In general, the phases of a singlet superconductor and a FM are incompatible, since the exchange field of the FM destroys the superconductivity by aligning the antiparallel spins of the electrons in singlet Cooper pairs. This pair-breaking effect makes the homogeneous coexistence of both phases almost impossible. On the other hand, the coexistence of FM and TSC states is more favorable, as the exchange field is only pair breaking when it is perpendicular to the Cooper pair spins. Therefore, one can expect interesting effects, since the orientation of the FM moment relative to the TSC vector order parameter is now a crucial variable. In addition to the pair breaking, spin-flip reflection processes at the interface with the FM scatter the triplet Cooper pairs between the spin ↑ and ↓ condensates, setting up a Josephson-like coupling between them . The resulting "spin Josephson effect" is manifested as a spontaneous spin current in the TSC normal to the TSC/FM interface. Both the pair breaking and spin Josephson coupling make significant contributions to the free energy of a TSC-FM junction through the proximity effect, interface electronic reconstruction, and the variation of the TSC gap. Although these contributions depend upon the direction of the FM exchange field, the two effects do not necessarily act constructively: while pair breaking is always absent for a moment perpendicular to the TSC vector order parameter, the effective Josephson phase difference can vanish for parallel and perpendicular configurations, depending on the orbital pairing state.
We examine a lattice model of the TSC-FM junction shown in Fig. 1. The lattice size is (L+1)×(L+1) with periodic boundary conditions imposed along the direction parallel to the interface. Indicating each site by a vector i=(ix,iy) with ix and iy being integers ranging from –L/2 to L/2, we consider the Hamiltonian 
H = Σ<ij>,σ tij (ciσ+ cjσ + h.c.) − μ Σiσ niσ − Σ<ij>∈TSC V(ni↑nj↓ + ni↓nj↑) − Σi∈FM h·si
Fig.1: Schematic diagram of the two-dimensional TSC-FM junction. The FM region is located at x<0, while the TSC is realized for x> 0. The magnetization M of the FM is collinear to the exchange field h and forms an angle Φ with the d-vector of the TSC, which defines the z-axis. We study TSC states with px, py, and px+ipy symmetry.[less]
Fig.1: Schematic diagram of the two-dimensional TSC-FM junction. The FM region is located at x<0, while the TSC is realized for x> 0. The magnetization M of the FM is collinear to the exchange field h and forms an angle Φ with the d-vector of the TSC, which defines the z-axis. We study TSC states with px, py, and px+ipy symmetry.
where ciσ is the annihilation operator of an electron with spin σ at the site i, niσ=ciσ+ciσ is the spin-σ number operator, and si=Σss’ cis+sss’ cis’ is the local spin density. The lattice is divided into three regions: the FM subsystem for ix<0, the TSC subsystem for ix>0, and the interface at ix=0. The chemical potential m is the same across the lattice. The order parameter of the TSC, the so-called d-vector, encodes the spin structure of the Cooper pairs: it is defined as d=(1/2)×(Δ1−Δ-1)x−(i/2)(Δ1+Δ-1)y+Δ0z, where ΔSz is the gap for triplet pairing with the z component of the spin Sz=−1, 0, 1. Since the TSC state is invariant under spin rotations about the d-vector, h can be restricted to the x-z plane, i.e., h=h(sin(Φ),0,cos(Φ)). We obtain a single-particle Hamiltonian HMF from Eq.(1) by decoupling the interaction term and solving self-consistently for the mean-field amplitudes Δij=<ci↑ci↓>. We then calculate the condensation energy EΔ of the TSC and the Gibbs free energy F of the junction.
In Fig. 2 we present the pairing amplitude profile near the interface for h=1.5, tint=1, and several different values of 0≤Φ≤π/2. This is determined by minimizing the Gibbs energy functional with respect to the pairing amplitudes for a fixed angle. Independent of the orbital symmetry, the proximity effect in the FM smoothly evolves from a monotonic decay at Φ=π/2 to a damped oscillating behavior at Φ=0. On the other hand, the pairing amplitude in the TSC side of the interface strongly depends upon both the angle Φ and the orbital symmetry of the TSC. For a TSC with px orbital symmetry, the pairing amplitude near the interface is reduced as the exchange field is tilted from parallel to perpendicular with respect to the d-vector (Fig. 2(a)); the opposite behavior is observed for a py TSC, although the effect is less pronounced (Fig. 2(b)). The chiral px±ipy TSC evidences both trends: decreasing Φ from π/2 to 0 enhances the real (px) part of the gap (Fig. 2(c)), but suppresses the imaginary (py) part (Fig. 2(d)). Competition between the two gap components enhances their variation with f compared to the time-reversal symmetric states.
The pair breaking due the spin-spin coupling alone cannot explain the different f dependence of the py and px gap profiles. This instead originates from the spin-flip reflection of triplet Cooper pairs at the interface with the FM, which is crucial for the spin Josephson effect. During this scattering process an incident Cooper pair acquires a spin- and orbital-dependent phase shift .
Fig. 3: Sketch of the most favorable magnetic (small red arrow) configurations with respect to the orbital symmetry and the d-vector (large blue arrow) of the TSC as well as the character of interface transparency.[less]
Fig. 3: Sketch of the most favorable magnetic (small red arrow) configurations with respect to the orbital symmetry and the d-vector (large blue arrow) of the TSC as well as the character of interface transparency.
Finally, we have calculated Gibbs free energy F directly from HMF. The minimum of F fixes the stable moment orientation. For the px orbital symmetry, the profile exhibits a single minimum at Φ=0 and a maximum at Φ=π/2, and vice versa for the py TSC. The stable magnetic orientation is therefore parallel (perpendicular) to the d-vector if the antinodes of the p-wave TSC gap are perpendicular (parallel) to the interface. Thus, the orbital pairing state in the bulk TSC plays a critical role in fixing the stable orientation of the magnetization in the FM, which is summarized by the sketch in Fig. 3. For the time-reversal symmetric gaps, the easy axis in the FM originates from the maximization of the TSC’s condensation energy. On the other hand, the orbital frustration of the condensation energy in a chiral TSC leads to a magnetic configuration with a first-order transition between the perpendicular and parallel configurations as a function of the exchange field .