Spin-orbit and anisotropic strain effects on the electronic correlations in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$

We present an implementation of the rotationally invariant slave boson technique as an impurity solver for density functional theory plus dynamical mean field theory (DFT+DMFT). Our approach provides explicit relations between quantities in the local correlated subspace treated with DMFT and the Bloch basis used to solve the DFT equations. In particular, we present an expression for the mass enhancement of the quasiparticle states in reciprocal space. We apply the method to the study of the electronic correlations in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ under anisotropic strain. We find that the spin-orbit coupling plays a crucial role in the mass enhancement differentiation between the quasi-one-dimensional $\alpha$ and $\beta$ bands, and on its momentum dependence over the Fermi surface. The mass enhancement, however, is only weakly affected by either uniaxial or biaxial strain, even across the Lifshitz transition induced by the strain.

Figure 1

Density of states projected on the $t_{2g}$ Wannier orbital, calculated using LDA, LDA+RISB, and LDA+DMFT(CTQMC).

Figure 2

(a) LDA density of states of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ projected on the Ru-$t_{2g}$ orbitals and within the energy window ${\mathcal{W}}_s$ considered in this paper to construct the Wannier orbitals. (b) Band structure as obtained with DFT and with DFT+RISB, including the SOC in both cases.

Figure 3

Quasiparticle weight on the Fermi surface calculated without including the spin-orbit interaction (a) or including it (b). (c) Projection of the $\alpha$ sheet's Bloch states at the Fermi surface on atomic multiplets, as a function of the angle ${\theta }_{\alpha }$ defined in (b). (d), (e) Idem as (c) for $\beta$ and $\gamma$ sheets as a function of the angle ${\theta }_{\beta \gamma }$.

Figure 4

(a) $h/d$ as a function of $a$, where $h$ is the height of oxygen octahedron and $d=a/2$ is the distance betwen the rutenium atom and its first neighbor oxygen. Empty circles and red filled squares correspond to uniaxial and biaxial stress, respectively. (b) Occupancy of Wannier orbitals according to LDA. Effective bandwidth of (c) $xz$, $yz$, and (d) $xy$ orbitals, as defined in the text.

Figure 5

Quasiparticle weight of the $\{xz,yz\}$ (a) and $xy$ orbitals (b) as a function of lattice parameter $a$ calculated using RISB at $T=5K$. Empty red and filled blue symbol datapoints are obtained without and with SOC, respectively. Triangle symbols are CTQMC results without SOC. (c), (d) Imaginary part of the self-energy calculated with CTQMC at different temperatures. The interaction parameters are $U=2.3\text{eV}$ and $J=0.4\text{eV}$.

Figure 6

Evolution of the renormalized Fermi surfaces (calculated with SOC) for $a$= 3.862, 3.890, $3.929\text{Å}$ (from left to right), showing that the system undergoes a Lifshitz transition close to $a$= 3.890 Å.

Figure 7

(a) Quasiparticle weight of the orbital $\{xz,yz\}$ as function of the stress ${\epsilon }_{xx}$ at $T=5\mathrm{K}$. (b) Quasiparticle weight of the orbital $xy$ as function of the stress ${\epsilon }_{xx}$. (c)–(e) Evolution of the renormalized Fermi surfaces for the percentage uniaxial stress ${\epsilon }_{xx}=0,-0.2$ and $-0.5\%$ (from left to right), showing that the system undergoes a Lifshitz transition close to ${\epsilon }_{xx}=0.2\%$ around the $(\frac{\pi }{2a},0)$ point.