# Inhomogeneous time-reversal symmetry breaking in
$\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$

Roland Willa  Institute for Theory of Condensed Matter, Karlsruhe Institute of
Technology, Karlsruhe 76131, Germany  Matthias Hecker  Institute for Theory of
Condensed Matter, Karlsruhe Institute of Technology, Karlsruhe 76131, Germany
Rafael M. Fernandes  School of Physics and Astronomy, University of Minnesota,
Minneapolis, 55455 MN  Jörg Schmalian  Institute for Theory of Condensed
Matter, Karlsruhe Institute of Technology, Karlsruhe 76131, Germany  Institute
for Quantum Materials and Technologies, Karlsruhe Institute of Technology,
Karlsruhe 76021, Germany

###### Abstract

We show that the observed time-reversal symmetry breaking (TRSB) of the
superconducting state in $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$ can be
understood as originating from inhomogeneous strain fields near edge
dislocations of the crystal. Specifically, we argue that, without strain
inhomogeneities, $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$ is a single-
component, time-reversal symmetric superconductor, likely with
$d_{x^{2}-y^{2}}$ symmetry. However, due to the strong strain inhomogeneities
generated by dislocations, a slowly-decaying sub-leading pairing state
contributes to the condensate in significant portions of the sample. As it
phase winds around the dislocation, time-reversal symmetry is locally broken.
Global phase locking and TRSB occur at a sharp Ising transition that is not
accompanied by a change of the single-particle gap and yields a very small
heat capacity anomaly. Our model thus explains the puzzling absence of a
measurable heat capacity anomaly at the TRSB transition in strained samples,
and the dilute nature of the time-reversal symmetry broken state probed by
muon spin rotation experiments. We propose that plastic deformations of the
material may be used to manipulate the onset of broken time-reversal symmetry.

##  I Introduction

Establishing the symmetry of the Cooper pair wave function is the pivotal step
to understand a superconductor. It not only determines the macroscopic
phenomenology, but it also narrows down the microscopic mechanism of the
pairing state. Arguably the strongest evidence that cuprate-based high-
temperature superconductors are governed by an electronic mechanism is the
observation of the $d_{x^{2}-y^{2}}$ symmetry of the pair wave function
Wollman1993 ; Tsuei1994 . $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$ is a
layered perovskite superconductor that is iso-structural to La2CuO4 Maeno1994
. Given the crossover from incoherent to coherent transport as function of
temperature, $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$ is—just like the
cuprates—governed by strong electronic correlations Georges2011 . However,
early on it was advocated that the more appropriate analogue for the origin
and symmetry of the pairing state might be 3He Rice1995 ; Baskaran1996 .
Indeed, in distinction to the cuprates, broken time-reversal symmetry below
$T_{c}$ was observed in muon spin relaxation Luke1998 ; Grinenko2020  and
polar Kerr effect Xia2006  measurements, reminiscent of the $p_{x}+ip_{y}$
pairing state of the A-phase of 3He.

Recently, several experiments have forced the community to reexamine widely
accepted beliefs about this fascinating material. NMR measurements revealed
that $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$ is in fact a singlet
superconductor Pustogow2019 ; Ishida2020 ; Chronister2020 . Applying uniaxial
strain leads to a separate onset of superconductivity at $T_{c}$ and of time-
reversal symmetry breaking (TRSB) at $T_{{\scriptscriptstyle\mathrm{TRSB}}}$
Grinenko2020 . Combined with the observed jump of certain elastic constants
across $T_{c}$ Ghosh2020 ; Benhabib2020 , these observations have led to the
proposal that $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$ is a two-component
singlet superconductor, with $T_{{\scriptscriptstyle\mathrm{TRSB}}}$
signalling the condensation of the second component Mazin2005 ; Romer2019 ;
Suh2020 ; Kivelson2020 ; Willa2020 ; Scaffidi2020 . Meanwhile, there is no
indication of a heat capacity anomaly at
$T_{{\scriptscriptstyle\mathrm{TRSB}}}$ Li2020 , in contrast to the sharp jump
seen at $T_{c}$. More generally, there is no experimental evidence that time-
reversal symmetry breaking is a bulk effect—in fact, muon spin rotation
($\mu$SR) experiments find variations of the effect for distinct samples and
signatures consistent with diluted magnetic moments Grinenko2020 . This calls
into question the widely accepted view that
$\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$ is a multi-component
superconductor.

Figure 1:  (a) Edge dislocation in $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$
inducing local breaking of time-reversal symmetry due to inhomogeneous strain.
Near the dislocation, large strain fields mix a primary $d_{x^{2}-y^{2}}$
pairing state with other sub-leading symmetry channels, e.g. a
$g_{xy(x^{2}-y^{2})}$ pairing state Kivelson2020 . As a result, $d\pm g$
pairing emerges on opposite sides of the dislocation, forcing the relative
phase $\varphi(\boldsymbol{r})$ to wind in between, thus breaking time-
reversal symmetry locally. (b) Time-reversal symmetry breaking appears in the
bulk system as a stand-alone phase transition where the local phase windings
are locked through the long-range decay of strain fields that couple
dislocations.

In this paper, we argue that the pairing states of
$\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$ and the cuprate superconductors
display significantly more parallels than previously thought. Specifically, we
argue that $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$ without strain
inhomogeneities is a single-component, time-reversal symmetric superconductor
with a $d_{x^{2}-y^{2}}$ pairing state. This is consistent with recent
quasiparticle interference measurements Sharma2020  that strongly favor such a
pairing state, despite the onset of TRSB. The recently discussed phase diagram
of strained $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$, where
superconductivity occurs in the vicinity of a state with magnetic order,
likely a spin-density wave, Grinenko2020  is also suspiciously similar to the
generic cuprate phase diagram with a state of antiferromagnetic order
vanishing near superconductivity.

In our theory, time-reversal symmetry breaking of
$\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$ is a consequence of strong strain
inhomogeneities that locally break the crystalline symmetry and, due to the
large mechanical stresses, strongly couple to a sub-leading pairing state. We
analyze the nucleation of the corresponding pairing wave-function and its non-
trivial phase windings near edge dislocations of the crystal. In particular,
we solve for the pairing state near a single edge dislocation and proceed with
a coarse-grained model of many such dislocations. The first part of the theory
demonstrates that inhomogeneous strain induces local phase windings at a
temperature below $T_{c}$, see Fig. 1(a), the second shows the emergence of
global phase locking. This yields a sharp phase transition where time-reversal
symmetry is globally broken, see Fig. 1(b), albeit with a weak heat-capacity
anomaly. It hence explains the absence of a calorimetric signature at
$T_{{\scriptscriptstyle\mathrm{TRSB}}}$ in strained samples Li2020 , which is
strong evidence against any scenario where breaking of time-reversal symmetry
is associated with the opening of a quasiparticle pairing gap.

Despite the similar pairing state, an important difference between
$\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$ and the cuprates is that at least
one other pairing state must be sufficiently close in energy to contribute to
the condensate near strong local strain inhomogeneities. The idea that a
second competing superconducting instability is likely present in this
material was proposed before both on phenomenological and microscopic grounds
Zhang2018 ; Romer2019 ; Kivelson2020 ; Willa2020 ; Scaffidi2020 . In
distinction to these investigations, the sub-leading instability of our theory
does not need to be fine-tuned to have approximately the same $T_{c}$ value as
that of the leading $d$-wave instability. Moreover, because the dislocation
induces strain in all symmetry channels, TRSB occurs independently of the sub-
leading state’s symmetry—$s$-wave or $g$-wave. Finally, in our scenario, TRSB
does not correspond to the homogeneous condensation of the second pairing
state, but to a collective phenomenon in which the phase windings of the sub-
leading pairing wave-function induced near a dislocation phase-lock globally.
A direct consequence of the assumption of competing superconducting
instabilities is that, in a clean sample, the sub-leading pairing state should
lead to a Bardasis-Schrieffer excitation Bardasis1961  in the electronic Raman
spectrum. Finally, strain engineering of
$\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$, e.g. via plastically deformation
of the sample, which is known to impact dislocations in other perovskites
Hameed2020 , may be used to create and manipulate time-reversal symmetry
broken superconducting states.

##  II Results

###  II.1 Overview of the experimental constraints on the pairing state

Before we present our theory we summarize some of the relevant experimental
observations and their puzzling implications that motivated our proposal.
Excluding accidental degeneracies, homogeneous TRSB at $T_{c}$ may only occur
for a superconducting order parameter that has at least two components, i.e.,
transforms according to a two- or higher-dimensional irreducible
representation of the symmetry group Sigrist1991 . For
$\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$, with $D_{4h}$ point group, this
leaves only two options: $E_{u}$ triplet pairing that transforms like
$p_{x}\pm ip_{y}$ or the $E_{g}$ singlet pairing of the type $d_{xz}\pm
id_{yz}$. The most recent NMR measurements strongly favor singlet pairing
Pustogow2019 ; Ishida2020 ; Chronister2020 . Furthermore, $\mu$SR experiments
under uniaxial stress report a clear splitting between the superconducting
transition temperature $T_{c}$ and the onset temperature for broken time-
reversal symmetry, $T_{{\scriptscriptstyle\mathrm{TRSB}}}\\!<\\!T_{c}$
Grinenko2020 . For such a two-component order parameter, uniaxial stress
should indeed lift the degeneracy. In this picture, the strain-favored
component first condenses to a single-component state at $T_{c}$, while the
strain-disfavored component mixes-in with relative phase $\pm\pi/2$ at
$T_{{\scriptscriptstyle\mathrm{TRSB}}}$. Numerous theories for
$\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$ consider either $p_{x}\pm ip_{y}$
triplet pairing Rice1995 ; Baskaran1996 ; Nomura2000 ; Wang2013 ; Raghu2010 ;
Scaffidi2014  or, given the recent NMR results of Refs. Pustogow2019 ;
Ishida2020 ; Chronister2020 , $d_{xz}\pm id_{yz}$ singlet pairing Mazin2005 ;
Huang2019 ; Suh2020 .

There is however one very robust thermodynamic argument against either of
these mean-field scenarios (see also appendix A): recent heat-capacity
measurements in strained samples Li2020  see no jump at
$T_{{\scriptscriptstyle\mathrm{TRSB}}}$ and a large discontinuity at $T_{c}$.
Within mean-field theory, the ratio of the heat-capacity jumps at $T_{c}$ and
$T_{{\scriptscriptstyle\mathrm{TRSB}}}$ are generally related to the slopes
$\frac{dT_{c}}{d\epsilon_{B_{1g}}}$ and
$\frac{dT_{{\scriptscriptstyle\mathrm{TRSB}}}}{d\epsilon_{B_{1g}}}$, see Ref.
Grinenko2020 . If the magnitudes of the slopes are assumed to be approximately
equal for small strain, the heat-capacity jumps must essentially be the same.
In the more realistic case, as determined by the experimental strain-
temperature phase diagram, where
$\big{|}\frac{dT_{{\scriptscriptstyle\mathrm{TRSB}}}}{d\epsilon_{B_{1g}}}\big{|}<\frac{dT_{c}}{d\epsilon_{B_{1g}}}$,
the jump at the lower transition must even be the largest one. This is in
sharp contrast to the measurements of Ref. Li2020 , which reports no anomaly
at the lower transition temperature, estimating an upper bound for the second
discontinuity at $T_{{\scriptscriptstyle\mathrm{TRSB}}}$ of less that $5\%$ of
the discontinuity $\Delta C(T_{c})$ at $T_{c}$. This finding provides strong
thermodynamic evidence against a BCS-type transition of either $E_{g}$ or
$E_{u}$ symmetry. More generally, the absence of a heat-capacity jump at
$T_{{\scriptscriptstyle\mathrm{TRSB}}}$ argues against any scenario where the
opening of a pairing gap affects the entropy of electronic quasiparticles. In
addition, the quadratic dependence of $T_{c}$ with strain Hicks2014 ;
Watson2018  and the absence of a jump in the elastic constant in the $B_{1g}$
symmetry channel Ghosh2020  constitute additional evidence against a two-
component order parameter with $E_{g}$ or $E_{u}$ symmetry. For further
discussions about the puzzles related to the chiral $E_{g}$ or $E_{u}$ pairing
states, see Refs. Mineev2007 ; Kallin2012 ; Mackenzie2017 ; Kivelson2020 .

Given this strong evidence against a symmetry-protected, two-component order
parameter, the notion of an accidental (near)-degeneracy of two single-
component order parameters seems a very compelling approach Zhang2018 ;
Romer2019 ; Kivelson2020 ; Willa2020 ; Scaffidi2020 . This is rooted in
significant microscopic evidence that electronic pairing mechanisms give rise
to several competing superconducting states Schnell2006 ; Cho2013 ; Zhang2018
; Romer2019 . A range of arguments is brought forward in Ref. Kivelson2020
supporting the degeneracy between $\psi_{d}=d_{x^{2}-y^{2}}$ and
$\psi_{g}=g_{xy(x^{2}-y^{2})}$ pairing. This state is further supported by the
observed jump of the elastic constant in the $B_{2g}$ symmetry channel
Ghosh2020 ; Benhabib2020 , see also Ref. Okuda2002 , and by the report of
vertical line nodes Hassinger2017 . Other candidates include the near
degeneracy between $\psi_{d}=d_{x^{2}-y^{2}}$ and a nodal $s$-wave state
$\psi_{s}$ Romer2019  or the degeneracy between singlet and triplet states in
a model that focuses on the quasi one-dimensional parts of the electronic
structure Scaffidi2020 . Notice, an implication of the small heat-capacity
signature at the lower transition is that the amplitude of this second
component must be exceedingly small; see appendix A

That the primary pairing state is indeed unconventional is most plainly
demonstrated by the system’s strong sensitivity to non-magnetic disorder
Mackenzie2003 ; Mackenzie1998 . It is further supported by thermal
conductivity measurements Hassinger2017 , which report vertical nodal lines,
and by the recent quasiparticle interference measurements of Ref. Sharma2020 .
It is also in line with numerous microscopic theories for unconventional
pairing in this compound Schnell2006 ; Cho2013 ; Gingras2019 ; Roising2019 ;
Sharma2020 ; Scaffidi2020 ; Husain2020 . Our approach—partially inspired by
Ref. Kivelson2020 —relies on the existence of a secondary order parameter that
is reasonably close in energy and that admixes to the primary phase in the
presence of inhomogeneous strain. However, in distinction to Ref. Kivelson2020
, we neither require an almost perfect degeneracy of both states for
unstrained homogeneous samples, nor do we expect a second homogeneous mean-
field like transition anywhere below the onset of superconductivity at
$T_{c}$. Instead, we argue that the breaking of time-reversal symmetry is
exclusively a consequence of strain inhomogeneities in the material.

The presence of inhomogeneous strain fields was clearly established in
scanning SQUID microscopy measurements on
$\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$ single crystals Watson2018 . The
strong coupling of edge dislocations to superconductivity was revealed in Ref.
Ying2013 , where the local superconducting transition temperature in the
vicinity of a lattice dislocation can reach almost twice the value of the
bulk. Finally, we note that there is really no evidence at this point that the
TRSB transition in $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$ is a mean-field
homogeneous phase transition of the bulk system. The $\mu$SR signal shows a
broad distribution of local magnetic moments, somewhat similar to what is
often observed in spin glasses Luke1998 ; Grinenko2020 . In fact, the authors
of Ref. Grinenko2020  explicitly state that the internal field is thought to
arise at edges, defects, and domain walls. In addition, the splitting of
$T_{c}-T_{{\scriptscriptstyle\mathrm{TRSB}}}$ determined in Ref. Grinenko2020
varies in magnitude _and_ sign between different samples. These observations
guided us to investigate the role of inhomogeneous strain fields. We will
first analyze the behavior in the vicinity of a single edge dislocation. In a
second step we will formulate an effective model to describe the case of many
dislocations.

###  II.2 Time-reversal symmetry breaking at a single dislocation

Figure 2: Monte-Carlo simulation to minimize the Ginzburg-Landau free energy,
Eq. (1), for the two order parameters in the presence of a dislocation. (a)
Strain-induced onset of the imaginary part of the order parameter $\psi_{g}$
at four locations (see inset) around a dislocation. Because $\psi_{d}$ is
already present at $T<T_{c}$, this onset _locally_ breaks time-reversal
symmetry at $T_{{\scriptscriptstyle\mathrm{TRSB}}}^{\star}$. Note that both
these temperatures are much larger than the temperature $T_{c}^{<}$ at which
the homogeneous $g$-wave order parameter would onset. (b)-(c) Order parameter
distribution near the dislocation for fixed temperature [red arrow in (a)] and
for $\lambda_{0}=5$ and $\lambda_{x}=8$ respectively, see Eq. (7).

In what follows, we consider a Ginzburg-Landau theory for the superconducting
order parameter $\psi=(\psi_{d},\psi_{g})$, where the two components belong to
different one-dimensional irreducible representations, i.e. they do not
transform into each other under a crystalline symmetry. As discussed, we have
in mind a primary order in the $d_{x^{2}-y^{2}}$ channel. For a homogeneous
system this order parameter alone would be sufficient; the secondary component
only becomes important once we include strain inhomogeneities. Following Ref.
Kivelson2020 , this second component is assumed to be $g_{xy(x^{2}-y^{2})}$.
We later discuss how our results change for different sub-leading pairing
states, such as $s$-wave.

In an otherwise clean system, such as superconducting
$\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$, inhomogeneous strain is most
prominently induced by dislocations, i.e. a linear crystallographic defect, as
shown in Fig. 1. To study the impact of dislocation-induced strain on
superconductivity we consider the Ginzburg-Landau free-energy density

$\displaystyle f=f_{0}+f_{\mathrm{disl}},$ (1)

where the dislocation-free contribution $f_{0}$ reads

$\displaystyle f_{0}$
$\displaystyle=\frac{1}{2}\psi^{\dagger}(r_{0}\tau_{0}-\delta
T\,\tau_{z})\psi+\frac{u_{+}}{8}(\psi^{\dagger}\tau_{0}\psi)^{2}+\frac{u_{-}}{8}(\psi^{\dagger}\tau_{z}\psi)^{2}$
$\displaystyle\quad-\frac{v}{8}(\psi^{\dagger}\tau_{x}\psi)^{2}+\frac{w}{4}(\psi^{\dagger}\tau_{0}\psi)(\psi^{\dagger}\tau_{z}\psi)+f_{\mathrm{grad}}.$
(2)

Here, $r_{0}=T-T_{0}$ is the effective temperature, with $T_{0}$ setting a
transition temperature scale. The parameters $u_{\pm}$, $v$, and $w$ are
associated with the different symmetry-allowed quartic terms. The conditions
$0<w^{2}<u_{+}u_{-}$ ensure the stability of the functional. The Pauli
matrices $\tau_{i}$ act in the space of the two-component order parameter
$\psi$. Within a mean-field approach, the system described by Eq. (2) develops
a uniform $d$-wave state below $T_{c}\equiv T_{0}+\delta T$, where it picks a
global $U(1)$ phase $\Phi$. Owing to the invariance of the free energy under
the transformation $\psi_{d}\to\psi_{d}$ and $\psi_{g}\to\psi_{g}^{*}$, the
relative phase $\varphi$ of the gap function

$\displaystyle\Psi=e^{i\Phi}(|\psi_{d}|+e^{i\varphi}|\psi_{g}|),$ (3)

becomes a $\mathbb{Z}_{2}$ Ising degree of freedom that can undergo its own
transition. In fact, a finite expectation value of the pseudospin

$\displaystyle\sigma=\mathrm{sign}(\sin\varphi),$ (4)

marks the onset of the time-reversal symmetry broken state. The condensation
of the corresponding degree of freedom plays an important role in our
subsequent analysis, albeit not for a homogeneous order parameter.

A uniform $g$-wave component $\psi_{g}$ would emerge below $T_{c}^{<}\equiv
T_{0}-\delta T\frac{u_{+}+w}{u_{-}+w}$, with, for $v>0$, a phase
$\varphi=\pm\pi/2$ relative to $\psi_{d}$. In the fine-tuned system where
$\delta T=0$, analyzed in Ref. Kivelson2020 , both order parameters develop at
the same temperature. In this work we consider $\delta T\sim T_{c}$
sufficiently large such that the homogeneous, perfect crystal only displays
one phase transition with order parameter $\psi_{d}$. Still, the expansion Eq.
(2) implicitly assumes that the sub-leading channel $\psi_{g}$ is much closer
to condensation than any other pairing channel. We include the gradient terms

$\displaystyle
f_{\mathrm{grad}}=\frac{\kappa_{\parallel}}{2}\big{(}|\partial_{x}\psi|^{2}+|\partial_{y}\psi|^{2}\big{)}+\frac{\kappa_{\perp}}{2}|\partial_{z}\psi|^{2},$
(5)

where $\kappa_{\parallel}>\kappa_{\perp}$ reflects the electronic anisotropy
of the material. In principle, there are additional symmetry-allowed gradient
terms that mix the two components but that will not qualitatively alter the
findings of our analysis.

We now analyze the effect of a single edge dislocation on the pairing state
described by Eq. (2). Edge dislocations occur even in otherwise exceptionally
clean materials and lead to strong local strain fields, yielding associated
stress values of several GPa Hull1984 . These strain fields change the local
energetics of the superconductor and locally break the tetragonal lattice
symmetry, mixing different pairing channels. The strain field in the vicinity
of an edge dislocation is given in standard textbooks Landau1986 . It is
characterized by the Burgers vector $\boldsymbol{b}=b\,\boldsymbol{\hat{b}}$
and the unit vector $\boldsymbol{\hat{t}}$ tangential to the dislocation line.
The strain tensor
$\epsilon_{\alpha\beta}\\!=\\!(\partial_{\alpha}u_{b}+\partial_{\beta}u_{\alpha})/2$
is evaluated from the displacement field $\boldsymbol{u}$. For an edge
dislocation, the latter is given by

$\displaystyle\\!\boldsymbol{u}$
$\displaystyle=\frac{\boldsymbol{b}}{2\pi}\Big{[}\theta\\!+\\!\frac{A}{2}\sin(2\theta)\Big{]}+\frac{\boldsymbol{\hat{t}}\times\boldsymbol{b}}{2\pi}[A\cos^{2}(\theta)\\!+\\!B\log(\rho)]$
(6)

where $\rho$ and $\theta$ are the polar coordinates in the plane perpendicular
to the dislocation and the parameters $A$ and $B$ can be expressed in terms of
elastic constants Landau1986 . While the electronic system is very
anisotropic, the elastic deformations induced by a dislocation are expected to
be similar to those in an isotropic elastic system. The key implications of
Eq. (6) for the strain tensor $\epsilon_{\alpha\beta}$ are that its magnitude
decays like $1/\rho$, while the sign of the tensor elements depends on the
orientation of $\boldsymbol{b}$ and $\boldsymbol{\hat{t}}$, similarly to the
case of an electric field generated by a dipole. Quite generally then, one
finds tension ($\epsilon_{\alpha\beta}>0$) on one side of the dislocation and
compression ($\epsilon_{\alpha\beta}<0$) on the other side, following a
typical dipolar angular dependence, see Ref. Landau1986 . Based on the results
of Ref. Ying2013 , we consider a Burgers vector
$\boldsymbol{b}=b\boldsymbol{\hat{z}}$, see Fig. 1. Dislocations form loops or
end at the sample surfaces. In what follows we consider a segment that can be
considered locally as a straight dislocation line. In our specific analysis,
we opt for a tangent
$\boldsymbol{\hat{t}}=(\boldsymbol{\hat{x}}+\boldsymbol{\hat{y}})/\sqrt{2}$
along the crystallographic $[110]$ direction. Except for
$\boldsymbol{\hat{t}}=\boldsymbol{\hat{x}}$ (or $\boldsymbol{\hat{y}}$)—in
which case $\epsilon_{xy}$ vanishes—any generic direction of
$\boldsymbol{\hat{t}}$ yields a similar behavior

Figure 3:  Amplitude (circle size) and phase (color) of the strain-induced
$g$-wave component in proximity to a dislocation. Here we consider
$\lambda_{x}$ to be the dominant strain-superconductivity coupling, as shown
in Fig. 2(b). Inhomogeneous TRSB occurs when the phase winding (right, middle
and bottom) is energetically favored over the vanishing of the $g$-wave
component. Away from the dislocation the magnitude of the $g$-component is
small and a direct sign-change is always preferred (right, top, cut at
$z=25b$). Both degenerate solutions—distinguished by the pseudospin $\sigma$,
see Eq. (4), and reflecting the Ising symmetry of the TRSB order—are shown.

The coupling between the dislocation and the superconducting order parameter
can be cast in the form

$\displaystyle
f_{\mathrm{disl}}=\frac{1}{2}\sum_{\alpha=0,x,z}\lambda_{\alpha}\varepsilon^{\alpha}(\boldsymbol{r})\psi^{\dagger}(\boldsymbol{r})\tau_{\alpha}\psi(\boldsymbol{r}).$
(7)

Hereby $\varepsilon^{\alpha}(\boldsymbol{r})$ denotes the projection of the
spatial variation of the strain field to distinct symmetry sectors
($\alpha=0,x,z$) and $\lambda_{\alpha}$ are the associated coupling constants.
The $A_{1g}$ strain components $\varepsilon^{0}$ and $\varepsilon^{z}$ are
linear combinations of $\epsilon_{xx}+\epsilon_{yy}$ and $\epsilon_{zz}$. As
such, they locally change the superconducting transition temperature and the
degree of degeneracy between the $d$\- and $g$-wave components. The
experiments of Ref. Ying2013  demonstrated that a large coupling of this kind
is present in SrRu2O4. In our case, the component $\varepsilon^{x}$
corresponds to a $B_{2g}$ strain, $\varepsilon^{x}\equiv\epsilon_{xy}$, which
breaks the tetragonal lattice symmetry near the dislocation to a degree that
both order parameters belong to the same irreducible representation and,
hence, emerge simultaneously. The correspondence
$\varepsilon^{x}\leftrightarrow\epsilon_{xy}$ is dictated by the product
representation $B_{2g}\\!=\\!B_{1g}\otimes A_{2g}$ of the order parameters,
see Eq. (7). Given the low symmetry of the dislocation-induced strain for a
generic orientation of $\boldsymbol{\hat{t}}$ one expects that all strain
components are finite. The pronounced strain dependence of the bulk
superconducting transition Hicks2014 ; Watson2018 ; Grinenko2020  and the near
doubling of the local onset of superconductivity near dislocations Ying2013
are evidence for strong coupling constants $\lambda_{\alpha}$. It is
instructive to investigate the mechanism of local TRSB for pure strain
couplings $\propto\lambda_{\alpha}$. For either $\lambda_{0}$ or $\lambda_{z}$
the breaking of time-reversal symmetry is dictated by the quartic term
$-v(\psi^{\dagger}\tau_{x}\psi)^{2}/8$ in Eq. (2). This implies that the
strain-induced $g$-wave component develops with a phase $\pm\pi/2$ relative to
the leading order. In distinction, the coupling $\propto\lambda_{x}$ mixes the
two order parameters in a way that on opposite sides of the dislocation we
find $d+g$ and $d-g$ pairing states, since the strain changes from tensile to
compressive as one traverses the dislocation. To connect the $d+g$ and $d-g$
regions, the $g$-wave component can either vanish or establish a local time-
reversal symmetry broken state $d+e^{i\varphi(\boldsymbol{r})}g$ where the
phase $\varphi$ gradually winds from $0$ to $\pi$. The system opts for the
second possibility when the strain term $\pm\lambda_{x}\cos\varphi$ dominates
over the gradient terms in Eq. (5). The mechanism is reminiscent, albeit
qualitatively different, from other cases discussed in the literature where
time-reversal symmetry breaking superconductivity takes place at twin
boundaries of the crystal Callin1997 ; Sigrist1998 .

Independent of the strain coupling, the onset of a non-trivial phase winding
takes place at a well-defined temperature
$T_{{\scriptscriptstyle\mathrm{TRSB}}}^{\star}$. Indeed, the problem of
determining the onset of phase-winding near a dislocation can be recast as a
problem of finding the condition under which a bound state emerges in a
Schrödinger-type problem Massih2011 ; see appendix B. The onset temperature of
local TRSB is then larger than the temperature where a homogenous $\psi_{g}$
component would appear, $T_{{\scriptscriptstyle\mathrm{TRSB}}}^{\star}\geq
T_{c}^{<}$. Importantly, $f+f_{\mathrm{disl}}$ is still invariant under the
transformation $\psi_{d}\to\psi_{d}$ and $\psi_{g}\to\psi_{g}^{*}$ and hence,
the $\mathbb{Z}_{2}$ Ising degree of freedom remains intact. Thus, at
$T_{{\scriptscriptstyle\mathrm{TRSB}}}^{\star}$ the system locally picks one
of two degenerate solutions, solely distinguished by the Ising pseudospin
$\sigma=\pm 1$, see Eq. (4). A large system with many dislocations undergoes a
TRSB transition at $T_{{\scriptscriptstyle\mathrm{TRSB}}}$ once it locks all
its Ising pseudospins $\sigma_{i}$ in a ferromagnetically ordered state and
_globally_ breaks the $\mathbb{Z}_{2}$ symmetry. This point will be further
explored in the next section, where we will show that
$T_{{\scriptscriptstyle\mathrm{TRSB}}}\lesssim
T_{{\scriptscriptstyle\mathrm{TRSB}}}^{\star}$.

We have solved the coupled non-linear equations $\delta
f/\delta\psi_{d}=\delta f/\delta\psi_{g}=0$ for the total free energy Eq. (1)
via a stochastic annealing approach. All numerical results are obtained from a
Monte-Carlo simulation of a discretized system with $100\times 100$ lattice
sites (in units of $b$) with $(u_{+}\\!+\\!u_{-})/2\\!=\\!v\\!=\\!1$,
$(u_{-}\\!-\\!u_{+})/2\\!=\\!w\\!=\\!0.5$, $\kappa_{\parallel}\\!=\\!10$,
$\kappa_{\perp}\\!=\\!2$. A realistic description of the strain fields is
achieved with the simple expression
$\varepsilon^{\alpha}(\boldsymbol{r})=\varepsilon^{\alpha}(\rho,\theta)=b\cos(\theta)/\rho$.
The spatial distribution of the order parameters near the dislocation is shown
in Fig. 2 for the limiting cases where the strain-superconductivity coupling
is purely determined by $\lambda_{x}$ [Fig. 2(a,b)] and by $\lambda_{0}$ [Fig.
2(c)], respectively. Fig. 3 illustrates both options of connecting the $d\pm
g$ regions: near (far from) the dislocation the connection occurs via phase
winding (vanishing) of $\psi_{g}$. Furthermore, it also shows the two
degenerate solutions $\sigma=\pm 1$ for winding the phase between the $d+g$
and $d-g$ regions.

Figure 4: (a) Evolution of $T_{{\scriptscriptstyle\mathrm{TRSB}}}^{\star}$
upon varying the interaction parameters $\lambda_{0}$, $\lambda_{z}$, and
$\lambda_{x}$. Recall that $T_{c}^{<}$ is the temperature at which a
homogeneous $\psi_{d}$ component would onset, resulting in a homogeneous time-
reversal symmetry broken state. The analytic results (grey curve) indicate
that $T_{{\scriptscriptstyle\mathrm{TRSB}}}^{\star}(\lambda_{0,z})$ deviates
quadratically away from $T_{c}^{<}$. Meanwhile,
$T_{{\scriptscriptstyle\mathrm{TRSB}}}^{\star}(\lambda_{x})$ deviates from
$T_{c}^{<}$ only above a threshold value. (b) Phase diagram as function of
homogeneous $B_{1g}$ strain and temperature. It is evaluated for
$\lambda_{x}=8.75$ and for
$T_{c}(0)=T_{{\scriptscriptstyle\mathrm{TRSB}}}^{\star}(0)$. The analytic
relation $T_{c}(\epsilon)-T_{c}(0)=(\delta T^{2}+\epsilon^{2})^{1/2}$ is shown
in black.

The dependence of $T_{{\scriptscriptstyle\mathrm{TRSB}}}^{\star}$ on the
magnitude of the coupling constants $\lambda_{x}$ and $\lambda_{0}$ is shown
in Fig. 4(a). For sufficiently large strain couplings one can induce local
TRSB pairing above the bulk temperature, a consequence of the fact that
pairing can be boosted locally. In Fig. 4(b) we analyze the change of the bulk
transition temperature $T_{c}$ and of the local TRSB onset temperature
$T_{{\scriptscriptstyle\mathrm{TRSB}}}^{\star}$ as a function of the in-plane
strain $\epsilon_{B_{1g}}$. Fig. 4 was obtained for the specific coupling
strength $\lambda_{x}\\!=\\!8.75$ after fine-tuning $T_{c}\approx
T_{{\scriptscriptstyle\mathrm{TRSB}}}^{\star}$ at zero applied strain. The
superconducting transition temperature changes quadratically with strain, as
is required for a single-component order parameter.
$T_{{\scriptscriptstyle\mathrm{TRSB}}}^{\star}$ on the other hand is weakly
affected by strain values that are much smaller than the local, dislocation-
induced strain. Even the largest applied strain field in modern experiments
amounts to stress values that are an order of magnitude smaller than typical
edge-dislocation stresses Hull1984 . This may explain why the experiments in
Ref. Grinenko2020  did not observe a significant change for the TRSB onset
temperature with applied strain along the $[100]$ direction.

It is also interesting to discuss what happens if strain is applied along the
$[110]$ direction (i.e. $B_{2g}$ strain). For our scenario of inhomogeneous
local TRSB at $T_{{\scriptscriptstyle\mathrm{TRSB}}}^{\star}$, the phase
diagram of Fig. 4(b) would remain essentially unchanged. However, for the
scenario of homogeneous TRSB taking place at $T_{c}^{<}$, significant changes
are expected. In particular, because $B_{2g}$ strain couples the $\psi_{d}$
and $\psi_{g}$ order parameters bi-linearly, one would generally expect a
stronger, linear enhancement of $T_{c}$ and of the separation between $T_{c}$
and the TRSB transition temperature with increasing $B_{2g}$ strain.

To discuss what happens in the presence of an external magnetic field $H_{z}$
along the $z$-direction, we note that it transforms according to the $A_{2g}$
irreducible representation and that it is odd under time reversal.
Consequently, there is an additional term in $f_{\mathrm{disl}}$

$\displaystyle f_{\mathrm{disl}}\rightarrow
f_{\mathrm{disl}}-H_{z}\lambda_{y}\varepsilon^{y}(\boldsymbol{r})\psi^{\dagger}(\boldsymbol{r})\tau_{y}\psi(\boldsymbol{r})/2,$
(8)

where $\varepsilon^{y}=\epsilon_{x^{2}-y^{2}}$ is the projection of the strain
field onto the $B_{1g}$ channel. This coupling is crucial to explain the
finite Kerr signal observed in Ref. Xia2006 , as it allows one to train
different regions of broken time-reversal symmetry with a magnetic field. The
condition of a broken vertical mirror symmetry to achieve a finite Kerr signal
is naturally fulfilled near the dislocation.

Let us finally discuss what would happen for different combinations of single-
component order parameters. Consider the case $\psi=(\psi_{d},\psi_{s})$ with
a second component transforming as $A_{1g}$, i.e. $s$-wave. Then, the
symmetry-mixing strain in Eq. (7) is given by
$\varepsilon^{x}=\epsilon_{x^{2}-y^{2}}$ since $B_{1g}\\!=\\!B_{1g}\otimes
A_{1g}$. Other than that, our analysis would be largely unchanged. On the
other hand if one considered the case of $d_{x^{2}-y^{2}}$ and $d_{xy}$
states, no strain mixing term $\varepsilon^{x}$ is allowed, because
$A_{2g}\\!=\\!B_{1g}\otimes B_{2g}$ and there is no $A_{2g}$ strain in the
$D_{4h}$ point group. The same is true for a combination of $s$-wave with
$g$-wave pairing. Hence, the presented scenario applies to combinations of $d$
with either $g$\- or $s$-wave states. The time-reversal symmetry-broken state
of all combinations of single-component order parameters can be trained with a
magnetic field $H_{z}$, since the strain fields induced by the dislocation
always contain a component that couples $H_{z}$ with the two components of
$\psi$.

Thus far, we established the occurrence of phase winding near a single
dislocation and determined the onset temperature scale
$T_{{\scriptscriptstyle\mathrm{TRSB}}}^{\star}$ for this behavior. The local
breaking of time-reversal symmetry—marked by a non-vanishing
$|\psi_{g}|\sin(\varphi)$—takes place in a finite region around the core of
the dislocation. In the next section we allow for the Ising variable $\sigma$
to vary in space along the dislocation line and between different edge
dislocations.

###  II.3 Collective excitations and global phase transition for many
dislocations

Figure 5:  Mapping of the physical system to an effective coarse-grained
model. Panel (a) shows the Ising domain walls associated with the Ising TRSB
order parameter of Eq. 4, which proliferate along a dislocation line according
to the 1D Ising model. Panel (b) illustrates the coupling between multiple
dislocations, which can form open lines connecting the surfaces of the sample
of closed loops. As a result, as shown in panel (c), the critical behavior of
the model is the same as that of an anisotropic 2D Ising model with exchange
parameters $J$ and $J/\eta$, with $\eta\gg 1$.

In this section we analyze the case of many coupled dislocations. We find that
this leads to a sharp TRSB phase transition in the Ising universality class at
a temperature $T_{{\scriptscriptstyle\mathrm{TRSB}}}$. Given the underlying
Ising symmetry, it is however possible that the phase variable undergoes a
thermally excited kink $\varphi\rightarrow 2\pi n-\varphi$ along the
dislocation line. In terms of the Ising variable $\sigma$, see Eq. (4), the
kink corresponds to a domain wall, as illustrated in Fig. 5(a). In a coarse-
grained approach, the statistical mechanics of a single dislocation maps to a
one-dimensional Ising chain

$\displaystyle H_{1}$ $\displaystyle=-J\sum_{i}^{N}\sigma_{i}\sigma_{i+1},$
(9)

where the dislocation is divided along its length $L$ in $N=L/\ell$ segments
of typical length $\ell$. Each segment is associated with a local Ising spin
$\sigma_{i}$ and a binding energy per unit length $J$. The characteristic
values

$\displaystyle\ell\sim(\kappa_{\parallel}\kappa_{\perp})^{1/2}/\lambda_{\alpha}b,$
(10)

and $J\sim T_{{\scriptscriptstyle\mathrm{TRSB}}}^{*}$ follow from the
aforementioned solutions of the Schrödinger equations, see appendix B. For the
one-dimensional Ising model it holds that $\langle\sigma\rangle=0$ at any
finite temperature. The density of kinks per unit length,
$n_{\mathrm{kink}}\sim-\ell^{-1}\log[\tanh(2J/T)]$, is exponentially small at
low $T$. Shorter dislocation loops, with $N<e^{2J/T}$, have negligible
probability for kinks and are essentially ordered. However, averaging over
many uncoupled loops gives rise to a vanishing global order parameter
$\langle\sigma\rangle$.

To account for the coupling between distinct dislocations, we note that the
magnitude of local TRSB decays exponentially as $e^{-\rho/\ell}$ in the
direction transverse to the dislocation. Then, two dislocation segments at a
distance $\rho$ will interact via the exponentially small coupling
$J^{\prime}(\rho)=Je^{-\rho/\ell}$. From an analysis of the gradient terms of
the Ginzburg-Landau expansion follows that the sign of this interaction is
ferromagnetic. The many-dislocation problem is therefore governed by the
coarse-grained Hamiltonian

$\displaystyle
H=\sum_{m}H_{1}^{m}+\sum_{m,m^{\prime}}\sum_{i,j}J^{\prime}(\rho_{i,j})\sigma_{i}^{m}\sigma_{j}^{m^{\prime}}.$
(11)

The superscripts $m$, $m^{\prime}$ label the dislocation, and $\rho_{i,j}$
denotes the distance between the segments $i$ (of dislocation $m$) and $j$ (of
dislocation $m^{\prime}$).

Owing to the exponential dependence of the coupling $J^{\prime}(\rho)$, one
can further map the problem to an anisotropic Ising model with coupling
constant $J$ in one direction and a single coupling constant $J/\eta$ in the
orthogonal directions. Here $\eta=e^{\bar{\rho}/\ell}$ with $\bar{\rho}$ the
typical value for the shortest distance of two neighboring dislocations. This
is shown in panels (b) and (c) of Fig. 5. The anisotropic Ising model develops
a finite order parameter $\langle\sigma\rangle$ below the transition
temperature

$\displaystyle T_{{\scriptscriptstyle\mathrm{TRSB}}}\sim
J/\ln(\eta)\sim(\ell/\bar{\rho})T_{{\scriptscriptstyle\mathrm{TRSB}}}^{\star}.$
(12)

This result follows regardless of whether the coupling via $J^{\prime}$ leads
to a two- or three-dimensional network of dislocations. It is a consequence of
the one-dimensional Ising model being at its lower critical dimension. The
above expression implicitly assumes a weak coupling between nearby
dislocations, i.e. $\ell<\bar{\rho}$. In that case the transition temperature
$T_{{\scriptscriptstyle\mathrm{TRSB}}}$ is smaller than, but potentially
comparable to, the local ordering temperature
$T_{{\scriptscriptstyle\mathrm{TRSB}}}^{\star}$. While the ordering of single
dislocations is hindered by kinks along the loop, the weak dislocation-
dislocation coupling locks the global Ising variable and leads to a global
symmetry breaking.

Figure 6:  Entropy change $C(T)/T$ per Ising pseudospin (in units of
$k_{\scriptscriptstyle B}/J$) of the anisotropic Ising model with
$J^{\prime}/J=0$ (red) and $0.01$ (blue). This illustrates our approach to
estimate the total weight of the heat-capacity anomaly at the transition for
finite $J^{\prime}$ from the transferred calorimetric weight in
$C_{\mathrm{Ising},d=1}/T$.

The heat-capacity anomaly associated with the lifting of the $\mathbb{Z}_{2}$
symmetry is much smaller in magnitude than the one due to the opening of the
single particle gap at the bulk superconducting transition temperature
$T_{c}$. To show this, we estimate the change of the electronic specific heat
at the superconducting transition as $\Delta C(T_{c})\sim
S_{\mathrm{el}}(T_{c})\sim T_{c}/\epsilon_{\scriptscriptstyle F}$, with
$S_{\mathrm{el}}$ the electronic contribution to the entropy and
$\epsilon_{\scriptscriptstyle F}$ the Fermi energy [the Boltzmann constant
$k_{\scriptscriptstyle B}$ is set to unity]. In contrast, the heat capacity
per Ising spin of an anisotropic two-dimensional Ising model follows from
Onsager’s solution and is shown in Fig. 6. The extent of the calorimetric
anomaly at $T_{\rm TRSB}$ [blue shaded area in Fig. 6] corresponds to the
heat-capacity transfer [dark red area in Fig. 6]. To estimate the magnitude of
this effect, we use the approximate form

$\displaystyle\delta
S_{{\scriptscriptstyle\mathrm{TRSB}}}\approx\int_{0}^{T_{{\scriptscriptstyle\mathrm{TRSB}}}}\frac{C_{\mathrm{Ising},d=1}(T^{\prime})}{T^{\prime}}dT^{\prime},$
(13)

with the heat capacity of the one-dimensional Ising model

$\displaystyle
C_{\mathrm{Ising},d=1}=\frac{a^{2}c}{\bar{\rho}^{2}\ell}\frac{J^{2}}{T^{2}[\cosh(J/T)]^{2}}.$
(14)

The expression in Eq. (13) accounts for the total red area in Fig. 6. Here $a$
and $c$ are the in-plane and out-of-plane lattice constants. The small
prefactor $a^{2}c/(\bar{\rho}^{2}\ell)$ accounts for the number of Ising
pseudospins per unit cell and is needed to compare with the electronic
contribution. Using Eq. (10) and the dimensionless coupling
$\bar{\lambda}_{\alpha}=\lambda_{\alpha}/T_{c}$ we find
$\ell=(a/\bar{\lambda}_{\alpha})(\epsilon_{\scriptscriptstyle F}/T_{c})^{2}$
and obtain the ratio

$\displaystyle\frac{\delta
S_{{\scriptscriptstyle\mathrm{TRSB}}}}{S_{\mathrm{el}}(T_{c})}\sim\frac{1}{S_{\mathrm{el}}(T_{c})}\frac{a^{2}c}{\bar{\rho}^{2}\ell}\frac{\log\eta}{\eta^{2}}\sim\bar{\lambda}_{\alpha}^{2}\frac{a}{\bar{\rho}}\Big{(}\frac{T_{c}}{\epsilon_{\scriptscriptstyle
F}}\Big{)}^{3}e^{-2\bar{\rho}/\ell}.$ (15)

As anticipated, the entropy associated with the collective phase locking, and
thus to the TRSB transition, is—in addition to the two small prefactors
$(T_{c}/\epsilon_{\scriptscriptstyle F})^{3}$ and
$(a/\bar{\rho})$—exponentially suppressed.

Our conclusion of a global Ising transition at
$T_{{\scriptscriptstyle\mathrm{TRSB}}}$ can also be reached by considering
disorder that is caused by dislocations at random positions and with random
Burgers-vector orientations SAKivelson . From Eq. (7) follows that dislocation
strain acts like a uniaxial random field that couples to
$\cos[\varphi(\boldsymbol{r})]$, with the relative phase
$\varphi(\boldsymbol{r})$, see Eq. (3). The corresponding $XY$-model with
random uniaxial field is a well-studied problem in the statistical mechanics
of disordered systems, with several rigorous results Aharony1978 ; Minchau1985
; Feldman1998 ; Feldman1999 ; Crawford2013 ; Crawford2014 ; Bera2014 . It
undergoes an Ising transition with order parameter
$\langle\cos[\varphi(\boldsymbol{r}])\rangle\neq 0$, fully consistent with our
analysis. This demonstrates that our conclusions concerning the nature of the
transition as clean Ising transition are robust despite the random nature of
the dislocation problem.

##  III Discussion

In this paper we proposed an alternative origin for the observed breaking of
time-reversal symmetry in $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$ due to
inhomogeneous strain fields that locally couple a single-component primary
order parameter to a sub-leading pairing state. While the material without
strain inhomogeneities would only display the primary order, which is likely
of $d_{x^{2}-y^{2}}$ symmetry, strong strain fields mix these symmetry-
distinct pairing states. A promising candidate for such strain inhomogeneities
are those originated near edge dislocations as shown in Fig. 1. Local stress
values near such dislocations are easily in the regime of several GPa, leading
to a strong order-parameter mixing. Compressive and tensile strain on opposite
sides of the dislocation are then shown to lead to local TRSB via phase-
winding of the induced pairing wave-function. The main appeal and ultimate
motivation for this proposal is the absence of a mean-field behavior of the
heat capacity at $T_{{\scriptscriptstyle\mathrm{TRSB}}}$ under homogeneous
strain, while such behavior occurs at $T_{c}$. A scenario which attributes
TRSB to the opening of an additional pairing gap would directly affect the
entropy of electronic quasiparticles. The fact that no heat-capacity jump is
experimentally detected at $T_{{\scriptscriptstyle\mathrm{TRSB}}}$ is a strong
evidence against any such scenario.

There are certain similarities, but also important differences between our
theory and the description of Ref. Kivelson2020  advocating an accidental
degeneracy of homogeneous $d$\- and $g$-wave pairing. In both cases two
symmetry distinct states are relevant and time-reversal symmetry breaking is a
consequence of the relative phase between these two states. In both
descriptions one has to fine-tune
$T_{{\scriptscriptstyle\mathrm{TRSB}}}\approx T_{c}$ for unstrained samples.
There are, however, significant distinctions. First, in our approach it is in
principle possible to observe time-reversal symmetry broken states via local
probes even above $T_{c}$. This effect is a consequence of the local
enhancement of pairing near dislocations, as reported by experiments Ying2013
. Second, our findings suggest that TRSB may be engineered by controlling the
dislocation density during sample preparation, which in principle allows for a
clearer separation between $T_{c}$ and
$T_{{\scriptscriptstyle\mathrm{TRSB}}}$. Evidence for a sample-dependence of
$T_{{\scriptscriptstyle\mathrm{TRSB}}}$, and even of
$T_{{\scriptscriptstyle\mathrm{TRSB}}}>T_{c}$, was reported in Ref.
Grinenko2020 . Third, we do not expect a strong heat-capacity anomaly at
$T_{{\scriptscriptstyle\mathrm{TRSB}}}$. In our description, time-reversal
symmetry breaking is a phase locking process where comparatively few degrees
of freedom are involved and no gap-opening of single particle states occurs.
As a result, the transition leaves weak signatures in the heat-capacity signal
and hence, naturally explains the absent heat-capacity anomaly at
$T_{{\scriptscriptstyle\mathrm{TRSB}}}$ found in strained samples Li2020 . In
the case of a homogeneous $d\\!+\\!ig$ state Kivelson2020  one would expect a
jump in the heat capacity at the second bulk transition. Finally, a prediction
that may be used to falsify our theory in future experiments is the change of
the transition temperature $T_{c}$ with uniaxial strain $\epsilon_{B_{1g}}$
and $\epsilon_{B_{2g}}$. We expect that $T_{c}$ changes quadratically with
strain for both symmetries, while Ref. Kivelson2020  clearly predicts a
quadratic variation with $\epsilon_{B_{1g}}$ but a linear change with
$\epsilon_{B_{2g}}$. An investigation of $T_{c}(\epsilon_{B_{2g}})$ should be
able to discriminate between the two scenarios.

An important observation that concerns the pairing state of
$\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$ is the discontinuity of the elastic
constant $C_{66}$ at the superconducting transition temperature Ghosh2020 ;
Benhabib2020 , see also Ref. Okuda2002 . Elastic constants associated with
strain that transforms trivially, i.e. according to $A_{1g}$, are expected to
be discontinuous at any superconducting transition Testardi1975 . However,
elastic constants like $C_{66}$ or $C_{11}-C_{12}$, corresponding to $B_{1g}$
or $B_{2g}$ strain, respectively, are continuous for a single-component order
parameter. A jump of these elastic constants then clearly signals a multi-
component order parameter Fernandes2013 . Nevertheless, they should not be
taken as irrevocable evidence for a homogeneous two-component order parameter.
Given the slowly decaying dislocation-induced $g$-wave component, the system
is according to our theory essentially a two-component order parameter for
large parts of the sample. Our analysis of single dislocations shows clearly
that the regions of the sample with local $d\pm g$ pairing, that should affect
the elastic constant jump, are much larger than the regions near the core of
the dislocation, where time reversal symmetry is broken. This gives rise to a
discontinuity of $C_{66}$ while $C_{11}-C_{12}$ should be continuous or at
best have a much smaller jump consistent with Refs. Ghosh2020 ; Benhabib2020 ;
Okuda2002 . Quantitatively, the scattering of elastic waves off dislocations
is strong Li2017_a ; Li2017_b  and gives rise to corrections of the elastic
constants of the order of $10^{-2}$ Grimwall1999 . These effects are much
larger than the largest observed relative magnitudes of the discontinuities of
Ref Ghosh2020 , which are of the order of $10^{-4}$. Finally, the size of the
reported jump of $C_{66}$ in different measurements (compare Refs. Ghosh2020
and Benhabib2020 ) varies significantly, by about two orders of magnitude.
While this may partially be due to the different experimental techniques used,
sample-to-sample variations of the effect follow very naturally within our
theory.

Our proposed scenario for the origin of TRSB in
$\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$ could also be tested by experiments
in plastically deformed samples. In contrast to elastic strain, plastic strain
is known to create, move, and combine dislocations. It was recently observed
that the transition temperature of another superconducting perovskite,
$\mathrm{Sr}\mathrm{Ti}\mathrm{O}_{3}$, increased substantially upon
application of plastic strain, a behavior attributed to the formation of self-
organized dislocation structures Hameed2020 . In
$\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$, an enhancement of the dislocation
density by plastic strain would likely result in a larger TRSB signal. Even if
the dominant effect of plastic deformation is to combine and merge the
existing dislocations, one would still expect a significant change of the TRSB
signal. Manipulating this unconventional superconductor via plastic
deformation may therefore be utilized to strain engineer the degree of time-
reversal symmetry breaking in the sample.

###### Acknowledgements.

We are grateful to Erez Berg, Martin Greven, Clifford Hicks, Steven A.
Kivelson, Avraham Klein, You-Sheng Li, Andrew P. Mackenzie, Brad Ramshaw, and
Andrew Chang Yuan for fruitful discussions. R.W., M.H., and J.S. were
supported by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) - TRR 288-422213477 Elasto-Q-Mat (project B01). R.M.F was
supported by the U.S. Department of Energy through the University of Minnesota
Center for Quantum Materials, under Grant No. DE-SC-0016371.

## References

  * (1) D. A. Wollman, D. J. Van Harlingen, W. C. Lee, D. M. Ginsberg, and A. J. Leggett, _Experimental determination of the superconducting pairing state in YBCO from the phase coherence of YBCO-Pb dc SQUIDs,_ Phys. Rev. Lett. 71, 2134 (1993).
  * (2) C. C. Tsuei, J. R. Kirtley, C. C. Chi, Lock See Yu-Jahnes, A. Gupta, T. Shaw, J. Z. Sun, and M. B. Ketchen, _Pairing Symmetry and Flux Quantization in a Tricrystal Superconducting Ring of YBa 2Cu3O7-δ, _Phys. Rev. Lett. 73, 593 (1994).
  * (3) Y. Maeno, H. Hashimoto, and K. Yoshida, S. Nishizaki, T. Fujita, J. G. Bednorz, and F. Lichtenberg, _Superconductivity in a layered perovskite without copper_ , Nature (London) 372, 532 (1994).
  * (4) J. Mravlje, M. Aichhorn, T. Miyake, K. Haule, G. Kotliar, and A. Georges, _Coherence-Incoherence Crossover and the Mass-Renormalization Puzzles in Sr 2RuO4_, Phys. Rev. Lett. 106, 096401 (2011).
  * (5) T. M. Rice and M. Sigrist, _$\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$ : an electronic analogue of 3He?, _J. Phys. Condens. Matter 7, L643 (1995).
  * (6) G. Baskaran, _Why is Sr2RuO4 not a high Tc superconductor? Electron correlation, Hund’s coupling and p-wave instability_ , Physica B 223–224, 490 (1996).
  * (7) G. M. Luke, Y. Fudamoto, K. M. Kojima, M. I. Larkin, J. Merrin, B. Nachumi, Y. J. Uemura, Y. Maeno, Z. Q. Mao, Y. Mori, H. Nakamura, and M. Sigrist, _Time-reversal symmetry-breaking superconductivity in $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$, _Nature (London) 394, 558–561(1998).
  * (8) V. Grinenko, S. Ghosh, R. Sarkar, J.-C. Orain, A. Nikitin, M. Elender, D. Das, Z. Guguchia, F. Brückner, M. E. Barber, J. Park, N. Kikugawa, D. A. Sokolov, J. S. Bobowski, T. Miyoshi, Y. Maeno, A. P. Mackenzie, H. Luetkens, C. W. Hicks, H.-H. Klauss, _Split superconducting and time-reversal symmetry-breaking transitions, and magnetic order in $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$ under uniaxial stress, _arXiv:2001.08152 __
  * (9) J. Xia, Y. Maeno, P. T. Beyersdorf, M. M. Fejer, and A. Kapitulnik, _High Resolution Polar Kerr Effect Measurements of $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$: Evidence for Broken Time-Reversal Symmetry in the Superconducting State, _Phys. Rev. Lett. 97, 167002 (2006).
  * (10) A. Pustogow, Y. Luo, A. Chronister, Y.-S. Su, D. A. Sokolov, F. Jerzembeck, A. P. Mackenzie, C. W. Hicks, N. Kikugawa, S. Raghu, E. D. Bauer, and S. E. Brown, _Constraints on the superconducting order parameter in $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$ from oxygen-17 nuclear magnetic resonance, _Nature 574, 72–75 (2019).
  * (11) K. Ishida, M. Manago, K. Kinjo, and Y. Maeno, _Reduction of the 17O Knight Shift in the Superconducting State and the Heat-up Effect by NMR Pulses on $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$, _J. Phys. Soc. Jpn. 89, 034712 (2020).
  * (12) A. Chronister, A. Pustogow, N. Kikugawa, D. A. Sokolov, F. Jerzembeck, C. W. Hicks, A. P. Mackenzie, E. D. Bauer, S. E. Brown, _Evidence for even parity unconventional superconductivity in $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$_, arXiv:2007.13730
  * (13) S. Ghosh, A. Shekhter, F. Jerzembeck, N. Kikugawa, D. A. Sokolov, M. Brando, A. P. Mackenzie, C. W. Hicks, B. J. Ramshaw, _Thermodynamic Evidence for a Two-Component Superconducting Order Parameter in $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$,_ arXiv:2002.06130.
  * (14) S. Benhabib, C. Lupien, I. Paul, L. Berges, M. Dion, M. Nardone, A. Zitouni, Z. Q. Mao, Y. Maeno, A. Georges, L. Taillefer, C. Proust, _Jump in the $c_{66}$ shear modulus at the superconducting transition of $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$: Evidence for a two-component order parameter_, arXiv:2002.0591.
  * (15) H. G. Suh, H. Menke, P. M. R. Brydon, C. Timm, A. Ramires, and D. F. Agterberg, _Stabilizing even-parity chiral superconductivity in Sr 2RuO_4, Phys. Rev. Res. 2, 032023(R) (2020).
  * (16) S. A. Kivelson, A. C. Yuan, B. J. Ramshaw, R. Thomale, _A proposal for reconciling diverse experiments on the superconducting state in $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$,_ npj Quantum Mater. 5, 43 (2020).
  * (17) A. T. Rømer, D. D. Scherer, I. M. Eremin, P. J. Hirschfeld, and B. M. Andersen, _Knight Shift and Leading Superconducting Instability from Spin Fluctuations in Sr 2RuO_4, Phys. Rev. Lett. 123, 247001 (2019).
  * (18) R. Willa, _Symmetry-mixed bound-state order: extended degeneracy of $(d\\!+\\!ig)$-superconductivity in $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$_, arXiv:2005.04124.
  * (19) T. Scaffidi, _Degeneracy between even- and odd-parity superconductivity in the quasi-1D Hubbard model and implications for $Sr_{2}RuO_{4}$,_ arXiv:2007.13769.
  * (20) I. Zutic and I. Mazin, _Phase-Sensitive Tests of the Pairing State Symmetry in Sr 2RuO4_, Phys. Rev. Lett. 95, 217004 (2005).
  * (21) Y.-S. Li, N. Kikugawa, D.A. Sokolov, F. Jerzembeck, A.S. Gibbs, Y. Maeno, C.W. Hicks, M. Nicklas, A.P. Mackenzie, _High sensitivity heat capacity measurements on Sr2RuO4 under uniaxial pressure_ , arXiv:1906.07597
  * (22) R. Sharma, S. D. Edkins, Z. Wang, A. Kostin, C. Sow, Y. Maeno, A. P. Mackenzie, J. C. S. Davis, and V. Madhavan, _Momentum-resolved superconducting energy gaps of $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$ from quasiparticle interference imaging, _Proc. Natl. Acad. Sci. 117 5222 (2020).
  * (23) L.-D. Zhang, W. Huang, F. Yang, and H. Yao, _Superconducting pairing in $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$ from weak to intermediate coupling_, Phys. Rev. B 97, 060510(R) (2018).
  * (24) A. Bardasis and J. R. Schrieffer, _Excitons and Plasmons in Superconductors,_ Phys. Rev. 121, 1050 (1961).
  * (25) M. Sigrist and K. Ueda, _Phenomenological theory of unconventional_ superconductivity _,_ Rev. Mod. Phys. 63, 239 (1991).
  * (26) T. Nomura and K. Yamada _, Perturbation Theory of Spin-Triplet Superconductivity for $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$_, J. Phys. Soc. Jpn. 69, 3678 (2000).
  * (27) Q. H. Wang, C. Platt, Y. Yang, C. Honerkamp, F. C. Zhang, W. Hanke, T. M. Rice, and R. Thomale _, Theory of superconductivity in a three-orbital model of $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$, _Europhys. Lett. 104, 17013 (2013).
  * (28) S. Raghu, A. Kapitulnik, and S. A. Kivelson, _Hidden Quasi-One-Dimensional Superconductivity in $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$,_ Phys. Rev. Lett. 105, 136401 (2010).
  * (29) T. Scaffidi, J. C. Romers, and S. H. Simon, _Pairing symmetry and dominant band in $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$,_ Phys. Rev. B 89, 220510(R) (2014).
  * (30) W. Huang, Y. Zhou, and H. Yao, _Exotic Cooper pairing in multiorbital models of $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$_, Phys. Rev. B 100, 134506 (2019).
  * (31) C. W. Hicks, D. O. Brodsky, E. A. Yelland, A. S. Gibbs, J. A. N. Bruin, M. E. Barber, S. D. Edkins, K. Nishimura, S. Yonezawa, Y. Maeno, and A. P. Mackenzie, _Strong Increase of T c of $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$ Under Both Tensile and Compressive Strain_, Science 344, 283 (2014).
  * (32) C. A. Watson, A. S. Gibbs, A. P. Mackenzie, C. W. Hicks, and K. A. Moler, _Micron-scale measurements of low anisotropic strain response of local T c in $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$,_ Phys. Rev. B 98, 094521 (2018).
  * (33) V. P. Mineev, _Broken time-reversal symmetry in the superconducting state of $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$, _Phys. Rev. B 76, 212501 (2007).
  * (34) C. Kallin, _Chiral p-wave order in $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$._ Rep. Prog. Phys. 75, 042501 (2012).
  * (35) A. P. Mackenzie, T. Scaffidi, C. W. Hicks, and Y. Maeno, _Even odder after twenty-three years: the superconducting order parameter puzzle of $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$_, npj Quantum Materials 2, 40 (2017).
  * (36) I. Schnell, I. I. Mazin, and A. Y. Liu, _Unconventional superconducting pairing symmetry induced by phonons_ , Phys. Rev. B 74, 184503 (2006).
  * (37) W. Cho, R. Thomale, S. Raghu, and S. A. Kivelson, _Band structure effects on the superconductivity in Hubbard models_ , Phys. Rev. B 88, 064505 (2013).
  * (38) N. Okuda, T. Suzuki, Z. Mao, Y. Maeno, and T. Fujita, _Unconventional Strain Dependence of Superconductivity in Spin-Triplet Superconductor $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$_, J. Phys. Soc. Jpn. 71, 1134 (2002).
  * (39) E. Hassinger, P. Bourgeois-Hope, H. Taniguchi, S. R. de Cotret, G.Grissonnanche, M. S. Anwar, Y. Maeno, N. Doiron-Leyraud, and L. Taillefer, _Vertical line nodes in the superconducting gap structure of Sr 2RuO2_, Physical Review X 7, 011032 (2017).
  * (40) A. P. Mackenzie and Y. Maeno, _The superconductivity of $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$ and the physics of spin-triplet pairing_, Rev. Mod. Phys. 75, 657 (2003).
  * (41) A. P. Mackenzie, R. K. W. Haselwimmer, A. W. Tyler, G. G. Lonzarich, Y. Mori, S. Nishizaki, and Y. Maeno, _Extremely Strong Dependence of Superconductivity on Disorder in Sr 2RuO2,_ Phys. Rev. Lett. 80, 161 (1998); Phys. Rev. Lett. 80, 3890 (1998).
  * (42) O. Gingras, R. Nourafkan, A.-M. S. Tremblay, and M. Côté, _Superconducting Symmetries of $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$ from First-Principles Electronic Structure,_ Phys. Rev. Lett. 123, 217005, (2019).
  * (43) H. S. Røising, T. Scaffidi, F. Flicker, G. F. Lange, and S. H. Simon, _Superconducting order of $Sr_{2}RuO_{4}$ from a three-dimensional microscopic model_, Phys. Rev. Research 1, 033108 (2019).
  * (44) A. A. Husain, M. Mitrano, M. S. Rak, S. I. Rubeck, H. Yang, C. Sow, Y. Maeno, P. E. Batson, and P. Abbamonte, _Coexisting Fermi Liquid and Strange Metal Phenomena in $Sr_{2}RuO_{4}$_, arXiv:2007.06670.
  * (45) Y. A. Ying, N. E. Staley, Y. Xin, K. Sun, X. Cai, D. Fobes, T. J. Liu, Z. Q. Mao, and Y. Liu, Enhanced spin-triplet superconductivity near dislocations in $\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$, Nature Comm. 4, 2596 (2013).
  * (46) D. Hull and D. J. Bacon, _Introduction to Dislocations_ 3rd edn. (Pergamon, Oxford, (1984).
  * (47) L. D. Landau and E. M. Lifshitz, _Theory of Elasticity,_ Pergamon, New York, (1986).
  * (48) D. L. Feder, A. Beardsall, A. J. Berlinsky, and C. Kallin, _Twin boundaries in d-wave superconductors_ , Phys. Rev. B 56, R5751(R) (1997).
  * (49) W. Belzig, C. Bruder, and M. Sigrist, _Quasiclassical Theory of Twin Boundaries in High- $T_{c}$ Superconductors_, Phys. Rev. Lett. 80, 4285 (1998).
  * (50) A. R. Massih, _Second-phase nucleation on an edge dislocation, Philosophical Magazine_ , 91:31, 3961 (2011), DOI: 10.1080/14786435.2011.598883.
  * (51) L. R. Testardi, _Structural instability and superconductivity in A-15 compounds_ , Rev. Mod. Phys. 47, 637 (1975).
  * (52) R. M. Fernandes and A. J. Millis, _Nematicity as a Probe of Superconducting Pairing in Iron-Based Superconductors_ , Phys. Rev. Lett. 111, 127001 (2013).
  * (53) M. Li, W. Cui, M. S. Dresselhaus, G. Chen, _Canonical quantization of crystal dislocation and electron-dislocation scattering in an isotropic medium_ , New J. Phys. 19, 013033 (2017).
  * (54) M. Li, Z. Ding, Q. Meng, J. Zhou, Y. Zhu, H. Liu, M. S. Dresselhaus, and G. Chen, _Nonperturbative Quantum Nature of the Dislocation-Phonon Interaction_ , Nano Lett. 17, 1587 (2017).
  * (55) G. Grimwall, _Thermophysical properties of materials_ , North-Holland (1999).
  * (56) S. Hameed, D. Pelc, Z. W. Anderson, A. Klein, R. J. Spieker, M. Lukas, Y. Liu, M. J. Krogstad, R. Osborn, C. Leighton, R. M. Fernandes, and M. Greven, _Enhanced superconductivity in plastically deformed strontium titanate_ , arXiv:2005.00514.
  * (57) S. A. Kivelson, private communication.
  * (58) A. Aharony, _Spin-flop multicritical points in systems with random fields and in spin glasses,_ Phys. Rev. B 18, 3328 (1978).
  * (59) B. J. Minchau and R. A. Pelcovits, _Two-dimensional XY model in a random uniaxial field_ , Phys. Rev. B 32, 3081 (1985).
  * (60) D. E. Feldman, _Exact zero-temperature critical behaviour of the ferromagnet in the uniaxial random field,_ J. Phys. A: Math. Gen. 31, L177 (1998).
  * (61) D. E. Feldman, _Critical behavior of a degenerate ferromagnet in a uniaxial random field: Exact results in a space of arbitrary dimension,_ Journ. of Expt. and Theor. Phys. 88, 1170–1178 (1999).
  * (62) N. Crawford, _Random field induced order in low dimension,_ Europhys. Lett. 102, 36003 (2013).
  * (63) N. Crawford, _Random Field Induced Order in Low Dimension I_ , Commun. Math. Phys. 328, 203–249 (2014).
  * (64) A. Bera, D. Rakshit, M. Lewenstein, A. Sen(De), U. Sen, and J. Wehr, _Classical spin models with broken symmetry: Random-field-induced order and persistence of spontaneous magnetization in the presence of a random field,_ Phys. Rev. B 90, 174408 (2014).

##  Appendix A Heat capacity of two degenerate superconductors

The primary motivation for our analysis was the absence of a signature in the
heat capacity at the time-reversal symmetry breaking transition of strained
samples Li2020 , where $\mu$SR experiments find that $T_{c}$ and
$T_{{\scriptscriptstyle\mathrm{TRSB}}}$ are distinct transitions Grinenko2020
. In this appendix we briefly summarize the heat-capacity anomalies for
homogeneous two-component order parameters. We use the Ginzburg-Landau
expansion of Eq. (2). For $\delta T=0$ one should distinguish between an order
parameter that belongs to a two-dimensional irreducible representation, such
as $E_{g}$ or $E_{u}$, and two accidentally degenerate oder parameters that
each belong to one-dimensional irreducible representations. From the
perspective of the heat capacity, the former is a special case of the latter
where the quartic coupling constant $w$ in Eq. (2) vanishes.

For degenerate order parameters holds $\delta T=0$, and $T_{c}=T_{0}$. The
discontinuity of the heat capacity is

$\displaystyle\frac{\Delta C(T_{0})}{T_{0}}=\frac{u_{-}}{u_{+}u_{-}-w^{2}}.$
(16)

Let us now analyze finite $\delta T$, i.e. the degeneracy is lifted. The upper
transition temperature is $T_{c}=T_{0}+\delta T$ with heat-capacity jump

$\displaystyle\frac{\Delta C_{1}(T_{c})}{T_{c}}=\frac{1}{u_{+}+u_{-}+2w}.$
(17)

The lower transition temperature is
$T_{c}^{<}=T_{0}-\frac{u_{+}+w}{u_{-}+w}\delta T$ and the corresponding heat
capacity jump is

$\displaystyle\frac{\Delta
C_{2}(T_{c}^{<})}{T_{c}^{<}}=\frac{(u_{-}+w)^{2}}{u_{-}u_{+}-w^{2}}\frac{\Delta
C_{1}(T_{c})}{T_{c}}.$ (18)

Let us first focus on the case when $\delta T$ lifts the degeneracy of a two-
dimensional irreducible representation. In this situation it holds $w=0$. It
then follows that the ratio of the heat capacities at the two transitions
equals the ratio of the slopes of the transition temperatures

$\displaystyle\frac{\Delta C_{2}(T_{c}^{<})}{\Delta
C_{1}(T_{c})}=\frac{u_{-}}{u_{+}}=\frac{\frac{dT_{c}}{d\delta
T}}{\left|\frac{dT_{c}^{<}}{d\delta T}\right|}.$ (19)

If both temperatures split evenly, the heat-capacity jumps at the two
transitions have to be the same. If the upper transition departs more rapidly
away from $T_{0}$—which seems to be the case for
$\mathrm{Sr}_{2}\mathrm{Ru}\mathrm{O}_{4}$—then the lower heat-capacity jump
should be the larger one. This is in sharp disagreement with experiments of
Ref. Li2020 . More freedom is left if two symmetry-distinct order parameters
are degenerate by accident and $w\neq 0$ is allowed. Now a negligible second
calorimetric jump is possible if $w=-u_{-}$. However, this condition implies
that the second component, i.e. the $g$-wave order parameter, develops
negligible amplitude, i.e. superconductivity prefers to order with $\psi_{d}$
finite but $\psi_{g}\approx 0$.

##  Appendix B Schrödinger bound-state problem

To illustrate the mapping of the local TRSB to the appearance of a bound-state
in a Schrödinger-type problem we focus on the case with pure $\lambda_{0}$
coupling. Once the system has developed a bulk $d$-wave component $\psi_{d}$
(we assume $\psi_{d}$ to be real) the Ginzburg-Landau equation for the
imaginary part $\psi_{g,i}\equiv\text{Im}[\psi_{g}]$ reads

$\displaystyle\kappa\nabla^{2}\psi_{g,i}=(r_{0}+\delta
T+\lambda_{0}\varepsilon^{0})\psi_{g,i}+[(u_{+}+u_{-})/2]\psi_{g,i}^{3}$ (20)

For simplicity we chose $\kappa\equiv\kappa_{\|}=\kappa_{\perp}$. The
parameter $\kappa$ relates to the superconducting coherence length
$\xi\sim(\epsilon_{\scriptscriptstyle F}/T_{c})a$ via
$\kappa\sim\xi^{2}T_{c}$. The cubic term in $\psi_{g,i}$ may be neglected when
identifying the transition temperature; this term is relevant to normalizing
the bound-state wave-function. The problem can then be cast in the form of a
Schrödinger equation

$\displaystyle\mathcal{H}\Psi=E\Psi$ (21)

with $\mathcal{H}=-\nabla^{2}+V(\boldsymbol{r})$,
$V(\boldsymbol{r})=\lambda_{0}\varepsilon^{0}/\kappa$, and $E=-(r_{0}+\delta
T)/\kappa$. As strain decays with a power-law $b/r$ away from the dislocation,
the potential is of Coulomb-type, i.e.
$V(\boldsymbol{r})=(\ell/r)\cos(\theta)$, with a characteristic length-scale
$\ell^{2}=\kappa/b\lambda_{0}$. For $\kappa_{\|}\neq\kappa_{\perp}$, $\kappa$
should be replaced by the geometric average
$(\kappa_{\|}\kappa_{\perp})^{1/2}$ as done in the expression for $\ell$ given
in Eq. (10). The above Schrödinger problem has a well-defined ground-state
energy $E_{0}$, where a bound-state is realized. As temperature is lowered,
the effective energy $E$ increases until reaching $E_{0}$. This defines
$T=T_{{\scriptscriptstyle\mathrm{TRSB}}}^{\star}$.