The equal-spin pairing (ESP) state, the so-called A-like phase, of superfluid 3 He in aerogels is studied theoretically in the Ginzburg-Landau (GL) region by examining thermodynamics, and the resulting equilibrium phase diagram is mapped out. We find that the ABM pairing state with presumably quasi long-ranged superfluid order has a lower free energy than the planar and "robust" states and is the best candidate of the A-like phase with a strange lowering of the polycritical point (PCP) observed experimentally.
PACS numbers:It is an important subject on liquid 3 He in aerogel [1,2] to understand whether or not the pairing symmetry and other details of superfluid states are changed due to impurity effects provided by the aerogel background. At present, it will be clear that the B-like phase is, just as in bulk 3 He, the BW pairing state with superfluid long range order (LRO), while it is a matter under debate whether or not the equal-spin pairing (ESP) state, the so-called A-like phase, near the superfluid transition temperature T c (P ) is the ABM state.By assuming that the orientation of the l-vector in the ABM pairing state should lose its LRO due to the impurity disorder, Volovik [3] has speculated that the A-like phase in zero magnetic field should be a glass state with the ABM pairing but with no superfluid rigidity (i.e., ρ s (q = 0) = 0). Fomin [4] has argued with no free energy calculation that another ESP state, "robust" state, should be the A-like phase in aerogel. Among experimental suggestions on the A-like state, the lowering [1, 2, 5] of the polycritical pressure (PCP) at which T c (P ) and the A-B transition line T AB (P ) merge is a key observation: Although the range T c − T AB of the equilibrium A-like state at a fixed pressure P above the bulk PCP is narrower in aerogels with lower porosity, the pressure range over which this state is detected seems to be wider with decreasing the porosity [2,5]. In this paper, we examine candidates of the A-like phase in equilibrium by calculating the superfluid free energy. We find that, due primarily to effects of quenched disorder on the amplitude of the pair-field (superfluid order parameter) A µ,i , the ABM state is the best candidate of the A-like state with the above-mentioned properties.To derive the Ginzburg-Landau (GL) free energy functional microscopically, we start from the quasiparticle hamiltonian with an effective interaction mediated by spin fluctuations [6] and an impurity term H imp = d 3 r σ ψ † σ (r) u(r) ψ σ (r), where u(r) is an impurity potential with zero mean. For a moment, impurity scattering effects inducing inhomogenuities of A µ,i will be neglected. If the impurity effects on the quasiparticles are included just through their relaxation time, the GL functional is, as usual, expressed [7] by assuming a p-wave pairing interaction as the sum of H 0 and H gr , whereandis the mean field transition temperature in bulk at a fixed P , and τ −1 = 2πN (0) |u p−p ′ | 2 ang is a relaxation rate, where the overbar and · · · ang denote the avera...