A theoretical description of the new phases of liquidHe3

This paper reviews the theory of anisotropic superfluid phases and its application to the new A and B phases of liquid $^{3}\mathrm{He}$. It is tutorial in nature and advanced formal techniques are avoided; even the formalism of second quantization is not required. After an initial discussion of the Fermi-liquid theory of Landau and its application to the normal phase of liquid $^{3}\mathrm{He}$, the idea of instability against formation of Cooper pairs is introduced. The effective interaction in liquid $^{3}\mathrm{He}$ is considered, with emphasis on the spin-dependent interaction arising from virtual spin polarization of the medium ("spin fluctuation exchange"). Next, a self-contained discussion of the "weak-coupling" BCS theory as applied to anisotropic superfluids is given, with special attention to the "Ginzburg-Landau" region close to the transition temperature. Formulas are derived for the specific heat, spin susceptibility, normal density tensor, and static spin-dependent correlation properties of superfluids with both singlet and triplet pairing: In the triplet case the ideas of "spin superfluid velocity" and "spin superfluid density" are also introduced. After a preliminary comparison of the weak-coupling theory with experiment, it is shown that feedback effects due to the modification, by formation of Cooper pairs, of the effective interaction connected with spin fluctuation exchange can produce results which are qualitatively different from those of the weak-coupling theory. An attempt is made to reformulate recent graph-theoretical treatments of this phenomenon in a more elementary language, and considerations based on possible invariant forms of the free energy are also introduced. The properties of the so-called Anderson-Brinkman-Morel and Balian-Werthamer states, which are commonly identified with $^{3}\mathrm{He}$-A and B, respectively, are studied in detail. Next, the effects which tend to orient the Cooper pair wave function in a given experimental situation are discussed; in this context the form of the free energy terms arising from spatial variation of the wave function is explored. A semiphenomenological theory of the nuclear magnetic resonance properties is developed and applied in particular to the case of unsaturated cw resonance; the analogy with the Josephson effect is emphasized. The question of relaxation and linewidths is also briefly discussed. A partial account is given of the theory of finite-wavelength collective oscillations, with particular reference to first, second, and fourth sound and spin waves. The splitting of the A-normal transition in a magnetic field is considered, with special attention to the possibility it offers of testing theories of the "spin fluctuation" type. Finally, a brief assessment is made of the extent to which the current experimental data support the conventional identification of $^{3}\mathrm{He}$-A and B and the spin fluctuation theory, and some outstanding problems and possibilities are outlined. Subjects not discussed include "first-principles" theories of the effective interaction in $^{3}\mathrm{He}$ collective excitations in the "collisionless" regime, and the problem of ultrasonic absorption, "orbit waves," and the theory of the kinetic coefficients.