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It is assumed that the phase of IN> has been so chosen such that the matrix elements of l/J are real. The important feature to note is that states of different N have definite relative phases (instead of random relative phases as in an ensemble). The states labeled by different cP are not independent, however, because they are not orthogonal to each other:

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(13.56) are essentially orthogonal, and thus may be considered independent. Accordingly, in each of our coarse-grained cells, the values of cP and N are to be averaged over respective intervals ~cP and tiN, which are small on a macroscopic scale, and at the same time satisfy (13.56). The variables cP and N are then semiclassical coordinates for the coarse-grained cell, which we call the "superfluid coordinates." Their values are defined only to accuracies within the "phasespace" cells shown in Fig. 13.6. When the superfluid coordinates are uniform throughout the system, the superfluid is in equilibrium. Any variations in the coordinates induce a flow of

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'This is an uncertainty relation similar to that between energy and time (t1 E t1 t ::> n), there being no operator whose eigenvalue is cp.

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SPECIAL TOPICS IN STATISTICAL MECHANICS

Fig. 13.6 Number of particles Nand superflUI': phase </> obey uncertainty relation Ii N Ii</> ~ 1. Sho\\!: here are minimal cells in "phase space."

the superfluid, which tends to restore uniformity. This is because any nonun!' formity in density will obviously cause the energy to rise above its absolute minimum. A nonuniform phase cp will also make the energy rise, because, while the system is invariant under a global gauge transformation (i.e., constant CP), it 1not invariant under a local one (i.e., space-dependent cp.) If the number of particles in a cell changes by !::.N during the time inter\ai !::.t, then the energy of the cell is uncertain by an amount !::.E ::::: Ii/!::.t. Thus

(13.57 1

from which we obtain

dcp aE Ii- = = JL dt aN dN aE Ii-= dt acp

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where JL is the chemical potential. Another way to derive these equations is to note that the energy E( cp, N) of a cell is an effective classical Hamiltonian, where licp and N are canonically conjugate coordinates. The equations above are the Hamiltonian equations of motion. These are Anderson's equations* governing superfluid flow. Although we have derived them at absolute zero, they continue to apply at finite temperatures, with the partial derivatives taken to be adiabatic derivatives.

Vortex Motion

The superfluid order parameter (o/(x) > must be a continuous function of x. In particular, its phase cp(x) must change by at most an integer multiple of 27T when we traverse a closed loop in the superfluid. This means that the vorticity in

P. W. Anderson, Rev. Mod. Phys. 38, 298 (1966). This paper contains applications of these equations.

SUPERFLUIDS

superfluid flow must be quantized:

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(13.59)

(n=0,1,2, ... )

Obviously n = 0 if the closed loop can be continuously shrunken to a point. The values n 4= 0 occur if the superfluid flows in a multiply connected region, for example, between two cylinders. The multiply connected region may also be created by the liquid itself: The superfluid can be expelled from a self-formed tube filled with normal fluid, and will flow around this tube with nonzero vorticity, i.e., n 4= 0 in (13.59). The tube, called the vortex core, could be a line whose ends terminate on boundary surfaces or it could form a ring. The excitation is called a vortex line in the first case, and a vortex ring in the second. Both types of vortex motion have been observed experimentally. * The coarse-grained description of the superfluid obviously breaks down in the vicinity of the vortex core. The core diameter is a phenomenological parameter that can be calculated only by considering the microscopic dynamics. In the approximation of incompressible superfluid flow in the presence of vorticity, the equations for the superfluid velocity are