bm{k}n\downarrow}\right)$ are
the Kramers-degenerate eigenvectors of the $n$-th band and
$V_{-\bm{k}n}=V_{\bm{k}n}$. We make the following gauge choice for the
pseudospins:

$\displaystyle\begin{aligned}
V_{\bm{k}n}^{{\dagger}}(\mathbbm{1}\otimes\mathrm{i}\upsigma_{y})V_{\bm{k}n}^{*}&=\mathrm{i}\upsigma_{y},\\\
V_{\bm{k}n}^{{\dagger}}(\mathbbm{1}\otimes\upsigma_{z})V_{\bm{k}n}&=s_{z}\upsigma_{z},\\\
V_{\bm{k}n}^{{\dagger}}(\mathbbm{1}\otimes\upsigma_{x})V_{\bm{k}n}&=s_{x}\upsigma_{x}+\delta_{xz}\upsigma_{z},\end{aligned}$
(28)

where $s_{z},s_{x},\delta_{xz}\in\mathbb{R}$. This is the closest one can make
the pseudospins look like spins. In general $\delta_{xz}$ is not zero, nor are
the $\delta_{yx},\delta_{yz}$ from
$V_{\bm{k}n}^{{\dagger}}(\mathbbm{1}\otimes\upsigma_{y})V_{\bm{k}n}=s_{y}\upsigma_{y}+\delta_{yx}\upsigma_{x}+\delta_{yz}\upsigma_{z}$.
However, in SRO the only regions where $\delta_{xz},\delta_{yx},\delta_{yz}$
are substantially different from zero is at the nesting of the $\alpha$,
$\beta$, and $\gamma$ bands at $k_{1}=\pm k_{2}$ [Figure 1]. The explanation
for this is the fact that spin-orbit coupling most strongly affects the band
structure there.

Using the tight-binding model of SRO [Appendix B], we have explored the
orientation of the $\bm{d}_{\bm{k}n}$-vectors on the $\alpha$, $\beta$, and
$\gamma$ Fermi sheets. Everywhere except near the $k_{1}=\pm k_{2}$ nesting of
the sheets, we find that symmetric spin-orbit matrices from 1D irreps have
$\bm{d}_{\bm{k}n}$ pointing along $\pm\bm{\hat{z}}$, whereas
$\\{\Gamma_{1}\mid\Gamma_{2}\\}$ from $E_{g}^{+}$ always have in-plane
$\bm{d}_{\bm{k}n}$. So the non-suppressed $A_{2u}$ and $B_{2u}$ from Table 6b
(b) have $\bm{d}_{\bm{k}n}\parallel\bm{\hat{z}}$. Moreover, among odd pairings
not made of body-centered $d_{a}(\bm{k})$, $A_{1u}$ and $B_{1u}$ pairings have
$\bm{d}_{\bm{k}n}\parallel\bm{\hat{z}}$ and $E_{u}$ pairings have in-plane
$\bm{d}_{\bm{k}n}$. Given that body-centered $\\{d_{1}\mid d_{2}\\}\in E_{u}$
have horizontal line nodes, on the one hand, and that the spin susceptibility
is intimately related to the orientation of the Balian-Werthamer
$\bm{d}$-vector, on the other, this information may prove to be useful in
further narrowing down the odd-pairing SC candidates.

##  VI Conclusion

This paper was motivated by the measurements of the elastocaloric effect of
Sr2RuO4 under strain reported in Ref. [63]. The elastocaloric effect measures,
with high accuracy, the entropy derivative $\partial
S(\varepsilon,T)/\partial\varepsilon$. Above $T_{c}$, the elastocaloric effect
revealed a pronounced maximum in the entropy as function of strain
$\varepsilon$. As demonstrated in Ref. [63], this maximum of $S(\varepsilon)$
can be fully accounted for by the DOS enhancement that occurs when the Fermi
energy crosses the Van Hove points near the lines
$\left(0,\pm\frac{\pi}{a},k_{3}\right)$. Below $T_{c}$, the entropy maximum
was found to transform into a minimum. This is only possible if the states
near the saddle points of the electronic dispersion open a gap as one enters
the SC state. Hence, with rather minimal modelling, it is possible to obtain
information about the momentum-space structure of the SC gap from a
thermodynamic measurement.

In order to draw more detailed conclusions about the allowed pairing states,
we performed a symmetry analysis for a three-dimensional, three-band
description of SRO. Here we focus primarily on even-parity states, given the
strong evidence for even parity in NMR measurements [35, 36, 37]. From a
simple two-dimensional perspective, one would conclude that the SC state must
open a gap at the Van Hove points $\left(\pm\frac{\pi}{a},0\right)$ and
$\left(0,\pm\frac{\pi}{a}\right)$. However, to distinguish the relevant
pairing states, in particular those of the 2D irreducible representation
$E_{g}$ that transform like $\\{d_{xz}\mid d_{yz}\\}$, we must include the
third momentum direction. It is well known that the energy dispersion of SRO
is strongly anisotropic. Indeed, our analysis shows that the energy scale
below which the three-dimensionality of the Fermi surface becomes important is
about one kelvin, fully consistent with magneto-oscillation experiments [2].
We also show that the saddle points deviate by very small amounts
$\mathop{}\\!\delta{k_{\text{VH},2}}\ll\frac{2\pi}{a}$ from the lines
$\left(\pm\frac{\pi}{a},0,k_{3}\right)$ and
$\left(0,\pm\frac{\pi}{a},k_{3}\right)$. However, this need not be the case
for the SC state. While the single particle spectrum of SRO is highly
anisotropic, it is possible that many-body interactions that are responsible
for the SC pairing couple different layers more efficiently. Hence, at least
in principle, one should not exclude a strong dependence of the gap function
on $k_{3}$; such dependence is crucial for the $\\{d_{xz}\mid d_{yz}\\}$-wave
pairing states.

With these insights, we then turned to the symmetry analysis of potential
pairing states. If one assumes for a moment that the crystal structure of SRO
is simple tetragonal, one is left with only two possible even pairing states,
namely, the $s$-wave state of $A_{1g}$ symmetry and the $d_{x^{2}-y^{2}}$-wave
state of $B_{1g}$ symmetry. Given that fine-tuning is required for $s$-wave
pairing to be consistent with the pair-breaking role of impurities [27,
*Mackenzie1998-E, 29, 30, 31], $d_{x^{2}-y^{2}}$-wave pairing would then
appear to be the only natural pairing candidate. However, Sr2RuO4 is a body-
centered tetragonal compound. The corresponding symmetry analysis now allows,
in addition to $d_{x^{2}-y^{2}}$-wave pairing, for a $\\{d_{xz}\mid
d_{yz}\\}$-wave state of $E_{g}$ symmetry like the one given in Eq. (25).

Our analysis does, however, allow us to exclude $d_{xy}$-wave pairing states
that transform like $B_{2g}$ and $g_{xy(x^{2}-y^{2})}$-wave pairing states
that transform like $A_{2g}$ as sole pairing states. Such states may at best
be subleading contenders that could be added to the pairing wavefunction at
fine-tuned points of accidental degeneracy. In addition, we can exclude
$\\{d_{xz}\mid d_{yz}\\}$-wave pairing that is exclusively of the type given
in Eq. (26). The nature of our argument does not allow us to more precisely
quantify how large these subleading $d_{xy}$-wave or
$g_{xy(x^{2}-y^{2})}$-wave contributions are because they vanish precisely
where the elastocaloric experiment is most sensitive: at the Van Hove lines.
Thus, while the elastocaloric measurements do not allow for a unique
determination of the superconducting order parameter symmetry, they do
constrain the available options. To finally resolve the nature of
superconductivity in Sr2RuO4 requires a better understanding of the origin of
time-reversal symmetry breaking and of the orientation of line nodes.

###### Acknowledgements.

We are grateful to Markus Garst for helpful discussions. This work was
supported by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) – TRR 288-422213477 Elasto-Q-Mat project A05 (R.V.), project A07
(G.P. and J.S.), and project A10 (C.H. and A.P.M.). A.R. acknowledges support
from the Engineering and Physical Sciences Research Council (grant numbers
EP/P024564/1, EP/S005005/1, and EP/V049410/1). Research in Dresden benefits
from the environment provided by the DFG Cluster of Excellence ct.qmat (EXC
2147, project ID 390858940).

##  Appendix A Gell-Mann matrices

We use the following unconventional choice for the nine Gell-Mann matrices:

$\displaystyle\Lambda_{0}$ $\displaystyle=\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&0\end{pmatrix},$ $\displaystyle\Lambda_{1}$
$\displaystyle=\begin{pmatrix}0&1&0\\\ 1&0&0\\\ 0&0&0\end{pmatrix},$ (29)
$\displaystyle\Lambda_{2}$ $\displaystyle=\begin{pmatrix}0&-\mathrm{i}&0\\\
\mathrm{i}&0&0\\\ 0&0&0\end{pmatrix},$ $\displaystyle\Lambda_{3}$
$\displaystyle=\begin{pmatrix}1&0&0\\\ 0&-1&0\\\ 0&0&0\end{pmatrix},$ (30)
$\displaystyle\Lambda_{4}$ $\displaystyle=\begin{pmatrix}0&0&0\\\ 0&0&0\\\
0&0&\sqrt{2}\end{pmatrix},$ (31)

and:

$\displaystyle\Lambda_{5}$ $\displaystyle=\begin{pmatrix}0&0&1\\\ 0&0&0\\\
1&0&0\end{pmatrix},$ $\displaystyle\Lambda_{6}$
$\displaystyle=\begin{pmatrix}0&0&-\mathrm{i}\\\ 0&0&0\\\
\mathrm{i}&0&0\end{pmatrix},$ (32) $\displaystyle\Lambda_{7}$
$\displaystyle=\begin{pmatrix}0&0&0\\\ 0&0&1\\\ 0&1&0\end{pmatrix},$
$\displaystyle\Lambda_{8}$ $\displaystyle=\begin{pmatrix}0&0&0\\\
0&0&-\mathrm{i}\\\ 0&\mathrm{i}&0\end{pmatrix}.$ (33)

They are normalized so that
$\operatorname{tr}\Lambda_{\mu}\Lambda_{\nu}=2\updelta_{\mu\nu}$.

##  Appendix B Tight-binding model of SRO

Within the effective tight-binding model of SRO based on the $t_{2g}$ orbitals
of Ru, the point group operation $g\in D_{4h}$ acts on electrons according to:

$\displaystyle\hat{\mathcal{U}}^{{\dagger}}(g)\psi_{\bm{k}}\hat{\mathcal{U}}(g)$
$\displaystyle=M(g)\otimes S(g)\psi_{R(g^{-1})\bm{k}},$ (34)

where $\psi_{\bm{k}}$ are column vectors of fermionic destruction operators in
the basis
$\left(d_{yz\uparrow},d_{yz\downarrow},d_{zx\uparrow},d_{zx\downarrow},d_{xy\uparrow},d_{xy\downarrow}\right)^{\intercal}$,
$\hat{\mathcal{U}}(g)$ are the Fock-space symmetry operators, and $R,M,S$ are
unitary representations of $D_{4h}$ whose generators are listed in Table 7.
Time-reversal $\Theta$ acts like:

$\displaystyle\Theta^{-1}\psi_{\bm{k}}\Theta$
$\displaystyle=(\mathbbm{1}\otimes\mathrm{i}\upsigma_{y})\psi_{-\bm{k}},$ (35)

where $\mathbbm{1}$ is the $3\times 3$ identity matrix and $\upsigma_{\mu}$
are the Pauli matrices.

Since there is only one ruthenium atom per a body-centered unit cell, the
tight-binding Hamiltonian takes the form:

$H_{0}=-\sum_{\bm{R},\bm{\delta}}\psi_{\bm{R}+\bm{\delta}}^{{\dagger}}\left[T_{\bm{\delta}}\otimes\mathbbm{1}+\mathrm{i}\sum_{\mu=1}^{3}\Lambda_{\bm{\delta};\mu}\otimes\upsigma_{\mu}\right]\psi_{\bm{R}},$
(36)

where $\bm{R},\bm{\delta}$ go over the body-centered tetragonal lattice whose
primitive lattice vectors are:

$\bm{a}_{1}=a\bm{\hat{e}}_{1},\;\;\bm{a}_{2}=a\bm{\hat{e}}_{2},\;\;\bm{a}_{3}=\tfrac{1}{2}\\!\left(a\bm{\hat{e}}_{1}+a\bm{\hat{e}}_{2}+c\bm{\hat{e}}_{3}\right).$
(37)

The Hamiltonian is hermitian only when
$T_{-\bm{\delta}}=T_{\bm{\delta}}^{{\dagger}}$ and
$\Lambda_{-\bm{\delta};\mu}=-\Lambda_{\bm{\delta};\mu}^{{\dagger}}$. Point
group symmetries constrain and relate different hopping amplitudes:

$\displaystyle M^{{\dagger}}(g)T_{\bm{\delta}}M(g)=T_{R(g^{-1})\bm{\delta}},$
(38) $\displaystyle M^{{\dagger}}(g)\Lambda_{\bm{\delta};\mu}M(g)=\det
R(g)\sum_{\nu=1}^{3}R_{\mu\nu}(g)\Lambda_{R(g^{-1})\bm{\delta};\nu}.$ (39)

To ensure time-reversal invariance, all matrix elements must be made real,
i.e., $T_{\bm{\delta}}^{*}=T_{\bm{\delta}}$ and
$\Lambda_{\bm{\delta};\mu}^{*}=\Lambda_{\bm{\delta};\mu}$.

Table 7: The generators of the representations $R$, $M$, and $S$ of the point group $D_{4h}$. $C_{4z}$ is a rotation by $\pi/2$ around $z$. $C_{2x}^{\prime}$ and $C_{2d}^{\prime\prime}$ are rotations by $\pi$ around $x$ and the diagonal $x+y$, respectively. $P$ is parity. $g$ | $R(g)$ | $M(g)$ | $S(g)$
---|---|---|---
$C_{4z}$ | $\begin{pmatrix}0&-1&0\\\ 1&0&0\\\ 0&0&1\end{pmatrix}$ | $\begin{pmatrix}0&1&0\\\ -1&0&0\\\ 0&0&-1\end{pmatrix}$ | $\displaystyle\frac{\upsigma_{0}-\mathrm{i}\upsigma_{z}}{\sqrt{2}}$
$C_{2x}^{\prime}$ | $\begin{pmatrix}1&0&0\\\ 0&-1&0\\\ 0&0&-1\end{pmatrix}$ | $\begin{pmatrix}1&0&0\\\ 0&-1&0\\\ 0&0&-1\end{pmatrix}$ | $\displaystyle-\mathrm{i}\upsigma_{x}$
$C_{2d}^{\prime\prime}$ | $\begin{pmatrix}0&1&0\\\ 1&0&0\\\ 0&0&-1\end{pmatrix}$ | $\begin{pmatrix}0&-1&0\\\ -1&0&0\\\ 0&0&1\end{pmatrix}$ | $\displaystyle-\mathrm{i}\frac{\upsigma_{x}+\upsigma_{y}}{\sqrt{2}}$
$P$ | $-\mathbbm{1}$ | $\mathbbm{1}$ | $\displaystyle\upsigma_{0}$
Table 8: Upper part: the values of our tight-binding model parameters in
$\mathrm{meV}$ according to Ref. [117]. The parameters set to zero have not
been considered in Ref. [117]. Lower part: the parameters in $\mathrm{meV}$
according to other references. Those parameters not listed vanish.

$t_{1}$ | $t_{2}$ | $t_{3}$ | $t_{4}$ | $t_{5}$ | $t_{6}$ | $t_{7}$ | $t_{8}$ | $t_{9}$
---|---|---|---|---|---|---|---|---
$27.8$ | $257.8$ | $-22.4$ | $13.6$ | $3.2$ | $-35.5$ | $0$ | $-4.7$ | $0$
$t_{10}$ | $t_{11}$ | $\bar{t}_{1}$ | $\bar{t}_{2}$ | $\bar{t}_{3}$ | $\bar{t}_{4}$ | $\bar{t}_{5}$ | $\bar{t}_{6}$ | $\bar{t}_{7}$
$0$ | $-2.4$ | $356.8$ | $126.3$ | $-1.0$ | $17.0$ | $22.3$ | $0$ | $0$
$\mu_{\text{1D}}$ | $\mu_{\text{2D}}$ | $t_{i1}$ | $t_{i2}$ | $t_{i3}$ | $t_{i4}$ | $t_{j}$ | $\eta_{1}$ | $\eta_{2}$
$286.9$ | $351.9$ | $-2.0$ | $7.8$ | $0$ | $0$ | $2.7$ | $59.2$ | $59.2$

$t_{1}$ | $t_{2}$ | $\bar{t}_{1}$ | $\bar{t}_{2}$ | $\bar{t}_{4}$ | $\mu_{\text{1D}}$ | $\mu_{\text{2D}}$ | $t_{i1}$ | $\eta_{1}$ | $\eta_{2}$ | Ref.
---|---|---|---|---|---|---|---|---|---|---
$16$ | $145$ | $81$ | $39$ | $5$ | $122$ | $122$ | $0$ | $32$ | $32$ | [120]
$9$ | $88$ | $80$ | $40$ | $5$ | $109$ | $109$ | $0$ | $35$ | $35$ | [121]
$13$ | $165$ | $119$ | $49$ | $0$ | $178$ | $176$ | $21$ | $0$ | $0$ | [130]

Symmetries that map $\bm{\delta}$ to itself constrain the forms of the hopping
amplitudes. For the eight closest $\bm{\delta}$ of SRO, we find that:

$\displaystyle T_{\bm{0}}=\begin{pmatrix}\mu_{\text{1D}}&0&0\\\
0&\mu_{\text{1D}}&0\\\
0&0&\mu_{\text{2D}}\end{pmatrix},\;\;T_{\bm{a}_{1}}=\begin{pmatrix}t_{1}&0&0\\\
0&t_{2}&0\\\ 0&0&\bar{t}_{1}\end{pmatrix},$ (40) $\displaystyle
T_{\bm{a}_{1}+\bm{a}_{2}}=\begin{pmatrix}t_{3}&t_{i1}&0\\\ t_{i1}&t_{3}&0\\\
0&0&\bar{t}_{2}\end{pmatrix},\;\;T_{\bm{a}_{3}}=\begin{pmatrix}t_{4}&t_{i2}&t_{j}\\\
t_{i2}&t_{4}&t_{j}\\\ t_{j}&t_{j}&\bar{t}_{3}\end{pmatrix},$ (41)
$\displaystyle T_{2\bm{a}_{1}}=\begin{pmatrix}t_{5}&0&0\\\ 0&t_{6}&0\\\
0&0&\bar{t}_{4}\end{pmatrix},\;\;T_{2\bm{a}_{1}+\bm{a}_{2}}=\begin{pmatrix}t_{7}&t_{i3}&0\\\
t_{i3}&t_{8}&0\\\ 0&0&\bar{t}_{5}\end{pmatrix},$ (42) $\displaystyle
T_{2(\bm{a}_{1}+\bm{a}_{2})}=\begin{pmatrix}t_{9}&t_{i4}&0\\\
t_{i4}&t_{9}&0\\\
0&0&\bar{t}_{6}\end{pmatrix},\;\;T_{3\bm{a}_{1}}=\begin{pmatrix}t_{10}&0&0\\\
0&t_{11}&0\\\ 0&0&\bar{t}_{7}\end{pmatrix}.$ (43)

Among these closest and thus largest $T_{\bm{\delta}}$, only $T_{\bm{a}_{3}}$
connects different layers, reflecting the high anisotropy of SRO. Moreover, it
is only through $T_{\bm{a}_{3}}$ that the body-centered periodicity of SRO is
felt on the level of the one-particle Hamiltonian. The on-site spin-orbit
coupling takes the form:

$\displaystyle\Lambda_{\bm{0};1}=\begin{pmatrix}0&0&0\\\ 0&0&-\eta_{1}\\\
0&\eta_{1}&0\end{pmatrix},\;\;\Lambda_{\bm{0};2}=\begin{pmatrix}0&0&\eta_{1}\\\
0&0&0\\\ -\eta_{1}&0&0\end{pmatrix},$ (44)
$\displaystyle\Lambda_{\bm{0};3}=\begin{pmatrix}0&-\eta_{2}&0\\\
\eta_{2}&0&0\\\ 0&0&0\end{pmatrix}.$ (45)

Off-site ($\bm{k}$-dependent) spin-orbit coupling we shall not include,
although one should keep in mind that some [16] have found that it has a large
effect on the preferred Cooper pairing, even when small.

For our analysis, we have used the tight-binding parameter values of Ref.
[117], which they found by fitting to the ARPES-based tight-binding $17$-band
model of Ref. [131]. Their tight-binding parameter values are reproduced in
Table 8. The hopping amplitudes of Refs. [117] and [16] are broadly in
agreement, as one would expect given that both were fitted to Ref. [131].
However, the hoppings of both [117, 16] are by a factor of two or so larger
than those of Refs. [120, 121, 130], which are also ARPES-derived; see Table
8. Although all these models give the correct shapes for the Fermi sheets,
find that the $\gamma$ band is responsible for over $50\text{\,}\mathrm{\char
37\relax}$ of the normal-state DOS, and predict a roughly
$20\text{\,}\mathrm{\char 37\relax}$ increase in the DOS at Van Hove strain,
consistent with our entropy data [Figure 2], the predicted values for the
total DOS differ by a factor of two. Only Ref. [120] has checked that their
model gives a total DOS ($g\approx 17$ states per $\mathrm{eV}$ per body-
centered tetragonal unit cell) that is consistent with the experimentally
measured Sommerfeld coefficient
$\gamma=(\pi^{2}/3)Rg\approx$40\text{\,}\mathrm{mJ}\text{/}\text{(}{\mathrm{K}}^{2}\text{\,}\mathrm{mol}\text{)}$$
[86, 87, 88], where $R$ is the molar gas constant. The main takeaway is that
the various estimates cited in the main text might be off by a factor of two,
which is still sufficient for our purposes and does not impact our argument in
any way.

In momentum space, the tight-binding Hamiltonian equals:

$\displaystyle H_{\bm{k}}$
$\displaystyle=-\sum_{\bm{\delta}}\left[T_{\bm{\delta}}\otimes\mathbbm{1}+\mathrm{i}\sum_{\mu=1}^{3}\Lambda_{\bm{\delta};\mu}\otimes\upsigma_{\mu}\right]\mathrm{e}^{-\mathrm{i}\bm{k}\bm{\cdot}\bm{\delta}}$
(46)
$\displaystyle=\begin{pmatrix}\epsilon_{\text{1D}}(\bm{k})&\epsilon_{i}(\bm{k})&\epsilon_{j}(\bm{k})\\\
\epsilon_{i}(\bm{k})&\tilde{\epsilon}_{\text{1D}}(\bm{k})&\tilde{\epsilon}_{j}(\bm{k})\\\
\epsilon_{j}(\bm{k})&\tilde{\epsilon}_{j}(\bm{k})&\epsilon_{\text{2D}}(\bm{k})\end{pmatrix}\otimes\upsigma_{0}+\begin{pmatrix}0&\mathrm{i}\eta_{2}\upsigma_{3}&-\mathrm{i}\eta_{1}\upsigma_{2}\\\
-\mathrm{i}\eta_{2}\upsigma_{3}&0&\mathrm{i}\eta_{1}\upsigma_{1}\\\
\mathrm{i}\eta_{1}\upsigma_{2}&-\mathrm{i}\eta_{1}\upsigma_{1}&0\end{pmatrix},$
(47)

where
$\tilde{\epsilon}_{\text{1D}}(k_{1},k_{2},k_{3})=\epsilon_{\text{1D}}(k_{2},k_{1},k_{3})$,
$\tilde{\epsilon}_{j}(k_{1},k_{2},k_{3})=\epsilon_{j}(k_{2},k_{1},k_{3})$,
and:

$\displaystyle\epsilon_{\text{1D}}(\bm{k})$
$\displaystyle=-\mu_{\text{1D}}-2t_{1}\cos\kappa_{1}-2t_{2}\cos\kappa_{2}-4t_{3}\cos\kappa_{1}\cos\kappa_{2}-8t_{4}\cos\tfrac{1}{2}\kappa_{1}\cos\tfrac{1}{2}\kappa_{2}\cos\tfrac{1}{2}\kappa_{3}-2t_{5}\cos
2\kappa_{1}-2t_{6}\cos 2\kappa_{2}$ $\displaystyle\qquad-4t_{7}\cos
2\kappa_{1}\cos\kappa_{2}-4t_{8}\cos\kappa_{1}\cos 2\kappa_{2}-4t_{9}\cos
2\kappa_{1}\cos 2\kappa_{2}-2t_{10}\cos 3\kappa_{1}-2t_{11}\cos 3\kappa_{2},$
(48) $\displaystyle\epsilon_{\text{2D}}(\bm{k})$
$\displaystyle=-\mu_{\text{2D}}-2\bar{t}_{1}\left(\cos\kappa_{1}+\cos\kappa_{2}\right)-4\bar{t}_{2}\cos\kappa_{1}\cos\kappa_{2}-8\bar{t}_{3}\cos\tfrac{1}{2}\kappa_{1}\cos\tfrac{1}{2}\kappa_{2}\cos\tfrac{1}{2}\kappa_{3}-2\bar{t}_{4}\left(\cos
2\kappa_{1}+\cos 2\kappa_{2}\right)$
$\displaystyle\qquad-4\bar{t}_{5}\left(\cos
2\kappa_{1}\cos\kappa_{2}+\cos\kappa_{1}\cos
2\kappa_{2}\right)-4\bar{t}_{6}\cos 2\kappa_{1}\cos
2\kappa_{2}-2\bar{t}_{7}\left(\cos 3\kappa_{1}+\cos 3\kappa_{2}\right),$ (49)
$\displaystyle\epsilon_{i}(\bm{k})$
$\displaystyle=4t_{i1}\sin\kappa_{1}\sin\kappa_{2}+8t_{i2}\sin\tfrac{1}{2}\kappa_{1}\sin\tfrac{1}{2}\kappa_{2}\cos\tfrac{1}{2}\kappa_{3}+8t_{i3}\left(\cos\kappa_{1}+\cos\kappa_{2}\right)\sin\kappa_{1}\sin\kappa_{2}+4t_{i4}\sin
2\kappa_{1}\sin 2\kappa_{2},$ (50) $\displaystyle\epsilon_{j}(\bm{k})$
$\displaystyle=8t_{j}\sin\tfrac{1}{2}\kappa_{1}\cos\tfrac{1}{2}\kappa_{2}\sin\tfrac{1}{2}\kappa_{3}.$
(51)

Above $\bm{k}=(k_{1},k_{2},k_{3})$, $\kappa_{1}=ak_{1}$, $\kappa_{2}=ak_{2}$,
and $\kappa_{3}=ck_{3}$.

The coupling to strain, needed for Figure 1, was taken from the Supplementary
information of Ref. [63]. The dispersion of the $\gamma$ band near the Van
Hove line $\left(0,\frac{\pi}{a},k_{3}\right)$, provided in the main text in
Eqs. (9) and (18), was found by diagonalizing $H_{\bm{k}}$ with the parameter
values of Ref. [117].

Table 9: A sample of possible pairing wavefunctions $d_{a}(\bm{k})$ categorized according to the transformation rule (56). The irrep subscripts $g$ and $u$ mean even and odd under parity, respectively. The two-component $\\{d_{1}(\bm{k})\mid d_{2}(\bm{k})\\}$ transform according to the $\rho^{(E)}(g)$ given in Eq. (55). $\bm{k}=(k_{1},k_{2},k_{3})$ and $\kappa_{1}=ak_{1}$, $\kappa_{2}=ak_{2}$, $\kappa_{3}=ck_{3}$. Highlighted red are those wavefunctions that are periodic under body-centered translations, but not under simple tetragonal translations. $A_{1g}$ |  $1$, $\cos\kappa_{1}+\cos\kappa_{2}$, $\cos\kappa_{3}$, $\cos\kappa_{1}\cos\kappa_{2}$
---|---
$A_{2g}$ | $\left(\cos\kappa_{1}-\cos\kappa_{2}\right)\sin\kappa_{1}\sin\kappa_{2}$
$B_{1g}$ | $\cos\kappa_{1}-\cos\kappa_{2}$
$B_{2g}$ |  $\sin\kappa_{1}\sin\kappa_{2}$, ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sin\tfrac{1}{2}\kappa_{1}\sin\tfrac{1}{2}\kappa_{2}\cos\tfrac{1}{2}\kappa_{3}}$
$E_{g}$ | $\begin{matrix}\left\\{\sin\kappa_{2}\sin\kappa_{3}\mid-\sin\kappa_{1}\sin\kappa_{3}\right\\},\\\ {\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\left\\{\cos\tfrac{1}{2}\kappa_{1}\sin\tfrac{1}{2}\kappa_{2}\sin\tfrac{1}{2}\kappa_{3}\mid-\sin\tfrac{1}{2}\kappa_{1}\cos\tfrac{1}{2}\kappa_{2}\sin\tfrac{1}{2}\kappa_{3}\right\\}}\end{matrix}$
$A_{1u}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\left(\cos\kappa_{1}-\cos\kappa_{2}\right)\sin\tfrac{1}{2}\kappa_{1}\sin\tfrac{1}{2}\kappa_{2}\sin\tfrac{1}{2}\kappa_{3}}$
$A_{2u}$ |  $\sin\kappa_{3}$, ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\cos\tfrac{1}{2}\kappa_{1}\cos\tfrac{1}{2}\kappa_{2}\sin\tfrac{1}{2}\kappa_{3}}$
$B_{1u}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sin\tfrac{1}{2}\kappa_{1}\sin\tfrac{1}{2}\kappa_{2}\sin\tfrac{1}{2}\kappa_{3}}$
$B_{2u}$ | $\left(\cos\kappa_{1}-\cos\kappa_{2}\right)\sin\kappa_{3}$
$E_{u}$ | $\begin{matrix}\left\\{\sin\kappa_{1}\mid\sin\kappa_{2}\right\\},\\\ \left\\{\left(\cos\kappa_{1}-\cos\kappa_{2}\right)\sin\kappa_{1}\mid-\left(\cos\kappa_{1}-\cos\kappa_{2}\right)\sin\kappa_{2}\right\\},\\\ {\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\left\\{\sin\tfrac{1}{2}\kappa_{1}\cos\tfrac{1}{2}\kappa_{2}\cos\tfrac{1}{2}\kappa_{3}\mid\cos\tfrac{1}{2}\kappa_{1}\sin\tfrac{1}{2}\kappa_{2}\cos\tfrac{1}{2}\kappa_{3}\right\\}}\end{matrix}$

##  Appendix C Superconducting states of SRO

For the purpose of classifying even-frequency pairings, it is sufficient to
consider the static case because the two behave the same symmetry-wise [132].
Odd-frequency pairings are beyond the scope of this article. On the mean-field
level, static zero-momentum SC is described by a pairing term in the
Hamiltonian of the form:

$\displaystyle H_{\text{SC}}$
$\displaystyle=\sum_{\bm{k}\alpha\beta}\psi_{\bm{k}\alpha}^{{\dagger}}\Delta_{\alpha\beta}(\bm{k})\psi_{-\bm{k}\beta}^{{\dagger}}+\mathrm{h.c.},$
(52)

where $\alpha,\beta$ are spin-orbit indices. Because of the fermionic
anticommutation, the SC gap matrix $\Delta_{\alpha\beta}(\bm{k})$ satisfies
the exchange property:

$\displaystyle\Delta^{\intercal}(\bm{k})$ $\displaystyle=-\Delta(-\bm{k}),$
(53)

where ⊺ is transposition.

If the pairing were conventional, all point group operations would be
preserved and
$\hat{\mathcal{U}}^{{\dagger}}(g)H_{\text{SC}}\hat{\mathcal{U}}(g)=H_{\text{SC}}$
would hold for all $g\in D_{4h}$, giving the constraint
$U^{{\dagger}}(g)\Delta\\!\left(R(g)\bm{k}\right)U^{*}(g)=\Delta(\bm{k})$,
where $U(g)\equiv M(g)\otimes S(g)$. Unconventional pairing is classified by
the way it breaks this constraint:

$\displaystyle U^{{\dagger}}(g)\Delta_{a}\\!\left(R(g)\bm{k}\right)U^{*}(g)$
$\displaystyle=\sum_{b=1}^{\dim\lambda}\rho_{ab}^{(\lambda)}(g)\Delta_{b}(\bm{k}).$
(54)

Here, $\lambda$ is an irrep of $D_{4h}$, $a,b$ are indices internal to the
irrep, and $\rho_{ab}^{(\lambda)}$ are the corresponding matrices. Only for
the 2D irreps $E_{g/u}$ are there multiple possible $\rho_{ab}^{(\lambda)}$.
We choose (cf. representation $R$):

$\displaystyle\begin{aligned} \rho^{(E)}(C_{4z})&=\begin{pmatrix}0&-1\\\
1&0\end{pmatrix},&\rho^{(E)}(C_{2x}^{\prime})&=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix},\\\ \rho^{(E)}(C_{2d}^{\prime\prime})&=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},&\rho^{\left(E_{g/u}\right)}(P)&=\pm\begin{pmatrix}1&0\\\
0&1\end{pmatrix}.\end{aligned}$ (55)

Table 10: Spin-orbit matrices $\Gamma_{a}$ categorized according to the transformation rule (57) and (anti-)symmetry (58). The irrep subscript $g$ means even under parity. The irrep superscripts $\pm$ are the values of $s$ [Eq. (58)], so $+$ ($-$) means that the matrices are (anti-)symmetric under transposition. The matrices are written in terms of $[\mu,\nu]\equiv\Lambda_{\mu}\otimes\upsigma_{\nu}(\mathrm{i}\upsigma_{y})$, where the Gell-Mann matrices $\Lambda_{\mu}$ have been listed in Appendix A. Subtractions of pairs $[\mu,\nu]$ represent subtractions of the respective matrices. The two-component $\\{\Gamma_{1}\mid\Gamma_{2}\\}$ transform according to the $\rho^{(E)}(g)$ given in Eq. (55). Highlighted blue are the singlet and triplet pairings with trivial orbital structures, typical of one-band SC. $A_{1g}^{-}$ |  ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}[0,0]}$, $[2,z]$, ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}[4,0]}$, $[6,y]-[8,x]$
---|---
$A_{2g}^{-}$ | $[6,x]+[8,y]$
$B_{1g}^{-}$ |  $[3,0]$, $[6,y]+[8,x]$
$B_{2g}^{-}$ |  $[1,0]$, $[6,x]-[8,y]$
$E_{g}^{-}$ |  $\left\\{[2,y]\mid-[2,x]\right\\}$, $\left\\{[7,0]\mid-[5,0]\right\\}$, $\left\\{[6,z]\mid[8,z]\right\\}$
$A_{1g}^{+}$ | $[5,y]-[7,x]$
$A_{2g}^{+}$ |  ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}[0,z]}$, $[2,0]$, ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}[4,z]}$, $[5,x]+[7,y]$
$B_{1g}^{+}$ |  $[1,z]$, $[5,y]+[7,x]$
$B_{2g}^{+}$ |  $[3,z]$, $[5,x]-[7,y]$
$E_{g}^{+}$ | $\begin{matrix}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\left\\{[0,x]\mid[0,y]\right\\}},\left\\{[1,y]\mid[1,x]\right\\},\left\\{[3,x]\mid-[3,y]\right\\},\\\ {\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\left\\{[4,x]\mid[4,y]\right\\}},\left\\{[5,z]\mid[7,z]\right\\},\left\\{[8,0]\mid-[6,0]\right\\}\end{matrix}$

To construct a $\Delta_{a}(\bm{k})$ that properly transforms according to Eq.
(54), we need to combine the momentum dependence and spin-orbit structure in
just the right way. This is accomplished [112, 113, *Kaba2019-E, 115] by first
separately classifying pairing wavefunctions and spin-orbit matrices (Tables 9
and 10), and then combining them according to a set of rules (Table 11). Let
us emphasize that the emergent SC order parameter that enters Ginzburg-Landau
theory belongs to the irrep determined by the total SC gap
$\Delta_{a}(\bm{k})$ according to Eq. (54), and not to the irreps of its
momentum or spin-orbit parts.

Pairing wavefunctions $d_{a}(\bm{k})$ are classified according to:

$\displaystyle d_{a}\\!\left(R(g)\bm{k}\right)$
$\displaystyle=\sum_{b=1}^{\dim\lambda}\rho_{ab}^{(\lambda)}(g)d_{b}(\bm{k}).$
(56)

All $d_{a}(\bm{k})$ should be made periodic, just like $\Delta_{a}(\bm{k})$.
If we call $\kappa_{1}=ak_{1}$, $\kappa_{2}=ak_{2}$, and $\kappa_{3}=ck_{3}$,
the primitive translations of a body-centered tetragonal lattice map
$(\kappa_{1},\kappa_{2},\kappa_{3})$ to
$(\kappa_{1}+2\pi,\kappa_{2},\kappa_{3}-2\pi)$,
$(\kappa_{1},\kappa_{2}+2\pi,\kappa_{3}-2\pi)$, and
$(\kappa_{1},\kappa_{2},\kappa_{3}+4\pi)$. Conventionally, we also make
$d_{a}(\bm{k})$ real so that TRSB is seen through imaginary coefficients
preceding $d_{a}(\bm{k})$. Examples of pairing wavefunctions are provided in
Table 9.

Table 11: Direct sum decompositions of the direct products between irreps of $D_{4h}$. Since all $\Gamma$ irreps are even, the parity of the $d$ irrep and direct product irrep are the same so we have suppressed their $g/u$ subscripts. All $E$ irreps transform according to the same 2D representation (55). $\otimes$ | $A_{1}\\{d\\}$ | $A_{2}\\{d\\}$ | $B_{1}\\{d\\}$ | $B_{2}\\{d\\}$ | $E\\{d_{1}\mid d_{2}\\}$
---|---|---|---|---|---
$A_{1g}\\{\Gamma\\}$ | $A_{1}\\{\Gamma d\\}$ | $A_{2}\\{\Gamma d\\}$ | $B_{1}\\{\Gamma d\\}$ | $B_{2}\\{\Gamma d\\}$ | $E\\{\Gamma d_{1}\mid\Gamma d_{2}\\}$
$A_{2g}\\{\Gamma\\}$ | $A_{2}\\{\Gamma d\\}$ | $A_{1}\\{\Gamma d\\}$ | $B_{2}\\{\Gamma d\\}$ | $B_{1}\\{\Gamma d\\}$ | $E\\{\Gamma d_{2}\mid-\Gamma d_{1}\\}$
$B_{1g}\\{\Gamma\\}$ | $B_{1}\\{\Gamma d\\}$ | $B_{2}\\{\Gamma d\\}$ | $A_{1}\\{\Gamma d\\}$ | $A_{2}\\{\Gamma d\\}$ | $E\\{\Gamma d_{1}\mid-\Gamma d_{2}\\}$
$B_{2g}\\{\Gamma\\}$ | $B_{2}\\{\Gamma d\\}$ | $B_{1}\\{\Gamma d\\}$ | $A_{2}\\{\Gamma d\\}$ | $A_{1}\\{\Gamma d\\}$ | $E\\{\Gamma d_{2}\mid\Gamma d_{1}\\}$
$E_{g}\begin{Bmatrix}\Gamma_{1}\\\ \Gamma_{2}\end{Bmatrix}$ | $E\begin{Bmatrix}\Gamma_{1}d\\\ \Gamma_{2}d\end{Bmatrix}$ | $E\begin{Bmatrix}\Gamma_{2}d\\\ -\Gamma_{1}d\end{Bmatrix}$ | $E\begin{Bmatrix}\Gamma_{1}d\\\ -\Gamma_{2}d\end{Bmatrix}$ | $E\begin{Bmatrix}\Gamma_{2}d\\\ \Gamma_{1}d\end{Bmatrix}$ | $\begin{matrix}A_{1}\\{\Gamma_{1}d_{1}+\Gamma_{2}d_{2}\\}\oplus A_{2}\\{\Gamma_{1}d_{2}-\Gamma_{2}d_{1}\\}\\\ \oplus B_{1}\\{\Gamma_{1}d_{1}-\Gamma_{2}d_{2}\\}\oplus B_{2}\\{\Gamma_{1}d_{2}+\Gamma_{2}d_{1}\\}\end{matrix}$

When it comes to spin-orbit matrices $\Gamma_{a}$, notice that
$U(P)=\mathbbm{1}$ leaves the matrix part of Eq. (54) invariant. This means
that all spin-orbit matrices are even.133 We classify them according to:

$\displaystyle U^{{\dagger}}(g)\Gamma_{a}U^{*}(g)$
$\displaystyle=\sum_{b=1}^{\dim\lambda}\rho_{ab}^{(\lambda)}(g)\Gamma_{b},$
(57)

where $U(g)=M(g)\otimes S(g)$. Given the transposition in Eq. (53), it is
natural to further categorize $\Gamma_{a}$ according to (anti-)symmetry:

$\displaystyle\Gamma_{a}^{\intercal}$ $\displaystyle=s\Gamma_{a},$ (58)

where $s\in\\{\pm 1\\}$. We shall also ensure time-reversal invariance:

$\displaystyle(\mathbbm{1}\otimes\mathrm{i}\upsigma_{y})\Gamma_{a}^{*}(\mathbbm{1}\otimes\mathrm{i}\upsigma_{y})$
$\displaystyle=\Gamma_{a}^{\intercal},$ (59)

so that TRSB manifests itself through imaginary prefactors. As the basis of
the orbital part of $\Gamma_{a}$, we use Gell-Mann matrices $\Lambda_{\mu}$
[Appendix A]. The spin-orbit matrices we write in terms of these:

$\displaystyle\Gamma_{a}$
$\displaystyle\sim\sum\Lambda_{\mu}\otimes\upsigma_{\nu}(\mathrm{i}\upsigma_{y}).$
(60)

Given that $\Lambda_{\mu}^{{\dagger}}=\Lambda_{\mu}$, written thusly
$\Gamma_{a}$ automatically satisfy time-reversal invariance (59). In three-
band systems, there are in total $4\cdot 3^{2}=36$ possible $\Gamma_{a}$, of
which $15$ are antisymmetric and $21$ are symmetric. The categorization of all
$\Lambda_{\mu}\otimes\upsigma_{\nu}(\mathrm{i}\upsigma_{y})\equiv[\mu,\nu]$ is
given in Table 10.

1111footnotetext:  Odd spin-orbit matrices occur when the conduction bands mix
parities, as happens (e.g.) for topological band structures.

SC gap matrices $\Delta(\bm{k})$ are constructed by combining pairing
wavefunctions $d_{a}(\bm{k})$ and spin-orbit matrices $\Gamma_{a}$. Because of
the exchange property (53), we may only combine even $d_{a}(\bm{k})$ with
antisymmetric $\Gamma_{a}$, or odd $d_{a}(\bm{k})$ with symmetric
$\Gamma_{a}$. Now consider a $\\{d_{a}(\bm{k})\\}\in\lambda_{d}$ and
$\\{\Gamma_{a}\\}\in\lambda_{\Gamma}$, where $\lambda_{d}$ and
$\lambda_{\Gamma}$ are irreps. The object
$\Delta_{ab}(\bm{k})\equiv\Gamma_{a}d_{b}(\bm{k})$ then transforms according
to the $\lambda_{\Gamma}\otimes\lambda_{d}$ representation:

$\displaystyle U^{{\dagger}}(g)\Delta_{ab}\\!\left(R(g)\bm{k}\right)U^{*}(g)=$
(61) $\displaystyle\hskip
28.45274pt=\sum_{a^{\prime}=1}^{\dim\lambda_{\Gamma}}\sum_{b^{\prime}=1}^{\dim\lambda_{d}}\rho_{aa^{\prime}}^{(\lambda_{\Gamma})}(g)\rho_{bb^{\prime}}^{(\lambda_{d})}(g)\Delta_{a^{\prime}b^{\prime}}(\bm{k}).$

Since we want to construct SC gap matrices that transform according to
_irreducible_ representations [Eq. (54)], we decomposed $\Delta_{ab}(\bm{k})$
into irreducible parts with the help of Table 11. The most general
$\\{\Delta_{a}(\bm{k})\\}$ belonging to irrep $\lambda_{\Delta}$ is then given
by a sum over all possible $\\{d_{a}(\bm{k})\\}\in\lambda_{d}$ and
$\\{\Gamma_{a}\\}\in\lambda_{\Gamma}$ such that
$\lambda_{\Delta}\in\lambda_{\Gamma}\otimes\lambda_{d}$.

For example, let us construct SC gap matrices belonging to $B_{1g}$. In Table
11 every row has a $B_{1}$, meaning antisymmetric $\Gamma_{a}$ belonging to
every irrep could be used. Combining
$[0,0]=\Lambda_{0}(\mathrm{i}\upsigma_{y})\in A_{1g}^{-}$ and
$\cos\kappa_{1}-\cos\kappa_{2}\in B_{1g}$ gives a
$\Delta(\bm{k})=\Lambda_{0}(\mathrm{i}\upsigma_{y})\left(\cos\kappa_{1}-\cos\kappa_{2}\right)\in
B_{1g}$, but so do many others:

$\displaystyle A_{1g}^{-}\otimes B_{1g}\colon$
$\displaystyle\quad\left(\Lambda_{6}\upsigma_{y}-\Lambda_{8}\upsigma_{x}\right)(\mathrm{i}\upsigma_{y})\left(\cos\kappa_{1}-\cos\kappa_{2}\right),$
$\displaystyle A_{2g}^{-}\otimes B_{2g}\colon$
$\displaystyle\quad\left(\Lambda_{6}\upsigma_{x}+\Lambda_{8}\upsigma_{y}\right)(\mathrm{i}\upsigma_{y})\sin\kappa_{1}\sin\kappa_{2},$
$\displaystyle B_{1g}^{-}\otimes A_{1g}\colon$
$\displaystyle\quad\Lambda_{3}(\mathrm{i}\upsigma_{y})\cos\kappa_{1}\cos\kappa_{2},$
$\displaystyle B_{2g}^{-}\otimes A_{2g}\colon$
$\displaystyle\quad\Lambda_{1}(\mathrm{i}\upsigma_{y})\left(\cos\kappa_{1}-\cos\kappa_{2}\right)\sin\kappa_{1}\sin\kappa_{2},$
$\displaystyle E_{g}^{-}\otimes E_{g}\colon$
$\displaystyle\quad\Lambda_{2}\left(\upsigma_{x}\sin\kappa_{1}-\upsigma_{y}\sin\kappa_{2}\right)(\mathrm{i}\upsigma_{y})\sin\kappa_{3},$

etc. The most general $\Delta(\bm{k})\in B_{1g}$ is a linear superposition of
all of these.

##  Appendix D Van Hove line-gapping SC states

In Figures 6, 7, and 8, we have plotted the Fermi surface-projections of a
number of Van Hove line-gapping even SC states from Table 6b. These have been
constructed by combining the six $A_{1g}^{-}$ and $B_{1g}^{-}$ spin-orbit
matrices [Table 10] with the lowest order $A_{1g}$, $B_{1g}$, and $E_{g}$
pairing wavefunctions [Table 9]. $\Delta_{\bm{k}}$ constructed from the highly
suppressed $E_{g}^{-}$ spin-orbit matrices aren’t shown. Of all the possible
superpositions in the case of $E_{g}$ pairing, we have shown the chiral ones
as they are the most interesting because of the various evidence [52, 53, 54,
55, 56, 57, 58] indicating TRSB. The most general Van Hove line-gapping
$\Delta_{\bm{k}}$ belonging to $A_{1g}$, $B_{1g}$, or chiral $E_{g}$ is a
superposition of the shown ones, plus higher order harmonics.
$\kappa_{1}=ak_{1}\in[-\pi,\pi]$, $\kappa_{2}=ak_{2}\in[-\pi,\pi]$,
$\kappa_{3}=ck_{3}\in[-2\pi,2\pi]$, and
$d_{(x\pm\mathrm{i}y)z}(\bm{k})=\left(\sin\frac{\kappa_{1}}{2}\cos\frac{\kappa_{2}}{2}\pm\mathrm{i}\cos\frac{\kappa_{1}}{2}\sin\frac{\kappa_{2}}{2}\right)\sin\frac{\kappa_{3}}{2}$.
In the $\gamma$ sheet plots, the Van Hove lines
$\left(\pm\frac{\pi}{a},0,k_{3}\right)$ and
$\left(0,\pm\frac{\pi}{a},k_{3}\right)$ have been highlighted red. Even though
the projections of some $\Delta_{\bm{k}}$ onto the $\gamma$ band might be
small (shaded blue) near the Van Hove lines (e.g., Figure 6 (b)), they are
only exactly zero at a certain $\kappa_{3}$ for the $\Delta_{\bm{k}}\in E_{g}$
that have horizontal nodes at $\kappa_{3}=0,\pm 2\pi$.

(a) $\Delta_{\bm{k}}=\Lambda_{0}(\mathrm{i}\upsigma_{y})$

(b) $\Delta_{\bm{k}}=\Lambda_{2}\upsigma_{3}(\mathrm{i}\upsigma_{y})$

(c) $\Delta_{\bm{k}}=\Lambda_{4}(\mathrm{i}\upsigma_{y})$

(d)
$\Delta_{\bm{k}}=\left(\Lambda_{6}\upsigma_{y}-\Lambda_{8}\upsigma_{x}\right)(\mathrm{i}\upsigma_{y})$

(e)
$\Delta_{\bm{k}}=\Lambda_{3}(\mathrm{i}\upsigma_{y})\left(\cos\kappa_{1}-\cos\kappa_{2}\right)$

(f)
$\Delta_{\bm{k}}=\left(\Lambda_{6}\upsigma_{y}+\Lambda_{8}\upsigma_{x}\right)(\mathrm{i}\upsigma_{y})\left(\cos\kappa_{1}-\cos\kappa_{2}\right)$

Figure 6: Projections onto the Fermi sheets of a number of Van Hove line-
gapping SC states $\Delta_{\bm{k}}$ belonging to the $A_{1g}$ irrep. See the
text for details.

(a)
$\Delta_{\bm{k}}=\Lambda_{0}(\mathrm{i}\upsigma_{y})\left(\cos\kappa_{1}-\cos\kappa_{2}\right)$

(b)
$\Delta_{\bm{k}}=\Lambda_{2}\upsigma_{3}(\mathrm{i}\upsigma_{y})\left(\cos\kappa_{1}-\cos\kappa_{2}\right)$

(c)
$\Delta_{\bm{k}}=\Lambda_{4}(\mathrm{i}\upsigma_{y})\left(\cos\kappa_{1}-\cos\kappa_{2}\right)$

(d)
$\Delta_{\bm{k}}=\left(\Lambda_{6}\upsigma_{y}-\Lambda_{8}\upsigma_{x}\right)(\mathrm{i}\upsigma_{y})\left(\cos\kappa_{1}-\cos\kappa_{2}\right)$

(e) $\Delta_{\bm{k}}=\Lambda_{3}(\mathrm{i}\upsigma_{y})$

(f)
$\Delta_{\bm{k}}=\left(\Lambda_{6}\upsigma_{y}+\Lambda_{8}\upsigma_{x}\right)(\mathrm{i}\upsigma_{y})$

Figure 7: Projections onto the Fermi sheets of a number of Van Hove line-
gapping SC states $\Delta_{\bm{k}}$ belonging to the $B_{1g}$ irrep. See the
text for details.

(a)
$\Delta_{\bm{k}}=\Lambda_{0}(\mathrm{i}\upsigma_{y})d_{(x\pm\mathrm{i}y)z}(\bm{k})$

(b)
$\Delta_{\bm{k}}=\Lambda_{2}\upsigma_{3}(\mathrm{i}\upsigma_{y})d_{(x\pm\mathrm{i}y)z}(\bm{k})$

(c)
$\Delta_{\bm{k}}=\Lambda_{4}(\mathrm{i}\upsigma_{y})d_{(x\pm\mathrm{i}y)z}(\bm{k})$

(d)
$\Delta_{\bm{k}}=\left(\Lambda_{6}\upsigma_{y}-\Lambda_{8}\upsigma_{x}\right)(\mathrm{i}\upsigma_{y})d_{(x\pm\mathrm{i}y)z}(\bm{k})$

(e)
$\Delta_{\bm{k}}=\Lambda_{3}(\mathrm{i}\upsigma_{y})d_{(x\pm\mathrm{i}y)z}(\bm{k})$

(f)
$\Delta_{\bm{k}}=\left(\Lambda_{6}\upsigma_{y}+\Lambda_{8}\upsigma_{x}\right)(\mathrm{i}\upsigma_{y})d_{(x\pm\mathrm{i}y)z}(\bm{k})$

Figure 8: Projections onto the Fermi sheets of a number of chiral Van Hove
line-gapping SC states $\Delta_{\bm{k}}$ belonging to the $E_{g}$ irrep. See
the text for details.

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