Strain-Induced Time Reversal Breaking and Half Quantum Vortices Near a Putative Superconducting Tetra-Critical Point in Sr $ 2 $ Ruo $

Strain-induced time reversal breaking and half quantum vortices near a putative superconducting tetra-critical point in Sr2RuO4

Andrew C. Yuan,1 Erez Berg,2 and Steven A. Kivelson1 1Department of Physics, Stanford University, Stanford, CA 93405, USA 2Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 7610001, Israel It has been shown [1] that many seemingly contradictory experimental findings concerning the superconducting state in Sr2RuO4 can be accounted for as resulting from the existence of an assumed tetra-critical point at near ambient pressure at which dx2−y2 and gxy(x2−y2) superconducting states are degenerate. We perform both a Landau-Ginzburg and a microscopic mean-field analysis of the effect of spatially varying strain on such a state. In the presence of finite xy shear strain, the superconducting state consists of two possible symmetry-related time-reversal symmetry (TRS) preserving states: d ± g. However, at domain walls between two such regions, TRS can be broken, resulting in a d + ig state. More generally, we find that various natural patterns of spatially varying strain induce a rich variety of superconducting textures, including half-quantum fluxoids. These results may resolve some of the apparent inconsistencies between the theoretical proposal and various experimental observations, including the suggestive evidence of half-quantum vortices [2].

I. INTRODUCTION

If it happens that superconducting (SC) orders with two distinct symmetries are comparably favorable for some microscopic reason, it is possible to have a two-parameter phase diagram (e.g. T and isotropic strain, 0) that exhibits a multicritical point (e.g. at = ? and T = T ?) at which the transition temperatures, Tc, of the two different or- (a) (b) ders coincide, as shown in Fig. (1a). For instance, a change from s± to d-wave pairing is thought to occur FIG. 1. (a). Schematic phase diagram as a function of as a function of doping in certain Fe-based supercon- two parameters - taken here to be isotropic strain (0) ductors [3], and it was recently conjectured that the and T , in the neighborhood of a tetra-critical point at which the transition temperatures to SC states with d- layered perovskite Sr2RuO4 (SRO) [4] under ambient conditions is “accidentally” close to such a multi- wave (i.e. B1g) and g-wave (i.e. A2g) coincide. (b). A similar schematic phase diagram – now where the x-axis critical point involving either dx2−y2 and g-wave [1] signifies symmetry breaking shear strain, shear – for a or dxy and s-wave [5] pairing. Even though both or- system which at zero strain favors a SC order parameter ders by themselves transform as one dimensional ir- the transforms according to a two dimensional irrep., px reducible representations (irreps) of the point-group and py (i.e. Eu symmetry). symmetries, near such a multi-critical point the system can exhibit a variety of features usually associated with multi-component SC order that trans- (A2g) pairing channels, where, as in Fig. (1a), for form according to a higher dimensional irrep (e.g. the uniform case the coexistence regime is a d+ig SC p a -wave). Conversely, for a SC order parameter with spontaneously broken time-reversal-symmetry that transforms according to a 2d irrep, the point (TRS). We study this problem in the mean-field of zero shear-strain, shear = 0, can be viewed as a approximation, both from an effective field theory T special case of such a multi-critical point in the - (Landau-Ginzburg/non-linear sigma model) and a

arXiv:2106.00935v1 [cond-mat.supr-con] 2 Jun 2021 shear plane, as shown in Fig. (1b). In both cases, microscopic perspective. Following a similar line of the response of the different components of the SC reasoning as in Ref. [6] (where TRS breaking near order parameter to specific components of the strain dislocations was investigated), we show that inhomo- tensor can produce a variety of novel effects. geneous strain can lead to a highly inhomogeneous Specifically, the proximity of a multicritical point SC state in which TRS breaking is strongly manifest implies that even small amplitude spatial variations only along a network of domain-walls separating re- of the strain field can locally stabilize different dis- gions in which the local strain favors one or another tinct forms of superconducting order in different do- TRS preserving combination of the d and g order mains. In this paper, we treat the case of a tetracriti- parameters. However, this only occurs when the un- cal point involving dx2−y2 (B1g) and gxy(x2−y2)-wave strained system is close enough to the tetra-critical 2 point - in a sense that we make precise. We also matter of principle, possible routes to reconcile both show that appropriate strain patterns can induce an these observations with the conjectured theoretical order parameter texture with a spontaneous frac- scenario. tional magnetic flux that equals a half superconducting flux quantum in under a range of circumstances. • Despite the fact that the crystals involved in The results we obtain are quite general as they fol- these experiments are paragons of crystalline low largely from symmetry considerations. As an ap- perfection, the only plausible interpretation of plication of these ideas, we explore their implications the specific heat data is that the TRS break- for the still vexed problem of settling the symmetry ing involves a small fraction of the electronic of the superconducting state in SRO [4]. Here, there degrees of freedom. This can be naturally are a number of proposals, each of which can plau- accounted for by the theoretically expected sibly account for a subset of the experimental ob- extreme sensitivity of the SC order to local servations. Triplet pairing of any sort has seemingly strain, and the fact that the TRS breaking been ruled out by recent NMR experiments [7–9]. d+ig order arises only in a network of domain- ? wall-like regions at which 0 and shear Moreover, recent ultra-sound measurements [10, 11], | − | | | taken at face value, require a two-component order are vanishingly small. Moreover, so long as the parameter arising from the system in the absence typical magnitude of the inhomogeneous strain of strain being accidentally tuned to a multi-critical is larger than an applied strain, no significant T T point involving dxy & s or dx2−y2 & g wave pairing. splitting between c and trsb is expected. However, there are at least two sets of phenomena The fact that a half-quantum vortex can be that seemingly challenge these theoretical proposals: • the ground-state in the presence of a suitable • Below a critical temperature, Ttrsb, the SC strain texture similarly opens the possibility phase appears to break time-reversal symme- that the experimental evidence of fractional try [12–14] where at zero shear strain Ttrsb vortices in SRO is likewise consistent with the ≈ Tc, while in the presence of substantial B1g propsed scenario. shear strain these two transitions are split, such that Ttrsb < Tc. However, specific heat measurements under the same circumstances II. GINZBURG-LANDAU THEORY show no signature of the TRS breaking transition [15, 16]. An additional constraint on A. Setup theory is the recent observation [17] that Tc can be depressed with the application of hy- We consider a Ginzburg-Landau free energy den- drostatic pressure (i.e. compressive uniform sity [1] with order parameter (OP) fields ∆T = strain < 0) without producing a detectable 0 (D,G) in the presence of an external magnetic field splitting between T and T - an observa- c trsb B~ = ~ A~, given by tion that was declared to rule out any the- ∇ × ory based on an accidental degeneracy be- B2 tween two symmetry-distinct superconducting F = V2 + V4 + K + (1) orders. 2 † † V2 = α0∆ ∆ + ∆ (α τ ) ∆ • A somewhat complicated experiment on · 1 † 2 β1 † 2 β3 † 2 mesoscale crystals adduced evidence of the ex- V4 = [∆ ∆] + [∆ τ1∆] + [∆ τ3∆] 2 2 2 istence of a topological excitation capable of 0 β3 † † admitting a half-quantum of magnetic flux [2]. + [∆ τ3∆][∆ ∆] This has been argued [18] to constitute direct 2 κ evidence that the SC state is a chiral p + ip K = ( i~ A~)∆ 2 state, despite the contrary evidence from the 2 | − ∇ − | 0 NMR studies [7–9]. Although additional ev- κ † + (i∂x Ax)∆ τ1 [( i∂y Ay)∆] + c.c. idence of half quantum vortices has been re- 2 − − − cently reported [19], it is still possible that Where τ is the vector of Pauli matrices. there is an alternative explanation for the ob- The complex scalar fields D,G are normalized so servation that does not involve fractional vor- that the stiffness constants are equal, i.e., κ = κ tices. However, taking the result at face value d g κ, and that quartic isotropic coupling constant β =≡ presents us with the need to identify a route 0 1. We have used units such that the Cooper pair to fractional vortices in a singlet SC. charge 2e = 1. The quantity α = (α1, 0, α3) repre- In this context, our present results provide, as a sents the effect of local strain as a two-component 3

vector, where α3 = α3(x, y) is proportional to the (VA): deviation of the isotropic (A1g) strain from its criti- ? F = Kκ + Kκ0 + Knˆ (3) cal value 0 and α1 = α1(x, y) is proportional to the ~ ~ ˜ ˜ shear (B2g, i.e., xy) strain. The free energy at A = 0 β1 2 β3 2 + V0( ∆ ) + α˜ nˆ + [n1] + [n3] + ... respects TRS, has a U(1) symmetry associated with | | · 2 2 the overall superconducting phase and, for α1 = 0, a κ˜ 2 Kκ = ~ χ + ~a A~ D4h point group symmetry that involves both space 2 ∇ − and the order parameters. For example, under a ro- 0 Y [nˆ ∂µnˆ]1 tation by π/2, x y and y x, D D and Kκ0 =κ ˜ n1 ∂µχ + aµ × Aµ − 2n − G G. The quartic→ terms determine→ − → the − favored µ=x,y 1 form→ of the ordered state: β > 0 favors coexistence 2 3 κ˜ ~ nˆ of non-zero d and g pairing and β > 0 favors the 1 Knˆ = ∇ TRS breaking combinations d ig. The microscopic 2 2 ± BdG calculations reported in Sec. (IIIA) yield β1 & 0 [nˆ ∂xnˆ]1 [nˆ ∂ynˆ]1 ∂xnˆ ∂ynˆ β3 > 0. +κ ˜ n1 × × , 2n1 2n1 − 2 · 2 st where [nˆ ∂µnˆ]1 = n2∂µn3 n3∂µn2 is the 1 com- × − 2 ponent of the vector nˆ ∂µnˆ, whileκ ˜ ∆ κ, 2 0 × 2 ˜ 4 ≡ | | α˜ ∆ α1, 0, α3 + β3 ∆ , βj ∆ βj, and ... signiﬁes≡ | | termsh that would| | comei from≡ | | higher order terms in the Ginzburg-Landau theory. Importantly, B. Non-linear sigma model ~a is the Berry connection associated with the motion ofn ˆ on the Bloch sphere: In the special case, αj = βj = 0 for all j > 0 and † 1 ~ 0 0 ~a Z Z (4) β3 = 0, κ = 0, the free energy has a global SU(2) ≡ i ∇ symmetry that relates the two components of the The corresponding Berry curvature is related to the order parameter. Not too close to T , and to the ex- c Pontryagin density tent that this symmetry is not too strongly broken, the variations of the magnitude of the order param- 1 ~ ~a = µν [nˆ (∂µnˆ ∂ν nˆ)] (5) eter are unimportant. This allows us to associate ∇ × 2 · × the relevant order parameter values with a point on a Bloch sphere nˆ S2 via the isomorphism between 1 3 ∈ C. Ground State CP and S /U(1), i.e.,

Let us first determine the ground state of a system in the presence of a uniform strain vector α using ∆ = ∆ eiχZ (2) the Ginzburg-Landau free energy (1). For calcula- | |  θ β0 cos e−iφ/2 tional convenience, we will consider the case 3 = 0  2  and β = β = β > 0 (in which case the free en- Z =   1 3 θ ergy is invariant under a U(1) symmetry associated sin e+iφ/2  2 with rotations of nˆ around n2). The general case is † analyzed in Appendix (VB). n = Z τ Z i i The potential term V of the Ginzburg-Landau free energy can be rewritten as a sum of two terms, V 2 α α m Here, defining χd and χg to be the phases of D and 2 = ∆ ( 0 + ) 1 −| | · G respectively, χ = 2 [χd + χg] and φ = [χg χd] 1 4 2 − V4 = ∆ 1 + βm (6) are the overall and relative SC phases and θ = 2| | 2 arctan[ G / D ]. Note that OPs ∆ which differ only | | | | where m is the projection of the normalized vector by their global phases χ are mapped onto the same p 2 2 nˆ onto the eˆ1-eˆ3 plane and m m = n + n , point on the Bloch sphere. ≡ | | 1 3 where nj nˆ eˆj. The potential is minimized when ≡ · More generally, to the extent that it is possible to m points in the same direction as α so that the ignore variations in the magnitude of ∆, the problem potential term is given by reduces to a non-linear sigma model with a weakly 2 1 4 2 broken U(1) SO(3) symmetry, derived in Appendix V = ∆ (α0 + αm) + ∆ 1 + βm , (7) × −| | 2| | 4

where α α . Since 0 m 1, the values of ∆ and m that≡ | minimize| V≤are≤ uniquely determined:| | for α0 α/β 0, ≥ ≥ 2 α ∆ = α0, m = , (8) | | βα0 which corresponds to two distinct values nˆ (related 2 by time-reversal symmetry) with n2 = √1 m . ± − For α/β > α0 α, we have ≥ − α + α ∆ 2 = 0 , m = 1, (9) | | 1 + β

so that n2 = 0. Finally, ∆ = 0 for α < α0. | | − Henceforth, we restrict our attention to the case FIG. 2. Domain Walls. The figure is the Bloch sphere α0 > 0 so ∆ > 0 and that the nature of the ground representation of possible transitions across the domain | | state is determined by the value of α/βα . wall where α1(y) → ±α1 as y → ±∞, and α3(y) ≡ 0 ¯ In the limit of small strain α βα0 so that m 0, α3 > 0 is constant. The blue arrow represents a TRS ∼ preserving¯ transition through a pure d state. The green nˆ points in the eˆ2 direction, which corresponds to ± arrow represents a general TRSB transition restricted to a TRSB d ig state. Conversely, if α βα0 so that m = 1,± the Bloch vector nˆ points in≥ the same a 2D plane intersecting the Bloch sphere, the angle of which relative to the eˆ -eˆ -plane is denoted by ω. In α 1 3 direction as as denoted by the solid black dot in particular, if ω = π/2, then only the relative phase φ Fig. (2). This corresponds to a TRS preserving changes from φ = π → 0. state determined by the the local strain. From now on, when we discuss a situation in which the strain is nonzero, we shall implicitly assume that α βα0 Below we discuss the derivation of δχ in several ≥ and thus the uniform ground state is as in (9). cases. This is done by minimizing the free energy in Eq. (3) in the presence of a given strain texture. To be concrete, we will consider the case in which D. Domain walls α (y) = α is a constant and α (y) changes sign 3 ¯ 3 1 across the domain wall such that α1(y) α1 as → ±¯ We now consider the behavior of the order param- y with α > βα0 [20]. The result can be → ±∞ ¯ eter ∆ in the presence of spatially varying strain. As expressed as a trajectory on the Bloch sphere, as it simplifies the analysis, we will do this in the con- shown in Fig. (2), where the arrow indicates the text of the non-linear sigma model (3). To begin direction of evolution as y varies from to + . −∞ ∞ with, we consider a domain wall separating a region To begin with, consider some general results that at y < 0 in which the shear strain favors d g pair- follow without explicit calculation: ing (α < 0) from a region at y > 0 that favors− d + g 1 • Away from the multicritical point: Con- (α1 > 0). We consider the system to be translation- sider the case in which α3 > βα0. In this case, ally invariant in the directions parallel to the domain D is everywhere larger¯ than G and even at wall. the| | “center” of the domain wall,| defined| as the Far from the domain wall, the relative phase φ(y) point y = 0 where α1(0) = 0, there is no local and amplitudes are determined by the strain. Thus, tendency to TRSB. The optimal order parame- at long distances, the only property of the order pa- ter texture lies in the eˆ1-eˆ3 plane - as indicated rameter texture that can depend on the nature of by the blue trajectory in Fig. (2). Here the G the domain wall is the change in the global phase, component of the order parameter vanishes at δχ . If we choose the global phase such that the or- y = 0 and is negative on one side and positive y der parameter is real at , across the domain on the other. Obviously, the analogous consid- → −∞ d g wall the order parameter must change from to erations apply for α < βα , with the role of eiδχ d g − 3 0 ( + ). D and G interchanged.¯ − In this case, TRS is If TRS is preserved everywhere (i.e. if the order preserved everywhere, and δχ = π. parameter can be chosen to be real), then the only possible values of δχ are 0 and π. Any other value • Broad domain wall near the multi- of δχ requires TRSB on the domain wall, which con- critical point: If α3 < βα0, then near the sequently means that there must be two symmetry center of the domain|¯ wall,| the local terms in related optimal values, δχ. the free energy favor a TRSB solution, d ig. ± ± 5

0.0 Fig. (2). When this plane is perpendicular to eˆ2, TRS is preserved (blue line) while all other trajec- tories break TRS. A more complete solution of the 0.1 problem does not result in qualitative changes in the − conclusions. With these simplifications, the domain α /βα 3 0 wall energies ∆F can be computed analytically (see δχ/π 0.2 0.20 Appendix (VC)). − 0.60 We can then find the plane ω with minimum 0.65 domain wall energy for each set of values α1, α3 ¯ ¯ 0.3 and compute the corresponding change in global − 1.00 phase δχ, as shown in Fig. (3). For α3 βα0 ¯ ≥ 0 4 (i.e., way from the multi-critical point), TRS is pre- α1/βα0 served everywhere, such that δχ = 0. Conversely, if α3 < βα0, (near the multi-critical point), TRS FIG. 3. Global phase. The change δχ in the global phase breaking¯ near the domain wall is possible, yielding across a narrow domain wall where the xy component 0 < δχ < π/2. of the strain, α1, is discontinuous. The domain wall is | | characterized by α (y) = α sign(y), while α (y) = α 1 ¯ 1 3 ¯ 3 is constant. δχ is shown as a function of α1/βα0. The different curves correspond to different fixed¯ values of E. Topological Point Defects the uniform component of the strain, α /βα . ¯ 3 0 The domain walls we have discussed are natural strain patterns that are plausibly generic in real ma- Moreover, if the strain fields vary slowly on the terials. In addition, because there are two compo- scale of the superconducting coherence length, nents of the strain-dependent vector, α, one can also then the order parameter will be well approx- conceive of vortex-like defects with point-like cores imated by the uniform state corresponding to in 2d or line-like cores in 3d. Here, we consider a the local value of the strain. Thus, the order pattern of strain such that along any path encircling parameter texture is not confined to the eˆ1-eˆ3 the origin, α(~r ) rotates by 2π. Given such a strain plane, as shown by the green trajectory. This pattern, we can use the non-linear sigma model to implies that both components of the order pa- explore the properties of the resulting SC order parameter remain non-zero everywhere, and thus rameter texture that results. that δφ = δχg δχd = π. However, how much of this phase− change is± accommodated by changing the phase of D or G depends on ener- 1. Vorticity and associated flux getics; if the D wave order is everywhere dominant, then δχd 0 and hence δχ π/2, while ≈ ≈ Far from the defect core, the form of the SC order if the D and G are of nearly equal strength, parameter (up to its overall phase) is essentially de- then δχg δχg and hence δχ 0. Clearly, termined by the local pattern of strain. Regardless for intermediate| | ≈ | cases,| 0 < δχ <≈ π/2. | | the SC order parameter texture, the current density, J~ = δF/δA~, can be computed exactly from the • Narrow domain wall near the multicrit- − ical point: Here, the nature of the solution nonlinear sigma model (3) since variations in mag- depends on a host of microscopic details. Since nitude ∆ do not couple to the vector potential A~. | | “narrow” and “broad” refer to the width of the From this it follows that domain wall relative to the superconducting Jx =κ ˜[∂xχ + ax Ax] (10) coherence length, and given that the supercon- − ducting coherence length diverges as T Tc, [nˆ ∂ nˆ] → +κ ˜0n ∂ χ + a y 1 A at least near Tc this is likely the most physi- 1 y y × y − 2n1 − cally relevant situation. We will thus treat this case more explicitly below. and similarly for Jy. Far from the core under most ~ ~ As an explicit model of a narrow domain wall, let circumstances, J 0; indeed, beyond a London penetration depth→ it vanishes exponentially. Thus, α1(~r ) = α1 sign(y) where α1 > 0 and α3(~r ) α3 0 be a constant.¯ We consider¯ paths in which≡ ¯ as ≥y we can invert Eq. (10) to obtain an expression for ~ goes from to , nˆ(y) follows a trajectory that A in terms of the SC order parameter texture valid lies in a plane−∞ ω intersecting∞ the Bloch sphere - of wherever J~ is negligible. Then, by integrating the the sorts illustrated by the different colored paths in vector potential A~ along a contour C that encloses 6

the origin at a distance, we obtain an expression (modulo an additive integer) for the enclosed ﬂux Φ in units of the superconducting ﬂux quantum Φ0 = h/2e:

Φ Ω κ0 I = [nˆ ∂µnˆ]1Tµν drν (11) Φ0 4π − 4π C × 0 −1 κ n1 κ T = 0 κ κ n1 (a) where Ω is the solid angle enclosed by the contour of nˆ on the Bloch sphere. Note that when [nˆ ∂µnˆ]1 = 0 along C, the second term vanishes and thus× the flux quantum captured is expressed entirely in terms of the Berry phase Ω/4π. In particular, since strain stabilizes a TRS preserving state, we will typically be interested in situations in which nˆ eˆ2 = 0 far from the defect core, insuring that this condition· is satisfied. In this case, when- ever nˆ follows a trajectory that encircles the origin (as in Fig. (4b)), there must be an associated half- flux quantum of flux bound to the defect.

2. Example of a strain-induced half-quantum fluxoid (b) We now consider an explicit version of such a strain texture (Fig. (4a)), consisting of four do- FIG. 4. A strain induced half quantum fluxoid. (a) Real space contour around a topological point defect, along mains separated by domain walls that intersect at which the strain vector α winds by 2π. The dashed the origin. Thus, we take α (x, y) = α F (x) and 3 ¯ 3 3 cross shows the location of a π mismatch in the phase α1(x, y) = α1F1(y), where Fj(r) = Fj( r), and ¯ − − of the order parameter. D + g in the x > 0, y > 0 Fj(r) 1 as r . We further assume that quadrant represents order parameter with a dominant → → ∞ αj > βα0 – i.e. far away from the origin we are D component and a smaller G component, and similarly in¯ the large strain regime where the preferred order in the other quadrants. (b) Corresponding trajectory parameter is set by the strain and TRS is preserved. of the order superconducting order parameter along the As indicated by the labels, the D component is en- Bloch sphere. TRS is preserved along the path. Each hanced compared the G component for x > 0. For segment of the contour is color coded so that the same x < 0, the G component is favored. The combina- colors in the bottom and top figure correspond to each other. tion d + g is favored for y > 0, while d g is favored for y < 0. − Let us now consider a closed path C encircling the origin (Fig. (4a)), where the preferred SC state (up to the global phase) is denoted in each quadrant, e.g., D + g denotes a TRS preserving SC state At an intuitive level, the same conclusion can be with dominant D component. Since the contour C is reached by considering the nature of the order pa- far away from the origin, the domain walls between rameter texture along the various line segments in quadrants are narrow, and the strain is always suffi- Fig. (4a). Since it is energetically favorable to keep ciently large such that TRS breaking is never favored the dominant piece of the SC order parameter uni- locally. Fig. (4b) shows the trajectory of the order form, the overall phase (δχ) is constant along any parameter on the Bloch sphere, in which each seg- of the segments other than the red one, along which ment of the contour is color coded to correspond to the dominant portion of the order parameter changes that in the top diagram. We then see that the OP sign, favoring δχ = π. Of course, this change in ∆ wraps around by 2π while being confined to the phase will in actuality be spread out along the en- eˆ1-eˆ3 plane and thus [nˆ ∂µnˆ]1 = 0 along the con- tire path, but this argument captures the π phase tour. Eq. (11) thus implies× that this strain texture mismatch along the close path that results in a half captures a half quantum of flux. quantum vortex. 7

III. MICROSCOPIC ANALYSIS Let the full Hamiltonian Hfull = H0 + H1 be de- ﬁned on a 2D square lattice. The free Hamilto- nian H0 is characterized by the (single-band) TRS- Zero Strain 1.50 preserving hopping matrix t so that

X 0 † H0 = [t(~r , ~r ) + µ] c 0 c (12) − ~r s ~rs ~r 0,~r,s

d + ig and the attractive (pairing) interaction term H1, as-

g sumed to be spatially uniform, is III d

X X † H1 = λτ Pτ (~r )Pτ (~r ) (13) g − I II τ=d,g ~r † X 0 † † P (~r ) = fτ (~r ~r )c 0 0 [iσ2]s0sc τ − ~r s ~rs ~r 0,s,s0 0.90 1.20 1.80 d where λτ > 0 with τ = d or g encodes the strength of (a) the interaction in the designated symmetry channels. Here fτ are TRS preserving form factors that trans- I I 1 0.08 form according to the requisite distinct irreducible |D| representations of the point-group symmetry: |G| / 1 2 2 amplitudes fd(~r ) = δ(r = 1) x y (14) 0 0.00 4 − gd0 gs II III 3√3 2 2 1 0.08 fg(~r ) = δ(r = √5)[xy(x y )] 32 − where the factor of the Kronicker-δ in each expression is 1 when ~r connects, respectively, ﬁrst and 0 0.00 0.15 0.00 0.15 0.15 0.00 0.15 fourth nearest-neighbor sites. The hopping matrix

gd0 gd0 elements can likewise be expressed in terms of these (b) and the local strain as

FIG. 5. (a) Phase diagram of the microscopic Hamil- t(~r + ~r 0, ~r ) = δ(r0 = 1) + tδ(r0 = √2) (15) tonian, Eq. (12,13), as a function of the interaction 0 0 + gs(~r ) fs(~r ) + gd0 (~r ) fd0 (~r ) strengths λd, λg, calculated within self-consistent mean- field theory, in the absence of strain. Near degeneracy where g (~r ) and g 0 (~r ) parameterize, respectively, λ ∼ λ , the yellow portion denotes the TRSB d + ig s d d g the spatial profile of the isotropic and shear strain, state, while the cyan and purple regions represents a and is the symmetrization of the product term, pure d or pure g state, respectively. 3 representative 0 1 0 0 e.g., gs(~r ) fs(~r ) [gs(~r ) + gs(~r + ~r )]fs(~r ). point (I, II, III), one in each phase, are used in further ≡ 2 calculations. (b) The top two panels depict the evolu- Here, fs(~r ) and fd0 (~r ) are form factors with isotropic tion of the uniform state at point (I) with shear gd0 and and shear [xy] symmetry of the underlying lattice, isotropic gs strain. The dashed line represents the rela- which we take to be tive phase φ given by the left y-axis, while the solid (red, green) lines represent the amplitudes |D|, |G|, given by fs(~r ) = δ(r = 2), fd0 (~r ) = δ(r = √2) [xy] (16) the right y-axis. Similarly, the bottom two panels represent the evolution of the uniform state for sample points We construct a mean-field BCS trial Hamiltonian (II) and (III) under shear gd0 strain. H, given by X H = (~r, ~r )c† c (17) We now address these same issues from a more ~rs ~rs 0 T microscopic perspective in the context of Bardeen- ~r ,~r,s h i Cooper-Schrieffer (BCS) mean-field theory. Specifi- X 0 † † + ∆(~r , ~r )c~r 0↑c~r ↓ + h.c. cally, we solve the self-consistency equations for a ~r 0,~r generic 2D Hamiltonian with attractive d and g- wave interactions, and with spatially varying band- The full self-consistency field equations (SCFs) can structure parameters that encode the same patterns then be derived by extremizing the resulting varia- of spatially varying strain discussed above. tional free energy in the standard fasion – details are 8

given in Appendix (VD). It should be noted that in the same way as in the top panel. As required order to guarantee that the results satisfy the equa- by symmetry, the component that vanished in tion of continuity, i.e., ~ J~ = 0, it is generally insuf- the absence of strain exhibits an initial linear ficient to only solve the∇· SCFs for the gap function - increase in magnitude with increasing strain. both and ∆ must be determined self-consistently However, in this case, the relative phase is a (see AppendixT (VE) for a proof). discontinuous function of strain; the ground- state always preserves TRS and jumps from being d + g to d g as the sign of the strain A. Uniform states changes. − We relate these results to the Ginzburg-Landau To begin with, we study the uniform case tuned theory as follows: In the absence of strain, the close to the multi-critical point. In Fig. (5a) we system is tuned to the tri-critical condition α1 show the mean-field ground-state phase diagram of ≈ α3 0 when λd λg. Moreover, since the sys- the microscopic model defined above as a function of tem≈ exhibits d + ig≈ order in this case, the requi- the pairing interactions, λd and λd, in the absence site inequalities β1 > 0, β3 > 0 are automatically of “strain” (i.e. for gs = gd0 = 0), for t = 0.4 and satisfied. The shear strain can be identified with for the chemical µ chosen so that the mean electron α1 gd0 . Conversely, even relatively small values of density per site is n 0.3. There are three dis- ∝ ≈ λd λg / λd +λg > 0.1 are enough to produce a suf- tinct phases in this case: a pure d wave phase for λd | − | | | ficiently large value of α3 such that even at T = 0, sufficiently larger than λg, a pure g wave phase for α > β/ α0 . Also notice that we have taken rel- sufficiently large λd, and a relatively narrow coexis- atively| | strong| | interactions; this condition becomes tence phase centered at the the line λd = λg. The exponentially more restrictive the weaker the overall latter phase has a relative phase φ = π/2, i.e., it coupling, λd + λg . The extent to which the system is a d ig phase, for all parameters studied± here. | | ± needs to be “fine-tuned” near to the tetra-critical To illustrate the effect of shear strain, we chose point is quantified as the narrowness of this coexis- representative points in the phase diagram indicated tence phase in the zero-temperature phase diagram. by the three points in Fig. (5a), and explore the evolution of the ground-state order upon application of uniform strain, i.e. non-zero gd0 or gs. Shown in Fig. B. Domain Walls (5b) are the magnitude of the d and g components D G of the order parameter, and , as well as the We next address the domain wall behavior for φ | | | | relative phase, , for these three cases: a system near and away from the multi-critical point. To circumvent the difficulty of large coher- • The top two panels of Fig. (5b) show the evolution with strain of the case in which we ence lengths at small interaction strength, we use λ . λ . are most interested - the strain-free ground large interaction strengths d = 3 2 and g = 4 9, state has d ig pairing. As the shear strain, chosen such that in the absence of strain, the uni- ± form system is near the multi-critical point, i.e.., in g 0 , is varied, D and G remain compara- d d ig x ble, although both| | increase| | slightly, roughly a + state. We then introduce an -dependent 2 shear strain with a domain wall along the y-axis, in proportion to gd0 - which is a density of states effect. More| dramatically,| the relative along which the system is translationally invariant. g 0 . x/l phase evolves smoothly, up to a critical value As a specific example, we take d = 0 4 tanh( ) where l = 5 and x L = 50. We take gs(x) to be at which TRS is restored, i.e. where φ reaches | | ≤ either 0 or π, which marks the point of a tran- constant. sition to d + g or d g pairing respectively. In Fig. (6) (top) shows the profiles of the order pa- φ x contrast, as a function− of the isotropic strain rameters and relative phase as a function of in g g , the evolution from the d+ig state to a pure the case s = 0, i.e., near the multi-critical point. s φ π/ g or pure d state involves a change of the rela- twists through 2 passing through the domain tive amplitudes D and G , while the relative wall, indicating the TRS is broken there. Fig. (6) phase φ = π/2| remains| | constant.| (bottom) shows a TRS preserving domain wall for ± gs = 0.3, i.e., away from the multi-critical point. • The lower two of Fig. (5b) represent the shear strain evolution under conditions in which at zero strain either the d or g component is ab- C. Half quantum vortex sent. In both cases, the component that is dominant at zero strain remains dominant; in- Finally, to construct a tractable situation in which deed, its overall magnitude increases in much the previously discussed phenomenological analysis 9

d + g g d g d d + g gs = 0 1 1.5 2 2 |D| |G| / gs |D| / 1 1 gdd amplitudes

|G| 0.4

0 0.0 amplitudes gs = 0.3 0.0 0 1.5 0 0 gd0 0.4 0.4 200 100 0 100 200 -200 0 200 (a) 0.4 50 0 50

n3 1 0.0 50 0 50 x position

FIG. 6. Domain Walls. The top panel shows the behavior of the SC order parameter across a domain wall at which the shear strain gd0 (x) changes sign when the system is detuned from the multi-critical point such that gs(x) = 0.3.The bottom panel shows the same sort of n2 n1 domain wall for a system near the multi-critical point so that gs(x) = 0. The spatial variation of gd0 , shown in the mini-plot, is of the form gd0 (x) = 0.4 tanh(x/l) where l = 5 and |x| ≤ L = 50. The dashed lines represent the relative phase φ with values given by the left y-axis, while the solid lines are the amplitudes of the D and G components with values given by the right y-axis.

(b)

FIG. 7. Half quantum vortex pinned by strain texture. (a) The behavior of the OP ∆ = (D,G) as a function leads to a half-quantum vortex, we consider the pre- of x. The two components of the strain, gs(x) ∝ α3 and g 0 (x) ∝ α , are shown on the right. The domain vious system on an infinite cylinder with periodic d 1 walls are of the form ± tanh(x/l) with l = 5, and the boundary conditions in the x-direction and transla- cylinder’s circumference is 400 sites. The dashed line tional invariance along the y-direction. The circum- represents the relative phase φ given by the left vertical ference of the cylinder (in the x-direction) is 400 axis, while the solid lines represent that amplitudes |D| sites. We then vary the microscopic terms [gs(x) and |G| (right vertical axis). The bottom horitzontal axis and gd0 (x)] corresponding the evolution of α1 and represents the x position, while the top horizontal axis α3 shown in Fig. (4). Solving the full SCF equa- is labeled with the uniform state corresponding to the tions, we obtain D(x) and G(x) as shown in Fig. given strain texture at that position. (b). Trajectory (7). of the OP ∆ = (D,G) projected onto the unit Bloch sphere (blue line). The arrow represent the direction of the contour. The gray circle line emphasizes the eˆ1-eˆ3 The results corroborate the expected behavior plane. from the Ginzburg-Landau theory. Notice that the x-labels on top of Fig. (7a) indicate the expected state at each position along the contour. Most im- IV. CONCLUSIONS portantly, the relative phase φ winds by 2π around the cylinder, from which it follows that δχ = π. This result is translated into a trajectory on the Bloch In this paper we have primarily explored the effect sphere in Fig. (7b). Note that the trajectory of the of inhomogeneous strain on the superconducting OP order parameter is close to the eˆ1-eˆ3 plane, but devi- textures of a system tuned close to a tetra-critical ates from it somewhat in parts of the trajectory, in- point at which two different superconducting com- dicating that TRS is broken in certain regions along ponents have equal Tc’s. Such a tetracritical point the path. Correspondingly, in this calculation, we can arise due to symmetry - when in the absence of expect the flux induced by the strain pattern to de- strain the two components form a two dimensional viate slightly from Φ0/2 [see Eq. (11) and the dis- irreducible representation of the point group sym- cussion that follows]. metries. However, we have primarily focused on the 10 case in which the tetracritical point arises from tun- breaking transition and the possibility of half quan- ing a symmetry preserving parameter near to a crit- tum vortices, even though some aspects of the actual ical value - a situation that might arise accidentally experiment [2] - for instance the dependence on an in a small subset of superconducting materials. in-plane field - still need be addressed. In this situation, even relatively small local strain As mentioned in the introduction, a key issue con- can readily detune the system from the tetracritical cerns the strain-induced splitting between the SC and the TRS breaking transitions. It has been found condition sufficiently that a single component OP 2 2 that x y (B1g) shear strain can produce a signif- is locally favored. However, if on average the sys- − tem is near enough to this critical value, it is also icant increase in Tc, with a small decrease in Ttrsb natural to find domain walls between regions of dif- [15] - i.e. a split transition - while hydrostatic pres- ferent dominant strain in which the system is ap- sure (which produces A1g strain) can produce a pro- proximately tetra-critical. This, in turn, can lead to nounced depression of Tc but no detectable split- TRS breaking on such domain walls, and thus to a ting of the transition [17]. These observations are state which globally breaks TRS but in which the trivially accounted for if one assumes that α3 has a symmetry breaking is locally significant only on a strong (albeit quadratic) dependence on shear strain network of such domain walls. We have also shown but only weakly dependent on isotropic strain while that appropriate patterns of inhomogeneous strain α0 depends on both components of the strain. can bind a half-quantum vortex. How stringent a condition this places on the isotropic strain dependence of α depends on the These results may have interesting implications 3 magnitude and character of the strain inhomo- for a number of materials that show evidence for ei- geneities - i.e. the width of the SC transition. To ther an exact or a near-degeneracy of two SC orders, the extent that we can ignore the effect of xy (B ) including UPt [21–23], URu Si [24, 25], UTe [26], 2g 3 2 2 2 shear strain (i.e. for α 0), it follows that so long doped Bi Se [27, 28], certain Fe based superconduc- 1 2 3 as there are regions of d≈and regions of g wave SC, tors [29–31], and of course SRO. Most importantly, there must be domain walls between them at which such a near degeneracy necessarily implies an en- TRS breaking can arise. Thus, a spilt transition hanced sensitivity to variations in local strain, which will be apparent only when the mean value of α is can lead to a variety of otherwise unexpected behav- 3 greater than its variance. iors. Note, however, that half quantum vortices and The present considerations are encouraging in the other topological defects can also arise as dynamical sense that they illustrate a plausible explanation of a excitations in multi-component superconductors in set of previously puzzling experiments in SRO. How- the absence of any strain effects [32–36]. ever, it is important to reiterate that this analysis It is worth noting that while our results are quite sheds no insight of what is probably the most vex- general, they are obtained within mean-field theory, ing aspect of the proposed scenario: why are these and neglect thermal fluctuations of the supercon- two symmetry distinct forms of SC order nearly de- ducting order parameter. These can lead to inter- generate with one another without need of any fine esting effects close to the Tc in multi-component su- tuning? perconductors [37–40]. This study was undertaken with the SC state of SRO in mind. It is well established that the SC state ACKNOWLEDGMENTS is highly strain-sensitive. There are also a variety of experimental observations - ultra-sound anomalies We acknowledge extremely helpful discussions key among them - that are most naturally consis- with Clifford Hicks, Catherine Kallin, Tony Leggett, tent with an assumed near degeneracy between a Andy MacKenzie, and Brad Ramshaw. SAK and d and g wave SC component. However, a variety AY were supported, in part, by NSF grant No. of other experimental results appear, at first, dif- DMR-2000987 at Stanford. EB was supported by ficult to reconcile with this scenario [2, 14, 17, 41]. the European Research Council (ERC) under grant The present results suggest a route to understanding HQMAT (Grant Agreement No. 817799), the Israel- some of these additional observations. This includes US Binational Science Foundation (BSF), and a Re- a suppressed thermodynamic signature of the TRS search grant from Irving and Cherna Moskowitz.

[1] Steven A Kivelson, Andrew C Yuan, BJ Ramshaw, Sr2RuO4,” npj Quantum Mater 5 (2020). and Ronny Thomale, “A proposal for reconciling di- [2] J Jang, DG Ferguson, V Vakaryuk, Raﬃ Budakian, verse experiments on the superconducting state in SB Chung, PM Goldbart, and Y Maeno, “Observa- 11

tion of half-height magnetization steps in Sr2RuO4,” 167002 (2006). Science 331, 186–188 (2011). [14] Francoise Kidwingira, JD Strand, DJ Van Harlin- [3] Christian Platt, Ronny Thomale, Carsten Hon- gen, and Yoshiteru Maeno, “Dynamical supercon- erkamp, Shou-Cheng Zhang, and Werner Hanke, ducting order parameter domains in Sr2RuO4,” Sci- “Mechanism for a pairing state with time-reversal ence 314, 1267–1271 (2006). symmetry breaking in iron-based superconductors,” [15] Vadim Grinenko, Shreenanda Ghosh, Rajib Sarkar, Phys. Rev. B 85, 180502(R) (2012). Jean-Christophe Orain, Artem Nikitin, Matthias [4] Andrew P Mackenzie, Thomas Scaffidi, Clifford W Elender, Debarchan Das, Zurab Guguchia, Felix Hicks, and Yoshiteru Maeno, “Even odder after Brückner, Mark E Barber, et al., “Split supercon- twenty-three years: The superconducting order pa- ducting and time-reversal symmetry-breaking tran- rameter puzzle of Sr2RuO4,” npj Quantum Materi- sitions in Sr2RuO4 under stress,” Nature Physics , als 2, 1–9 (2017). 1–7 (2021). [5] Astrid T Rømer, Andreas Kreisel, Marvin A Müller, [16] Y-S Li, R Borth, CW Hicks, AP Mackenzie, and PJ Hirschfeld, Ilya M Eremin, and Brian M Ander- M Nicklas, “Heat-capacity measurements under uni- sen, “Theory of strain-induced magnetic order and axial pressure using a piezo-driven device,” Review splitting of Tc and Ttrsb in Sr2RuO4,” Physical Re- of Scientific Instruments 91, 103903 (2020). view B 102, 054506 (2020). [17] Vadim Grinenko, Debarchan Das, Ritu Gupta, [6] Roland Willa, Matthias Hecker, Rafael M. Fernan- Bastian Zinkl, Naoki Kikugawa, Yoshiteru Maeno, des, and JörgSchmalian, “Inhomogeneous time- Clifford W. Hicks, Hans-Henning Klauss, Manfred reversal symmetry breaking in Sr2RuO4,” arXiv e- Sigrist, and Rustem Khasanov, “Unsplit super- prints , arXiv:2011.01941 (2020), arXiv:2011.01941 conducting and time reversal symmetry breaking [cond-mat.supr-con]. transitions in Sr2RuO4 under hydrostatic pressure [7] Andrej Pustogow, Yongkang Luo, Aaron Chro- and disorder,” (2021), arXiv:2103.03600 [cond- nister, Y-S Su, DA Sokolov, Fabian Jerzembeck, mat.supr-con]. Andrew Peter Mackenzie, Clifford William Hicks, [18] Anthony J Leggett and Ying Liu, “Symmetry Naoki Kikugawa, Srinivas Raghu, et al., “Con- Properties of Superconducting Order Parameter in straints on the superconducting order parameter Sr2RuO4,” Journal of Superconductivity and Novel in Sr2RuO4 from oxygen-17 nuclear magnetic res- Magnetism , 1–27 (2020). onance,” Nature 574, 72–75 (2019). [19] Xinxin Cai, Brian M. Zakrzewski, Yiqun A. Ying, [8] Kenji Ishida, Masahiro Manago, Katsuki Kinjo, and Hae-Young Kee, Manfred Sigrist, J. Elliott Ort- Yoshiteru Maeno, “Reduction of the 17O Knight mann, Weifeng Sun, Zhiqiang Mao, and Ying Liu, shift in the superconducting state and the heat-up “Magnetoresistance oscillation study of the half- effect by NMR pulses on Sr2RuO4,” Journal of the quantum vortex in doubly connected mesoscopic Physical Society of Japan 89, 034712 (2020). superconducting cylinders of Sr2RuO4,” arXiv e- [9] Aaron Chronister, Andrej Pustogow, Naoki Kiku- prints , arXiv:2010.15800 (2020), arXiv:2010.15800 gawa, Dmitry A Sokolov, Fabian Jerzembeck, Clif- [cond-mat.supr-con]. ford W Hicks, Andrew P Mackenzie, Eric D Bauer, [20] Note that the same sort of analysis can be applied and Stuart E Brown, “Evidence for even parity un- to a domain wall across which α3(y) changes sign conventional superconductivity in Sr2RuO4,” arXiv with α1 = constant, e.g. between a region of D + g preprint arXiv:2007.13730 (2020). pairing (indicating dominant D and subdominant g) [10] Sayak Ghosh, Arkady Shekhter, F Jerzembeck, to d + G pairing (dominant G). N Kikugawa, Dmitry A Sokolov, Manuel Brando, [21] D.S. Jin, S.A. Carter, B. Ellman, T.F. Rosenbaum, AP Mackenzie, Clifford W Hicks, and BJ Ramshaw, and D.G. Hinks, “Uniaxial-stress anisotropy of the “Thermodynamic evidence for a two-component su- double superconducting transition in UPt3,” Physi- perconducting order parameter in Sr2RuO4,” Na- cal review letters 68, 1597 (1992). ture Physics 17, 199–204 (2021). [22] G.M. Luke, A. Keren, L.P. Le, W.D. Wu, Y.J. Ue- [11] Siham Benhabib, C Lupien, I Paul, L Berges, mura, D.A. Bonn, Louis Taillefer, and J.D. Garrett, M Dion, M Nardone, A Zitouni, ZQ Mao, Y Maeno, “Muon spin relaxation in UPt3,” Physical review A Georges, et al., “Ultrasound evidence for a letters 71, 1466 (1993). two-component superconducting order parameter in [23] ER Schemm, WJ Gannon, CM Wishne, Sr2RuO4,” Nature physics 17, 194–198 (2021). WP Halperin, and Aharon Kapitulnik, “Ob- [12] G Ml Luke, Y Fudamoto, KM Kojima, MI Larkin, servation of broken time-reversal symmetry in the J Merrin, B Nachumi, YJ Uemura, Y Maeno, heavy-fermion superconductor UPt3,” Science 345, ZQ Mao, Y Mori, et al., “Time-reversal symmetry- 190–193 (2014). breaking superconductivity in Sr2RuO4,” Nature [24] T Yamashita, Y Shimoyama, Y Haga, TD Matsuda, 394, 558–561 (1998). E Yamamoto, Y Onuki, H Sumiyoshi, S Fujimoto, [13] Jing Xia, Yoshiteru Maeno, Peter T Beyersdorf, A Levchenko, T Shibauchi, et al., “Colossal ther- MM Fejer, and Aharon Kapitulnik, “High resolu- momagnetic response in the exotic superconductor tion polar Kerr effect measurements of Sr2RuO4: URu2Si2,” Nature Physics 11, 17–20 (2015). Evidence for broken time-reversal symmetry in the [25] E.R. Schemm, R.E. Baumbach, P.H. Tobash, superconducting state,” Physical review letters 97, F. Ronning, E.D. Bauer, and A. Kapitulnik, “Evi- 12

dence for broken time-reversal symmetry in the su- Kim, “Stability of Half-Quantum Vortices in px + perconducting phase of URu2Si2,” Physical Review ipy Superconductors,” Phys. Rev. Lett. 99, 197002 B 91, 140506(R) (2015). (2007). [26] Ian M. Hayes, Di S. Wei, Tristin Metz, Jian [35] Fengcheng Wu and Ivar Martin, “Majorana Zhang, Yun Suk Eo, Sheng Ran, Shanta R. Saha, Kramers pair in a nematic vortex,” Phys. Rev. B John Collini, Nicholas P. Butch, Daniel F. Agter- 95, 224503 (2017). berg, Aharon Kapitulnik, and Johnpierre Paglione, [36] Sarah B Etter, Wen Huang, and Manfred Sigrist, “Weyl Superconductivity in UTe2,” (2020), “Half-quantum vortices on c-axis domain walls in arXiv:2002.02539 [cond-mat.str-el]. chiral p-wave superconductors,” New Journal of [27] K Matano, M Kriener, K Segawa, Y Ando, and Physics 22, 093038 (2020). Guo-qing Zheng, “Spin-rotation symmetry breaking [37] Mark H. Fischer and Erez Berg, “Fluctuation and in the superconducting state of CuxBi2Se3,” Nature strain effects in a chiral p-wave superconductor,” Physics 12, 852–854 (2016). Phys. Rev. B 93, 054501 (2016). [28] Liang Fu, “Odd-parity topological superconductor [38] Yue Yu and S. Raghu, “Effect of strain inhomogene- with nematic order: Application to CuxBi2Se3,” ity on a chiral p-wave superconductor,” Physical Re- Phys. Rev. B 90, 100509(R) (2014). view B 100, 094517 (2019). [29] RM Fernandes, AV Chubukov, and J Schmalian, [39] Matthias Hecker and JörgSchmalian, “Vestigial ne- “What drives nematic order in iron-based supercon- matic order and superconductivity in the doped ductors?” Nature physics 10, 97–104 (2014). topological insulator CuxBi2Se3,” npj Quantum [30] Vadim Grinenko, Rajib Sarkar, K Kihou, CH Lee, Materials 3, 1–6 (2018). I Morozov, S Aswartham, B Büchner, P Chekhonin, [40] Chang-woo Cho, Junying Shen, Jian Lyu, W Skrotzki, K Nenkov, et al., “Superconductivity Omargeldi Atanov, Qianxue Chen, Seng Huat with broken time-reversal symmetry inside a su- Lee, Yew San Hor, Dariusz Jakub Gawryluk, perconducting s-wave state,” Nature Physics 16, Ekaterina Pomjakushina, Marek Bartkowiak, 789–794 (2020). Matthias Hecker, Jörg Schmalian, and Rolf [31] T Böhm,A.F. Kemper, B. Moritz, F. Kretzschmar, Lortz, “Z3-vestigial nematic order due to super- B. Muschler, H.M. Eiter, R. Hackl, T.P. Devereaux, conducting fluctuations in the doped topological D.J. Scalapino, and Hai-Hu Wen, “Balancing act: insulators NbxBi2Se3 and CuxBi2Se3,” Nature evidence for a strong subdominant d-wave pairing Communications 11, 3056 (2020), arXiv:1905.01702 channel in Ba0.6K0.4Fe2As2,” Physical Review X 4, [cond-mat.supr-con]. 041046 (2014). [41] KD Nelson, ZQ Mao, Y Maeno, and Ying Liu, [32] GE Volovik and VP Mineev, “Line and point singu- “Odd-parity superconductivity in Sr2RuO4,” Sci- larities in superfluid 3He,” JETP lett 24 (1976). ence 306, 1151–1154 (2004). [33] M. Sigrist, T. M. Rice, and K. Ueda, “Low-field [42] Volker Bach, Elliott H Lieb, and Jan Philip Solovej, magnetic response of complex superconductors,” “Generalized Hartree-Fock theory and the Hub- Phys. Rev. Lett. 63, 1727–1730 (1989). bard model,” Journal of statistical physics 76, 3–89 [34] Suk Bum Chung, Hendrik Bluhm, and Eun-Ah (1994).

V. APPENDIX

A. Derivation of nonlinear sigma model Therefore, we can write

Here, we review an efficient method of deriving ( i∂µ Aµ)∆ = (∂µχ + aµ Aµ Hµ) ∆ (20) the nonlinear sigma model in Eq. (3). Suppose that − − − − the OP ∆ = ∆(q) depends smoothly on parameter q. 1 Hµ = (nˆ ∂µnˆ) τ (21) Then it’s change ∆(q) ∆(q +δq) can be expressed 2 × · as an infinitesimal rotation→ along with a change in the Berry phase γ, i.e., Using the (anti-)commutation relations of the Pauli matrices, we can then efficiently derive the nonlinear iδγ δq ∆(q + δq) = e U(q) ∆(q) (18) sigma model, e.g.,

δq Where U(q) exp( iH(q)δq) SU(2) is the in- 2 ≡ − ∈ † 2 ∆ 2 ﬁnitesimal rotation with ∆ H ∆ = | | tr H (τ0 + nˆ τ ) (22) µ 2 µ · 2 2 ∂µnˆ 1 = ∆ (23) H(q)δq = (nˆ δnˆ) τ (19) | | 2 2 × · 13

Similarly, we have 2. Full generality: V is stable and superconducting

∆†H ∆ = 0 (24) µ In full generality, we still require that the † ∂xnˆ ∂ynˆ Ginzburg-Landau energy is stable, i.e., the poten- ∆ Hxτ1Hy∆ = n1 (25) < − 2 · 2 tial term V in the limit where ∆ . This → ∞ | | → ∞ † 1 corresponds to the condition ∆ Hµτ1∆ = [nˆ ∂xnˆ]1 (26) < 2 × 02 4β3 β > 0 (32) − 3 B. General ground state Similarly, we are only concerned with nontrivial superconducting ground states, which corresponds to Here, we present the exact solution of the ground the condition state of the general Ginzburg-Landau free energy in 0 α 2α0β3 α3β3 > 0 (33) the presence of a uniform strain vector . To provide − a more intuitive picture of the general ground state, this section ﬁrst treats the less restrictive case where Therefore, our general ground state solution is sub- 0 ject to the 2 conditions above. The potential term β1 = β3 but β3 = 0. We then present the ground state6 in full generality. of the Ginzburg-Landau free energy can be rewritten It should be noted, however, that the uniform as ground state does not depend on the kinetic terms 2 K and thus we can always renormalize D,G so that V2 = ∆ (α0 + α1n1 + α3n3) (34) 0 −| | β3 = 0. 1 4 2 2 0 V4 = ∆ (1 + β1[n1] + β3[n3] + β n3) (35) 2| | 3

0 1. β1 6= β3 and β3 = 0 By our stability condition (32), the potential term V is a strongly joint-convex function in terms of 2 2 2 0 ∆ , ∆ n1 and ∆ n3. Therefore, convex opti- In the case where β1 = β3 are positive and β3 = | | | | | | 0, the potential term of6 the Ginzburg-Landau free mization guarantees that the ground state of the energy can be rewritten as uniform system is unique and given by

2 0 V = ∆ (α + α n + α n ) (27) 2 2α0β3 α3β3 2 0 1 1 3 3 ∆ = 2 − > 0 (36) −| | 4β β02 1 4 2 2 | | × 3 3 V4 = ∆ (1 + β1[n1] + β3[n3] ) (28) −0 2| | 2 2α3 β3α0 ∆ n3 = 2 − 02 (37) | | × 4β3 β Where nˆ is the Bloch vector corresponding to the − 3 2 α1 OP ∆. It’s then clear that the potential V is a ∆ n1 = (38) 2 2 β strongly joint-convex function of ∆ , ∆ n3 and | | 1 2 | | | | ∆ n1. Therefore, convex optimization guarantees 2 2 that| | the ground state of the uniform system is unique Provided that [n1] + [n3] 1. Conversely, in the case where the above solution≤ is not feasible, i.e., and given by 2 2 [n1] + [n3] > 1, the unique ground state is given 2 αi by ∆ = α0, ni = (29) | | βiα0 2 2 2 2 Provided that [n1] + [n3] 1. Conversely, in the ∆ = 02 (39) ≤ | | 4(β3 + λ) β3 case where the above solution is not feasible, i.e., − 2 2 02 (β3 + λ)λ [n1] + [n3] > 1, the unique ground state is given 2(β3 + λ)α0 α3β3 + 4α3 0 by × − β3 0 2(1 λ)α3 β α0 α0 2 3 2 ∆ n3 = 2 − − 02 (40) ∆ = (30) | | × 4(β3 + λ) β | | 1 λ − 3 − α1 2 αi 2 ∆ ni = (31) ∆ n1 = (41) | | βi + λ | | β1 + λ

2 2 2 2 Where λ [0, 1) is chosen so that [n1] + [n3] = Where λ 0 is chosen so that [n1] + [n3] = 1. In 1. In particular,∈ the ground state corresponds to a particular,≥ the ground state corresponds to a Bloch Bloch vector nˆ which is in the eˆ1-eˆ3 plane and thus vector nˆ which is in the eˆ1-eˆ3 plane and thus does does not break TRS. not break TRS. 14

C. Narrow Domain Wall Calculations Using the boundary conditions θ ψ and θ˙ 0 as y and that θ = 0 at y = 0, we→ see that → → ∞ In this section, we present detailed calculations 8α of a narrow domain wall between the transition of θ˙2 = ¯ (1 cos(θ ψ)) (48) the TRS states d g d + g. To be more con- κ − − r crete, let us consider− the→ case where α (y) = α θ˙ 4α θ ψ 3 ¯ 3 = ¯ sin − (49) is constant, while α1(y) = α1 sign(y) changes sign 2 − κ 2 ¯ as y = + . The quartic terms are set ψ to β =−∞β = →β >∞ 0, β0 = 0 as before and the θ = ψ 4 arctan tan e−y/ξ (50) 1 3 3 − 4 strain vector magnitude satisfies α βα0 so that ¯ ≥ the uniform ground state at y is denoted by a p → ±∞ Where ξ = κ/4α is the characteristic length of the Bloch vector nˆ pointing in the same direction as the ¯ 2 domain wall. We can similar solve for the transition strain vector α and that the magnitude ∆ = α0. ¯ | | in the case where y 0 so that in general, Without loss of generality, we shall also assume that ≤ α , α > 0 so that the angle ψ of the strain vector ¯ 1 ¯ 3 ψ α α(sin ψ, 0, cos ψ) satisfies 0 < ψ < π/2 as shown θ = ψ 4 arctan tan e−|y|/ξ (51) in¯ ≡ Fig.¯ (2). − 4 π We then calculate the domain wall energy ∆F of φ = (1 sign y) (52) possible transition paths as illustrated by the blue 2 − and green arrows in Fig. (2) where the magnitude ∆ 2 of the OP ∆ is assumed to be constant and We can then calculate the domain wall energy | | ∆F = F F0 for y 0 where F0 = F [ψ] is the = α0. In particular, the blue arrow represents a TRS − ≥ preserving transition, while the green arrow denotes Ginzburg-Landau free energy of the uniform ground a TRSB transition restricted to a 2D plane at angle state, i.e., ω with respect to the eˆ3-axis. Z +∞ ∆Ftrs = [F [χ, θ, φ] F0]dy (53) 0 − 1. TRS preserving transition !2 Z +∞ θ˙ = κα0 dy (54) 2 We shall first consider a TRS preserving transition 0 Z ψ as described by the blue path in Fig. (2). Notice κα0 θ ψ dθ that we implicitly assumed that ψ < π/2. If on = sin − (55) ξ 0 − 2 2 the other hand, ψ > π/2, then we would consider 2κα ψ the complement path wrapping from underneath the = 0 sin2 (56) Bloch sphere. Since the transition preserves TRS so ξ 4 that the Bloch vector is in the eˆ1-eˆ3 plane, we can restrict the azimuthal angle φ 0 for y 0 and φ π for y < 0. Therefore,≡ we can rewrite≥ the 2. TRSB transition Ginzburg-Landau≡ free energy in terms of polar angle θ and average phase χ, i.e., if y 0 so that θ 0, Let us now consider the special TRSB transition ≥ ≥ then as described by the path with ω = π/2 in Fig. (2), so that the polar angle θ of the Bloch vector nˆ remains F V V K = 2 + 4 + (42) constants and = ψ < π/2, while the azimuthal an- 2 V2 = ∆ (α0 + α cos (θ ψ)) (43) gle φ twists π 0. Therefore, we can rewrite the −| | ¯ − → 1 4 Ginzburg-Landau free energy in terms of azimuthal V4 = + ∆ (1 + β) (44) angle φ and average phase χ, i.e., if y 0 so that 2| | ≥  2 0 φ π/2, then ˙ ! ≤ ≤ κ 2 2 θ K = ∆ χ˙ +  (45) 2 | | 2 F = V2 + V4 + K (57) 2 V2 = ∆ (α0 + α1 sin ψ cos φ + α3 cos ψ) (58) It’s then clear that χ = const and that −| | 1 4 2 2 V4 = + ∆ (1 + β(1 sin ψ sin φ)) (59) κ θ¨ 2| | − = α sin(θ ψ) (46)  2  ˙ ! 2 2 ¯ − κ 2 2 φ 8α K = ∆ χ˙ + χ˙φ˙ cos ψ (60) θ˙2 + ¯ cos(θ ψ) = const (47) 2 | | 2 − κ − 15

By the Euler-Lagrange equations, it’s then clear that if the average phase χ satisfies the initial condition χ(y ) = 0, then → −∞ φ π χ = − cos ψ (61) 2 In particular, the average phase χ changes by ∆χ = (π/2) cos ψ as φ twists π 0. We can similarly − → solve for φ using the boundary condition φ,˙ φ 0 as y + so that → → ∞ κ φ¨ = sin φ(α βα0 cos φ) (62) 2 2 ¯ − s φ˙ 1 φ φ = sin 1 cos2 (63) 2 −ξ 2 − 2 p Where = βα0/α and ξ = κ/4α is the characteristic length of the¯ domain wall. Notice¯ that by our ground state discussion (9), the finite strain α satisfies 1 and thus the square root is well-defined.¯ We can≤ then solve the first order differential equation so that ! 1 η φ = 2 arccos p − (64) (1 + η)2 4η − Where √1 η = C exp y − (65) − 2ξ π = φ(η = C) (66) 2 We can then calculate the domain wall energy ∆F = F F0 for y 0 where F0 = F [ψ] is the Ginzburg-Landau− free≥ energy of the uniform ground state, i.e., FIG. 8. General TRSB Domain Wall. The top figure is the Bloch sphere representation of a general TRSB +∞ Z transition through the domain wall α1(y) = α1 sign(y). ∆Ftrsb = [F [χ, θ, φ] F0]dy (67) ¯ − The green arrow which representing the transition path 0 is assumed to be in a plane at angle ω with respect to the !2 Z +∞ ˙ ˆ ˆ 2 φ e1-e3 plane. The bottom figure is the transition rotated = κα0 sin ψ dy (68) of angle ω about the +x-axis in the clockwise direction. 0 2 κα = 0 sin2 ψ (69) ξ s 3. General TRSB transition Z π/2 φ φdφ sin 1 cos2 (70) × 2 − 2 2 0 Let us finally consider a general TRSB transi- This can be solved analytically if necessary with the tion as described by the green path in Fig. (2). substitution x = cos (φ/2). In particular, in the ex- To simplify the problem, we will only consider general TRSB transitions which occur in a 2D plane as treme large strain limit α βα0, the domain wall energy is approximated by¯ shown in Fig. (8), where the transition plane is at an arbitrary angle ω with respect to the vertical plane. 2κα π If ω = 0, then the transition corresponds to the TRS ∆F 0 sin2 ψ sin2 (71) ≈ ξ 8 preserving transition (blue arrow), and if ω = π/2, 16

p then it corresponds to the special TRSB transition. Where = βα0/α 1 and ξ = κ/4α is the charac- Since the Ginzburg-Landau free energy has a global teristic length of¯ the≤ domain wall. It should¯ be noted SU(2)-symmetry, we can rotate the system about that the equation can be solved analytically for ar- the +x-axis of angle ω in the clockwise direction bitrary 1, albeit in implicit form, i.e., f(ϕ) = y so that the transition occurs in a plane parallel to for some≤ function f. It should be noted that the xy-plane and thus can be parametrized in terms of explicit form ϕ = ϕ(y) is not necessary to compute azimuthal angle φ, while the polar angle θ is held the domain wall energy ∆F . Indeed, we have fixed, as shown in Fig. (8). This corresponds to Z +∞ ˜ U using the rotated order parameter ∆ = ∆ where ∆F = [F [χ, θ, φ] F0]dy (82) U SU(2) corresponds to the described rotation R. 0 − ∈ 2 We then rewrite the Ginzburg-Landau free energy Z +∞ ˙ ! 2 φ so that if α = Rα is the rotated strain vector and = κα0 sin θ dy (83) ϕ π/2 ˜φ is the¯ azimuthal angle relative to the 0 2 ≡ − +y-axis, then κα0 = sin2 θ (84) ξ F = V2 + V4 + K (72) Z $/2 q 2 2 2 2 V2 = ∆ (α0 + α cos θ) (73) sin x 1 ( sin ω) sin ($ x)dx −| | ¯ × 0 − − ∆ 2α sin2 θ cos (ϕ $) ¯ − | | − In the extreme large strain limit α βα0, the do- 1 4 ¯ V4 = + ∆ (1 + β) (74) main wall energy is approximated by 2| | 2 1 sin θ cos θ 2κα0 2 2 $ ∆ 4β (cos ϕ cos $)2 ∆F sin θ sin (85) − 2| | cos ψ − ≈ ξ 4 " # κ ϕ˙ 2 K = ∆ 2 χ˙ 2 + +χ ˙ϕ˙ cos θ (75) 2 | | 2 D. Full self-consistent equations θ φ Here is held fixed and transitions in a manner In this section, we derive the full self-consistency such that field equations (SCFs) for the full Hamiltonian π ϕ φ = $ 0 $ (76) Hfull = H0 + H1 defined in Eq. (12), (13). Let ≡ 2 − − → → us first write the corresponding BCS Hamiltonian As y goes from 0 + , where θ, $ are H as −∞ → → ∞ determined by the boundary condition α nˆ(y X 0 † ˜ ∝ → H = (~r , ~r )c 0 c (86) + ), i.e., T ~r s ~rs ∞ ~r 0,~r,s sin ψ = sin θ sin $ (77) X † † + (∆~r 0,~r c 0 c + h.c.) (87) cos ψ cos ω = sin θ cos $ (78) ~r ↑ ~r ↓ ~r 0,~r ψ ω θ cos sin = cos (79) h i c † ˆ ↑ ˆ ∆ = c↑ c↓ H † , H = T† ¯ (88) In particular, θ = π/2, $ = ψ for the TRS preserv- c↓ ∆ ing transition, and θ = ψ, $ = π/2 for the special −T TRSB transition. To simplify notation, let us introduce the notation By the Euler-Lagrange equations, it’s then clear 1~r = ~r ~r for the projection operator and the 1- | ih | that if the global phase χ satisfies the initial condi- particle density matrices (1-pdms)[42] of the BCS tion χ(y ) = 0, then Hamiltonian H, i.e., → −∞ 0 † † φ 1 π γ(~r , ~r ) = c c~r 0↑ = c c~r 0↓ (89) χ = cos θ + $ cos θ (80) h ~r ↑ i h ~r ↓ i 2 − 2 2 0 α(~r , ~r ) = c~r ↓c~r 0↑ = c~r ↑c~r 0↓ (90) Hence, the average phase χ changes by ∆χ = h i −h i $ cos θ as y = . We can similarly solve The Hartree-Fock energy Hfull = H0 + H1 can h i h i h i for− ϕ so that −∞ → ∞ then be computed via Wick’s theorem so that ϕ˙ 1 ϕ $ H0 = 2 tr( γ) (91) = sin − (81) h i T 2 −ξ 2 X X † † H1 = 2 λτ tr 2fτ α 1~r αf 1~r (92) s h i − τ ϕ $ τ=d,g ~r 2 ω 2 $ 1 ( sin ) sin − + † † × − 2 + fτ γ1~r γfτ 1~r + γ1~r fτ γfτ 1~r 17

The self-consistency field equations (SCFs) can then In this case, the SCFs (93) are reduced to be derived from extremizing the variational free energy δF δ Hfull T δSH = 0 and written in op- ~ ~ X ~ ≡ h i − (k) = t(k) λτ fτ (k) fτ γ + h.c) (102) erator form T − τ=d,g J K X X † = t λτ (fτ 1~r fτ γ1~r + h.c.) (93) 2 ~ 2 T − + γ fτ + fτ (k) γ τ=d,g ~r ∈Λ J | | K | | J K † † ~ X ~ † +1~r fτ γfτ 1~r + fτ 1~r γ1~r fτ ∆(k) = 4 λτ fτ (k) αf (103) − τ X X † τ=d,g J K ∆ = 2 λτ (fτ 1~r αf 1~r + tr) (94) − τ τ=d,g ~r Where we use the notation as the average value ··· † summed over ~k-space, i.e., J K Where tr implies the transpose term = 1~r fτ α1~r fτ = † 1~r αfτ 1~r fτ , and we used the fact that α, fτ are sym- T T Z d2~k metric, i.e., α = α, fτ = fτ . The proposed format ~ h = 2 h(k) (104) of writing the SCFs in operator form has the advan- J K (−π,π]2 (2π) tage of working in any basis set, and in particular, we can work in real space ~r or momentum space Notice that the specific form of our ∆-function (102), | i ~k . This will simplify our algebra later on. The cor- i.e., ∆(~k) = Df (~k) + Gf (~k) for complex constants | i d g responding BdG equations can similarly be written D,G implies that the SCFs indeed yield a two- as component OP theory. Similarly, the BdG equations are reduced to γ = un u† +v ¯(1 n )vT (95) E E ! − ~ 1 βE ~ 1 (k) ~ α = u tanh v† + transpose (96) γ(k) = 1 + T n(k) (105) −2 2 2 E(~k) ! 1 (~k) where nE is the diagonal matrix with entries of the n ~k βE −1 + 1 T 1 ( ) Fermi distribution nE = (e + 1) and u, v are 2 − E(~k) − chosen so that the following unitary matrix W diag- ! ˆ ∆(~k) βE(~k) onalizes the first quantized Hamiltonian H, i.e., α(~k) = tanh (106) −2E(~k) 2 E 0 u v¯ Hˆ = W W †,W = − (97) 0 E v u¯ ~ βE(~k) −1 − Where n(k) = (e + 1) is the Fermi distribution of the Bogoliubov quasi-particles with disper- Notice that the BdG equations (95) can also be writ- sion relation ten in the more compact form [42] q ~ ~ 2 ~ 2 n 0 E(k) = (k) + ∆(k) (107) Γ = W E W † (98) T | | 0 1 nE − Where E. ∇ · J = 0 in BCS theory γ α 1 Γ = † = (99) Let H[ψ] denote the BCS Hamiltonian with pa- α 1 γ¯ eβHˆ − + 1 rameters ψ = (ξ, ∆). Let the particular choice of parameter ϕ be such that H[ϕ] satisifes the full self- consistency equations with respect to the full Hamil- 1. Uniform system in momentum space tonian Hfull, i.e.,

The full SCFs can be further simpliﬁed in a uni- ∂ 1 ∂ Hfull ψ = S[ψ]) (108) form system, so that the 1-pdms γ, α (89), hopping ∂ψ h i β ∂ψ ~ ψ=ϕ ψ=ϕ matrix t and form factors fτ are diagonalized in k- space. In particular, Where S[ψ] is the von-Neumann entropy of H[ψ] de- ﬁned by S = tr (ρ log ρ) where ρ is the Gibbs dis- ~ † † − γ(k) = c~ c~ = c~ c~ (100) tribution of the BCS Hamiltonian H[ψ], and ψ h k↑ k↑i h k↓ k↓i h· · ·i ~ is the thermal average at temperature T with respect α(k) = c ~ c~ = c~ c ~ (101) h −k↓ k↑i −h k↑ −k↓i to the BCS Hamiltonian H[ψ]. 18

We shall subsequently show that at any tempera- The ﬁrst term is = 0 since the trace is invariant ture and every lattice site ~r , the charge density ρ(~r ) under unitary transforms. Setting s = 0, we see is constant in BCS theory, i.e., that

∂ρ(~r ) = 0 (109) ∂t t=0 As a corollary, we can apply the continuity equation ∂Hfull(s) P 0 ∂tρ(~r ) = J(~r ) 0 J(~r , ~r ) to obtain − ∂s ∇ · ≡ ~r s=0 ϕ ∂ X 0 1 −βH[ϕs] J(~r ) ϕ J(~r , ~r ) ϕ = 0 (110) = tr Hfull e (118) h∇ · i ≡ h i Zϕ ∂s ~r 0∈Λ s=0 ∂ 1 −βH[ϕs] Indeed, let us introduce the notation for gauge = tr Hfulle (119) ∂s Zϕ transformation at lattice site ~r , i.e., if M is an op- s=0 ∂ erator, e.g., the full Hamiltonian Hfull or the BCS = Hfull ϕs (120) Hamiltonian H[ψ], then deﬁne ∂s s=0 h i

∂ϕs ∂ isρ(~r ) −isρ(~r ) = Hfull ψ (121) M(s) = e Ae , s R (111) ∂s ∂ψ ∈ s=0 ψ=ϕ h i

In this case, notice that 1 ∂ϕs ∂ = S[ψ] (122) β ∂s s=0 ∂ψ ψ=ϕ H[ϕ](s) = H[ϕ(s)] (112) 1 ∂ = S[ϕs] (123) where ϕ(s) denote the parameters β ∂s s=0

00 0 (~r 00, ~r 0)(s) = (~r 00, ~r 0)eis(δ(~r ,~r )−δ(~r ,~r )) (113) T T 00 0 ∆(~r 00, ~r 0)(s) = ∆(~r 00, ~r 0)eis(δ(~r ,~r )+δ(~r ,~r )) (114)

Notice that S[ϕs] is independent of s since the trace Also notice that the BCS partition function is invariant under unitary gauge transforms. Hence, Z ϕ s s [ ( )] is independent of since the trace is invari- the right-hand-side is = 0 and thus we arrive at the ant under unitary gauge transforms and thus statement ∂Hfull(s) ∂s ϕs 1 ∂Hfull(s) −βH[ϕs] = tr e (115) ∂ρ(~r ) ∂Hfull(s) Zϕ ∂s = = 0 (124) ∂t − ∂s ∂ t=0 ϕ s=0 ϕ 1 −βH[ϕs] = + tr e Hfull(s) (116) Zϕ ∂s ∂ 1 −βH[ϕs] tr Hfull(s) e (117) − Zϕ ∂s