Physica 90B (1977) 96-106 © North-Holland Publishing Company COLLISIONLESS COLLECTIVE MODES IN SUPERFLUID 3He * P. WOLFLE Institut flit Theoretische Physik, Teehnische Universitiit Mi~nchen, D-8046 Garching, West Germany A systematic and complete discussion of the collectivemodes of the BWand ABM states of 3He is given for the colfision- less regime. In addition to the gapless modes associated with the spontaneous breakdown of symmetries there exist a num- ber of pair vibration modes with nonzero energygap in the limit of wave vector tending to zero. The temperature depend- ence and dispersion of these modes as well as the possibility of exciting them is discussed. 1. Introduction The collective modes of a many-body system are of interest because they give detailed information about the macroscopic order in the system as well as the microscopic dynamics. This is in particular true for the superfluid phases of 3He, where several ifew collec- tive modes appear due to the complex structure of the order parameter (for a review see ref. 2). Some of these modes have been studied already by several authors [3-15] employing a number of different techniques, so that the interrelation of these results is not always clear. It therefore seems worth while to give a systematic discussion of all collective modes that can appear in the model superfluid states introduced by Balian- Werthamer (BW) and Anderson-Brinkman-Morel (ABM), currently thought to represent aHe-B and aHe-A, respectively. This is done in the framework of weak-coupling theory. A systematic incorporation of strong-coupling effects in the dynamics seems to be hopeless at this point, even within the somewhat sim- pler spin fluctuation model. However, it is almost cer- tain that the structure of the collective modes in each state is not changed by strong coupling effects. Further- more, there is reason to believe that a good part of the strong-coupling corrections may be absorbed into re- * Part of a talk presented at the Symposiumon Superfluid 3He, Sussex University, 22-25 August 1976. The other parts on sound propagation at ultrahigh frequencies and the kinetic coefficient of second viscositywill be reviewed elsewhere [ll. normalized coupling constants and gap parameters. The collective modes of a many-body system may be classified according to the generating principles to which they owe their existence (table I). The most familiar class of modes comprises the hydrodynamic modes which are a consequence of the local conserva- tion laws for mass, momentum, etc. Whenever the system undergoes a continuous phase transition asso- ciated with a spontaneous breakdown of symmetry, there appear additional quasiconserved variables pos- sessing a gapless excitation spectrum due to the de- generacy of the ground state with respect to the cor- responding symmetry operation. These Goldstone modes may be hydrodynamic or nonhydrodynamic (collisionless), according to whether the thermal excita- tions of the system are in a state of local equilibrium or not. The third class of modes is generated by a quantum mechanical coherence property of the many- body wave function, which provides a certain rigidity of the equilibrium state against small deviations. The (diagonal or off-diagonal) self-consistent field, which is established in this way, may oscillate around equili- brium in many different modes, depending on the dy- namics of the system. In superfluid 3He all three types of modes occur, which makes it a particularly interest- ing system from this point of view. Only the coUisionless regime will be considered here, i.e. the part of the frequency-temperature plane where ~o is large and T is small. The collective modes in the opposite hydrodynamic regime are well under- stood. Their structure follows from phenomenological considerations [16-18]. 96