Simple Model for T_c and Pairing Symmetry Changes in Sr₂RuO₄ Under (100) Uniaxial Strain

Abstract

Uniaxial strain in the (100) direction has the effect of increasing the superconducting ${\mathrm{T}}_c$ in Sr₂RuO₄ from $1.5$ K to over 3 K. The enhanced $T_c$ corresponds to a Lifshitz transition in the Fermi surface topology of this unconventional superconductor. We model this using a simple two-dimensional one-band model for the $\gamma$ sheet of the Fermi surface. This reproduces the experimental $T_c$ results well if we assume a $d_{x^2-y^2}$ singlet pairing state. On the other hand, the triplet state $p_x+i{p}_y$ does not show any distinct peaks in ${\mathrm{T}}_c$ associated with the Lifshitz transition. A mixed symmetry state pairing of the form $d+ig$ can both describe the $T_c$ changes and show a distinct transition temperature for time-reversal symmetry breaking (TRSB).

1. Introduction

The unconventional superconductivity of ${Sr}_2$${RuO}_4$ has been intensively studied for more than two decades [1,2,3]. The material was initially of interest as a layered perovskite structurally similar to the high $T_c$ cuprate superconductors. Although the superconducting $T_c$ was limited to $1.5$ K, the fact that the normal state was clearly a Fermi liquid led to an early proposal by Rice and Sigrist that this material could be regarded as a solid state analogue of superfluid ³He [4]. Weak coupling spin-fluctuation-driven pairing was shown to lead to both spin singlet d-wave and spin triplet p-wave [5,6,7,8]. More recently, a better understanding of the normal state as a strongly correlated Hund metal has emerged from Dynamical Mean Field and Numerical Renormalization Group (NRG) methods. This correlated normal state also has instabilities towards various superconducting symmetry states of both singlet and triplet pairing [9,10,11,12].

Experimentally, it has also proved surprisingly difficult to determine the pairing state fully. The strong sensitivity of $T_c$ to non-magnetic impurity disorder [13] confirmed an unconventional pairing state. Evidence for a vertical line node in the gap emerged from both specific heat experiments [14] and thermal conductivity [15]. More recently, quasiparticle interference experiments uniquely pointed towards a line node structure consistent with a bulk $d_{x^2-y^2}$ pairing state [16]. However, it has proved difficult to also rule out completely the possibility of a horizontal line node [3]. A spin singlet state, such as d-wave, is consistent with the most recent Knight shift and spin susceptibility and experiments [17,18,19,20].

However, another set of experiments appear inconsistent with this $d_{x^2-y^2}$ pairing state, at least in its simplest form. Evidence for a time-reversal symmetry breaking state (TRSB) was first seen in $\mu$SR [21] and later in the Kerr effect [22]. TRSB requires a pairing state with two degenerate or near-degenerate components, and evidence for such degeneracy was reported in [23,24]. A possible d-wave state with TRSB would be the $E_g$ symmetry state, $d_{xz}+i{d}_{yz}$ [25]. However, this state requires by symmetry a horizontal line node in the plane $k_z=0$, not a vertical one. But detailed models suggest that this $d_{xz}+i{d}_{yz}$ state could have a vertical gap minimum on at least one of the Fermi surface sheets [20,25] in the location indicated by the quasiparticle interference experiments [16]. A very different route to reconciling the vertical line node with TRSB, suggested by Kivelson et al. [26], was a state of the form $d+ig$. This mixes two distinct symmetries of the BCS gap equation, $B_{1g}$ and $A_{2g}$, which are not normally degenerate in a tetragonal crystal, and so it is necessary to assume that, for microscopic reasons, they become degenerate or nearly degenerate because of specific features of the fundamental electronic structure of ${Sr}_2$${RuO}_4$.

An alternative set of constraints on the pairing state in bulk ${Sr}_2$${RuO}_4$ are provided by uniaxial strain experiments, especially the application of $\left(100\right)$ strain [27,28,29,30,31,32,33]. Remarkably, a uniaxial strain of about $0.6\%$, either compressive or expansive, is sufficient to increase the superconducting $T_c$ from $1.5$ K to about $3.5$ K [27,28]. The maximum $T_c$ coincides with a Lifshitz topological Fermi surface transition, in which the $\gamma$ Fermi surface sheet changes from a closed cylinder around $\Gamma$ to become open [29,34,35]. Furthermore, the $\left(100\right)$ uniaxial strain separates the superconducting state $T_c$ from the temperature where TRSB becomes apparent [32]. In principle, such a splitting would be expected in a pairing state of $E_g$ or $E_u$ character, such as $d_{xz}+i{d}_{yz}$ or $p_x+i{p}_y$, where the degeneracy of the order parameter in the tetragonal crystal becomes broken. However, such splitting of a symmetry required degeneracy would lead to a linear change in $T_c$ for small $\left(100\right)$ strains, which is not consistent with the quadratic behaviour observed [36]. On the other hand, an accidental degeneracy (such as $d+ig$) would most likely become split by either hydrostatic pressure or disorder, which is not seen in experiments [37]. Finally, it is also surprising that uniaxial compression along the c-axis does not significantly increase $T_c$, despite the fact that this strain should also move the Fermi surface towards a Lifshitz transition [38].

The purpose of this paper is to examine the effect of $\left(100\right)$ strain in ${Sr}_2$${RuO}_4$ using a simple theoretical framework suggested by Fermi liquid theory, inspired by the original Fermi liquid approach to the phases of superfluid ³He [39]. To apply these simple but general methods, we first simplify the Fermi surface to a two-dimensional one-band model describing the $\gamma$ Fermi surface sheet. With this model, we study the changes in $T_c$ induced by uniaxial strain for a variety of different pairing models. Under the assumption that the effective electron–electron pairing interaction is unchanged during the Lifshitz topological transition, the model becomes independent of specific microscopic details of the pairing interaction. This approach allows us to make a direct link between changes in $T_c$ and a pairing symmetry weighted density of states (DOS) at the Fermi level. The largest changes in $T_c$ (consistent with experiments) are seen for the states with the largest weighted DOS, while pairing states with gap nodes at the Lifshitz transition point show small or no increases in $T_c$. The simple model results are compared with both experiments and multi-band theories [40,41,42,43,44].

The structure of this paper is as follows. We first present our Fermi surface model, derived from fitting ab initio DFT band structures for strained and unstrained ${Sr}_2$${RuO}_4$. We develop the BCS-like gap equation for different pairing symmeties, relating $T_c$ to the appropriate weighted DOS. The main results for $T_c$ in each pairng channel of interest are presented in Section 3. Finally, we explore the case where two distinct pairing symmerties are degenerate in unstrained ${Sr}_2$${RuO}_4$ so that the unstrained material has a TRSB temperature identical, or close to, the superconducting $T_c$. We examine in detail the cases of $d+ig$ mixed symmetry pairing with such a near degeneracy.

2. Materials and Methods

2.1. Normal State Band Structure

The starting point for our work is a tight binding fit to the band structure as the system goes through the Lifshitz transition under $\left(100\right)$ applied strain. Using the PBE96 [45] DFT energy functional and the all-electron LAPW Elk code [46], we find that the zero strain optimised crystal structure parameters for Sr₂RuO₄ are given by $a=0.38895$ nm and $c=1.2922$ nm, which are close to the experimental values of $a=0.3862$ and $c=1.2722$ nm [47].

The internal Wykoff structure parameters are $z\left(O2\right)=0.1619$ and $z\left(Sr\right)=0.3528$, compared to $0.16210$ and $0.35298$, respectively [47]. This is comparable to many earlier DFT calculations, for example, Refs. [48,49]. The calculated Fermi surface and band structure are well known from previous work, but for completeness, they are shown here in Figure 1.

Under $\left(100\right)$ uniaxial strain, the Fermi surface distorts and passes through a Lifshitz topological transition, as shown in Figure 2. This shows the Fermi surface under either a $2\%$ compressive or expansive strain along the crystal a-axis, ${ϵ}_{xx}=\mp 0.02$. The corresponding b-axis strain was chosen as ${ϵ}_{yy}=\pm 0.008$ based on a Poisson ratio of $0.4$ [50]. As can be seen in Figure 2a, a $2\%$ compression along a leads to an opening of the Fermi surface along the b direction, as a result of lattice expansion in that direction. Conversely, Figure 2b shows that a $2\%$ expansion of a (with a corresponding reduction of b by about $0.8\%$) now opens up the $\gamma$ band Fermi surface cylinder along the a direction in the Brillouin zone. Figure 2c shows the band structure in the compressive strain case. In comparison to the unstrained band structure, Figure 1d, one can see that the band peak along the $\Gamma -M$$\Sigma$ line has moved below the Fermi level at point $\mathbf{k}=(0,\pi /b,0)$. This $\mathbf{k}$ point is, therefore, the point where a Lifshitz topological change in Fermi surface geometry occurs. This Lifshitz transition will also coincide with a corresponding van Hove peak in the density of states (DOS), which, in turn, will be expected to lead to changes in the superconducting transition temperature $T_c$, as observed experimentally [27].

The Fermi surface and band structure changes under strain shown in Figure 2 are consistent with both earlier DFT calculations and direct measurements using ARPES [30,35]. Although the DFT calculations do not include strong correlation effects, as, for example, studied in Ref. [41], the Fermi surface of both distorted and undistorted lattices evolves in a way consistent with experiments. While the unstrained bulk Fermi surface topology is well described, the DFT band structure in bulk is well known to overestimate the bandwidth by about a factor of 3 for the $\gamma$-band compared to experiments [1]. Our DFT total energy calculations also overestimate the stress corresponding to the critical strain to be about 2–3 GPa, compared to the experimental value of 0.75 GPa [32].

DFT + DMFT calculations [30] for strained Sr₂RuO₄ show that strong Coulomb correlations make the unstrained system closer to the Lifshitz transition than the DFT calculations, such as those shown in Figure 2, alone would predict.

2.2. $\gamma$-Band Tight Binding Model

The changes in the Fermi surface under strain, as shown in Figure 1 and Figure 2, show that among the three Fermi surface sheets, it is only the $\gamma$-band that changes significantly. Therefore, it seems natural to focus on superconductivity within the $\gamma$-band in developing a simple model of the dominant strain effects on $T_c$. The idea that only one of the three Fermi surface sheets in Sr₂RuO₄ is dominant in driving the system towards the superconductivity has been discussed for many years [51]. More recent theories [9,10,11,12] of unstrained Sr₂RuO₄ show that all bands are closely coupled in the superconducting state below $T_c$. Nevertheless, it is a plausible model to suggest that, specifically at $T_c$, one of the bands has a stronger tendency to Cooper pairing than the others, even if the dominant pairing interaction itself arises from a combination of spin fluctuations within all three bands. In this picture, the linearized gap equation at $T_c$ would have a dominant eigenvector which has the majority of its weight on the dominant band. The gap on the other bands would be smaller initially at $T_c$ but would grow rapidly below $T_c$ due to interband coupling within the non-linear gap equation. If the $\gamma$-band did not have such a dominant role in determining $T_c$, then it seems unlikely that a Fermi surface change affecting essentially only the $\gamma$-band could increase the critical temperature so significantly. In contrast to other multi-band theories [42,44] of strain-induced changes in $T_c$, we therefore focus on a $\gamma$-band-only model. A previous one-band theory [52] was based upon a generic model of a circular Fermi surface, rather than a realistic model of the $\gamma$-band.

With this motivation, we therefore make a tight binding fit to the $\gamma$-band of Sr₂RuO₄. The $\gamma$-band is predominantly derived from the $Ru$ $d_{xy}$ orbitals hybridized with neighbouring in-plane oxygen atoms. First, it is useful to directly examine the Bloch function at the Lifthitz transition point $\mathbf{k}=(\pi /a,0,0)$, as shown in Figure 3. In this wave function density plot, we can clearly see the $d_{xy}$ orbital symmetry at each Ru atom in the $a-b$ plane. We can also see the in-plane $p_y$ orbitals of the oxygen atoms at $(a/2,0,0)$, forming a $\pi$-bonded band with hopping along the $k_x$ direction. The corresponding $p_x$ orbitals of the oxygen atoms at $(0,a/2,0)$ cannot be seen, which is because these coincide with a line node of the wave function at $y=a/2$. This nodal line is required by symmetry. The nodal lines of the $d_{xy}$ Ru orbitals at $y=0$ and $y=a$ combined with translational symmetry along y ($k_y=0$) force an additional nodal line at $y=a/2$. There is no corresponding nodal line at $x=a/2$ because the Bloch wave crystal momentum $k_x=\pi /a$ changes the wave function sign between the $d_{xy}$ Ru orbital at $(0,0,0)$ and its neighbour along x at $(a,0,0)$, removing the need for an additional node between these atoms.

This pattern of nodal lines, shown in Figure 3, has an important implication for the Lifthitz topological transition with strain. The nodal lines prevent direct interplane hopping between $d_{xy}$ Ru orbitals in adjacent planes, making this Lifthitz topological transition point essentially a purely two-dimensional band structure feature. To see why, we can note that the nodal lines in Figure 3 imply that interlayer hopping is forbidden at this critical $\mathbf{k}$-point. This is because the pattern of nodes at the Ru atoms in the basal plane (two perpendicular nodal lines, $x=0$, $y=0$) is orthogonal to the pattern of nodes at the Ru atoms in the next layer of the body-centred tetragonal crystal structure. At $x=a/2$, $y=a/2$. We see in Figure 3 a horizontal line node at $y=a/2$, but no corresponding vertical line node along $x=a/2$. The wave function shown in the figure is, therefore, orthogonal to its corresponding counterpart in the neighbouring Ru planes at $z=\pm c/2$. Because of this, the Lifshitz transition wave vectors $\mathbf{k}=(\pi /a,0,0)$ and $\mathbf{k}=(0,\pi /a,0)$, exactly coincide with nodes in the c-axis interlayer hopping, giving the Lifshitz topological transition an almost purely two-dimensional character. It also follows that the corresponding peak in the density of states at the transition point will more closely correspond to a two-dimensional van Hove singularity (logarithmic divergence) rather than a three-dimensional one (square root singularities) expected in a typical three-dimensional band structure.

Considering only the $\gamma$-band, a standard tight-binding expansion of the band structure takes the form

$$\begin{array}{ccc}\hfill {\epsilon }_{\mathbf{k}}& =& {\epsilon }_0-2t_0\left(cos\left(k_{x}a\right)+cos\left(k_{y}a\right)\right)-4t^{\prime }\left(cos\left(k_{x}a\right)cos\left(k_{y}a\right)\right)\hfill \\ & & -8t_{\perp }\left(cos\left({\displaystyle \frac{k_{x}a}{2}}\right)cos\left({\displaystyle \frac{k_{y}a}{2}}\right)cos\left({\displaystyle \frac{k_{z}c}{2}}\right)\right)-2t_{z}cos\left(k_{z}c\right).\hfill \end{array}$$

(1)

As expected from the arguments given above, the nearest-neighbour interlayer hopping term vanishes when $k_x=\pi /a$ or $k_y=\pi /a$. The only remaining term for band dispersion along $k_z$ is direct hopping along the c-axis between second neighbour planes, indicated by the term $t_z$ above. This hopping is expected to be very small, both because of the large distance between these planes ($c=1.2$ nm, in contrast to the in-plane lattice constant of $a=0.39$ nm), and also because the Ru $d_{xy}$ orbital symmetry forbids any direct hopping to the p-orbitals of the apical oxygen. The extreme two-dimensionality of the band structure is apparent in Figure 1b, which is a direct c-axis projection of the Fermi surface. Some dispersion in the $\gamma$-band is visible at the Fermi surface in the $\Gamma -X$ direction (visible as a widening of the Fermi surface line here), but no such dispersion is visible for the $\gamma$ band near the Lifshitz transition points $\mathbf{k}=(\pi /a,0,0)$ and $\mathbf{k}=(0,\pi /a,0)$. The tight binding band structure fit using Equation (1) is shown in the blue line in Figure 1c, where the parameters were chosen to be ${\epsilon }_0=-393$ meV, $t_0=245$ meV, and $t^{\prime }=110$ meV. The effects of the $k_z$ dispersion in the $\gamma$-band are estimated by fitting the DFT band dispersion with $k_z$ along the lines $\mathbf{k}=(\pi /a,0,k_z)$ and $\mathbf{k}=(0.8\times \pi /a,0,k_z)$, giving $t_{\perp }\approx 6$ meV and $t_z\approx 3$ meV. These parameters are similar to the $d_{xy}$ hopping parameters found in previous fits of all three bands, e.g., Ref. [49]. The effects of inter-orbital hybridization between the Ru $d_{xy}$- and $d_{xz}$/$d_{yz}$-derived $\alpha$ and $\beta$ bands are strictly zero in the planes $k_z=0$ and $k_z=2\pi /c$ because the crystal lattice mirror planes $z=0$, $z=c/2$, etc. (in the absence of spin–orbit coupling). In the $k_z=0$ plane, as shown in the band structure in Figure 1d, spin–orbit coupling does have a significant effect on the bands along the $\Gamma -X$ direction; however, the effect of spin–orbit coupling is entirely concentrated in this part of the band structure and there is no significant spin–orbit effect on the band structure along the $\Sigma$ line, or near the Lifshitz transition points [48,53]. For this reason, we do not explicitly include a SOC term within the band fitting model, although the fitted band dispersion $\epsilon \left(\mathbf{k}\right)$ along the $\Gamma -X$ line in Figure 1d was calculated including SOC.

We fit the uniaxially strained band structure, Figure 2c, by simply assuming a small linear change in the nearest-neighbor hopping integrals. Considering only the $\gamma$-band, a standard tight-binding expansion of the band structure takes the form

$$\begin{array}{ccc}\hfill {\epsilon }_{\mathbf{k}}& =& {\epsilon }_0+\delta {\epsilon }_0-2\left((t_0+\delta t_x)cos\left(k_{x}a\right)+(t_0+\delta t_y)cos\left(k_{y}b\right)\right)\hfill \\ & & -4(t^{\prime }+\delta t^{\prime })\left(cos\left(k_{x}a\right)cos\left(k_{y}b\right)\right),\hfill \end{array}$$

(2)

where, for simplicity, we now drop the small $k_z$ dispersion terms. For the $2\%$ compressive strain corresponding to Figure 2a,c, we find that a good fit is obtained with $\delta {\epsilon }_0=+36$ meV, $\delta t_x=0.17t_0$, $\delta t_y=-0.4\times 0.17t_0$, and $\delta t^{\prime }\approx 0$. The factor of $0.4$ is consistent with the Poisson ratio of $0.4$ [50] used in the strained lattice structure. Again, this single-band fit is consistent with the $d_{xy}$ component of earlier multi-band tight binding models of strained Sr₂RuO₄, e.g., Refs. [28,40].

Equation (2) will provide our model of the $\gamma$-band structure changes as a function of strain. We will assume that the nearest-neighbour hopping parameters vary linearly with strain given by

$$\begin{array}{c}\hfill \delta t_x=8.5{ϵ}_{xx}{t}_0\\ \hfill \delta t_y=8.5{ϵ}_{yy}{t}_0=-0.4\delta t_x\end{array}$$

(3)

with $\delta t^{\prime }\approx 0$. The only remaining strain-dependent band parameter, $\delta {\epsilon }_0$, can be found for any given value of the strain by the assumption that the total band occupancy of the $\gamma$-band does not change significantly near the Lifshitz transition point. We therefore choose $\delta {\epsilon }_0$ to ensure a constant occupancy for the $\gamma$-band,

$$N_e=2\sum _{\mathbf{k}}f\left({\epsilon }_{\mathbf{k}}\right),$$

(4)

where $f\left({\epsilon }_{\mathbf{k}}\right)$ is the Fermi–Dirac function, and the Fermi energy is always set to zero. This simplifying assumption has also been used in earlier models, e.g., Ref. [28], and its accuracy is confirmed directly in ARPES experiments [35].

2.3. Fermi Liquid Theory and Pairing

Using the band structure model in Equation (2), we introduce superconductivity via the usual BCS gap equation,

$${\Delta }_{\mathbf{k}}=\sum _{{\mathbf{k}}^{\prime }}{V}_{\mathbf{k},{\mathbf{k}}^{\prime }}{\displaystyle \frac{{\Delta }_{{\mathbf{k}}^{\prime }}}{E_{{\mathbf{k}}^{\prime }}}}(1-2f\left(E_{{\mathbf{k}}^{\prime }}\right)).$$

(5)

Here, as usual, $E_{\mathbf{k}}=\sqrt{{\epsilon }_{\mathbf{k}}^2+{\left|{\Delta }_{\mathbf{k}}\right|}^2}$ and $f\left(E_{\mathbf{k}}\right)$ is the Fermi–Dirac function evaluated at eigenenergy $E_{\mathbf{k}}$. For simplicity, we omit spin indices, assuming that the pairing state is opposite spin pairing ($\uparrow \downarrow$) type, which is valid for all the singlet and chiral p-wave $p_x+i{p}_y$ states of interest. The case of opposite spin pairing ($\uparrow \uparrow$ and $\downarrow \downarrow$) is assumed to be ruled out for Sr₂RuO₄ by the Knight shift measurements [17,18,19,20].

We assume that the pairing interaction has its origin in the strong electronic correlations, rather than electron–phonon interaction, and corresponds to the effective interactions within a Landau Fermi liquid state, or Hund metal, derived from intra-atomic and inter-atomic spin fluctuations within the Ru d-bands. These spin fluctuations appear in the spin response functions as a broad incommensurate background as well as specific peaks associated with nesting vectors of the Fermi surface, such as $\mathbf{Q}=({\displaystyle \frac{2}{3}}\pi /a,{\displaystyle \frac{2}{3}}\pi /a)$ [5,6,7,8].

To analyse different Cooper pair symmetries, it is useful to redefine the pairing interaction into symmetry distinct pairing channels. In superfluid ³He, the Landau Fermi liqud parameters are described by various spherical harmonics [39]. Here, we must use the corresponding cubic harmonics of the crystal point group. Noting that the pairing interaction is a linear integral operator in $\mathbf{k}$-space, it is possible to rewrite $V_{\mathbf{k},{\mathbf{k}}^{\prime }}$ as a sum of diagonal terms corresponding to its eigenvectors,

$$V_{\mathbf{k},{\mathbf{k}}^{\prime }}=\sum _i{U}_i{\Gamma }_i\left(\mathbf{k}\right){\Gamma }_i\left({\mathbf{k}}^{\prime }\right).$$

(6)

The properties of the group representations will ensure that the eigenvectors ${\Gamma }_i\left(\mathbf{k}\right)$ can be classified according to the different irreducible group representations of the appropriate symmetry group. For body-centred tetragonal Sr₂RuO₄, the eigenfunctions ${\Gamma }_i\left(\mathbf{k}\right)$ can be chosen as real-valued functions for the cases of the $D_{4h}$ and $D_{2h}$ point groups for unstrained and strained crystals. The corresponding eigenvalue, $U_i$, is the effective pairing strength in that symmetry channel.

In general, the BCS gap function can be a superposition of eigenvectors,

$${\Delta }_{\mathbf{k}}=\sum _i{\Delta }_i{\Gamma }_i\left(\mathbf{k}\right).$$

(7)

However, at $T_c$, the BCS gap equation becomes linear and therefore $T_c$ is defined in terms of the eigenfunction with the largest eigenvalue. We recover the usual BCS expression for $T_c$,

$$1={\displaystyle \frac{U_i}{2}}{\int }_{-\infty }^{\infty }\rho \left(\epsilon \right){\displaystyle \frac{1}{\epsilon }}(1-2f\left(\epsilon \right))d\epsilon ,$$

(8)

but where the effective density of states is weighted by the eigenvector ${\Gamma }_i\left(\mathbf{k}\right)$ of the specific pairing channel which becomes non-zero at $T_c$,

$$\rho \left(\epsilon \right)=\sum _{\mathbf{k}}{\Gamma }_i{\left(\mathbf{k}\right)}^2\delta (\epsilon -{\epsilon }_{\mathbf{k}}).$$

(9)

Note that if there are two degenerate eigenvalues (either a degeneracy required by symmetry or accidental), Equation (8) remains valid to define $T_c$. However, the full non-linear BCS gap equation is then required to determine the nature of the gap function (e.g., $p_x+i{p}_y$ or $d+ig$, etc.) below $T_c$. For simplicity, in Equation (8), we neglect retardation, on the basis that the energy scale of magnetic fluctuations in a Hund metal, J, is large compared to the energy scale of the pairing and the effective J may even be comparable to the electronic quasiparticle bandwidth, which is renormalized down by a factor of 3 from the original DFT bandwidth [1].

In principle, we have an infinite set of eigenvectors for each symmetry channel. But if we assume a short-ranged pairing interaction in real space, e.g., from spin fluctuations between nearby $Ru$ d-orbitals, we can represent the gap function ${\Delta }_{\mathbf{k}}$ with the simplest Fourier expansion consistent with the relevant crystal point group symmetry. This then gives the set of functions given in

Table 1. Gap symmetry. The table shows the specific gap function forms considered in our calculations. The functions are labelled by the irreducible representations for the unstrained square lattice case, the $D_{4h}$ point group. For convenience, we omit normalization constants. The mixed symmetry case $d+ig$ assumes the accidental degeneracy or near-degeneracy of distinct symmetries. In both the $p_x+i{p}_y$ and $d+ig$ cases, breaking the degeneracy by strain implies that the temperature for TRSB will become distinct from $T_c$.

${\mathit{D}}_{4\mathit{h}}$ Symmetry Rep.	$\mathbf{\Gamma }\mathbf{\left(}\mathbf{k}\mathbf{\right)}$	Name
$B_{1g}$	$cos\left(k_{x}a\right)-cos\left(k_{y}b\right)$	d-wave
$E_u$	$sin\left(k_{x}a\right),sin\left(k_{y}b\right)$	chiral $p_x+i{p}_y$
mixed $B_{1g}$ $A_{2g}$	$cos\left(k_{x}a\right)-cos\left(k_{y}b\right),$
	$\left(cos\left(k_{x}a\right)-cos\left(k_{y}b\right)\right)sin\left(k_{x}a\right)sin\left(k_{y}b\right)$	$d+ig$

.

In Table 1, we have only included pairing states which can be obtained in a purely two-dimensional band structure. Therefore, we do not include pairing states of the $E_g$ form $d_{xz}+i{d}_{yz}$ [20,44]. We also omit triplet pairing states of $\uparrow \uparrow$ and $\downarrow \downarrow$ type, such as $B_{1u}$ and $B_{2u}$, which appear to be ruled out by Knight shift experiments [17,18,19,20].

3. Results

Figure 4 shows our main results for $T_c$ under different pairing models from Table 1. In Figure 4a, we show the corresponding expected changes in $T_c$ assuming $d_{x^2-y^2}$ pairing. We set the effective d-wave pairing interaction U to give $T_c=1.5$ K in the unstrained case. Applying uniaxial strain by changing the $t_x$ and $t_y$ hoppings, as described above, leads to maxima in $T_c$ as a function of strain for both compressive and expansive uniaxial $\left(100\right)$ strain. There is a slight asymmetry between the expansive and compressive $T_c$ maxima, but, in both cases, the maximum $T_c$ is $3.5$ K or higher, similar to the value seen experimentally [27]. As expected, the peaks in $T_c$ correspond to van Hove peaks in the d-wave-weighted density of states at the Fermi level, $\rho \left({\epsilon }_F\right)$.

Very different behaviour is seen in Figure 4b, in the case of $p_x+i{p}_y$ chiral p-wave pairing. There are two $T_c$ values corresponding to the two different basis functions $sin\left(k_{x}a\right)$ and $sin\left(k_{y}b\right)$. These are degenerate in the unstrained case. Then, under strain, one of these $T_c$ values increases while the other decreases. The change in $T_c$ is linear for small strains, leading to a cusp-like change in $T_c$ as a function of uniaxial strain. As noted above in the introduction, such a cusp-like feature is not seen experimentally [36]. Although the calculated maximum $T_c$ rises for both signs of the strain, there are no clearly defined peaks in $T_c$, and there is a significant asymmetry which is quite different from what is observed [27]. There are also no distinct van Hove-like peaks in the weighted DOS at the Fermi level.

The origin of the different behaviours for $T_c$ and $\rho \left({\epsilon }_F\right)$, shown in Figure 4, is explored more fully in Figure 5. The d-wave-weighted DOS shown in Figure 5a has a distinct van Hove peak, which passess through the Fermi energy at the critical value of strain where the Lifshitz transition occurs. The gap nodes of the d-wave function, shown in the inset, are located along the $\Gamma -X$ directions in the Brillouin zone, and so are far from the Lifshitz transition points $\mathbf{k}=(\pi /a,0,0)$ and $\mathbf{k}=(0,\pi /b,0)$. In contrast, for the $p_x$ and $p_y$ weighted DOS, no van Hove peaks are visible at all. Under compressive $\left(100\right)$ strain, the $p_x$ weighted DOS shows an increase at ${\epsilon }_F$, but no sharp peak. The $p_y$-weighted DOS at ${\epsilon }_F$ decreases with strain with no van Hove peak. Overall, these lead to the increase in $T_c$ for $p_x$ and the suppression of $T_c$ for $p_y$ visible in Figure 4b. The insets to Figure 5b,c show that the absence of any distinct van Hove peak in the weighted DOS is because of the nodal structres of the $p_x$ and $p_y$ functions. In this compressive strain case, where the Lifthitz transition occurs at $\mathbf{k}=(0,\pi /b,0)$, this point obviously lies on the nodal line of $p_x$ ($k_x=0$). The fact that this Lifshitz point is also a nodal point of $p_y$ is more subtle. $p_y$ has an obvious nodal line along $k_y=0$. However, the condition of periodicity in the extended Brillouin zone (${\Delta }_{\mathbf{k}}={\Delta }_{\mathbf{k}+\mathbf{G}}$, where $\mathbf{G}$ is a reciprocal lattice vector) forces an additional nodal line to occur at $k_y=\pi /b$, exactly running through the Lifshitz transition point. The effect of this node is clear in the inset to Figure 5c.

Overall, from the above discussion, it is clear that the $d_{x^2-y^2}$ pairing model provides a much better description of the $T_c$ changes under uniaxial $\left(100\right)$ strain than the chiral $p_x+i{p}_y$ model. However, the pure d-wave case has no TRSB. To include a state with TRSB, we therefore model a $d+ig$ pairing state of the general type proposed by Kivelson et al. [26]. The results for this mixed symmetry state are shown in Figure 6. We can see in the figure that the mixed symmetry pairing state retains the two distinct maxima in $T_c$, consistent with experiment. The second transition where the g-wave component first becomes non-zero is assumed degenerate with the d-wave case for the unstrained case, but then it is suppressed slightly for both compressive and expansive uniaxial strains. For both expansive and compressive strains, in our model, there is a second point of degeneracy where d and g pairing states cross, so that at larger strains, the higher $T_c$ state is a pure g-wave state. Near this crossover point, we can see competition between the two pairing channels where the one with the higher $T_c$ suppresses $T_c$ for the alternative pairing channel. The lack of a van Hove peak in the g-wave channel is apparent from its DOS and nodal structure, as shown in Figure 6b.

4. Discussion

The results presented here show that the changes in $T_c$ of ${Sr}_2$${RuO}_4$ under $\left(100\right)$ uniaxial strain are most closely similar to the experiments [27] for the d-wave case than other scenarios, such as $p_x+i{p}_y$ pairing. The p-wave model does not show distinct peaks in $T_c$ with strain, and has a cusp feature in $T_c$ in the small strain limit, which is not seen experimentally [36]. The pure d-wave case has no separate transition to TRSB, as found by [32]; however the $d+ig$ state, proposed by Kivelson et al. [26] does allow for both the peaks in $T_c$ at critical strains for the Lifshitz transition and a distinct TRSB temperature. While the broad picture in Figure 6a looks similar to these experiments, there are some differences of detail, namely, that the temperature for TRSB is effectively independent of strain experimentally, while, in our model, TRSB is suppressed slowly as a function of uniaxial strain.

Our simple one-band model of the $\gamma$-band is unable to capture more complex multi-band pairing scenarios, such as those proposed in Refs. [40,41,42,43,44]. By examining only two-dimensional pairing states, we are also unable to consider alternative pairing states such as $d_{xz}+i{d}_{yz}$ in our model. Despite these limitations, the simple model has some advantages. Firstly, the link between $T_c$ and a single weighted DOS is simple to understand in a one-band picture, but more complex in the case of multi-band pairing, since the relative weights of DOS on all three bands must be considered. Secondly, the one-band model has the advantage of being essentially parameter-free given the simple band model chosen. The only empirically chosen parameter is the choice of the Fermi liquid interaction parameter $U_i$ in the specific symmetry channel considered, chosen to give the observed $T_c=1.5K$ in the unstrained sample. In a multi-band model, there may be different parameters for intra-band pairing in each band as well as separate intra-band pairing parameters. In our model, the case of the $d+ig$ pairing scenario does require some fine-tuning of parameters to obtain the assumed degeneracy of both pairing states in the unstrained case. It remains to be seen if such degeneracy occurs in more detailed microscopic pictures. The fact that no splitting of this degeneracy is observed experimentally for disordered samples or under hydrostatic pressure [37] remains unexplained, and is beyond the scope of the present paper. The simple one-band van Hove model here is also unable to explain the lack of enhanced $T_c$ under $\left(001\right)$ compressive strain [38], as shown in detail in Ref. [54].