Persistent Currents in Many-Boson Systems

A theoretical study of persistent currents in liquid ${\mathrm{He}}^4$ is given, in which a system of $N$ interacting bosons confined to a cylindrical container is chosen as a model. It is argued that these currents are metastable and a general model-independent stability criterion valid for all temperatures is derived. In this formulation it is essential that there is a single-particle state ${\varphi }_c$ with nonzero angular momentum $l_c$ which is occupied by a macroscopic number [$O\mathrm{}\left(N\right)\mathrm{}$] of particles. The stability criterion is applied to the study of two soluble models, one, an independent-particle model studied for all temperatures, the other a quasiparticle model studied for $T=0\mathrm{}$°K. For both models persistent currents are possible at $T=0\mathrm{}$ provided that the interactions between particles are sufficiently strong and $l_c$ does not exceed a characteristic critical value $l_M$. The quantity $l_M$ is calculated for different choices of single-particle functions, and in particular for the solutions of a Hartree equation, which is relevant to a description of line vortices with quantized circulation. The density ${\rho }_c\mathrm{}\left(T\right)\mathrm{}$ of particles macroscopically occupying ${\varphi }_c$ at a temperature $T$, as well as the critical temperature $T^{*}$ above which no persistent currents are possible, are calculated for the independent-particle model.