# Still mystery after all these years
— Unconventional superconductivity of Sr2RuO4 —

Yoshiteru Maeno1 Shingo Yonezawa2,3 Aline Ramires4 1Toyota Riken – Kyoto
University Research Center (TRiKUC)1Toyota Riken – Kyoto University Research
Center (TRiKUC) Kyoto 606-8501 Kyoto 606-8501 Japan
2Department of Electronic Science and Engineering Japan
2Department of Electronic Science and Engineering Graduate School of
Engineering Graduate School of Engineering Kyoto University Kyoto University
Kyoto 615-8510 Kyoto 615-8510 Japan
3Department of Physics Japan
3Department of Physics Graduate School of Science Graduate School of Science
Kyoto University Kyoto University Kyoto 606-8502 Kyoto 606-8502 Japan
4Condensed Matter Theory Group Japan
4Condensed Matter Theory Group Paul Scherrer Institute Paul Scherrer Institute
CH-5232 Villigen PSI CH-5232 Villigen PSI Switzerland
Switzerland

###### Abstract

(Dated: )

This review describes recent significant research developments made on the
layered perovskite Sr2RuO4 and discusses current issues from both experimental
and theoretical perspectives. Since the discovery of superconductivity in
Sr2RuO4 in 1994, studies using high-quality single crystals quickly revealed
it to be an archetypal unconventional superconductor among strongly correlated
electron systems. In particular, it was thought that the spin-triplet chiral
$p$-wave superconducting state, which breaks time-reversal symmetry, was a
prominent possibility. In 2019, however, a new development overturned the past
experimental results, and spin-singlet-like behavior became conclusive.
Furthermore, innovation in uniaxial strain devices has stimulated researchers
to explore changes in the superconducting state by controlling the symmetry
and dimensionality of the Fermi surfaces and enhancing the superconducting
transition temperature $T_{\mathrm{c}}$ from 1.5 K to 3.5 K. A spin-singlet
chiral $d$-wave superconducting state is consistent with most of these recent
experimental results. Nevertheless, there are still unnatural aspects that
remain to be explained. The focus of this review is on unraveling this
mystery. Unlike many other unconventional superconductors, the normal state of
Sr2RuO4 exhibits typical Fermi liquid behavior. Nevertheless, to elucidate its
superconducting state, it may be essential to go beyond the traditional
framework of unconventional superconductivity and recast the theory explicitly
considering the multi-orbital aspects of its electronic states. In this
review, we describe the frontiers of superconductivity research in Sr2RuO4 and
discuss how the remaining issues may be resolved.

## CONTENTS

##  1 Introduction A

##  2 Unconventional superconductivity in multi-orbital systems B

###  2.1 Symmetry considerations B.1

###  2.2 Normal-state Hamiltonian in the orbital basis B.2

###  2.3 Superconducting order parameter B.3

###  2.4 Order parameter in the orbital basis B.4

##  3 Normal state properties C

###  3.1 Fermi-liquid properties C.1

###  3.2 Fermi surfaces C.2

###  3.3 Properties under strain and the Lifshitz transition
C.3

##  4 Spin susceptibility and the behavior near $H_{\mathrm{c2}}$
D

###  4.1 Summary of superconducting properties D.1

###  4.2 Revised spin susceptibility by NMR D.2

###  4.3 Behavior near the upper critical field D.3

##  5 Superconducting gap structure E

###  5.1 Low-energy quasiparticle excitations E.1

###  5.2 Thermal conductivity E.2

###  5.3 Field-angle-resolved specific heat E.3

###  5.4 Quasiparticle interference E.4

###  5.5 Mystery of the horizontal line node E.5

##  6 Phase sensitive experiments F

###  6.1 Superconducting thin films F.1

###  6.2 Quasiparticle tunneling F.2

###  6.3 Dynamical behavior of the 1.5-K phase F.3

###  6.4 Superconducting junctions with Ru inclusions F.4

###  6.5 Proximity effects into a ferromagnet F.5

###  6.6 SQUID and half quantum fluxoid (HQF) F.6

##  7 Two-Component Order Parameters G

###  7.1 Landau theory of phase transitions G.1

###  7.2 Two-component order parameters from ultrasound
G.2

##  8 Time-reversal symmetry breaking H

###  8.1 Enhancement of $T_{\mathrm{c}}$ by strain H.1

###  8.2 Split and unsplit transitions by $\mu$SR H.2

###  8.3 Searching for the second transition H.3

###  8.4 Other experiments in search of TRS breaking H.4

###  8.5 Mystery of the horizontal line node, revisited H.5

##  9 Theoretical Review I

###  9.1 Previous and recent theories I.1

###  9.2 Accidental degeneracy I.2

###  9.3 Symmetry protected degeneracy I.3

###  9.4 Extrinsic TRS breaking I.4

##  10 Discussion: Re-evaluation of the available results in a new perspective
J

###  10.1 Paradigm shift from the chiral $p$-wave scenario J.1

###  10.2 Unresolved mysteries and how they may be solved
J.2

##  11 Conclusion K

##  Appendix A Derivation of the normal state Hamiltonian L

##  Appendix B From the orbital-spin basis to the band-pseudospin basis M

##  Appendix A Introduction

The study of unconventional superconductivity plays a central role in modern
fundamental research on condensed matter. Among archetypal examples is the
superconductivity in Sr2RuO4, discovered in 1994 [1], that is a layered
perovskite isostructural to a high-temperature copper oxide superconductor as
shown in Fig. 1 (a). Studies using high-quality single crystals grown by a
floating-zone method (Fig. 1 (b)) quickly revealed that this is a strongly
correlated electron system and clearly exhibits unconventional
superconductivity. In particular, the spin-triplet chiral $p$-wave
superconducting state, which breaks time-reversal symmetry, was considered a
prominent candidate state, and extensive experimental and theoretical studies
on Sr2RuO4 contributed to widening the recognition of the concepts of spin-
triplet order parameters (OPs) and time-reversal-symmetry breaking (TRS
breaking) in the field of superconductivity [2, 3].

Figure 1: (a) Crystal structure of Sr2RuO4, a body-centered tetragonal
structure with space group $I4/mmm$ (No. 139). Adopted from Ref.
Kinjo2022.Science.376.397 ($\copyright$ 2022 The American Association for the
Advancement of Science). (b) Single crystal of Sr2RuO4 grown by a floating-
zone method with an infrared image furnace. Shiny flat surface is the basal
facet parallel to the RuO2 plane. The feed rod (top) has a diameter of 5.5 mm.

Although the normal state of Sr2RuO4 is well-understood and quantitatively
characterized as a highly-correlated multi-band system, the observed
superconducting properties have not allowed conclusive determination of its
superconducting OP. In the last decade, dramatic changes in the
superconducting properties under uniaxial strains have provided game-changing
clues toward identifying the OP: the superconducting transition temperature
$T_{\mathrm{c}}$ systematically increases from 1.5 K to 3.5 K. [5] This is a
controllable realization of the previously observed phenomenon in the eutectic
crystals of Sr2RuO4 containing ruthenium-metal micro-domains, referred as “3-K
phase” superconductivity. [6] In 2019, it was revealed that previous NMR
Knight-shift results, which had been considered as strong evidence for spin-
triplet pairing, had a technical problem due to overheating by the radio
frequency (RF) pulses used in the measurements [7]. These new results forced
the community to re-examine previous interpretations. Nevertheless, the
consensus has not been reached yet on the superconducting OP of Sr2RuO4. This
paper reviews recent progress, especially since previous reviews [eg. Refs.
Maeno2011.JPhysSocJpn.81.011009, Mackenzie2017.npjQuantumMater.2.40] were
published, and presents the current status and open questions. We also revisit
previous key experimental results and discuss their implication from the
standpoint of recent progress.

The scope of this review is as follows. In Sec. 2, we provide a theoretical
background with a particular emphasis on the roles of crystalline symmetry and
multiple orbitals in unconventional superconductivity. This section is
intended as a self-contained tutorial on modern aspects of the theory of
superconductivity. Depending on their background, readers may first go
directly to Sec. 3 and come back to Sec. 2 whenever necessary. Extending the
introductory theory of Sec. 2, Appendices A and B provide a description of the
normal state including the orbital degrees of freedom and discuss the explicit
relation between orbital and band bases.

In Sec. 3, we review the normal state properties of Sr2RuO4. Its highly
correlated three-band Fermi surfaces have been quantitatively well-
characterized by state-of-the-art angle-resolved photoemission spectroscopy
(ARPES) and band-structure calculations based on dynamical mean-field theory
(DMFT). Dramatic changes in the electronic states under uniaxial strain that
induce a Lifshitz transition are also discussed.

In Sec. 4, the revised spin-susceptibility and the first-order superconducting
transition near the upper critical field $H_{\mathrm{c2}}$ are discussed. In
addition, the evidence for the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state
is introduced. These results support a spin-singlet-like pairing state. In
Sec. 5, results on the superconducting gap structure deduced from a variety of
experimental probes are summarized. A controversy over the presence of a
horizontal line node suggested by the spin-singlet scenario is briefly
discussed. Section 6 summarizes the results and interpretation of phase-
sensitive experiments in junctions and microstructures, as well as the current
status of thin-film growth.

Section 7 introduces the theoretical basis and key results of ultrasound
experiments corroborating the scenario of a two-component OP. Section 8
discusses the TRS breaking in the superconducting state. Muon spin resonance
($\rm{\mu}$SR) under uniaxial strain has revealed a splitting of the
superconducting transition into phase with and without TRS. According to
complementary $\rm{\mu}$SR studies under hydrostatic pressure or disorder,
such splitting does not occur as long as the tetragonal crystalline symmetry
is preserved, suggesting that the two components must be symmetry related.
Nevertheless, the splitting of the superconducting and TRS breaking
transitions has not been probed by specific heat nor elastocaloric
measurements.

In Sec. 9, we review recent theoretical models toward identifying the OP of
the superconducting state of Sr2RuO4. Among the models, we describe in some
detail a spin-triplet, even-parity, orbital-antisymmetric pairing model, which
leads to a spin-singlet-like susceptibility consistent with the experiments.
Finally, Sec. 10 summarizes the unresolved mysteries with a table of candidate
superconducting symmetries (Table 12) and discusses how the current
controversies may be resolved. Sec. 11 concludes this review.

##  Appendix B Unconventional superconductivity in multi-orbital systems

Superconductivity is considered “multi-band” when it emerges in metals with an
electronic structure derived from multiple atomic orbitals (or other internal
degrees of freedom) contributing to several bands close to the Fermi energy.
Multi-band superconductivity has been a topic of theoretical research in the
context of conventional superconductors such as MgB2 [10, 11], in which bands
develop superconducting order parameters of different magnitudes; and in the
context of unconventional superconductors such as Fe-pnictides [12], in which
the superconducting order parameters develop with opposite signs in different
Fermi surfaces. For a long time, it was generally thought that there was no
fundamental difference between superconductivity emerging from a single- or
multi-band electronic structure, and superconductivity within multi-band
systems was understood from the perspective of multiple weakly-coupled single-
band systems. The picture started to change with the discovery of Fe-based
superconductors, for which the multi-orbital description was necessary to
properly encode all possible interaction parameters, including interorbital
Coulomb interaction and Hund’s coupling. Similar treatment for Sr2RuO4 has
recently brought us many insights. In this review, we aim at highlighting that
the treatment of superconductivity in complex, multi-orbital electronic
systems can be better understood from the microscopic point of view by
constructing models and superconducting order parameters in the orbital-spin
basis, and not directly in the band-pseudospin basis (the definition of these
bases is given in the following subsections).

Below, we take the reader over the necessary steps for the construction of a
reliable normal-state Hamiltonian of a multi-orbital system using as example
Sr2RuO4, and discuss the classification of the superconducting order
parameters. We start highlighting the symmetries of the crystalline structure
of Sr2RuO4, introducing the necessary concepts of group theory in Sec. B.1.
Such treatment has already been recognized as essential in the past, but it is
becoming increasingly important in modern theory of superconductivity and
especially to properly interpret various experimental results. We discuss the
most general normal-state Hamiltonian based on the microscopic orbital and
spin degrees of freedom in Sec. B.2, giving the details of its derivation in
AppendixL. In Sec. B.3, we discuss the general classification of
superconducting order parameters. In Sec. B.4 we extend the superconducting
order parameter classification to explicitly include the orbital degree of
freedom, and discuss in detail the explicit relation between the orbital and
band bases in AppendixM.

###  B.1 Symmetry considerations

The crystalline structure of Sr2RuO4, as depicted in Fig. 1, has multiple
symmetries. The set of symmetry operations (translations, rotations,
reflections, inversion, and combinations of these) that leave a crystalline
structure invariant form a mathematical structure known as a _space group_. If
we impose that one point must remain invariant under the symmetry operations,
we eliminate translations from this set and find what is called a _point
group_. Using the standard nomenclature from group theory, Sr2RuO4 is
associated with the space group $I4/mmm$ ($\\#$139) and point group $4/mmm$,
most commonly referred to as $D_{4h}$. This point group is composed of sixteen
symmetry operations:

  * •

$I$, the identity;

  * •

$C_{4z}$ and $C_{4z}^{-1}$, rotations by $\pm\pi/2$ around the $z$-axis;

  * •

$C_{2x}$, $C_{2y}$, and $C_{2z}$, rotations by $\pi$ around the $x$-, $y$-,
and $z$-axis, respectively;

  * •

$C_{2d}$ and $C_{2\bar{d}}$, rotations by $\pi$ around the diagonal $d$-axis
(x=y for z=0), and anti-diagonal $\bar{d}$-axis (x=-y for z=0), respectively;

  * •

$P$, inversion (or parity);

  * •

$S_{4}$ and $S_{4}^{-1}$, screw rotations consisting of $\pi/2$ rotations
along the $z$-axis followed by a $xy$-mirror plane reflection;

  * •

$\sigma_{h}$, $xy$-mirror plane reflection;

  * •

$\sigma_{xz}$ and $\sigma_{yz}$, two vertical mirror reflections on the $xz$-,
and $yz$-plane, respectively;

  * •

$\sigma_{d}$ and $\sigma_{\bar{d}}$, two vertical mirror planes on the $dz$\-
and $\bar{d}z$-plane, respectively.

The explicit form of the transformation of Cartesian coordinates under each of
these operations is given explicitly in Table 1. The existence of sixteen
symmetry operations can be understood from the independent symmetry under
inversion of coordinates $x\rightarrow-x$, $y\rightarrow-y$, and
$z\rightarrow-z$, which together account for $2^{3}=8$ transformations,
combined with the equivalence between the $x$ and $y$ axes, which allows for
symmetry transformations interchanging $x\leftrightarrow y$, which doubles the
number of transformations to $2\times 2^{3}=16$.

Table 1: Symmetry operations and corresponding Cartesian coordinate transformation for the $D_{4h}$ point group. The transformations should be read as $(x,y,z)\rightarrow(a,b,c)$, with $(a,b,c)$ given by the entries in the second column.  Operation | Coordinate Transformation
---|---
I | (x, y, z)
$C_{4z}$ | (-y, x, z)
$C_{4z}^{-1}$ | (y, -x, z)
$C_{2x}$ | (x, -y, -z)
$C_{2y}$ | (-x, y, -z)
$C_{2z}$ | (-x, -y, z)
$C_{2d}$ | (y, x, -z)
$C_{2\bar{d}}$ | (-y, -x, -z)
$P$ | (-x, -y, -z)
$S_{4}$ | (-y, x, -z)
$S_{4}^{-1}$ | (y, -x, -z)
$\sigma_{h}$ | (x, y, -z)
$\sigma_{xz}$ | (x, -y, z)
$\sigma_{yz}$ | (-x, y, z)
$\sigma_{d}$ | (y, x, z)
$\sigma_{\bar{d}}$ | (-y, -x, z)

These symmetry operations are divided into ten _conjugacy classes_. Conjugacy
classes correspond to sets of symmetry operations of the same “type”, that are
related by complementary symmetry operations through conjugation. For example,
$C_{2x}$ and $C_{2y}$ belong to the same conjugacy class as they are related
by conjugation with $C_{4z}$ as the conjugating element:
$C_{4z}.C_{2x}.C_{4z}^{-1}=C_{2y}$. Here the dot $(.)$ corresponds to the
composition of operations, which are usually applied from right to left.
Examining all possible conjugation relations, the ten conjugacy classes in
$D_{4h}$ can be identified as: $\mathbf{I}=\\{I\\}$,
$\mathbf{2C_{4z}}=\\{C_{4z},C_{4z}^{-1}\\}$, $\mathbf{C_{2z}}=\\{C_{2z}\\}$;
$\mathbf{2C_{2x}}=\\{C_{2x},C_{2y}\\}$;
$\mathbf{2C_{2d}}=\\{C_{2d},C_{2\bar{d}}\\}$; $\mathbf{P}=\\{P\\}$;
$\mathbf{2S_{4}}=\\{S_{4},S_{4}^{-1}\\}$;
$\boldsymbol{\sigma_{h}}=\\{\sigma_{h}\\}$;
$\boldsymbol{2\sigma_{v}}=\\{\sigma_{xz},\sigma_{yz}\\}$;
$\boldsymbol{2\sigma_{d}}=\\{\sigma_{d},\sigma_{\bar{d}}\\}$. Here we label
the conjugacy classes by taking one representative element in the class (in
bold face), preceded by the number of symmetry operations in the class. Note
that the brackets correspond to the set of elements; these do not not indicate
anticommutation.

A well known theorem in group theory states that the number of conjugacy
classes determines the number of _irreducible representations_ , or “irreps”
for short [13, 14], so there are ten irreps in $D_{4h}$. Irreducible
representations can be thought of as families of objects or functions which
transform in a well defined manner when acted upon by the symmetry operations.
The first column of Table 2 labels the ten irreducible representations, which
carry information about how different objects and symmetries transform
according to symmetry operations in each class. We start considering inversion
symmetry. An object or a function that does not change under inversion is said
to be even parity, while one that picks up a minus sign under inversion is
said to be odd parity. The standard notation for the labelling of objects or
functions according to their parity follows from the German language, with a
subindex $g$ (_gerade_) for even and a subindex $u$ (_ungerade_) for odd
parity. Note that there are five even- and five odd-parity irreps in Table 2.

Table 2: Simplified character table for the $D_{4h}$ point group. The first column enumerates the ten irreps of $D_{4h}$, and the second column displays the character associated with the conjugacy class of the identity operation, which gives as entries the dimension of the corresponding irreps. The third to fifth columns give the character of three conjugacy classes associated with the three generators of the group, $C_{4z}$, $C_{2x}$, and $P$. The last column displays representative basis functions in Cartesian coordinates that transform according to the corresponding irrep. irrep | $\boldsymbol{I}$ | $\boldsymbol{2C_{4z}}$ | $\boldsymbol{2C_{2x}}$ | $\boldsymbol{P}$ | basis
---|---|---|---|---|---
$\boldsymbol{A_{1g}}$ | 1 | 1 | 1 | 1 | 1
$\boldsymbol{A_{2g}}$ | 1 | 1 | -1 | 1 | $xy(x^{2}-y^{2})$
$\boldsymbol{B_{1g}}$ | 1 | -1 | 1 | 1 | $x^{2}-y^{2}$
$\boldsymbol{B_{2g}}$ | 1 | -1 | -1 | 1 | $xy$
$\boldsymbol{E_{g}}$ | 2 | 0 | 0 | 2 | $\\{xz,yz\\}$
$\boldsymbol{A_{1u}}$ | 1 | 1 | 1 | -1 | $xyz(x^{2}-y^{2})$
$\boldsymbol{A_{2u}}$ | 1 | 1 | -1 | -1 | $z$
$\boldsymbol{B_{1u}}$ | 1 | -1 | 1 | -1 | $xyz$
$\boldsymbol{B_{2u}}$ | 1 | -1 | -1 | -1 | $z(x^{2}-y^{2})$
$\boldsymbol{E_{u}}$ | 2 | 0 | 0 | -2 | $\\{x,y\\}$

We can refine the irrep labelling considering how objects or functions
transform under all point group operations in a similar fashion. Before
continuing on this path, it is useful to point out that not all symmetry
operations in the group are independent. In particular,
$C_{2z}=C_{4z}.C_{4z}$, $C_{2d}=C_{2x}.C_{4z}$, $\sigma_{h}=P.C_{2z}$,
$S_{4}=C_{4z}.\sigma_{h}$, $\sigma_{xz}=P.C_{2y}$, and $\sigma_{d}=C_{2d}.P$.
These suggest that all elements in the group can be found by compositions of
three elementary operations (usually referred to as the _generators_ of the
group): $C_{4z}$, $C_{2x}$ and $P$. These relations allow us to simplify the
discussion and to uniquely classify the irreps in terms of the generators of
the group.

Continuing now with the labelling of the irreps, let’s consider the conjugacy
class labelled as $\mathbf{2C_{4z}}$. The function $f_{1}\sim z$ goes back
into itself under a $C_{4z}$ or a $C_{4z}^{-1}$ transformation, while the
function $f_{2}\sim x^{2}-y^{2}$ picks a minus sign. The standard notation to
label objects or functions according to how they transform with respect to
rotations along the principal axis of rotation (here the $z$-axis as it is the
only axis with 4-fold rotational symmetry) uses $A$ for the objects that go
back into themselves, and $B$ for objects that pick a minus sign under these
operations. If objects or functions transform differently under other symmetry
operations, these are distinguished by an extra numerical subindex. For
example, both $f_{2}\sim x^{2}-y^{2}$ and $f_{3}\sim xy$ functions pick a
minus sign under a $C_{4z}$ rotation, but transform differently under a
$C_{2x}$ rotation: $f_{2}$ goes back into itself, while $f_{3}$ gets a minus
sign. In this context, the irreps associated with $f_{2}$ and $f_{3}$ are
labelled as $B_{1g}$ and $B_{2g}$, respectively. One exception to this scheme
concerns higher-dimensional irreps. In these cases we do not have a single
object or function that transforms into itself up to a minus sign under the
point-group operations, but we have sets of objects or functions that
transform into each other under these operations. One example is the pair of
functions $f_{4}\sim x$ and $f_{5}\sim y$. Under $C_{4z}$ these functions
transform into each other (up to a minus sign), so this is an example of a
two-dimensional irrep. The standard label for two-dimensional irreps is the
letter $E$. Table 2, known as the _character table_ , summarizes this
discussion. The entries correspond to the properties of objects or functions
in a given symmetry class. Using the example of $f_{2}\sim x^{2}-y^{2}$ in
$B_{1g}$, the $-1$ entry in the fourth line of the third column tells us that
this function changes sign under $C_{4z}$ rotations. Note that for the one-
dimensional irreps the entries are always $\pm 1$, as these correspond to
single objects or functions that go back into themselves or pick a minus sign
under the symmetry operations. For the two-dimensional irreps, the entries are
zero or $\pm 2$. These entries are the traces of the two-dimensional matrices
corresponding to the symmetry operation acting on the space formed by the two
basis functions. For example, for the $E_{u}$ irrep, a $C_{2x}$ rotation takes
$x\rightarrow x$ and $y\rightarrow-y$, such that the matrix acting on the
basis $(x,y)$ in this case would be the Pauli matrix $\sigma_{3}$, with zero
trace. As a second example, inversion takes $x\rightarrow-x$ and
$y\rightarrow-y$, such that the corresponding matrix is minus the two-
dimensional identity matrix, with trace equal to $-2$. Note that the entries
in the second column, associated with the conjugacy class of the identity,
$\mathbf{I}$, gives us the dimension of the irrep.

Table 2 also gives examples of functions that transform according to each
irrep. The displayed functions correspond to the lowest order functions in
Cartesian coordinates that capture the symmetries associated with the
corresponding irreps. These can be thought of as _basis functions_ , and, in
principle, one should also consider higher order functions with the same
symmetry properties. Figure 2 gives a visual rendition of the lowest order
basis functions.

Group theory also gives us useful rules concerning the product of two
functions belonging to specific irreps. This is encoded in what is known as
the product table, displayed in Table 3. For example, the product of $f_{1}$
in $A_{2u}$ and $f_{2}$ in $B_{1g}$ gives a function $f_{6}\sim
z(x^{2}-y^{2})$ which belongs to $B_{2u}$. We note that for the product for
$(p,q)=(g,g)$, all the subscripts in this table become $g$. As we are going to
discuss below, this information is going to be key for the construction of
Landau theories for multi-component superconductors in presence of external
perturbations [15].

Figure 2:  Representative objects transforming according to the different irreps of $D_{4h}$. The pink and blue colours correspond to the sign of the lowest order basis functions in Cartesian coordinates indicated in the last column of Table 2 ($\copyright$ 2019 American Physical Society). Table 3: Simplified product table for the irreps of the $D_{4h}$ point group. The indices $(p,q,r)=\\{g,u\\}$. The product rules enforces $r=g$ for $(p,q)=\\{(g,g),(u,u)\\}$ or $r=u$ for $(p,q)=\\{(u,g),(g,u)\\}$. The order of the direct product is not relevant. For example: $A_{2u}\otimes B_{1g}=B_{1g}\otimes A_{2u}=B_{2u}$. | $\boldsymbol{A_{1p}}$ | $\boldsymbol{A_{2p}}$ | $\boldsymbol{B_{1p}}$ | $\boldsymbol{B_{2p}}$ | $\boldsymbol{E_{p}}$
---|---|---|---|---|---
$\boldsymbol{A_{1q}}$ | ${A_{1r}}$ | ${A_{2r}}$ | ${B_{1r}}$ | ${B_{2r}}$ | ${E_{r}}$
$\boldsymbol{A_{2q}}$ | ${A_{2r}}$ | ${A_{1r}}$ | ${B_{2r}}$ | ${B_{1r}}$ | ${E_{r}}$
$\boldsymbol{B_{1q}}$ | ${B_{1r}}$ | ${B_{2r}}$ | ${A_{1r}}$ | ${A_{2r}}$ | ${E_{r}}$
$\boldsymbol{B_{2q}}$ | ${B_{2r}}$ | ${B_{1r}}$ | ${A_{2r}}$ | ${A_{1r}}$ | ${E_{r}}$
$\boldsymbol{E_{q}}$ | ${E_{r}}$ | ${E_{r}}$ | ${E_{r}}$ | ${E_{r}}$ | ${A_{1r}}\oplus{A_{2r}}\oplus{B_{1r}}\oplus{B_{2r}}$

###  B.2 Normal-state Hamiltonian in the orbital basis

In this section, we construct the most general normal-state Hamiltonian to
describe Sr2RuO4, in the microscopic orbital-spin basis [16]. It is well known
that the important electronic degrees of freedom (DOF) for the description of
Sr2RuO4, are electrons in Wannier orbitals with $t_{2g}$ symmetry centered at
the Ru atoms. These Wannier orbitals are formed by a linear superposition of
Ru-d and O-p orbitals, but from here on we refer to them simply as d-electrons
and label them as $d_{yz}$, $d_{xz}$ and $d_{xy}$ [17], as depicted in Fig. 3.
Choosing the basis
$\Psi^{\dagger}_{\mathbf{k}}=(c_{yz\uparrow}^{\dagger},c_{yz\downarrow}^{\dagger},c_{xz\uparrow}^{\dagger},c_{xz\downarrow}^{\dagger},c_{xy\uparrow}^{\dagger},c_{xy\downarrow}^{\dagger})_{\mathbf{k}}$,
one can construct the most general three-orbital spinfull single-particle
Hamiltonian describing the normal state as:

$\displaystyle\mathcal{H}_{0}({\mathbf{k}})=\Psi^{\dagger}_{\mathbf{k}}H_{0}({\mathbf{k}})\Psi_{\mathbf{k}},$
(1)

with

$\displaystyle
H_{0}({\mathbf{k}})=\sum_{(a,b)}h_{ab}({\mathbf{k}})\lambda_{a}\otimes\sigma_{b},$
(2)

where $h_{ab}({\mathbf{k}})$ are thirty-six functions of momenta
${\mathbf{k}}$. Here $\lambda_{a=1,...,8}$ are the Gell-Mann (GM) matrices,
and $\lambda_{0}$ the three-dimensional identity matrix standing for the
orbital DOF, and $\sigma_{b=1,2,3}$ are Pauli matrices, and $\sigma_{0}$ the
two-dimensional identity matrix standing for the spin DOF. Note that the
Hermitian nature of the Hamiltonian constraints the functions
$h_{ab}({\mathbf{k}})$ to be real, given the choice of Hermitian basis
matrices.

Figure 3:  Representation of $t_{2g}$ orbitals on the square lattice formed by
the Ru ions in Sr2RuO4. The blue and yellow colors correspond to opposite
signs of the orbital wave functions.

The explicit form of the Pauli and GM matrices is, respectively:

$\displaystyle\sigma_{1}=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix},\hskip
2.84544pt\sigma_{2}=\begin{pmatrix}0&-i\\\ i&0\end{pmatrix},\hskip
2.84544pt\sigma_{3}=\begin{pmatrix}1&0\\\ 0&-1\\\ \end{pmatrix},$

and

$\displaystyle\lambda_{1}=\begin{pmatrix}0&1&0\\\ 1&0&0\\\
0&0&0\end{pmatrix},\hskip 5.69046pt\lambda_{2}=\begin{pmatrix}0&0&1\\\
0&0&0\\\ 1&0&0\end{pmatrix},$
$\displaystyle\lambda_{3}=\begin{pmatrix}0&0&0\\\ 0&0&1\\\
0&1&0\end{pmatrix},\hskip 5.69046pt\lambda_{4}=\begin{pmatrix}0&-i&0\\\
i&0&0\\\ 0&0&0\end{pmatrix},$
$\displaystyle\lambda_{5}=\begin{pmatrix}0&0&-i\\\ 0&0&0\\\
i&0&0\end{pmatrix},\hskip 5.69046pt\lambda_{6}=\begin{pmatrix}0&0&0\\\
0&0&-i\\\ 0&i&0\end{pmatrix},$
$\displaystyle\lambda_{7}=\begin{pmatrix}1&0&0\\\ 0&-1&0\\\
0&0&0\end{pmatrix},\hskip
5.69046pt\lambda_{8}=\frac{1}{\sqrt{3}}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&-2\end{pmatrix}.$ (4)

Note that the indices for the Pauli matrices convey some notion of
directionality considering a quantization axis for the spin DOF, with the
indices $1$, $2$, and $3$ generally also read as $x$, $y$, and $z$,
respectively. Conversely, the indices for the GM matrices do not bring such a
strong physical intuition and should be simply taken as labels. Note though,
that GM matrices with indices 0, 7, and 8 are diagonal; GM with indices 1, 2,
and 3 are off-diagonal but symmetric (can be thought of as three-dimensional
versions of the Pauli matrix $\sigma_{1}$); while GM matrices with indices 4,
5, and 6 are off-diagonal and anti -symmetric (can be thought of as three-
dimensional versions of the Pauli matrix $\sigma_{2}$).

Requiring the Hamiltonian to be invariant under inversion and time-reversal,
we find restrictions on the allowed pairs of indices $(a,b)$ which label the
functions $h_{ab}({\mathbf{k}})$ and the basis matrices
$\lambda_{a}\otimes\sigma_{b}$. Inversion symmetry is implemented as
$P=\lambda_{0}\otimes\sigma_{0}$ accompanied by
${\mathbf{k}}\rightarrow-{\mathbf{k}}$, and time-reversal symmetry is
implemented by $\Theta=\lambda_{0}\otimes(i\sigma_{2})\mathcal{K}$, where
$\mathcal{K}$ stands for complex conjugation and is accompanied by
${\mathbf{k}}\rightarrow-{\mathbf{k}}$. Requiring
$PH_{0}(-{\mathbf{k}})P^{-1}=H_{0}({\mathbf{k}})$, imposes that all
$h_{ab}({\mathbf{k}})$ are even functions of momentum. Requiring $\Theta
H_{0}(-{\mathbf{k}})\Theta^{-1}=H_{0}({\mathbf{k}})$, implies that
$h_{ab}({\mathbf{k}})=\pm h_{ab}^{*}(-{\mathbf{k}})$, with the plus (minus)
sign for combinations of real GM matrices with $\sigma_{0}$ (three Pauli
matrices) and imaginary GM matrices with the three Pauli matrices
($\sigma_{0}$). Together, the combination of Hermiticity, inversion symmetry,
and time-reversal symmetry imposes that only the combination of the six real
GM matrices with $\sigma_{0}$ or the combination of the three imaginary GM
matrices with the three Pauli matrices, totalling fifteen basis matrices, are
symmetry allowed. These are summarized in the Table 4. A detailed explanation
of how these terms are derived can be found in Appendix L.

Table 4: List of the fifteen symmetry-allowed $(a,b)$ terms in the normal-state Hamiltonian. For each $(a,b)$, the basis function $h_{ab}({\mathbf{k}})$ must belong to the same irrep of $D_{4h}$ as the matrix $\lambda_{a}\otimes\sigma_{b}$, see details in Appendix L. The table gives the irrep associated to each term in the first column, and the associated physical process ( or “Type”) in the last column, the latter can be inferred by the explicit form of the Gell-Mann and Pauli matrices. For the two-dimensional irrep $E_{g}$, the entries are organized such that the first transforms as $yz$ and the second as $xz$. Here ${\mathbf{k}}$-SOC stands for momentum-dependent spin-orbit coupling (SOC). Irrep in $D_{4h}$ | $(a,b)$ | Type
---|---|---
$A_{1g}$ | $(0,0)$ | intraorbital hopping
$(8,0)$ | intraorbital hopping
$(4,3)$ | atomic SOC
$(5,2)-(6,1)$ | atomic SOC
$A_{2g}$ | $(5,1)+(6,2)$ |  ${\mathbf{k}}$-SOC
$B_{1g}$ | $(7,0)$ | intraorbital hopping
$(5,2)+(6,1)$ |  ${\mathbf{k}}$-SOC
$B_{2g}$ | $(1,0)$ | interorbital hopping
$(5,1)-(6,2)$ |  ${\mathbf{k}}$-SOC
$E_{g}$ | $\\{(3,0),-(2,0)\\}$ | interorbital hopping
$\\{(4,2),-(4,1)\\}$ |  ${\mathbf{k}}$-SOC
$\\{(5,3),(6,3)\\}$ |  ${\mathbf{k}}$-SOC

###  B.3 Superconducting order parameter

Superconductivity is usually introduced and classified within the context of a
single spin-degenerate electronic band [15]. In its simplest form, the
superconducting order parameter can be encoded in two-by-two matrices:

$\displaystyle\Delta_{1Band}({\mathbf{k}})=\begin{pmatrix}\Delta_{\uparrow\uparrow}({\mathbf{k}})&\Delta_{\uparrow\downarrow}({\mathbf{k}})\\\
\Delta_{\downarrow\uparrow}({\mathbf{k}})&\Delta_{\downarrow\downarrow}({\mathbf{k}})\end{pmatrix},$
(5)

where $\Delta_{\alpha\beta}({\mathbf{k}})$ indicate all possible types of spin
configurations of the superconducting pairs. Given fermionic antisymmetry, any
superconducting order parameter should follow

$\displaystyle\Delta({\mathbf{k}})=-\Delta^{T}(-{\mathbf{k}}).$ (6)

This identity indicates that if the order parameter is an even function of
momenta, it should be accompanied by an antisymmetric matrix. Among the two-
by-two matrices, the only antisymmetric matrix is the Pauli matrix
$\sigma_{2}$, which is associated with a spin-singlet configuration of the
superconducting pair. Conversely, if the order parameter is an odd function of
momenta, it should be accompanied by a symmetric matrix. Among the two-by-two
matrices, there are three symmetric matrices, the identity $\sigma_{0}$, and
the Pauli matrices $\sigma_{1}$ and $\sigma_{3}$. These three matrices can be
more conveniently written in the form of a three-dimensional vector
$i\sigma_{2}\boldsymbol{\sigma}=i\sigma_{2}(\sigma_{1},\sigma_{2},\sigma_{3})$.
These basis matrices for the order parameter encode the three distinct spin
configurations of the spin of the superconducting pair in the triplet sector.
More concisely, we can write the most general form for the superconducting
order parameter in the single band scenario as:

$\displaystyle\Delta_{1Band}({\mathbf{k}})=i\sigma_{2}[\phi({\mathbf{k}})+\mathbf{d}({\mathbf{k}})\cdot\boldsymbol{\sigma}],$
(7)

where

$\displaystyle\phi({\mathbf{k}})=[\Delta_{\uparrow\downarrow}({\mathbf{k}})-\Delta_{\downarrow\uparrow}({\mathbf{k}})]/2$
(8)

is an even function of momentum, and
$\mathbf{d}({\mathbf{k}})=(d_{x}({\mathbf{k}}),d_{y}({\mathbf{k}}),d_{z}({\mathbf{k}}))$
is a three-dimensional vector, usually referred to as the _d-vector_ , and an
odd function of momentum, with

$\displaystyle d_{x}({\mathbf{k}})$
$\displaystyle=-[\Delta_{\uparrow\uparrow}({\mathbf{k}})-\Delta_{\downarrow\downarrow}({\mathbf{k}})]/2,$
(9) $\displaystyle d_{y}({\mathbf{k}})$
$\displaystyle=-i[\Delta_{\uparrow\uparrow}({\mathbf{k}})+\Delta_{\downarrow\downarrow}({\mathbf{k}})]/2,$
$\displaystyle d_{z}({\mathbf{k}})$
$\displaystyle=[\Delta_{\uparrow\downarrow}({\mathbf{k}})+\Delta_{\downarrow\uparrow}({\mathbf{k}})]/2.$

Note that, in presence of inversion symmetry, the superconducting order
parameters can be classified as even or odd parity. As the spin degree of
freedom transforms trivially under inversion, we find that spin-singlet order
parameters are even parity and spin triplets are odd parity within the single-
band picture.

One can refine the classification of the single-band order parameter further
based on the point-group symmetry of the crystalline structure in terms of
irreps, as introduced in the context of the normal state Hamiltonian above.
The spin component of a spin-singlet superconducting order parameter always
transforms trivially under all point-group operations (namely, it goes back
into itself after each operation), so all information about the symmetry of
the order parameter (or its associated irrep) is encoded in the function
$\phi({\mathbf{k}})$. In principle, this is a periodic function of momentum in
reciprocal space, but usually the lowest order expansion in momentum is used
to describe it. For example, an order parameter with $A_{1g}$ symmetry in
$D_{4h}$ could have $\phi({\mathbf{k}})=d_{0}$, where $d_{0}$ is a constant.
Order parameters that are constant in momentum are usually referred to as _s-
wave_ , in analogy to atomic orbitals with $\ell=0$. This type of even-parity,
spin-singlet order parameter with a gap that transforms trivially under all
point-group operations (associated with the $A_{1g}$ irrep) is usually
referred to as _conventional_. Examples of this type of order parameters are
presented in the first row under “one-dimensional irreps” in Fig. 4.

Figure 4: Examples of superconducting order parameter candidates for Sr2RuO4.
The order parameters are divided into one dimensional/component (left) and
two-dimensional/component (right), and into time-reversal symmetry (TRS)
preserving ones (top) and TRS broken ones (bottom). The color legend on the
left gives information about the magnitude and phase of the gap. Note that the
superconducting gaps in the top half display only red and cyan colors,
indicating that the order parameter does not have a complex phase. For these
order parameters the darker regions indicate gap minima (for gaps of red color
only, such as $s$\- or ($g+s$)-wave), or gap nodes (for gaps that change sign
and have both red and cyan colors, such as ($d+s$)-, $d$-wave, and $p$-wave).
In contrast, the superconducting order parameters in the bottom half display
multiple colors, indicating the complex phase winding of the order parameter
around the Fermi surface, a manifestation of time-reversal symmetry breaking.
Note that some of these gaps do not have nodes, or dark regions, such as the
accidentally degenerate ($d+id$)-wave or the symmetry-protected chiral
$p$-wave, while other gaps do display nodes, such as the accidentally
degenerate ($d+ig$)-wave or the symmetry protected chiral $d$-wave. We note
that the interorbital pairing states described in the next subsection are
beyond this traditional picture based on a single-band classification of the
conventional ($s$-wave) and unconventional (non $s$-wave) SC states.

Spin singlets can also be _unconventional_ if the associated
$\phi({\mathbf{k}})$ transforms according to a non-trivial irrep (for Sr2RuO4,
with $D_{4h}$ point-group symmetry, these are $A_{2g},B_{1g}$, $B_{2g}$, or
$E_{g}$). For example, an even-parity order parameter with $B_{1g}$ symmetry
in $D_{4h}$ could have $\phi({\mathbf{k}})=d_{0}[\cos(k_{x}a)-\cos(k_{y}a)]$
(plus higher harmonics), where $a$ is the in-plane lattice constant.
Generally, this order parameter is also written as $\phi({\mathbf{k}})\approx
d_{0}(k_{x}^{2}-k_{y}^{2})$, and referred to as _d-wave_ , in analogy to the
nomenclature of atomic orbitals with angular momentum $\ell=2$. Note that both
the momentum-periodic function and its approximation in second order in
momenta have nodes along the diagonals $k_{x}=\pm k_{y}$, as imposed by the
symmetry of the $B_{1g}$ irrep. Superconducting order parameters with $B_{1g}$
and $B_{2g}$ symmetry are displayed in the second row under “one-dimensional
irreps” in Fig. 4.

Order parameters in the triplet sector have a more complex classification
which depend on the spin configuration (more details are given below), but all
should be accompanied by odd-momentum vectors $\mathbf{d}({\mathbf{k}})$. As
one example, we can mention the functions $\\{\sin(k_{x}a),\sin(k_{y}a)\\}$,
which form a basis for the two-dimensional irrep $E_{u}$. These periodic
momentum-dependent functions are usually approximated by $\\{k_{x},k_{y}\\}$,
and superconducting order parameters with these momentum dependence are
referred to as _p-wave_ , again in analogy to the nomenclature of atomic
orbitals with angular momentum $\ell=1$. Note that triplet order parameters in
the single band context are always odd with respect to inversion symmetry, so
these always classify as unconventional.

Both even- and odd-parity order parameter sectors can be associated with one-
and two-dimensional irreps. Order parameters associated with one-dimensional
irreps can have nodes but their amplitude does not reduce the symmetry of the
lattice (see order parameters in the left column of Fig. 4). In contrast,
order parameters associated with two-dimensional irreps have two components
that can appear in different configurations. If the order parameter develops
with a real superposition of the two components, the order parameter amplitude
breaks some of the lattice symmetries and is called _nematic_ (see the top two
order parameters in the right column in Fig. 4). More interestingly, the two
components can appear with a non-trivial relative phase and the order
parameter breaks time-reversal symmetry. If the relative phase if of $\pi/2$
and the amplitudes of the two components are the same, these are called
_chiral_ as these carry a finite angular momentum. Examples are the chiral
$p$-wave and the chiral $d$-wave order parameters, which can be called
_symmetry-protected time-reversal symmetry breaking_ order parameters.
Examples of order parameters associated with such states are depicted in the
bottom right box in Fig. 4. Note that chiral order parameters are not the only
type of superconducting order parameters that break time-reversal symmetry. If
two order parameters associated with different symmetries (or irreps) are
accidentally (or almost) degenerate, they can both nucleate with a non-trivial
relative phase, leading to _accidental time-reversal symmetry broken states_.
Examples of order parameters associated with such states are given in the
bottom left box in Fig. 4.

The introduction of the superconducting order parameter in the context of
single-orbital superconductors above suffices to discuss much of the
phenomenology of unconventional superconductors. Recently, an extension of
this classification based on multiple-orbitals has been pursued, and brought
us many insights toward the apparently contradicting phenomenology of Sr2RuO4.
We provide an introduction to this extended framework below. For the reader
not interested on the technical details, the following subsection can be
skipped as these concepts are not necessary for the discussion of the
experimental results presented in the next sections.

###  B.4 Order parameter in the orbital basis

Given the introductory discussion to the superconducting order parameter in
the context of a single orbital in Sec. B.3, below we extend the
classification of the superconducting order parameters for multi-orbital
systems from the perspective of the microscopic DOFs.

Using the same orbital-spin basis employed in the normal state Hamiltonian, we
can write the most general superconducting order parameter in terms of thirty-
six functions $d_{ab}({\mathbf{k}})$:

$\displaystyle\Delta({\mathbf{k}})=\sum_{a,b}d_{ab}({\mathbf{k}})\lambda_{a}\otimes\sigma_{b}(i\sigma_{2}).$
(10)

Here we factor out the matrix $(i\sigma_{2})$ such that a spin-singlet state
correspond to $b=0$, and spin triplets correspond to $b=\\{1,2,3\\}.$ In
analogy to the _d-vector_ parametrizing the triplet order parameter for a
single band superconductor, here we introduce a _d-tensor_ notation, with
$d_{ab}({\mathbf{k}})$ carrying two indices. We recover the simplest form of
the _d-vector_ and its properties [2] if we restrict the discussion to
diagonal GM matrices $\lambda_{a=\\{0,7,8\\}}$. In particular, $a=0$
corresponds to intraorbital pairing with the same amplitude for all orbitals.

Table 5: Basis matrices $\lambda_{a}\otimes\sigma_{b}(i\sigma_{2})$, labelled as $[a,b]$, associated with even-parity superconducting order parameters $\Delta({\mathbf{k}})$, as defined in Eq. 10. The irrep associated with even-parity superconducting order parameters is given by the product of the irreps for the basis matrices listed below and the irrep of the accompanying $d_{ab}({\mathbf{k}})$, which must be even parity. Here Inter (S/AS) corresponds to interorbital pairing which is symmetric/anti-symmetric under orbital exchange. Irrep | $[a,b]$ | Orbital | Spin
---|---|---|---
$A_{1g}$ | $[0,0]$ | Intra | Singlet
$[4,3]$ | Inter (AS) | Triplet
$[8,0]$ | Intra | Singlet
$[5,2]-[6,1]$ | Inter (AS) | Triplet
$A_{2g}$ | $[5,1]+[6,2]$ | Inter (AS) | Triplet
$B_{1g}$ | $[7,0]$ | Intra | Singlet
$[5,2]+[6,1]$ | Inter (AS) | Triplet
$B_{2g}$ | $[1,0]$ | Inter (S) | Singlet
$[5,1]-[6,2]$ | Inter (AS) | Triplet
$E_{g}$ | $\\{[3,0],-[2,0]\\}$ | Inter (S) | Singlet
$\\{[4,2],-[4,1]\\}$ | Inter (AS) | Triplet
$\\{[5,3],[6,3]\\}$ | Inter (AS) | Triplet

In order to satisfy fermionic antisymmetry, the order parameter should follow
Eq. 6, such that we can separate the functions $d_{ab}({\mathbf{k}})$ in even
or odd in momentum if the accompanying matrix product
$\lambda_{a}\otimes\sigma_{b}(i\sigma_{2})$ is anti-symmetric or symmetric
under transposition, respectively. Note that, as inversion acts trivially on
the orbital and spin DOFs, the parity of the order parameter is directly
determined by the parity of the $d_{ab}({\mathbf{k}})$ functions. There are
fifteen basis matrices associated with even-parity order parameters [these are
the same as the symmetry allowed matrices in the normal state Hamiltonian,
multiplied by ($i\sigma_{2}$)], and twenty one basis matrices associated with
odd-parity order parameters. These are summarized in Tables 5 and 6,
respectively. In these tables, we introduce the label $[a,b]$ corresponding to
the indexes in the basis matrices $\lambda_{a}\otimes\sigma_{b}(i\sigma_{2})$.
Note the rectangular brackets, $[a,b]$, label the basis matrices associated
with the superconducting order parameters, while the round brackets, $(a,b)$,
label the basis matrices present in the normal state Hamiltonian. In Table 5,
we can identify the order parameter with indices $[a,b]=[0,0]$ as an even-
parity spin-singlet order parameter that has the same amplitude for all
orbitals. In Table 6, we can identify the order parameter with indices
$[a,b]=[0,i=1,2,3]$ as odd-parity spin-triplet order parameters with the same
amplitude for all orbitals.

Table 6: Basis matrices $\lambda_{a}\otimes\sigma_{b}(i\sigma_{2})$, labelled as $[a,b]$, associated with odd-parity superconducting order parameters, $\Delta({\mathbf{k}})$, as defined in Eq. 10. Note that the irrep associated with the basis matrices is always even parity, as both the spin and the orbitals transform trivially under inversion. The irrep associated with odd-parity superconducting order parameters is given by the product of the irreps of the basis matrices listed below and the irrep of the accompanying $d_{ab}({\mathbf{k}})$, which must be odd parity. Irrep | $[a,b]$ | Orbital | Spin
---|---|---|---
$A_{1g}$ | $[2,2]-[3,1]$ | Inter (S) | Triplet
$A_{2g}$ | $[0,3]$ | Intra | Triplet
$[4,0]$ | Inter (AS) | Singlet
$[8,3]$ | Intra | Triplet
| $[2,1]+[3,2]$ | Inter (S) | Triplet
$B_{1g}$ | $[1,3]$ | Inter (S) | Triplet
$[2,2]+[3,1]$ | Inter (S) | Triplet
$B_{2g}$ | $[7,3]$ | Intra | Triplet
$[2,1]-[3,2]$ | Inter (S) | Triplet
$E_{g}$ | $\\{[0,1],[0.2]\\}$ | Intra | Triplet
$\\{[1,2],[1,1]\\}$ | Inter (S) | Triplet
$\\{[2,3],[3,3]\\}$ | Inter (S) | Triplet
$\\{[6,0],-[5,0]\\}$ | Inter (AS) | Singlet
$\\{[7,1],-[7,2]\\}$ | Intra | Triplet
$\\{[8,1],[8,2]\\}$ | Intra | Triplet

Note that the classification of order parameters from a microscopic
perspective brings some surprises [18, 19]. Focusing on even-momentum order
parameters, enumerated in Table 5, we find that these can be either
intraorbital spin-singlet, such as $[0,0]$, $[7,0]$, and $[8,0]$; symmetric
inter orbital spin singlet, such as $[1,0]$, $[2,0]$, and $[3,0]$; and
antisymmetric inter orbital spin triplet, such as $[4,3]$, $[5,1]$, $[5,2]$,
among others. The last scenario shows that, in the orbital-spin basis
classification, the one-to-one correspondence between parity and spin
configuration of the electrons in the pairs is lost. In the single-band
scenario this correspondence is guaranteed by fermionic anti-symmetry, as the
only matrix-part of the superconducting order parameter is associated with the
spin DOF. In case the matrix structure encodes other DOFs, the matrix can be
antisymmetrized w.r.t. the spin or other DOFs, generating the exotic
possibility of an even-parity spin-triplet superconductor if pairing is anti-
symmetric in the orbitals.

We can further classify the order parameters considering other point-group
transformations. The analysis for the superconducting order parameter takes
the same steps as the classification for the terms in the normal state
Hamiltonian. Note, though, that the superconducting order parameter does not
need to transform trivially under the point-group operations, as it can
spontaneously break some of the lattice symmetries, what characterizes it as
_unconventional_ [15]. The classification of the superconducting order
parameters is therefore much richer than the classification of the terms in
the normal state Hamiltonian. More precisely, considering the symmetry
operation $C_{4z}$, the basis matrices for the order parameter transforms as
$C_{4z}\lambda_{a}\otimes\sigma_{b}(i\sigma_{2})[C_{4z}^{-1}]^{*}$, with
$C_{4z}$ encoding the transformations in both the spin and orbital sectors
discussed above. Note the complex conjugation on the last factor [20, 18]. The
result of the analysis of the transformations of the basis matrices
$\lambda_{a}\otimes\sigma_{b}(i\sigma_{2})$ with respect to $C_{4z}$ and
$C_{2x}$ allows us to associate these with specific irreps of $D_{4h}$, as
shown in Tables 5 and 6. The specific form of $d_{ab}({\mathbf{k}})$ stems
from the underlying pairing interaction. For on-site interactions it is
momentum independent and associated with the $A_{1g}$ irrep. If associated
with long-range interactions, it can develop a higher angular momentum and be
associated with one of the non-trivial irreps. Note that, if the parameters
$d_{ab}({\mathbf{k}})$ are constants or belong to the trivial irrep, $A_{1g}$,
the irrep of the complete order parameter given by Eq. 10 is the listed irrep
in Tables 5 and 6. If the function $d_{ab}({\mathbf{k}})$ transforms
nontrivially under the point-group operations, the order parameter is
associated with the irrep corresponding to the product of the irrep of the
basis matrices and the irrep of the $d_{ab}({\mathbf{k}})$ function following
Table 3. For example, if the order parameter basis matrix is $[1,0]$,
associated with the $B_{2g}$ irrep, and the momentum dependent function
$d_{10}({\mathbf{k}})\propto k_{x}^{2}-k_{y}^{2}$ belongs to the $B_{1g}$
irrep, the complete order parameter
${\Delta}({\mathbf{k}})=d_{10}({\mathbf{k}})\lambda_{1}\otimes\sigma_{0}(i\sigma_{2})$
belongs to the $A_{2g}$ irrep.

Order parameters associated with a one-dimensional irreducible representation
are said to be _single component_ , while order parameters associated with
higher dimensional irreps are said to be _multi component_. In the context of
$D_{4h}$, order parameters associated with $A_{1g/u}$, $A_{2g/u}$, $B_{1g/u}$,
and $B_{2g/u}$ are single component, while order parameters associated with
the irreps $E_{g/u}$ are two components. These two types of order parameters
are distinguishable by experiments such as polar Kerr effect [21] and muon
spin resonance [22], which can probe time-reversal symmetry breaking in case
the two components develop with a complex relative phase, or by ultrasound
attenuation experiments, which can probe the two-component nature of the
superconducting order parameter by the coupling of gauge-invariant
combinations of the order parameter components to shear lattice vibrational
modes [23, 24], as discussed in detail in Sec. G.

Some of the order parameters, when constructed in the orbital basis, have
unexpected features, such as being an even parity spin triplet (necessarily
antisymmetric in the orbital DOF), or being nodal even though the order
parameter in the orbital basis is momentum-independent [25]. Comparing the
classification of superconducting order parameters in the single-band scenario
with order parameters in the orbital-spin basis we could say that the
classification in the latter case includes cases that are “beyond
unconventional”, see summary in Table 7.

Table 7: Proposed classification of superconducting order parameters in the orbital-spin basis. Here $b=1,2,3$. We label as _conventional_ superconducting order parameters that are spin singlets and even parity with a momentum-dependent function in $A_{1g}$ (generally a constant), corresponding to intraorbital pairing, which in the single-orbital scenario correspond to intraband pairing. We label as _unconventional_ intraorbital order parameters with $d_{ab}({\mathbf{k}})$ functions that do not belong to the $A_{1g}$ irrep. We label as _beyond unconventional_ order parameters associated with interorbital pairing, which can be symmetric (S) or anti-symmetric (AS) in the orbital degree of freedom. If the order parameter is symmetric in the orbitals, the association of spin singlet with even parity and spin triplet with odd parity is preserved, as it is known for the single band scenario. If the order parameter is anti-symmetric (AS) in the orbitals, spin singlets are associated with odd parity and spin triplets with even parity. *The case of $[7,0]$ is subtle, as the basis matrix belongs to $B_{1g}$, so generally this order parameter should be classified as unconventional, but if $d_{ab}({\mathbf{k}})$ also belongs to $B_{1g}$, the overall order parameter belongs to $A_{1g}$ and could be classified as _conventional_. Pairing | $d_{ab}({\mathbf{k}})$ | Spin | Orbital | Parity | Examples
---|---|---|---|---|---
Conventional | In $A_{1g}$ | Singlet | Intra | Even | $[0,0],[8,0]$
Unconventional | Not in $A_{1g}$ | Singlet | Intra | Even | $[0,0],[8,0]$
Any∗ | Singlet | Intra | Even | [7,0]
| Any | Triplet | Intra | Odd | $[0,b],[7,b],[8,b]$
Beyond Unconventional | Any | Singlet | Inter(S) | Even | $[1,0],[2,0],[3,0]$
Triplet | Inter(AS) | Even | $[4,b],[5,b],[6,b]$
Triplet | Inter(S) | Odd | $[1,b],[2,b],[3,b]$
Singlet | Inter(AS) | Odd | $[4,0],[5,0],[6,0]$

##  Appendix C Normal state properties

###  C.1 Fermi-liquid properties

Sr2RuO4 is an ideal material to examine the effects of strong correlations on
the electronic states of a multi-orbital, multi-band system. It maintains a
body-centered tetragonal crystal symmetry (point group $D_{4h}$, space group
$I4/mmm$) of a layered perovskite structure down to the lowest temperatures
(Fig. 1). Large-size ultra-pure single crystals that retain chemical stability
in air over many years are available [26, 27]. These characteristics have led
to consistent results among researchers using the same probing techniques.

The electronic structure of Sr2RuO4 is characterized by quasi-two-dimensional
(Q2D) Fermi surfaces derived from the ruthenium 4d-t2g orbitals, $d_{xy}$,
$d_{yz}$ and $d_{zx}$, hybridized with oxygen 2p-orbitals. Table 8 summarizes
the basic normal-state parameters. Details of quantum oscillations revealed
substantial enhancements of the thermodynamic cyclotron masses of 3.3
($\alpha$ band), 7.0 ($\beta$ band), and 16 ($\gamma$ band) over the bare
electron mass [28]. The in-plane resistivity shows the residual-resistivity
ratio (RRR) exceeding 1000, corresponding to the low-temperature mean-free-
path of more than one micron.

Table 8: Normal state properties of Sr2RuO4. Quantity | Symbol | Values | Remarks
---|---|---|---
Lattice parameters @300 K | $a=b$ | 0.387073(2) nm | Ref. Chmaissem1998.PhysRevB.57.5067.
| $c$ | 1.27397(1) nm |
Unit cell volume @300 K | $V$ | 0.190873(3) nm3 | containing two formula units
Density @300 K (15 K) | $\rho$ | 5.921 (5.956) g/cm3 | formula-unit atomic mass: 340.31
Linear thermal expansion | $\delta_{a}$ | $-2.28\%$ | [$L$(15 K)-$L$(300 K)]/$L$(300 K)
| $\delta_{c}$ | $-1.35\%$ |
Compressibility @300 K | $\kappa_{a}$ |  $2.24\times 10^{-3}$ GPa-1 | $-(dL/dP)/L_{\mathrm{0}}$
| $\kappa_{c}$ |  $2.56\times 10^{-3}$ GPa-1 |
Bulk modulus @300 K | $B$ | 142 GPa |
Sommerfeld coefficient | $\gamma_{\mathrm{e}}$ | 38.5 mJ/K2mol | Ref. NishiZaki2000JPhysSocJpn
Debye temperature | $\Theta_{\mathrm{D}}$ | 410 K |
|  |  | Ref. Mackenzie2003RMP, after core-diamagnetism subtraction
Molar magnetic susceptibility | $\chi_{\mathrm{spin}}$ |  $1.1\times 10^{-8}$ m3/mol |  $0.9\times 10^{-3}$ emu/mol
Volume magnetic susceptibility | $\chi_{\mathrm{spin}}^{\mathrm{vol}}$ | $2.0\times 10^{-4}$ |  $1.6\times 10^{-5}$ emu/cm3
Resistivity @300 K | $\rho_{a}$ | 120 $\mu\Omega\rm{cm}$ | Ref. Barber2019.
@2 K |  | ¡ 0.100 $\mu\Omega\rm{cm}$ | RRR ¿ 1200
@300 K | $\rho_{c}$ | $\sim 30~{}\rm{m}\Omega\rm{cm}$ | Ref. Yoshida1998.
@2 K |  | $\sim 2.0~{}\rm{m}\Omega\rm{cm}$ | RRR $\sim 15$

###  C.2 Fermi surfaces

Fermi surface parameters have been determined from quantum oscillations in
magnetic susceptibility (the de Haas van Alphen (dHvA) effect) and in
resistivity (the Shubnikov-de Haas (SdH) effect) as described in previous
reviews [2, 28]. These quantum oscillation measurements were able to determine
precise details of the three-dimensional dispersions of the quasiparticles in
all three Fermi surfaces (FSs). The Brillouin zone of Sr2RuO4 with three FSs
of $\alpha,\beta$, and $\gamma$ bands is shown in Fig. 5. [33]

Figure 5: Brillouin zone (BZ) and the calculated Fermi surfaces (FSs) of
Sr2RuO4 projected on the $k_{x}k_{y}$-plane. [33]. The solid lines show the BZ
boundaries of the body-centred tetragonal unit cell; the peripheral line is at
$k_{z}=0$ and the central square is at $k_{z}=\pi$. Under uniaxial strain
along the [100], inequivalent M points may be designated as M1 and M2 points
as shown in Fig. 9 below. The dashed green line is the 2D zone boundary of the
RuO2 sheet, which is often used in the literature on Sr2RuO4. The BZ points
are added to the figure from Ref. Jerzembeck2022 ($\copyright$ 2022 The
Authors). The width of the FS lines indicates the warping along $k_{z}$.

Recent high-resolution ARPES measurements and the fitting with dynamical mean-
field theory (DMFT) revealed quantitatively that the effective spin-orbit
coupling (SOC) is strongly enhanced from the density-functional theory (DFT)
values by electron correlations as shown in Fig. 6 (a), and the orbital mixing
induced by the enhanced SOC is substantial as clearly shown in Fig. 6 (b)
[17]. An estimated minimal mixing between $d_{xy}$ and $\\{d_{xz},d_{yz}\\}$
for the $\gamma$ and $\beta$ bands is 20% : 80% along the $\Gamma$M direction
with a monotonic increase to approximately 50% along the Brillouin zone (BZ)
diagonal $\Gamma$X. The multi-orbital character of each band is significant,
and the orbitals retain a considerable physical importance at low
temperatures. An important feature in the band structure shown in Fig. 6 (c)
is the presence of the Van Hove singularity (VHS) of the $\gamma$ band at the
M point, only 35 meV above $E_{\mathrm{F}}$ [33]3

Figure 6: (a) Fermi surface of Sr2RuO4 obtained from high-resolution ARPES
(blue). The calculated Fermi surface using DFT with enhanced SOC (gray) gives
an excellent agreement in fine de- tails of the observation ($\copyright$ 2019
American Physical Society). (b) Orbital character of the eigenstates along the
Fermi surface. On the $\gamma$ FS, the orbital contribution is almost equal
between $d_{xy}$ and $\\{d_{xz},d_{yz}\\}$ along the diagonal $\Gamma$X
direction; the contribution from $\\{d_{xz},d_{yz}\\}$ is substantial even
along the $\Gamma$M direction. (c) DFT band structure for the three bands. The
$\gamma$ band contains a VHS slightly above $E_{\mathrm{F}}$. Reproduced from
Ref. Tamai2019.

Cleaved surface of Sr2RuO4 is known to exhibit atomic reconstruction. Angle-
resolved photoemission spectroscopy (ARPES) measurements revealed a
$\sqrt{2}\times\sqrt{2}$ surface reconstruction [34], associated with 7.46∘
rotation and about $1^{\circ}$ tilting of the surface RuO3 octahedra. The
surface-to-bulk progression of the electronic structure was revealed by LDA+SO
(local-density approximation plus SOC) calculation shown in Fig. 7 that
matches well the ARPES results. Near the surface, the electronic structure is
expected to consist of a hole-like $\gamma$ FS, in contrast with the electron-
like $\gamma$ FS in the bulk. In addition, VHSs cross at all four M points
within a few layers from the surface. Recently, by using low-energy muon spin
spectroscopy, exotic surface magnetism has been found in the normal state of
Sr2RuO4, and interpreted as due to orbital loop currents. [35]

Figure 7: Surface reconstruction and the corresponding Fermi surfaces. (a)
Partial structure of the Sr2RuO4 near the surface highlighting the rotated
layer (top), broken symmetry layer (middle), and bulk-like layer (bottom) with
corresponding FSs. (b) Top view of the reconstructed surface of Sr2RuO4, with
the original undistorted (rotated) unit cell marked by solid (dashed) lines.
(c) FS of Sr2RuO4 with shifted chemical potential to match the slight averaged
hole doping of the surface region observed by ARPES. Surface and bulk
character are shown in red and blue, respectively. Reproduce from Ref.
Veenstra2013 ($\copyright$ 2013 American Physical Society).

###  C.3 Properties under strain and the Lifshitz transition

As a notable recent progress, it has become possible to control the Fermi-
surface topology of Sr2RuO4 in-situ by an innovative uniaxial-strain apparatus
[37, 38]. As depicted in Fig. 8, the apparatus consists of two sets of piezo
stacks, one compressing and the other stretching a sample crystal. The inner
and outer piezo units help compensating the differential thermal contraction
of the structural material on cooling. This Hicks-type uniaxial strain device
is now applied to control a variety of physical properties in quantum
materials. [39, 40, 41].

Figure 8: A schematic of the strain apparatus introduced by Hicks et al. [37,
38] Upon voltage application, the inner piezo stacks compress the sample,
while the outer pair of piezo stacks stretches the sample. The displacement of
a long piezo-stack unit is transferred to a sample with a much shorter length,
thereby amplifying the strain in the sample to over 1% ($\copyright$ 2014 The
American Association for the Advancement of Science).

Using this device, it was found that the $T_{\mathrm{c}}$ of Sr2RuO4 is
remarkably enhanced by both compression (negative strain) and expansion
(positive strain) along the [100] direction. In order to minimize the effect
of strain inhomogeneity especially near the edge of the sample,
$T_{\mathrm{c}}$ is usually measured by alternating-current (AC)
susceptibility using a small pickup coil placed near the center of the sample.
Under the compression it was demonstrated that the $T_{\mathrm{c}}$ is
enhanced from 1.5 K to 3.5 K with a maximum at about 0.44% strain [42]. In
contrast, $T_{\mathrm{c}}$ exhibits a mild linear dependence on strain along
the [110] direction, with a decreasing $T_{\mathrm{c}}$ under compression
[37].

As a distinct feature of the application of such uniaxial stress over the
hydrostatic pressure, the symmetry of the lattice and therefore the electronic
states of a material may be readily controlled in-situ by just varying the
voltage applied to the piezo stacks. The variation of the $T_{\mathrm{c}}$ of
Sr2RuO4 is understood in terms of the Lifshitz transition associated with the
Fermi-level crossing of the VHSs of the $\gamma$ band at two of the four
M-points in the BZ. Such a transition becomes possible as a consequence of the
breaking the tetragonal symmetry. With a thin sample glued on a substrate,
which is strongly compressed by differential thermal contraction, ARPES
measurements clearly confirm the strain-induced Lifshitz transition at the
critical strain $\varepsilon_{\mathrm{VHS}}\sim-$0.5% (Fig. 9) [43]. According
to a simple tight-binding model, only the $\gamma$ band shows a sharp peak in
the DOS at the Lifshitz transition; $\alpha$ and $\beta$ bands are affected
only weakly (Fig. 2 of Barber et al. [31]). However, the NMR measurements [44]
suggest that there are two effects: in addition to the DOS enhancement
associated with the $\gamma$ band Fermi energy passing the VHS points, the
uniform susceptibility, namely the Stoner factor, is also substantially
enhanced toward the Lifshitz transition.

As an excellent example of demonstrating the importance of the multi-band
character of Sr2RuO4, a novel bulk acoustic plasmon was recently found in
momentum-resolved electron energy-loss spectroscopy (M-EELS). [45] This zero-
gap plasmon is attributable to the oscillation of the electron density between
the $\beta$ and $\gamma$ bands and perhaps the first realization of the Pines’
demon predicted in 1956. [46]

Figure 9:  Unstrained and uniaxially strained Sr2RuO4. (a) and (c) are
schematic in-plane lattice configurations; (b) and (d) are Fermi surfaces
observed by ARPES. [43] ($\copyright$ 2019 The Authors).

##  Appendix D Spin susceptibility and the behavior near $H_{\mathrm{c2}}$

###  D.1 Summary of superconducting properties

The superconducting properties of Sr2RuO4 are summarized in Table 9. Note that
the $c$-axis coherence length $\xi_{\mathrm{GL}\parallel c}$ is as small as
1.1-1.6 nm, which is comparable to, but still larger than, the spacing between
neighboring RuO2 planes ($c/2=0.64$ nm). This means that, in spite of
substantially large out-of-plane anisotropy of $\Gamma\sim 60$, the interlayer
coherence of Cooper pairs is still maintained in Sr2RuO4. Indeed, in Sr2RuO4,
bulk superconductive properties are rather mean-field like, nearly free from
phenomena attributable to fluctuations. This is in clear contrast with Bi-
based cuprates [47] or ET-based organics [48], where two-dimensional
superconductivity with very weak out-of-plane coherence associated with
dominant superconducting fluctuations is realized.

The absolute value of the penetration depth has been reported using neutron
scattering (Ref. Riseman1998.Nature.396.242; $\lambda_{ab}=190$ nm), $\mu$SR
(Ref. Luke2000.PhysicaB.289-290.373; $\lambda_{ab}=190$ nm), and microwave
impedance measurements (Ref. Ormeno2006.PhysRevB.74.092504; $\lambda_{ab}=160$
nm). Here, $\lambda_{ab}$ corresponds to the decay length of in-plane current
as a function of in-plane distance, measured under $c$-axis magnetic fields.
Correspondingly, $\lambda_{c}$, the current decay length along the $c$-axis,
is obtained by $\lambda_{c}=\Gamma\lambda_{ab}$ using the intrinsic
superconducting anisotropy $\Gamma$. Note that different notations for
$\lambda_{c}$ are used among the literature. The Ginzburg–Landau (GL)
parameter $\kappa$ for field parallel to the $c$-axis is then written as
$\kappa_{ab}=\lambda_{ab}/\xi_{ab}=2.7$, which is close to the border between
the type-I and type-II superconductivity $\kappa=1/\sqrt{2}$. Thus, non-local
effects may need to be taken into account for this field direction. In
contrast, under in-plane fields, Sr2RuO4 behaves as a strongly type-II
superconductor with $\kappa_{c}$ in the range of 81 – 160.

Table 9:  Superconducting properties of Sr2RuO4 evaluated by taking into account the Pauli pair-breaking effect. Quantity | Symbol | Direction | Values | Remarks
---|---|---|---|---
Zero-field critical temperature | $T_{\mathrm{c}}$ | - | 1.5 K | Ref. Mackenzie1998.PhysRevLett.80.161
Thermodynamic critical field | $\mu_{0}H_{\mathrm{c}}$ | - | 0.0194 T | From $\mu_{0}H_{\mathrm{c}}^{2}/2=\int C_{\mathrm{e}}(T)dT-\gamma_{\mathrm{e}}T_{\mathrm{c}}^{2}/2$ [53]
SC condensation energy density | $E_{\mathrm{cond}}$ | - | 150 J/m3 | From $E_{\mathrm{cond}}=\mu_{0}H_{\mathrm{c}}^{2}/2$
Experimental upper critical field for $T\to 0$ | $\mu_{0}H_{\mathrm{c2}}$ | $H\parallel c$ | 0.075 T |
$H\parallel$ [100] | 1.45 T |
$H\parallel$ [110] | 1.50 T |
Estimated orbital limiting field for $T\to 0$ | $\mu_{0}H_{\mathrm{c2}}^{\mathrm{orb}}$ | $H\parallel c$ | 0.075 T |
$H\parallel ab$ | 3.0 – 4.5 T |
Intrinsic SC anisotropy | $\Gamma$ | - | 30 – 60 |
Estimated Pauli limiting field | $\mu_{0}H_{\mathrm{P}}$ | - | 1.4 T | From $E_{\mathrm{cond}}=\chi_{\mathrm{spin}}^{\mathrm{vol}}\mu_{0}H_{\mathrm{P}}^{2}/2$ [54]
Critical temperature of FOT | $T_{\mathrm{FOT}}$ | $H\parallel ab$ | 0.6 – 0.8 K | Ref. Yonezawa2013.PhysRevLett.110.077003,Yonezawa2014.JPhysSocJpn.83.083706,Kittaka2014.PhysRevB.90.220502
GL coherence length | $\xi$ | $\xi_{ab}$ | 66 nm | From $H_{\mathrm{c2\parallel c}}^{\mathrm{orb}}=\Phi_{0}/(2\pi\xi_{ab}^{2})$
$\xi_{c}$ | 1.1 – 1.6 nm | From $H_{\mathrm{c2\parallel ab}}^{\mathrm{orb}}=\Phi_{0}/(2\pi\xi_{ab}\xi_{c})$
London penetration depth | $\lambda$ | $\lambda_{ab}$ | 160 – 190 nm | In plane decay of in-plane current [51]
$\lambda_{c}$ | 11 – 15 $\mu$m | From $\lambda_{c}=\Gamma\lambda_{ab}$
GL parameter | $\kappa$ | $H\parallel c$ | 2.7 | From $\kappa_{ab}=\lambda_{ab}/\xi_{ab}$
$H\parallel ab$ | 81 – 160 | From $\kappa_{c}=(\lambda_{ab}\lambda_{c}/\xi_{ab}\xi_{c})^{1/2}=\Gamma\kappa_{ab}$

###  D.2 Revised spin susceptibility by NMR

Up to year 2019, the invariant spin susceptibility across $T_{\mathrm{c}}$
obtained from NMR Knight shift and polarized neutron scattering was considered
as direct evidence for spin-triplet pairing. Brown et al.at UCLA, while
measuring NMR signals of Sr2RuO4 with elevated $T_{\mathrm{c}}$ (the 3-K
phase) under uniaxial strain, realized that the measured Knight shift depends
on the pulsating conditions (S. Brown: private communication). Using the free-
induction decay (FID) method with a reduced RF-pulse energy and working with
enhanced $T_{\mathrm{c}}$ up to 3.5 K, they were able to recognize that
exceptionally high conductivity of Sr2RuO4 induces large self-heating due to
eddy current, thus requiring an unexpectedly low RF-pulse energy to keep the
sample below $T_{\mathrm{c}}$.

As shown in Fig. 10, they were able to reveal that the 1.5-K phase without
strain exhibits reduction in the Knight shift for the field along [100], by
reducing the RF energy sufficiently [7]. These results were immediately
reproduced by the Kyoto group [57], which confirmed a large reduction of the
spin-part of the oxygen Knight shift on cooling as shown in Fig. 11 (a). The
field dependence of the Knight shift for field along the [110] direction was
also measured [58]. After taking account of the expected quasi-particle
excitations as a function of the applied magnetic fields, these revised
results indicate that the reduction in the spin part of the Knight shift (Fig.
12) are as large as 90%, which is sufficient to make the spin-triplet
scenarios with and without the _d-vector_ pinning as well as the spin-triplet
helical scenario unlikely. The Kyoto group also characterized the process of
overheating in detail and deduced that in the previous setup the sample
crystal was heated to the normal state for several milliseconds after the RF
pulses (Fig. 11 (b)).

Figure 10:  Importance of using small NMR pulse energy to reveal the intrinsic
Knight shift. (a) NMR spectra for various pulse energy. Reduction of the
Knight shift at low temperature is observed only for small RF pulse energy
$E$. (b) Pulse-energy dependence of the Knight shifts of different oxygen
nuclear sites. Saturation of the Knight shift at low energy indicates the need
for reducing the RF pulse energy to that level. Reproduced from Ref.
Pustogow2019.Nature.574.72 ($\copyright$ 2019 The Authors). Figure 11: (a)
Temperature dependence of the Knight shift. (b) Characterization of the
heating effect by RF pulses used in the NMR. The pulse sequence consisting of
a strong heat pulse and the subsequent weak spin-echo pulses after varying
time $\tau_{\mathrm{HP}}$ is used to extract the time-evolution of the
temperature of the electron-spin system. The electron temperature exceeds
$T_{\mathrm{c}}$ immediately after large-energy pulses and recovers the
original temperature only after ca. 100 ms. Thus, the Knight shift measured
immediately after the main pulse gave erroneous information. The pulse-energy
values in Figs. 10 and this figure cannot be directly compared since their
effectiveness depends on individual experimental setup. Reproduced from Ref.
Ishida2020.JPSJ.89.034712 ($\copyright$ 2020 The Authors).  Figure 12:
Reduction of the Knight shift at low temperature under magnetic field in
different in-plane directions. The large reduction of the Knight shift in both
in-plane directions is incompatible with the $d$-vector pointing in the
$c$-direction (proposed spin-triplet chiral p-wave state), the $d$-vector
having two components in the $ab$-plane (spin-triplet helical p-wave state),
nor a spin-triplet state with the $d$-vector pinned in the $ab$ plane.
Reproduced from Ref. Chronister2021.PNAS.118.25 ($\copyright$ 2021 National
Academy of Sciences).

Let us note that the measurements of the spin-lattice relaxation time, $T_{1}$
(Fig. 13) are expected to suffer much less from such initial overheating
within 10 ms. The temperature of the nuclear spin always remains nearly at the
initial temperature as the $T_{1}$ time scale is much longer. It is 30 ms even
at $T_{\mathrm{c}}$. [57].

Figure 13:  Temperature dependence of the nuclear relaxation rate $1/T_{1}$.
Since $T_{1}$ is substantially longer than 10 ms even at $T_{\mathrm{c}}$,
this data does not suffer from the effect of overheating. Reproduced from Ref.
Ishida2000.PhysRevLett.84.5387 ($\copyright$ 2000 American Physical Society).

After the revised NMR results, polarized neutron scattering experiments were
repeated under magnetic fields parallel to [100] reduced from 1 T to 0.5 T
[60]. Fig. 14 compares previous and recent results. Although it was difficult
to resolve any change in the previous results at 1 T, a clear reduction of 34
$\pm$6% in the magnetic susceptibility was resolved at 0.5 T. These neutron
scattering results rule out the odd-parity chiral $p$-wave superconducting
state with _d-vector_ $\sim\hat{z}(k_{x}\pm ik_{y})$. A spin-triplet helical
pairing state may exhibit a substantial drop in the spin susceptibility [61].
However, more recent results of a large reduction of the NMR Knight shift
shown in Fig. 12 make the helical triplet scenario unlikely. [58]

Figure 14:  Magnetic susceptibility, $\chi=M(G)/H$, measured by polarized
neutron scattering at $G=$ (011) under small magnetic fields. (a) Temperature
dependence at different field values and directions, (b) Field dependence at
60 mK. (c) Field dependence of the scaled spin susceptibility. Bars on the
vertical scale show expected values by two models with confidence intervals.
Reproduced from. Ref. Petsch2020.PhysRevLett.125.217004 ($\copyright$ 2020
American Physical Society).

###  D.3 Behavior near the upper critical field

####  D.3.1 FFLO state near $H_{\mathrm{c2}}$

NMR remains an indispensable technique to identify the spin configuration of
the pairs in the superconducting state. With sufficiently small RF-pulse
energy, a new phase evidencing the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)
state near the upper critical fields was identified [4]. The FFLO state was
theoretically proposed 60 years ago as a superconducting state with finite-
momentum Cooper pairs. Under strong spin imbalance due to spin polarization,
electrons may pair with a finite total momentum, resulting in a spatial
oscillation of the superconducting OP. Such an exotic state breaking
translational symmetry is possible in an ultraclean system with a large Pauli
spin susceptibility. It has been reported only in a handful of superconductors
[62, 4].

As illustrated in Fig. 15, the FFLO state with an amplitude modulation in a
spin-singlet superconducting state leads to a spatial oscillation of the local
spin density. Thus, such an electronic fluid state with directional spin
modulation is considered as a spin-smectic state, in analogy to a smectic
liquid crystal. In such a state, the NMR Knight shift would be characterized
by a double-horn spectrum with one of the peak positions at the enhanced spin
susceptibility over the normal state value. Indeed, such spectra were observed
in Sr2RuO4, as shown in Fig. 16 (a). The phase diagram in Fig. 16 (b)
identifies the regions in the $H-T$ phase diagram for $H$//[100] where the
spin-smectic phase emerges. On cooling at $\mu_{0}H$ = 1.25 T for example, the
Knight shift first decreases on entering the SC state, but on further cooling
the peak splits into two, with one of the peaks reflecting the enhanced spin
polarization. From the comparison between the NMR spectrum and the
simulation[63], the modulation wavelength is estimated to be
$\lambda_{\mathrm{FFLO}}\sim 30~{}\xi_{ab}\sim 2.0~{}\mu$m. A crude
theoretical estimation of the modulation period gives a reasonable
correspondence: $\lambda_{\mathrm{FFLO}}=2\pi\hbar
v_{\mathrm{F}}/(g\mu_{\mathrm{B}}H)=2\pi^{2}\Delta\xi/(g\mu_{\mathrm{B}}H)=29.8~{}\xi=2.0~{}\mu$m.

Figure 15:  Anticipated features of the NMR spectra in the normal state,
homogeneous superconducting state, and superconducting smectic state (FFLO
state). Reproduced from Ref. Kinjo2022.Science.376.397 2022 The American
Association for the Advancement of Science). Figure 16:  (a) A Double-horn
structure in the NMR spectra. (b) Phase diagram under in-plane magnetic field,
revealing the superconducting smectic state near the upper critical fields.
Reproduced from Ref. Kinjo2022.Science.376.397 ($\copyright$ 2022 The American
Association for the Advancement of Science).

To obtain the definitive evidence of the FFLO state, it is quite important to
probe the FFLO phase boundary by other thermodynamic measurements. Imaging of
the spatial modulation of the SC order parameter directly with scanning
tunneling microscopy is a highly promising method. Since the FFLO state is
observed under magnetic fields of one order of magnitude smaller than in other
candidate materials, Sr2RuO4 offers an ideal platform to study such an exotic
SC state.

####  D.3.2 $H_{\mathrm{c2}}$ suppression and first-order transition

In Sr2RuO4, the behavior of the upper critical field, $H_{\mathrm{c2}}$, for
fields parallel to the $ab$-plane had been a long mystery. [8, 9] Now, with
the correct knowledge on the spin susceptibility [7, 57, 60], the
$H_{\mathrm{c2}}$ behavior is well understood by the Pauli pair breaking
effect. Below, we give a brief historical overview of these developments.

Before year 2000, it was already found that the temperature dependence of
$H_{\mathrm{c2}}$ for $H\parallel ab$ exhibits strong suppression [53, 30]:
the $H_{\mathrm{c2}}(T)$ curve at low temperature is substantially suppressed
compared to the ordinary Wertharmer-Helfand-Hohenberg (WHH) curve [64]
accounting for the orbital pair breaking mechanism. In contrast,
$H_{\mathrm{c2}}$ parallel to the $c$-axis matches well the WHH curve. This
has been reproduced in subsequent studies of specific heat [65] and AC
susceptibility [66, 67]. Related to this $H_{\mathrm{c2}}$ suppression,
thermal conductivity, specific heat, and magnetization have been reported to
exhibit rapid recovery toward normal-state values near $H_{\mathrm{c2}}$ [68,
69, 65, 70].

Similar suppression is also revealed in the polar angle $\theta$ dependence of
$H_{\mathrm{c2}}$ [65, 67]: when the field direction is tilted from the
$ab$-plane, the $H_{\mathrm{c2}}(\theta)$ curve is well fitted with the
ordinary anisotropic-mass model, except for the field direction within about
2∘ from the $ab$-plane. We should comment here that the noticeable kink in the
$H_{\mathrm{c2}}(\theta)$ curve reported in Ref. Deguchi2002 has not been
observed in subsequent studies [67, 54, 55]; but deviation from the
anisotropic-mass model near $H\parallel ab$ has been certainly reproduced.

These facts indicate that ordinary orbital pair breaking effect cannot explain
the behavior of $H_{\mathrm{c2}}$ along the $ab$-plane, and another pair
breaking mechanism must play a role. Given the substantial decrease of the
spin susceptibility [7, 57], we can now understand that the Pauli effect is
responsible for the $H_{\mathrm{c2}}$ suppression.

Figure 17: Superconducting phase diagrams of Sr2RuO4 revealed by specific-heat
measurements using an ultra-clean sample. First-order phase transition is
indicated by red squares and second-order transition by blue circles. (a)
$H$-$T$ phase diagram for $H\parallel ab$. (b) In-plane $H_{\mathrm{c2}}$ as a
function of the in-plane field angle $\varphi$. (c) Polar angle $\theta$
dependence of $H_{\mathrm{c2}}$. Reproduced from Ref.
Yonezawa2014.JPhysSocJpn.83.083706 ($\copyright$ 2014 The Authors).

In 2013, it has been revealed that the superconducting-normal transition at
$H_{\mathrm{c2}}$ is a first-order transition (FOT) when the field is very
close to the $ab$-plane and temperature is less than 0.8 K [54], as depicted
in Fig. 17. In the initial report, the magneto-caloric effect, i.e. change of
the sample temperature induced by magnetic field sweeps, revealed clear
divergent-like peak in the field derivative of the entropy $S$, together with
a small but clear hysteresis in $H_{\mathrm{c2}}$ in ultra-clean samples. The
FOT becomes an ordinary second-order transition by tilting the magnetic field
away from the $ab$-plane only by 2.5∘. The first-order nature has also been
reported in subsequent specific heat [55], magnetization [56], and magnetic-
torque studies [71]. The Pauli effect, in general, has been known to cause FOT
at $H_{\mathrm{c2}}$ at low temperatures [72]. Indeed, even before 2019,
several studies have correctly attributed the FOT to the Pauli effect [73],
admitting apparent inconsistency with the early reports on invariant spin
susceptibility. We should also comment that several attempts have been made to
explain the FOT and $H_{\mathrm{c2}}$ suppression without the spin
paramagnetism [74, 20, 75].

Here, it would be valuable to characterize and discuss the FOT in more detail.
Firstly, $H_{\mathrm{c2}}$ value for $H\parallel ab$ is comparable to the
Pauli-limiting field $H_{\mathrm{P}}$. Assuming only the Zeeman effect,
$H_{\mathrm{P}}$ is given by
$H_{\mathrm{P}}=H_{\mathrm{c}}/(\chi_{\mathrm{spin,N}}-\chi_{\mathrm{spin,SC}})^{1/2}$,
where $H_{\mathrm{c}}$ is the thermodynamic critical field
($\mu_{0}H_{\mathrm{c}}=19.4$ mT for Sr2RuO4) [53], and
$\chi_{\mathrm{spin,N}}$ and $\chi_{\mathrm{spin,SC}}$ are spin
susceptibilities in the normal and superconducting states. Using the
experimental value of the bulk susceptibility $\chi_{\mathrm{bulk}}=0.9\times
10^{-3}$ emu/mol [2] as $\chi_{\mathrm{spin,N}}$ and assuming
$\chi_{\mathrm{spin,SC}}=0$, we obtain $\mu_{0}H_{\mathrm{P}}=1.4$ T, which is
in close agreement with the experimental $H_{\mathrm{c2}}$. Moreover, the
assumption $\chi_{\mathrm{spin,SC}}=0$ is consistent with the NMR study
showing $\chi_{\mathrm{spin,SC}}$ extrapolating to nearly zero for the limit
of $H,T\to 0$. Secondly, the critical temperature $T_{\mathrm{FOT}}$, below
which the transition becomes FOT, is in the range 0.6–0.8 K (40–50% of the
zero-field $T_{\mathrm{c}}$), depending on literature [54, 55, 56, 71]. This
is comparable to the simple theoretical prediction, where $T_{\mathrm{FOT}}$
is 56% of the zero-field $T_{\mathrm{c}}$, when only the Pauli effect is taken
into account. The entropy jump $\Delta S/T$ at $H_{\mathrm{c2}}$ revealed by
magnetocaloric effect is $-4.7\pm 1.2$ mJ/K mol at 0.2 K [54], amounting to
$12.5\pm 3.2\%$ of the normal-state entropy
$S_{\mathrm{N}}/T=\gamma_{\mathrm{e}}=37.5$ mJ/K mol. This value is confirmed
by the specific heat data [55]. Furthermore, the observed magnetization jump
$\mu_{0}\Delta M$ of $0.074\pm 0.015$ mT at $H_{\mathrm{c2}}$ [56] satisfies
the Clausius-Clapeyron relation for first-order transitions $\Delta S/\Delta
M=-dH_{\mathrm{c2}}(T)/dT$. More recent NMR study revealed a similar 40%-jump
in the Knight shift [58] at the lowest measurement temperatures. Thus, the
jump in magnetization is dominated by spin paramagnetism and the contribution
of superconducting diamagnetism is rather minor. Finally, we should comment
that features of the FOT, such as the sharpness of the transition and the
existence of hysteresis, are highly dependent on sample quality. In samples
with slightly lower quality, these features are immediately blurred [54, 55].
This is analogous to the freezing of water, where supercooling (hysteresis)
effect is observed only for pure water stored in a clean container. This
explains why it took around 20 years to reveal the first-order nature of the
transition, although anomalies reported earlier [65, 70] suggested the FOT
nature.

####  D.3.3 Out-of-plane superconducting anisotropy

As discussed above, $H_{\mathrm{c2}}$ for the $ab$-plane is dominated by the
Pauli pair-breaking mechanism. Thus, the out-of-plane superconducting
anisotropy $\Gamma=\xi_{ab}/\xi_{c}$ should not be evaluated using the simple
formulae $H_{\mathrm{c2}\parallel c}(T=0)=\Phi_{0}/(2\pi\xi_{ab}^{2})$ and
$H_{\mathrm{c2}\parallel ab}(T=0)=\Phi_{0}/(2\pi\xi_{ab}\xi_{c})$ for orbital
pair-breaking cases, where $\Phi_{0}~{}\approx$ 2.07 mT$(\mu\mathrm{m})^{2}$
is the superconducting flux quantum. . This fact has been recognized before
the Pauli pair-breaking is identified as the origin of the $H_{\mathrm{c2}}$
suppression, and the intrinsic $\Gamma$ value was estimated by evaluating the
orbital-limited $H_{\mathrm{c2\parallel ab}}^{\mathrm{orb}}$ in various ways.
Fitting $H_{\mathrm{c2}\parallel ab}(T)$ data near $T_{\mathrm{c}}$ using the
WHH formula, $H_{\mathrm{c2\parallel ab}}^{\mathrm{orb}}$ for $T\to 0$ was
estimated to be $\mu_{0}H_{\mathrm{c2\parallel ab}}^{\mathrm{orb}}(0)\sim 2.5$
T [67]. This value corresponds to the intrinsic $\Gamma=33$, which is
substantially larger than the naive evaluation of 20 from the ratio
$H_{\mathrm{c2}\parallel ab}/H_{\mathrm{c2}\parallel c}$. Extrapolation of the
polar angle $\theta$ dependence of $H_{\mathrm{c2}}$ toward the $ab$-plane
yields a temperature-dependent $\Gamma=22$-$46$ [65, 67]. The ratio
$H_{\mathrm{c2}\parallel ab}/H_{\mathrm{c2}\parallel c}$ near $T_{\mathrm{c}}$
has also been used since $H_{\mathrm{c2}\parallel ab}$ should be close to
$H_{\mathrm{c2\parallel ab}}^{\mathrm{orb}}$ near $T_{\mathrm{c}}$. This
evaluation leads to the intrinsic $\Gamma$ value of $\sim 60$ in the limit of
$T\to T_{\mathrm{c}}$ [76]. Analysis of the magnetic torque near $H\parallel
ab$ also concluded $\Gamma\sim 60$ [56].

Intrinsic out-of-plane superconducting anisotropy has been microscopically
studied using small-angle neutron scattering (SANS) [77], which characterized
the vortex-lattice structure for $H$ along the $ab$-plane. For this field
configuration, observation of the vortex lattice Bragg peak is extremely
difficult since the field modulation due to vortices is as small as 3 $\mu$T
due to the long $\lambda_{c}$. This difficulty has been overcome by measuring
the field component perpendicular to the magnetic field that is intentionally
tilted by a few degrees away from the $ab$-plane. In addition, among the six
expected Bragg peaks, only the ones along the reciprocal $c^{\ast}$-axis are
detectable in principle. Nevertheless, by assuming the flux quantization
within the unit cell of the vortex lattice, the authors of Ref.
Rastovski2013.PhysRevLett.111.087003 succeeded in evaluating the intrinsic
anisotropy $\Gamma$, which amounts to 60 in the limit of
$H\parallel{ab}$-plane. This value is in close agreement with some of the
evaluations above. In addition, the field-angle dependence of the vortex form
factor does not follow the ordinary orbital-limited behavior. These findings
are consistent with the Pauli-limiting mechanism.

####  D.3.4 In-plane superconducting anisotropy

Compared with the large out-of-plane anisotropy, the in-plane superconducting
anisotropy is much weaker, as expected for tetragonal crystal symmetry.
Studies with accurate two-axis field control have reported that in-plane
anisotropy is nearly absent in $H_{\mathrm{c2}}$ above 0.8 K, whereas
noticeable anisotropy emerges below this temperature [78, 79, 67, 55]. For
$T\to 0$, $H_{\mathrm{c2}}$ along the [110] direction is larger than
$H_{\mathrm{c2}}$ along [100] by around 3%. As a function of in-plane angle
$\phi$, $H_{\mathrm{c2}}$ exhibits a four-fold sinusoidal oscillation [78, 79,
55] as exemplified in Fig. 17(b).

Interestingly, the temperature below which the in-plane $H_{\mathrm{c2}}$
anisotropy emerges coincides with the onset temperature of the first-order
transition $T_{\mathrm{FOT}}\simeq 0.8$ K (see Fig. 17(a)). This fact
indicates that the four-fold in-plane $H_{\mathrm{c2}}$ oscillation is
intimately related to the Pauli pair-breaking effect. Indeed, theories predict
the emergence of $H_{\mathrm{c2}}$ anisotropy as a consequence of the
formation of the FFLO state [80], which is indeed recently discovered by NMR
as described earlier in this section [4].

##  Appendix E Superconducting gap structure

###  E.1 Low-energy quasiparticle excitations

The superconducting gap structure in Sr2RuO4 has been studied from the
temperature dependence of quasiparticle excitations. Experiments show power-
law $T$ dependence in nuclear-lattice relaxation rate, electronic specific
heat, penetration depth and ultrasound attenuation, indicating the presence of
low energy excitations due to a nodal superconducting gap structure.

NMR/NQR nuclear-lattice relaxation rate $1/T_{1}$ is, roughly speaking,
proportional to the square of thermally excited quasiparticle density of
states at $T\ll T_{\mathrm{c}}$, whereas just below $T_{\mathrm{c}}$ it
provides information of the order parameter symmetry through existence or
absence of the coherence peak. In Sr2RuO4, absence of the coherence peak below
$T_{\mathrm{c}}$ shown in Fig. 13 clearly indicates non-$s$-wave
superconductivity [59]. Moreover, $1/T_{1}$ exhibits $T^{3}$ dependence at
$T\ll T_{\mathrm{c}}$ for clean samples whereas $1/T_{1}\propto T$ for less
clean samples. This is in good agreement with thermally excited quasiparticle
density of states in superconductors with nodes, together with impurity-
induced density of states.

The electronic specific heat coefficient, $C_{\mathrm{e}}/T$, in the
superconducting state provides another important measure of the density of
thermally-excited quasiparticles. Various studies revealed line-nodal behavior
($C_{\mathrm{e}}/T\propto T$) with negligible or small residual values for
$T\to 0$ [30, 81, 82]. Nodal superconductivity is also made evident by the
size of the specific-heat jump at $T_{\mathrm{c}}$, which is only 70% of the
normal-state electronic specific heat $\gamma T_{\mathrm{c}}$ and is much
smaller than the BCS-theory prediction of 143%. Such a reduction in the
specific-heat jump is a common feature in line-nodal weak-coupling
superconductors [83].

The penetration depth $\lambda(T)$ has been studied using various techniques.
Measurements using tunneling diode oscillators (28 MHz) reveals
$\Delta\lambda(T)\equiv\lambda(0)-\lambda(T)\propto T^{2}$ for samples with
$T_{\mathrm{c}}=1.3$ – 1.4 K, whereas $\Delta\lambda(T)\propto T^{3}$ for an
impure sample with $T_{\mathrm{c}}=0.8$ K [84]. Since $\lambda(T)^{-2}$ is
proportional to the so-called ”superfluid density”
$n_{\mathrm{s}}$/$m^{\ast}$, where $n_{\mathrm{s}}$ is the superconducting
carrier density and $m^{\ast}$is the effective mass, for $T\ll T_{\mathrm{c}}$
one can approximate the expected quasiparticle density to be
$n=(n_{0}-n_{\mathrm{s}})\propto T^{2}$. This is actually different from the
canonical line-nodal behavior ($\Delta\lambda(T)\propto T$ and $n\propto T$).
Considering the cleanness of the higher-$T_{\mathrm{c}}$ samples, the
quadratic $T$ dependence was interpreted as a combination of non-local
electrodynamic effect and the presence of line-nodal quasiparticles, rather
than impurity-induced filling of the gap node. Non-local effects emerge as the
electromagnetic response in a certain position in space is determined by
properties distant from that position, resulting in a weaker temperature
dependence of $\Delta\lambda(T)$ [85]. Indeed, the Ginzburg-Landau parameter
$\kappa_{\mathrm{GL}}$ of Sr2RuO4 amounts to only 2 – 3 for $H\parallel c$,
indicating that Sr2RuO4 in $c$-axis field is located near the border between
type-I and type-II superconductivity [86]. In this case, non-local effects
become significant below $T<\kappa_{\mathrm{GL}}\Delta_{0}/k_{\mathrm{B}}\sim
0.6T_{\mathrm{c}}$. Results of measurements using dielectric resonators
utilizing GHz-range electromagnetic waves using high-quality samples with
$T_{\mathrm{c}}\geq 1.4$ K are somewhat more complicated because of the non-
classical skin effect: due to the cleanness of the sample, the electron
scattering rate $1/\tau$ and the electromagnetic wave frequency $\omega$
become comparable (i.e. $\omega\tau>1$) and a non-trivial $\omega$ dependence
emerges [51]. Nevertheless, the conclusion is the same as that of experiments
in the MHz-range ($\Delta\lambda(T)\propto T^{2}$). In-field muon-spin
rotation/relaxation ($\mu$SR) technique has also been used to study the
temperature dependences of the penetration depth and superfluid density.
Aegerter et al.reported that $\Delta\lambda^{-2}(T)$ is well fitted either
with functions of $T$-linear or $T^{2.5}$ dependences [87]. Subsequently, Luke
et al. [50] reported $\lambda^{-2}(T)\propto 1-(T/T_{\mathrm{c}})^{\alpha}$
with $\alpha=2.78$. More recently, Khasanov et al. [88] reported the canonical
$T$-linear line-nodal dependence by $\mu$SR at low temperatures but the
fitting requires line-nodal gap with a higher angular harmonics. We also note
that the recent penetration-depth measurements with a scanning SQUID probe
[89] and by AC susceptibility [90] both show
$\lambda(T)-\lambda(0)~{}\propto~{}T^{2}$. The latter study explains the
observed behavior as due to non-local effect with vertical-line-node gap.

Ultrasound attenuation $\alpha(T)$ is also a sensitive probe to quasiparticle
excitations. In Sr2RuO4 a power-law dependence $\alpha(T)\propto T^{n}$ with
$n=1.4$ – 1.8 has been observed for all of the measured longitudinal (L) and
transverse (T) waves along [100] and [110] directions (L100: $c_{11}$ mode,
T100: $c_{66}$ mode, L110: $(c_{11}+c_{12}+2c_{66})/2$ mode, and T110:
$(c_{11}-c_{12})/2$ mode) down to 40 mK using samples with
$T_{\mathrm{c}}=1.37$ K [91]. Similar results with $\alpha(T)\propto T^{2}$
down to 170 mK have been observed in the $(c_{11}-c_{12})/2$ mode with a
sample with $T_{\mathrm{c}}\sim 1.4$ K [92]. In principle, ultrasound can also
give us information about nodal directions. However, the large anisotropy of
the power-law exponent expected for vertical-line nodes (i.e.
$\alpha(T)\propto T^{1.5}$ for ultrasound that couples well to quasiparticles
and $\alpha(T)\propto T^{3.5}$ for ultrasound that does not directly couple to
quasiparticles) has not been revealed [91]. This might be related to the large
attenuation anisotropy existing in the normal state. It is also worth
mentioning that an additional horizontal line node could explain the absence
of anisotropy in $\alpha(T)$.

###  E.2 Thermal conductivity

Thermal conductivity is a powerful and directional probe to investigate
quasiparticle excitations in the superconducting state. Around the year 2000,
a number of thermal-conductivity measurements have been performed. In-plane
thermal conductivity $\kappa_{ab}$, which is dominated by electronic carriers,
exhibits power-law temperature dependence $\kappa_{ab}/T\propto T$ at zero
field [69, 68]. Moreover, the residual thermal conductivity obtained by the
extrapolation $\kappa_{ab}/T$ for $T\to 0$ remains finite even for the clean
limit [93]. This is well interpreted as a “universal” thermal conductivity,
which originates from the perfect cancellation of the increase in the
quasiparticle density and in the impurity scattering rate for nodal
superconductors due to unitary scatterers [94]. These facts strongly support
the existence of line nodes.

The field dependence of the thermal conductivity can provide information about
another important aspect of superconducting gap structures. In nodal
superconductors, supercurrents flow around magnetic vortices causes a Doppler
shift of the quasiparticle energy spectrum. This shift then excites
quasiparticles around gap nodes or minima where the excitation energy is very
small. Such quasiparticles tend to delocalize out of vortex cores, and are
detectable via thermal conductivity as discussed below, or specific heat as
discussed in the next subsection. In particular, at low temperatures and
fields, it is expected that the quasiparticle density of states exhibits
$\sqrt{H}$ behavior. This phenomenon is also called the Volovik effect [95].
In Sr2RuO4, for temperatures down to 0.3 K, a non-monotonic field dependence
in $\kappa_{ab}$, i.e. a small decrease at low field followed by a $H$-linear
increase at higher fields, has been observed [69, 68]. The observed behavior
is interpreted as a result of a delicate balance between the increase in
quasiparticle excitations due to the Volovik effect and the increase in
scattering by vortices. The $c$-axis thermal transport $\kappa_{c}$, which is
dominated by phonons at finite temperature, is rather free from possible
cancellations of various mechanisms and should be solely sensitive to the
quasiparticle excitation density. Nevertheless, the $\sqrt{H}$ behavior at low
fields characteristic of the Volovik effect is not clearly seen even at 0.3 K
[79]. More recently, the gap structure has been studied using both $c$-axis
and $ab$-plane thermal transport [96]. In this study, the quasiparticle
excitation contribution is extracted by a careful $T\to 0$ extrapolation. The
extrapolated residual $\kappa/T$ in Fig. 18 exhibits $\sqrt{H}$-like rapid
increase for $\kappa_{c}$ in low fields along the $a$ axis. This rapid
increase indicates that the field-induced nodal quasiparticles have noticeable
$c$-axis component in their velocity and thus vertical line nodes are likely
to exist. This behavior is not compatible with the presence of only symmetry-
protected horizontal line nodes at $k_{z}=0$ or $\pi/2c$, where $c$-axis
velocity is zero. Therefore, the overall results indicate the existence of at
least vertical line nodes.

Figure 18: Magnetic-field dependence of thermal conductivity $\kappa$ divided
by temperature for the $T\to 0$ limit. For the field along the in-plane $a$
axis, $\kappa_{c}$ exhibits the $\sqrt{H}$-like rapid increase indicative of
nodal quasiparticles with the velocity with the $c$-axis component. Reproduced
from Ref. Hassinger2017.PhysRevX.7.011032 ($\copyright$ 2017 American Physical
Society).

Aiming to determine the position of the nodes, in-plane field-angle dependence
of the thermal conductivity has also been tested [68]. When quasiparticle
excitations due to the Volovik effect are dominant, the excitation is actually
field-direction dependent since the Doppler shift is proportional to the inner
product of the Fermi velocity and the superfluid current velocity, the latter
being perpendicular to the applied magnetic field [97]. Such field-orientation
dependent shift results in oscillations in the quasiparticle density of
states, which are again accessible via thermal conductivity or specific heat
as experimentally demonstrated in various unconventional superconductors [98,
99, 100]. In Sr2RuO4, a small four-fold anisotropy in $\kappa_{ab}$ ($\sim
0.5$% in peak-to-peak,
$\kappa_{ab}(H\parallel[110])<\kappa_{ab}(H\parallel[100])$), superimposed to
a relatively large two-fold anisotropy originating from the angle between the
thermal current and the magnetic field, has been detected in the field range
away from in-plane $H_{\mathrm{c2}}$ [68]. The observed four-fold component is
very small compared to theoretical predictions and thus it is concluded that
it originates from the anisotropy of $H_{\mathrm{c2}}$. In $\kappa_{c}$, in-
plane anisotropy is not detectable with the accuracy of $0.5$% [79], except
for fields very close to $H_{\mathrm{c2}}$. From the smallness of the in-plane
anisotropy, both studies favor the presence of horizontal line nodes [79, 68].

###  E.3 Field-angle-resolved specific heat

Information about the nodal structure has also been studied based on field-
angle-resolved calorimetry. The principle is similar to the Volovik effect for
the thermal transport described in the previous subsection; When magnetic
field is applied perpendicularly to the Fermi velocity at a line node,
quasiparticles are immediately excited leading to a $\sqrt{H}$ behavior in
$C(H)/T$ at low temperatures. Oscillation in the specific heat as a function
of magnetic-field angle originates from low-energy excitations around nodes.
Compared with thermal transport, specific heat is not a directional probe and
thus kinetic information of the quasiparticles is not detectable.
Nevertheless, it has several advantages, such as not being influenced by
various scattering mechanisms, or by extrinsic anisotropy due to applied
thermal current.

For Sr2RuO4, Deguchi et al.studied the field-strength and angle dependence of
the specific heat [101]. The field-strength dependence indeed shows $\sqrt{H}$
behavior at low fields. Moreover, the $T$ and $H$ dependence of the specific
heat under $H\parallel c$ obeys the scaling law for nodal superconductors.
Most importantly, a four-fold oscillation in the in-plane field-angle
dependence of the specific heat deep inside the superconducting state was
observed [101, 81]. The observed oscillation with specific heat smaller for
$H\parallel[100]$ than [110] is attributed to the Volovic effect due to
vertical gap minima located in the [100] directions on the $\gamma$ Fermi
surface. Note that the direction of the gap minima would be in agreement with
a chiral $p$-wave theory by Nomura et al. [102], but $45^{\circ}$ rotated from
that of the $d_{x^{2}-y^{2}}$ state. However, the gap in the $\gamma$ band for
even-parity superconductivity (indicated by recent NMR studies [7]) would be
large for the [100] direction in order to utilize the large density of states
near the van-Hove singularity. This is closely related to the large
enhancement of $T_{\mathrm{c}}$ under uniaxial strain along [100] [37, 5].
Indeed, the nodal directions are now discussed to be [110] in a quasiparticle
interference experiment [103] (discussed below) and in a orbital-singlet spin-
triplet pairing theory (discussed in Sec. I.3 and Appendix M).

Later, it was theoretically revealed that the specific-heat oscillation due to
the Volovik effect should exhibit a sign reversal at high temperatures or
fields [104], which is indeed experimentally observed in e.g. CeCoIn5 [105].
However, such $T$-induced sign reversal has not been resolved in Sr2RuO4 [101,
81]. This absence of sign reversal has been confirmed in a subsequent study by
Kittaka et al.performed down to 0.06 K [82]. Because of this absence, the
oscillation is attributed to a combination of horizontal line nodes and
substantial Fermi-velocity anisotropy ($v_{\mathrm{F}}\parallel[100]\ll
v_{\mathrm{F}}\parallel[110]$ for the $\gamma$ band due to the proximity to
the van-Hove singularity), rather than the canonical Volovik effect from
vertical line nodes.

To summarize these thermal measurements, the location of the node is not fully
understood, although the existence of nodal quasiparticles is consistently
revealed in a number of studies. The existence of vertical line nodes is
supported by the quasiparticle contribution obtained from the $\kappa(T\to 0)$
extrapolation. The in-plane field-angle-dependent specific heat was first
attributed to vertical line nodes in the $[100]$ direction but later
interpreted as horizontal line nodes with substantial Fermi-velocity
anisotropy. The near absence of in-plane anisotropy in $\kappa$ suggests
horizontal line nodes as well. The interpretation of the $\kappa$ anisotropy,
however, might be worth being revisited with updated knowledge. For example,
it is now empirically known that, in many superconductors, the experimentally
observed quasiparticle density-of-states modulation due to the Doppler shift
tends to be much smaller than the prediction by quasi-classical theories [99,
100]. The multi-band nature of many materials would add more complications
toward an accurate theoretical calculation. In addition, since the in-plane
$H_{\mathrm{c2}}$ in Sr2RuO4 is now known to be determined by the Pauli
effect, it is not trivial how a Pauli-limited $H_{\mathrm{c2}}$ anisotropy
affects the low-field $\kappa_{ab}$ anisotropy. Modern theoretical
methodologies taking into account the realistic band structures and plausible
superconducting gap models would be valuable to resolve these issues.

Figure 19: In-plane field-angle dependence of the specific heat of Sr2RuO4.
(a) Specific-heat oscillation reported in Ref.
Deguchi2004.PhysRevLett.92.047002. Here $C_{4}$ is the oscillatory component
in the electronic specific heat and $C_{\mathrm{N}}$ is the normal-state
electronic specific heat. (b) Specific-heat oscillation reported in Ref.
Kittaka2018.JPhysSocJpn.87.093703. Here $C_{\mathrm{e}}$ is the
$\phi$-dependent electronic specific heat while $C_{\mathrm{ave}}$ is its
avaraged value over $\phi$. They are essentially characterizing the same
quantity, but notice the differences in the definitions of the vertical axes
in (a) and (b). The results in (b) is consistent with those in (a), but
importantly extended to lower temperatures. (c) Field dependence of the
oscillation amplitude normalized by field-induced part of the electronic
specific heat, i.e. $A_{4}=\Delta
C_{\mathrm{e}}/[C_{\mathrm{e}}(H)-C_{\mathrm{e}}(H=0)]$, where $\Delta
C_{\mathrm{e}}$ is the oscillation amplitude, deduced from the data in (b).
(a) is reproduced from Refs. Deguchi2004.PhysRevLett.92.047002 ($\copyright$
2004 American Physical Society); (b) and (c) from
Kittaka2018.JPhysSocJpn.87.093703 ($\copyright$ 2018 The Authors).

###  E.4 Quasiparticle interference

Microscopic information on the nodal structure can also be obtained via
quasiparticle interference (QPI). The density of states oscillations around a
lattice defect observed by scanning tunneling microscopy (STM) reflect
quasiparticle scattering processes. The Fourier transform of the oscillation
pattern indicates the dominant scattering vectors at different energies. For
Sr2RuO4, STM studies have been quite challenging because the crystalline
surface tends to exhibit non-superconductive surface states originating from
RuO6 octahedral rotations. Nevertheless, this issue has been overcome by a
low-temperature high-vacuum cleaving technique [103, 89]. In these recent
studies, superconducting properties are investigated on a cleaved SrO plane.
Note that the result is sensitive to the $\alpha$ and $\beta$ Fermi surfaces,
which consist primarily of $d_{xz}$ and $d_{yz}$ orbitals with wavefunctions
extending out of the RuO2 plane; whereas information on the $\gamma$ Fermi
surface is not directly accessible via $c$-axis tunneling. In the
superconducting state, five distinct interference wavenumbers of Bogoliubov
quasiparticles have been observed. The results are well interpreted as line
nodes along the $k_{x}\pm k_{y}=0$ planes, corresponding to a gap of the
$d_{x^{2}-y^{2}}$ type in the $\alpha$ and $\beta$ bands.

Figure 20: QPI of Sr2RuO4. (a) Predicted QPI pattern assuming dominant
superconductivity in the $\alpha$ and $\beta$ bands as illustrated in the
panel (d). (b,c) Experimental QPI patterns measured at (b) $E=100~{}\mu$V and
(c) $0~{}\mu$V. The red and blue circles in (a-c) are dominant interference
features predicted in the calculation (a) and indeed detected experimentally
(b,c). (d) SC gap structure in the $\alpha$ (blue) and $\beta$ (red) bands
suggested in this study. Reproduced from Ref.
Sharma2020.ProcNatlAcadSci.117.5222 ($\copyright$ 2020 National Academy of
Sciences).

###  E.5 Mystery of the horizontal line node

Here we summarize the experimental studies on gap anisotropy in Sr2RuO4.
Almost all experiments are convincingly in agreement with non $s$-wave
unconventional superconductivity and a spin-singlet-like character. In
contrast, the detailed momentum dependence of the gap structure and the
presence and location of nodes have remained controversial. Some experiments
are in favor of a gap structure with vertical line nodes, other experiments
are in favor of horizontal line nodes. However, symmetry-protected horizontal
line node at $k_{z}=0$ naively implies that intralayer electrons cannot pair
and is thus unnatural for a quasi-2D system like Sr2RuO4. We will come back to
this mystery in Sec. H.5. Nevertheless, we comment here that the vertical and
horizontal nodal structures are not mutually exclusive. Thus, models with both
vertical and horizontal line nodes, as described in next chapters, could be
leading candidates to describe the apparent paradox.

##  Appendix F Phase sensitive experiments

Quasiparticle and Josephson tunneling experiments may provide crucial
information on the phase and magnitude of the superconducting energy gap. It
was recognized at an early stage that superconducting junctions of Sr2RuO4
with conventional superconductors exhibit unusual behavior. Unlike $d$-wave
superconductors with $d_{x^{2}-y^{2}}$ symmetry such as the high-temperature
superconductor YBa2Cu3O7-δ (YBCO), the critical current $I_{\mathrm{c}}$ of a
junction of Sr2RuO4 with a conventional superconductor shows irreproducibility
or dynamical behavior as long as the junction size is more than a few
micrometers [106]. It was also recognized that the $I_{\mathrm{c}}$ of a SQUID
device based on Sr2RuO4 exhibits behavior suggesting odd parity [107].

Thin films are used to investigate the superconducting junction properties of
many superconductors, but it is a challenge to obtain Sr2RuO4 thin films
exhibiting the intrinsic $T_{\mathrm{c}}$ of 1.5 K as for bulk crystals.
Nevertheless, there has been substantial recent progress in producing Sr2RuO4
superconducting thin films. Up to now, single crystals were mainly used for
junction studies and the focused ion beam (FIB) technique was used for micro-
fabrication of junctions and other micro-structured devices.

In this section, we review some key results from a variety of tunneling
experiments using Sr2RuO4 and, when necessary, attempt to reinterpret the
results from the perspeective of an even-parity superconducting state.
Additional details on tunneling experiments with Sr2RuO4 can be found in the
review papers by Liu and Mao ($\copyright$ 2015) [108], Leggett and Liu
($\copyright$ 2019) [109], and Anwar and Robinson ($\copyright$ 2021) [110].

###  F.1 Superconducting thin films

Perhaps the first report of successful thin film growth of Sr2RuO4 was in 2010
by using pulse laser deposition (PLD) [111]. Several groups reported improved
superconducting films by molecular beam epitaxy (MBE) and PLD as summarized in
Table 10. Uchida et al. [112, 113] concluded that suppression of the ruthenium
deficiency in the film is crucially important. Nair et al. [114] clarified
detailed conditions for a successful MBE growth. It is interesting that the
$T_{\mathrm{c}}$ becomes higher than 1.5 K probably because the (110)
substrate provides anisotropic in-plane strain to the Sr2RuO4 film: the misfit
strain at room temperature is $-$0.39% along the [001] direction of the
substrate and $-$0.16% along the [11̄0] direction of the substrate. Goodge et
al. [115] also suggest that ruthenium vacancies play a more important role
than inclusion of local higher-$n$ Srn+1RunO3n+1 syntactic intergrowths in
suppressing superconductivity.

Thin film growth by PLD has been improved recently. Garcia et al. [116]
identified mosaic twist as a key in-plane defect that suppresses
superconductivity. Thin films exhibiting a sharp resistive transition with
onset at 1.15 K and a transition width of 0.1 K were grown by PLD by Kim et
al. [117] By using single-crystalline Sr3Ru2O7 ($n=2$ member of the
Ruddlesden-Popper (R-P) series) as a target, they were able to induce a $n=2$
intergrowth near the substrate interface that blocks the formation of out-of-
phase boundaries between the LSAT substrate ($n=\infty$) and Sr2RuO4 ($n=1$)
film. Superconducting thin films by hybrid MBE using a metal–organic precursor
of ruthenium were also reported [118].

Various phase-sensitive experiments using these superconducting films are
awaited at present.

Table 10: Superconductive thin films of Sr2RuO4. Those reporting the zero-resistivity films are listed. LSAT: La1-xSrxAl1-yTiyO3. RRR = $R$(300 K)/$R$(4.2 K) is the residual resistivity ratio. Authors (Year) | Method | Substrate | RRR |  $T_{\mathrm{c}}$(onset)/$T_{\mathrm{c}}$(${R=0}$) | Remarks
---|---|---|---|---|---
Krockenberger et al.(2010) [111] | PLD | LSAT(001) | 82 | 0.9 K /0.6 K | 96 nm thick
Uchida et al.(2017) [112] | MBE | LSAT(001) | 30 | 1.1 K /0.8 K |
Uchida et al.(2019) [113] | MBE | LSAT(001) | - | 1.3 K /0.8 K |
Nair et al.(2018) [114] | MBE | NdGaO3(110) | 69 | 2.0 K /1.6 K | anisotropic in-plane misfit
Goodge et al.(2022) [115] | MBE | NdGaO3(110) | 69 | 2.0 K /1.6 K | 55 nm thick
Garcia et al.(2021) [116] | PLD | LSAT(001) | 100 | 1.1 K /0.9 K | single-crystalline Sr3Ru2O7 target
Kim JK et al.(2021) [117] | PLD | LSAT(001) | 34 | 1.15 K /1.05 K | single-crystalline Sr3Ru2O7 target
Choudhary et al.(2023) [118] | hybrid MBE | LSAT(001) | 22 | 0.95 K /0.75 K | metal-organic precursor

###  F.2 Quasiparticle tunneling

The gap structure determination with STM, suggesting vertical line nodes in
the gap, was already discussed in section E.4. Andreev bound states on the
surface of Sr2RuO4 were investigated with S/I/N junctions consisting of Au
deposited on the surface of Sr2RuO4 [119]. These junctions exhibit an
unusually broad zero-bias conductance peak (ZBCP) with the width comparable to
the gap width as shown in Fig. 21. This wide ZBCP is in sharp contrast with
the narrow ZBCP in YBCO, which represents the Andreev bound states formed at
the edge of the $d_{x^{2}-y^{2}}$ superconductor [120]. The broad ZBCP in the
Sr2RuO4 junctions was interpreted as due to chiral edge states, but later
argued that it could also be explained by helical edge states of a spin-
triplet helical superconductor [121, 122]. We add that there are several
recent theoretical studies examining the surface Andreev bound states and
tunneling spectra of Sr2RuO4 [123, 124, 125, 126].

Figure 21: Contrasting zero-bias conductance peaks (ZBCPs) in the
quasiparticle tunneling experiments reflecting topological edge states. In
Sr2RuO4, the broad ZBCP is interpreted as due to chiral or helical edge state;
the sharp ZBCP along the [110] direction in the high-$T_{\mathrm{c}}$ cuprate
YBCO is interpreted as due to the flat-band edge state. Reproduced from Ref.
Kashiwaya2011.PhysRevLett.107.077003 ($\copyright$ 2011 American Physical
Society).

The ZBCP in the enhanced-$T_{\mathrm{c}}$ phase of Sr2RuO4-Ru eutectic system
was investigated using junctions consisting of the bulk Sr2RuO4 and a micron-
size ruthenium metal inclusion [127]. It was found that the onset of the ZBCP
is at somewhat smaller values of $H(T)$ compared to the 3-K superconductivity
$H_{\mathrm{c2}}(T)$ (Fig. 22). One general criterion of ZBCP is the presence
of in-gap edge states. For example, for YBCO with flat-band edge states
associated with the vertical line nodes along the ¡110¿ directions, the ZBCP
becomes particularly strong along the ¡110¿ directions and absent only in the
¡100¿ directions [128, 129]. This behavior can be viewed in terms of the flat-
band topological edge state [130]. For chiral gapless SC states, the edge
states appears in all directions. So the interpretation of this experiment is
that it captured ZBCP spreading in all $k$-directions. Considering that the
ZBCP is a consequence of chiral superconducting OP, the observation of split
transition temperatures in Fig. 22 suggests that the time-reversal symmetry
breaking (TRS breaking) emerges in the $H-T$ region somewhat inside the
$H_{\mathrm{c2}}(T)$ of the 3-K superconductivity, as predicted by Kaneyasu et
al. [131] The splitting of $T_{\mathrm{c}}$ and the onset of TRS breaking is
an important current issue that will be discussed further in Sec. 8.

Figure 22: Emergence of the zero-bias conductance peak (ZBCP) in a Sr2RuO4/Ru
junction. (a) The onset of ZBCP is at field $H^{\ast}$ that is somewhat
smaller than $H_{\mathrm{c2}}(T)$. (b) The onset temperature of the ZBCP in
zero field. (c) In the field-temperature phase diagram, the onset of ZBCP
shown by the open symbols occurs in the $H-T$ region inside the
superconducting phase. Reproduced from Ref. Kawamura2005 ($\copyright$ 2005
The Authors).

###  F.3 Dynamical behavior of the 1.5-K phase

As another feature different from other superconducting junctions including
those of high-$T_{\mathrm{c}}$ cuprates, a dynamical behavior of tunneling
junctions consisting of Sr2RuO4 was first reported by Kidvingra et al. [106]
In that study, the critical current under external magnetic field, the
$I_{\mathrm{c}}$-vs-$H$ patterns, of in-plane S/N/S’ junctions of SRO/Cu/Pb
varies significantly with thermal cycling: changing from Fraunhofer-like to
more complicated patters. Hysteric patterns in the $H$-sweep, abrupt changes
in $I_{c}$, and time-dependent telegraphic noise behavior are also observed.
Such behavior was interpreted as due to the formation of superconducting
domains with different order-parameters, and ascribed in terms of chiral
domain boundaries of a chiral $p$-wave superconductor with the OP $p_{x}\pm
ip_{y}$. From today’s viewpoint, interpretations with $d\pm id$ states may
also be applicable. Such dynamical junction behavior was observed by other
groups as summarized in Fig. 23. [132, 133, 134, 135, 109] In junctions of
SRO/Ru/$s$-wave superconductor [135] (Figs. 23 (g and h)), the critical
current $I_{\mathrm{c}}$ emerging below 3 K is stable down to 1.5 K, but
becomes unstable when the same device is cooled below 1.5 K. This behavior
suggests phase competition due to superconducting domain formation in the
1.5-K phase. We will describe such behavior in more detail in the next
subsection.

Figure 23: Stable $I_{\mathrm{c}}$ above 1.5 K, and unstable behavior below
1.5 K in various junctions involving eutectic Sr2RuO4-Ru crystals. See text
for brief descriptions of these figures. Reproduced from Ref. Anwar2021
($\copyright$ 2021 The Authors).

###  F.4 Superconducting junctions with Ru inclusions

An unusual drop in $I_{\mathrm{c}}(T)$ on cooling across 1.5 K was
accidentally discovered in junctions of Sr2RuO4 with lead as shown in Fig.23
(a) and (b) [136]. It was later demonstrated that the essential junction
configuration for this behavior is Sr2RuO4/Ru/($s$-wave SC) involving eutectic
segregation of Ru metal platelets, Fig.23 (c) [133]. Due to surface
reconstruction of Sr2RuO4 (Fig. 7), a proper junction contact cannot be made
directly on the $ab$-plane. Metallic Ru inclusions exposed on the surface of
Sr2RuO4, instead, bridge between in-plane conduction of Sr2RuO4 and the
conduction of the $s$-wave superconductor. A topological competition between
proximity induced $s$-wave in a Ru micro-island and the superconductivity of
Sr2RuO4 surrounding the Ru micro-island explains the observed unusual behavior
of $I_{\mathrm{c}}$, provided that the 3-K and 1.5-K phases have different
phase-winding numbers [132]. Similar “topological junctions” of Sr2RuO4/Ru/Nb
in which only one or a few Ru micro-islands are surrounded by Sr2RuO4 (Fig.23
(d-h)) exhibit poor reproducibility among different cooldowns, hysteresis
behavior ascribable to the superconducting domain-wall motion [137], asymmetry
between positive and negative current directions (the superconducting diode
effect) [138], and time-dependent telegraphic noise only below 1.5 K [135].
These experiments highlight the qualitative difference of superconducting OPs
between the 1.5-K and 3-K phases of Sr2RuO4.

###  F.5 Proximity effects into a ferromagnet

Superconducting proximity effects into a ferromagnet have recently been
actively studied [139]. In particular, the so-called long-range proximity
effect by the injection of spin-triplet even-parity odd-frequency pairs into a
ferromagnet has been demonstrated experimentally by a number of groups as
reviewed in Ref. Eschrig2015.Rep.Prog.Phys.. To realize such a state using an
$s$-wave superconductor and a ferromagnet, an insertion of a non-uniformly
magnetized ferromagnetic layer is essentially required. In contrast, Anwar et
al. [141] reported that the long-range proximity effect into the ferromagnet
SrRuO3 is realized in epitaxial films of SrRuO3 grown on a Sr2RuO4 crystal as
a substrate, without any inhomogeneous magnetic layer. The tunneling spectra
of the Au/SrRuO3/Sr2RuO4 junctions suggest the presence of a superconducting
gap across SrRuO3 with a proximity coherence length of about 9 nm. This is
much longer than the coherence length of 1 nm expected for a spin-singlet
proximity effect into SrRuO3.

###  F.6 SQUID and half quantum fluxoid (HQF)

The superconducting phases of Sr2RuO4 have also been investigated using
microstructures such as Sr2RuO4-Ru-Nb superconducting quantum interference
devices (SQUIDs) as well as Sr2RuO4 micro-rings. These results strongly
suggest the formation of superconducting domains in Sr2RuO4 and the
competition between superconducting phases in Sr2RuO4 and the $s$-wave
superconductor.

After a series of developments of Sr2RuO4-based SQUIDs using the FIB
technique, Nago et al. [142] reported the evolution of superconducting paths
in direct-current SQUIDs (dc-SQUIDs) of Nb/Ru/Sr2RuO4 below 1 K. Their SQUID
consists of a bridge-shaped Nb structure formed on one Ru island surrounded by
Sr2RuO4. The evolution of $I_{\mathrm{c}}$ with temperature, especially the
collapse of the SQUID behavior below 0.4 K, corresponding to the onset of the
bulk superconductivity of Ru metal, lead to the conclusion that the phase
competition between chiral superconductivity in Sr2RuO4 and $s$-wave
superconductivity in Ru occurs.

After evidence for half-quantum fluxoid (HQF) was reported in magnetic torque
measurements on micro-rings of single-crystalline Sr2RuO4 samples [143], a few
groups reported quantum oscillation phenomena in similar micro-rings [144,
145, 146, 147].

Yasui et al. [144] studied Little-Parks oscillations in micro-rings of Sr2RuO4
crystals. The quantum oscillations were studied in square and circular rings
at temperatures slightly below $T_{\mathrm{c}}$, and a resistance oscillation
reflecting the oscillations of $T_{\mathrm{c}}$ were recorded. The field
dependence of the resistance was quantitatively consistent with those expected
from integer-fluxoid quantization. In addition, they observed features at half
the period ascribable to the formation of HQF. With improved fabrication of
micro-rings of a variety of sizes, oscillations of the critical current
ascribable to a spontaneous formation of a dc-SQUID with a pair of weak links
were observed, despite the apparently homogeneous nature of these rings [145]
(Fig. 24). Such behavior, found in narrow-arm micro-rings, was interpreted as
the spontaneous formation of a pair of superconducting domain walls associated
with distinct chiral domains on either side of the ring.

Magnetoresistance oscillations in micro-rings at rather different measurement
conditions were performed by Cai et al. [146, 147] They measured the
magnetoresistance at 0.3 K, much lower than $T_{\mathrm{c}}$ and under in-
plane magnetic field, typically 400 to 1000 Oe, much larger than the unit
period of flux quantization, with a small out-of-plane field of 20 Oe. The
large in-plane field was applied in order to stabilize the HQF states expected
in an equal-spin-paired spin-triplet state. Under such conditions, they
observed magnetoresistance oscillations with a large magnitude attributable to
the motions of vortices across the current (Fig. 25). They found additional
secondary peaks that they ascribed to HQF.

Figure 24: Micro-ring experiments exhibiting a spontaneous DC-SQUID behavior.
(a)-(d) The oscillations in the critical current $I_{\mathrm{c}}$ in
microrings indicate a spontaneous formation of weak links in the rings to from
a SQUID. (d) It is expected that superconducting domain walls form in a
certain range of the aspect ratio of the ring. Reproduced from Ref. Yasui2020
($\copyright$ 2021 The Authors). Figure 25: Micro-ring experiments exhibiting
huge resistance oscillations under magnetic field. (a) Before and (b) after a
microring of Sr2RuO4 is formed with FIB. (c) and (d) The observed oscillation
amplitudes are much greater than the expectation from the Little-Parks
oscillations. BeforReproduced from Ref. Cai2022 ($\copyright$ 2022 American
Physical Society).

##  Appendix G Two-Component Order Parameters

###  G.1 Landau theory of phase transitions

The Landau theory of phase transitions assumes that the transition from a
disordered to an ordered phase of matter can be captured by a free energy
written as a Taylor expansion on the respective order parameter. The formalism
only assumes that the free energy is analytic in the order parameter and that
it should obey all the symmetries of the Hamiltonian in the disordered (or
normal) state, such that the symmetry breaking occurs spontaneously.

In the context of superconductivity, the simplest form of the Landau theory
that describes the superconducting phase transition is written in terms of a
single-component order parameter as a complex number $\Psi$. The free energy
should not depend on the global phase of the order parameter, as global phases
of quantum states are not observable. As a consequence, only terms
proportional to $|\Psi|^{2}$ and its powers are allowed. In addition,
following the group theory discussion in Sec. B.3, if the order parameter is
single-component, it is associated with a one-dimensional irreducible
representation. Consequently $|\Psi|^{2}$ always belongs to the trivial irrep
$A_{1g}$, as can be seen from the product Table 3. In this light, the most
general free energy for a single component order parameter associated with a
one-dimensional irrep (labelled as $1D$) in an homogeneous system (neglecting
gradient terms) can be written as:

$\displaystyle
F_{1D}=F_{0}(T)+\alpha(T)|\Psi|^{2}+\frac{\beta}{2}|\Psi|^{4}+...$ (11)

where $F_{0}(T)$ is the free energy of the normal state, $\alpha$ and $\beta$
are phenomenological parameters, and the dots correspond to higher order terms
in $|\Psi|^{2}$. In particular, $\beta$ should be positive for the stability
of the ordered phase within this expansion up to fourth-order. We can write
explicitly $\alpha(T)=a(T/T_{\mathrm{c}}-1)$, with $a$ a positive
phenomenological constant, $T$ the temperature, and $T_{\mathrm{c}}$ the
superconducting critical temperature. Note that the factor $\alpha$ is
positive for $T>T_{\mathrm{c}}$ and negative for $T<T_{\mathrm{c}}$.
Minimizing the free energy with respect to $\Psi$ for $T>T_{\mathrm{c}}$ gives
us a single minimum for $\Psi=0$, while minimizing the free energy for
$T<T_{\mathrm{c}}$ gives us minima with amplitude of the order parameter
$|\Psi|=\sqrt{-\alpha(T)/\beta}$ and arbitrary phase.

As discussed in Section B.3, there are order parameters associated with multi-
dimensional irreps which are therefore multi-component. For these cases, the
free energy needs to be generalized, including new types of fourth-order