PH Y SIGA L RE VIE% 8 VOLUME 18, NUMBER 3 1 AUGUST 1978 Two possible types of superfluidity in crystals Mario Liu~ Bell Laboratories, Murray Hill, New Jersey 07974 (Received 1 September 1977) Using a modified procedure to derive hydrodynamics, it is shown from symmetry considerations that a superfluid velocity 8 which is invariant under a Galilean transformation will be driven by the temperature: v' + (kslm) QT = 0. A crystal with such a velocity is able to sustain a persistent entropy flux rather than a mass current and has a propagating mode connected to temperature fluctuations. On the other hand, only a crystal with a P which behaves like a true velocity under a Galilean transformation, will be able to sustain a persistent mass current and have vacancy propagation as its Goldstone mode. This spectrum differs, however, from that given by previous authors. I. INTRODUCTION Numerous theoretical investigations' ' of recent years have advanced the opinion that some form of superfluidity in crystals is a distinct possibility. As ide from microscopic approaches such as ex- plicitly constructing a Bose condensed wave func- tion which complies with a crystalline ordering, ~ ' there were also attempts to study the problem from macroscopic symmetry considerations. En- visioning a quantum crystal capable of sustaining a persistent (liquidlike) flow of defects, Andreev and Lifshitz derived its equations of motion in close analogy to the two-fluid hydrodynamics, " and predicted a Goldstone mode, the vacancy propagation, which has a spectrum similar to fourth sound in He II. In addit'ion, the velocities of elastic waves are found to be modified by a common factor proportional to the superfluid density, making it possible to detect superfluidity in a crystal by measuring the spectrum of elastic waves. Recently, Saslow' suggested that, due to the presence of the lattice as a preferred inertial frame the superfluid velocity v' shouldbe invariant u~der a Galil. ean transformation, i. e. , v' should transform as a velocity difference. This alters one of the basic assumptions of the hydrodynamics as derived by Andreev and Lifshitz and makes a re- derivation necessary. Saslow' found that the structure of the hydrodynamic equations remains the same, and the only modification occurs in the thermodynamic relation between the superfluid velocity and the momentum density. But that was enough to cancel the changes in. elastic waves: they now remain essentially unaltered going through the superfluid transition, making the measure- ment of elastic waves unsuitable for the detection of superfluidity. The purpose of the present paper is to show that (i) only a "Galilean" v'(i. e. , a v' which behaves as a true velocity under a Galilean transformation: v' -v'+ &5) will lead to superfluidity as characterized by a persistent mass flow; while (ii) an "invariant" v' (namely, v'-v', as employed by Saslow), will give rise to a different kind of superfluidity- a persistent (or super) entropy flow. We will see that the modification made by Saslow suffers from internal inconsistency, and that the Andreev and Lifshi. tz equations are the only possible ones to describe superfluidity. of mass flow in a crystal. (When rectified, Saslow s hydrodynamic equations are consistent with those of Andreev and Lifshitz. ) We shall also see that (iii) the collective modes as calculated by Andreev and Lifshitz are not gen- erally correct. This is easily recognized by the following argument. Since their equations of motion are given by generalizing two-fluid hydro- dynamics, we must be able to obtain the spectrum of first (and second) sound in a superfluid liquid by setting to zero the appropriate elastic coef- ficients. Now, because the elastic waves as cal- culated by them are changed by a common fac- tor, independent of the thermal expansion coef- ficient, one has to conclude that first sound, too, is changed by the same factor. This, of course, is in contradiction to the mell, -known fact that first sound is only altered to the extent that the thermal expansion coefficient is not neglected. ' By the same token, vacancy propagation can- not be given by a fourth-sound-like spectrum, but must be given by a generalization'of sec- ond-sound spectrum. (The error can be traced back to the fact that Andreev and Lifshitz employed a perturbation calculation. , in which the perturba- tion can be comparable to the zeroth-order term. ) Our results will show that the change in the elastic waves is still proportional to p', although mark- edly different for transverse and longitudinal excitations. They remain a valid indicator for any superfluidity in crystals. We then go on to explore (iv) the consequences 1165 1978 The American Physical Society