52 14 v 2 1 5 Ju n 19 99 String Picture of Bose-Einstein Condensation

A nonrelativistic Bose gas is represented as a grand-canoni cal ensemble of fluctuating closed spacetime strings of arbitrary shape and le gth. The loops are characterized by their string tension and the number of time s th y wind around the imaginary time axis. At the temperature where Bose-Einstei condensation sets in, the string tension, being determined by the chemical pot ential, vanishes, the system becomes critical, and the strings proliferate. A com parison with Feynman’s description in terms of rings of cyclicly permuted bosons sh ows that the winding number of a loop corresponds to the number of particles conta ined in a ring. E-mail: schakel@physik.fu-berlin.de 1 The first experimental realization of Bose-Einstein conden sation (BEC) in a cloud of weakly interactingRb atoms trapped in a magnetic field by a group in Boulder, Colorado [1], and the second definite observation of BEC in a s ystem of Na atoms at MIT [2], turned this field into one of the most active research areas in contemporary condensed matter physics. This recent burst of interest mot ivated us to consider once more the fundamental properties of BEC. At present there exist at least three different description s f this phenomenon. In the first, and intuitively probably the simplest one, BEC is u nderstood as the condensation of a finite fraction of the total number of particles pres ent in the zero-momentum ground state [3]. The second description, due to Bogoliubov [4], uses the methods of quantum field theory [5]. The formation of a Bose-Einstein co densate is here signaled by a nonzero expectation value of the field describing the par ticles in the system. The third description, due to Feynman [6], is based on path integ rals [7]. BEC is now understood as the occurrence of large rings containing many bo sons which are cyclicly permuted in imaginary time [8, 9]. Using a mapping onto a cert ain problem of classical statistical mechanics [10] to numerically evaluate t he path integral, Ceperley and Pollock [11] showed that this picture also provides a powerf ul computational tool with which even a strongly interacting system like superfluid He can be accurately described (for a review, see Ref. [12]). The classical system i s a chain of beads connected to springs. In this Letter we discuss an alternative, string picture of B EC. This description is, like Feynman’s, based on the spacetime approach to quantum m echanics, i.e., on path integrals. The specific path-integral representation we ar rive at was first put forward by Montroll and Ward [13]. Physically, it describes a grandcanonical ensemble of fluctuating closed spacetime strings of arbitrary shape and length. We study the critical behavior of this loop gas close to the temperature Tc where Bose-Einstein condensation sets in, and show that the loops proliferate below Tc. That is to say, the loop gas undergoes a Hagedorn transition at Tc. The critical behavior we found here is similar to that of vort ex loops in superfluids [14] and in type-II superconductors [15]. (For a detaile d account of the so-called dual approach to superfluidity and superconductivity in whi ch the vortex loops are the elementary excitations, the reader is referred to the textb ook y Kleinert [16].) Our starting point is the Lagrangian of a nonrelativistic fr ee scalar field theory in Euclidean spacetime x = (τ,x), with τ the imaginary time [5]: L = φ ( h̄∂τ − μ − h̄ 2m ∇ ) φ. (1) Here,φ is the field describing the scalar particles, ∂τ = ∂/∂τ , μ is the chemical potential which accounts for a finite particle number density, and −h̄∇2/2m, with m the mass of the particles, is the kinetic energy operator. Depen ding on whether the particles are bosons or fermions, the field φ is an ordinary commuting or an anticommuting Grassman field. We adopt the imaginary-time approach to thermal field theory , where the time variable τ is taken to be of finite extent 0 ≤ τ ≤ h̄β, with β the inverse temperature β = 1/kBT (kB is Bolzmann’s constant) [17]. As a result, the energy variab le conjugate toτ takes on discrete values. Depending on whether φ describes bosons or 1 fermions, the field is periodic or antiperiodic in τ , φ(h̄β,x) = ±φ(0,x), and can be expanded in a Fourier series as: φ(x) = 1 h̄β ∞