Equal-spin pairing state of superfluidHe3in aerogel

Abstract

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...

Pairing states of superfluidHe3in uniaxially anisotropic aerogel

“…As argued in Ref. [3], the scattering amplitude |u k | 2 in an aerogel, where u k is the Fourier transform of u(r), should have a highly anisotropic and complicated momentum dependence, reflecting local anisotropy in aerogels. However, as far as the aerogel is globally isotropic, the local anisotropy is not reflected in averaged quantities such as |u k | 2 , where the overbar denotes the average over the impurity configuration due to the aerogel.…”

“…In contrast, the bulk liquid 3 He has an isotropic Fermi surface, and hence, there is a degeneracy in the transition point between different pairing states at least when the fluctuation is neglected [2,3]. Recently, possible pairing states of superfluid 3 He in globally isotropic aerogels were examined by comparing the free energy in the Ginzburg-Landau (GL) region between different states, and it was found [3] that, after averaging over the quenched disorder brought by the aerogel structure, the pairing states to be realized are unaffected by the locally anisotropic scattering events due to the aerogel. However, situation may change if, as in the superconducting case mentioned above, a global anisotropy [4,5] is introduced externally in the aerogel structure.…”

“…Due to such an intrinsic and global anisotropy, a specific pairing symmetry with the highest temperature of Cooper instability is realized in the case of anisotropic superconductivity with no degeneracy between different pairing states. In contrast, the bulk liquid 3 He has an isotropic Fermi surface, and hence, there is a degeneracy in the transition point between different pairing states at least when the fluctuation is neglected [2,3]. Recently, possible pairing states of superfluid 3 He in globally isotropic aerogels were examined by comparing the free energy in the Ginzburg-Landau (GL) region between different states, and it was found [3] that, after averaging over the quenched disorder brought by the aerogel structure, the pairing states to be realized are unaffected by the locally anisotropic scattering events due to the aerogel.…”

“…In the stretched case, a 1d-like pairing state is favored at least near T c , and actually the pure polar pairing state should be realized accompanied by a 2nd order transition to a deformed ABM state at a lower temperature. This is a rare situation in which a new pairing state is expected to occur as a 3d superfluid phase of 3 He. This research was preliminarily reported elsewhere [7].…”

Stripe order in superfluid3He confined in narrow cylinders

“…In films of the 3 He-B phase, the surfaceinduced distortion of the superfluid gap and a resultant transition into the A/planar phase have been suggested [2][3][4][5][6]. In aerogels, pinning effects in the ABM pairing state have been discussed [7][8][9][10][11][12], and the polar pairing state which does not exist in the bulk is expected to appear in uniaxially stretched mediums [9,13]. Also in narrow cylinders, occurrence of the polar state is theoretically predicted [14], but so far, any signature of this state has not been observed in experiments [15][16][17][18][19].…”