Quasiclassical determination of the in-plane magnetic field phase diagram of superconducting ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$

We have carried out a determination of the magnetic field temperature $\left(H\text{−}T\right)$ phase diagram for realistic models of the high-field superconducting state of tetragonal ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ with fields oriented in the basal plane. This is done by a variational solution of the Eilenberger equations. This has been carried for spin-triplet gap functions with a $\mathit{d}$ vector along the $c$ axis (the chiral $p$-wave state) and with a $\mathit{d}$ vector that can rotate easily in the basal plane. We find that, using gap functions that arise from a combination of nearest- and next-nearest-neighbor interactions, the upper critical field can be approximately isotropic as the field is rotated in the basal plane. For the chiral $\mathit{d}$ vector, we find that this theory generically predicts an additional phase transition in the vortex state. For a narrow range of parameters, the chiral $\mathit{d}$ vector gives rise to a tetracritical point in the $H\text{−}T$ phase diagram. When this tetracritical point exists, the resulting phase diagram closely resembles the experimentally measured phase diagram for which two transitions are only observed in the high-field regime. For the freely rotating in-plane $\mathit{d}$ vector, we also find that an additional phase transition exists in the vortex phase. However, this phase transition disappears as the in-plane $\mathit{d}$ vector becomes weakly pinned along certain directions in the basal plane.

Figure 1

Upper critical fields for $ϵ=0$ for the fields along the (1,0,0) and (1,1,0) directions. The field in units $2{\pi }^2{T}_c^2∕\left(e{v}_F{v}_c\right)$.

Figure 2

Upper critical fields for $ϵ=\infty$ and for the fields along the (1,0,0) and (1,1,0) directions.

Figure 3

Upper critical fields for $ϵ=10$ and for the fields along the (1,0,0) and (1,1,0) directions. The two upper critical fields almost identical.

Figure 4

Phase boundaries for $ϵ=10$ and for the field along the (100) direction.

Figure 5

Specific heat as a function of temperature for a fixed field $H∕H_{c2}^0=0.21$ and for $ϵ=10$.

Figure 6

Phase diagram showing a tetracritical point. Between $T_c$ and the tetracritical temperature there are two phase transitions, shown in the inset, that are difficult to distinguish from each other.

Figure 7

Specific heat as a function of temperature for fixed magnetic fields.

Figure 8

Specific heat for $ϵ=10$ for the field along the (1,0,0) direction, with fixed $H∕H_{c2}^0=0.21$ for different values of $g_2$ (which is measured here in units $g_z$).