Absence of Fragmentation and Broken Spatial Symmetry in the Ground State of Dilute Bose Gases

We show rigorously that condensate fragmentation cannot occur in dilute Bose systems unless forced by exact specification of the values of conserved quantities. We also prove that, relatedly, the symmetric ground state of a dilute Bose gas with positive scattering length in a cylindrical trap is globally stable, and with negative scattering length locally stable, a result with nontrivial implications for the ground state structure in highly anisotropic traps. The discovery of Bose-Einstein condensation (BEC) in atomic gases of alkali atoms [1] has lead to re-examination of a number of novel yet unconfirmed ideas for Bose systems, including the possibility of “condensate fragmentation,” where the system has more than one macroscopic condensate. Nozières and Saint James [2], first putting forth this concept, noticed that fragmentation of a homogeneous dilute Bose gas with negative scattering length (a < 0) (an “attractive” Bose gas) is favored by the exchange term in the interaction energy, although ultimately prevented by other energy costs. Motivated by this observation, Wilkin et al. [3], studying an exceedingly weakly interacting harmonically confined Bose gas, found that rotation induced fragmentation. Recently, Elgarøy and Pethick [4] examining attractive Bose gases in cylindrical potentials for realistic interactions, concluded via a variational approach that even though a cylindrical trap can prevent the collapse of an attractive gas, 1 it cannot cause fragmentation within the family of states considered. Their argument, while strong, is limited to the variational family in question, and does not definitively eliminate the possibility of fragmentation. In a separate development, Rokhsar [5] pointed out that a usual 2π-vortex in a dilute Bose gas with positive s-wave scattering length (a “repulsive” Bose gas) contains a negative energy bound state with zero angular momentum. Transfer of a fraction of the atoms in the rotating condensate to the non-rotating bound state can lower the system energy, creating a fragmented state with different angular momentum components. He argued though that such a fragmented state is unstable against small non-cylindrically symmetric perturbations which cause tunneling between two fragmented condensates [5]. In this paper we study the stability of fragmented condensates and the condition for their existence, and clarify the relations between the various results of Refs. (3-5). We show that condensate fragmentation does not exist unless the system is required to have strict values of certain physical quantities (angular momentum, parity, etc). While our proof applies to general fragmentation, we first focus on fragmentation into angular momentum L̂z eigenstates in cylindrically symmetric systems; the physics of these systems illustrates the essential mechanisms destablizing the fragmented condensates, as well as the role of conservation laws. Fragmented condensates with no specific symmetry relation with the underlying potential are even more unstable. In cylindrically symmetric systems, it is important to distinguish two situations, that Lz is precisely specified, i.e., that the system is in an angular momentum eigenstate, or that Lz is specified only on average, 〈L̂z〉 (through the Hamiltonian H − ωL̂z, where ω is the rotational frequency about ẑ), corresponding to a grand canonical angular momentum ensemble. Fragmentation can only occur when the system has a definite Lz that is not an integral multiple of particle number N . In the latter situation, as normally considered in rotating superfluids, fragmentation cannot occur. As we show, the instability of fragmented condensates in rotating systems is entirely controlled by interaction effects and is therefore intrinsic: by reassembling different compo2 nents of the fragmented condensate into a single condensate, the system can always lower its energy through interference of three or more fragmented components. This mechanism differs fundamentally from the picture given by Rokhsar since the energy gain is through the interaction rather than the (arbitrarily small) symmetry breaking potential, and that of both Elgarøy-Pethick and Rokhsar in that it involves at least three fragmented components. Wilkin et al. find fragmentation because they seek ground states that are eigenstates of total Lz. However, in an angular momentum grand canonical ensemble, the ground state need not an angular momentum eigenstate. [6] The argument naturally calls for examination of possible spontaneous broken rotational symmetry in the ground state. We prove that the cylindrically symmetric condensate of a repulsive Bose gas is globally stable; however, we can only establish local stability for the ground state of an attractive Bose gas. Thus we show rigorously that the ground state is always unfragmented, and that broken rotational symmetry is absent for a > 0 and could only occur for a < 0 in the unlikely case that the system has a minimum separated from the symmetric ground state by an energy barrier. The proof illustrates clearly an intrinsic mechanism rendering fragmentation unfavorable, which should dissuade further attempts to look for ground state fragmentation. Furthermore, while the proof of stability of the cylindrical symmetric ground state may appear merely to have confirmed the expected, it does lead to non-trivial conclusions for highly anisotropic traps. Definition of fragmentation: The single particle ground state density matrix, ρ(x,x) = 〈ψ̂†(x)ψ̂(x′)〉, where ψ̂(x) is the field operator, is Hermitian in x and x, ρ̂ and thus can be expanded in terms of its eigenfunctions νj(x) and eigenvalues λj as ρ(x,x ) = ∑ j λjνj(x) νj(x ), where ∫ dxνi(x) νj(x) = δij . The ground state has a single condensate if only one eigenvalue λj is of order the number of particles N , or is fragmented if more than one eigenvalue is of order N . All particles in the ground state of a dilute Bose gas are essentially in the condensate, whether single or fragmented; to a good approximation ρ(x,x) = F ∑ j=1 Njνj(x) νj(x ), F ∑