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We now turn to the low-temperature limit of electrons in Landau levels. The electrons will tend to occupy the lowest available levels. As the magnetic field is decreased, each Landau level can accomodate fewer electrons because the degeneracy is decreased. Consequently, some electrons will be forced to jump up to a higher level. This causes the de Haas-Van Alphen effect, the oscillation of the low-temperature magnetic susceptibility as the magnetic field is decreased. To study this effect in a simple context we shall assume kT liw o, so that we can set T = O. We shall also ignore the motion in the z direction. * Our problem is to calculate the ground state energy of a two-dimensional electron system of total area L 2 in a uniform magnetic field H. We write the and their degeneracy g in the following notation: Landau levels

FIGURE 5.12 This close-up of the symbol NTRI illustrates the detail of how the ribbons twist to change direction.

j = 2/l oH(j + t), /lo = eli/2mc (11.83) H o = nhc/e g = NH/Ho , where n = N / L 2 is the number of electrons per unit area. The field H o is the value of H above which the Landau level can hold all the N particles. If H/Ho > 1, then all particles can be accommodated in the lowest Landau level, and the ground state energy per particle is

Eo/N = /loH, (H/Ho > 1) (11.84) If H < H o, then some particles will have to occupy higher levels. Suppose His such that the j lowest Landau levels are completely filled, the (j + l)th level is partially filled, and all higher levels are empty. The condition for H is

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A superfluid is a fluid that flows without dissipation-a phenomenon that never ceases to cause excitement, because it defies common sense and raises the hope that one might capitalize on it. The classic example is liquid He 4 in its low-temperature phase, which can flow through the tiniest cracks without viscosity. Equally remarkable is a superconducting metal in which" supercurrents" can be maintained without an applied emf for months, and presumably indefinitely in principle. In this case one thinks of the charge carriers (the Cooper pairs) as a superfluid. Other examples include spin-aligned atomic hydrogen, which experimenters can now create in the laboratory. In this chapter, we shall concentrate on liquid He 4 as the prototype of a superfluid. The essence of superfluidity, as we shall see, is the existence of a condensate characterizable by a nonvanishing superfluid order parameter.

Helium is the only substance known to resist solidification under atmospheric pressure down to the lowest temperatures observed, and presumably to absolute zero. To solidify it requires an external pressure of at least 25 atm. This happy circumstance furnishes us with two naturally occurring quantum liquids, He 4 and its less abundant isotope He 3 The qualitative reasons for the fluidity of helium are two: (a) The molecular interactions between He atoms are weak, as evidenced by the fact that He is a noble gas; (b) the mass of He is the smallest among the noble gases. These circumstances lead to a large zero-point motion of the He atoms, so that it becomes impossible to localize the atoms at well-defined lattice sites. To understand these reasons, we must first have a few facts. The potential energy v( r) between two He atoms separated by a distance r has been calcui"ted by Slater and Kirkwood* on the basis of the electronic

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*J. C. Slater and J. G. Kirkwood, Ph s. Rev. 37, 682 (1931).

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