Colloquium: Weakly interacting, dilute Bose gases in 2D

This colloquium surveys a number of theoretical problems and open questions in the field of two-dimensional dilute Bose gases with weak repulsive interactions. In contrast to three dimensions, in two dimensions the formation of long-range order is prohibited by the Bogoliubov-Hohenberg theorem, and Bose-Einstein condensation is not expected to occur. Nevertheless, experimental indications supporting the formation of a condensate in low-dimensional systems have recently been obtained. This unexpected behavior appears to be due to the nonuniformity introduced into a system by the external trapping potential. Theoretical predictions, made for homogeneous systems, require therefore careful reexamination. A number of popular theoretical treatments of the dilute weakly interacting Bose gas are presented and their regions of applicability are discussed. The possibility of Bose-Einstein condensation in a two-dimensional gas, the validity of the perturbative $t$-matrix approximation, and the diluteness condition are issues also discussed in detail.

Figure 1

(Color online) Column density of a cloud with trapped noninteracting bosons along the $z$ direction. The total density is a superposition of condensate density $n_0$ and a thermal distribution of noncondensed particles $n_T$.

Figure 2

(Color online) Length scale in the Bogoliubov problem: correlation length ${\lambda }_c$. $k_c\sim 1∕{\lambda }_c$ separates the “free” particle behavior from the linear dispersion region. Zigzag lines denote the residual interaction between particles.

Figure 3

Bethe-Salpeter equation for the two-particle scattering vertex $\Gamma$.

Figure 4

(Color online) One-particle density matrix $\rho \left(r\right)=⟨{\psi }^{†}\left(0\right)\psi \left(r\right)⟩$ in two dimensions. Two characteristic length scales $R_{\theta }$ and ${\lambda }_c$ are shown, $R_{\theta }\gg {\lambda }_c$. At large distances ${\lambda }_c\ll L\ll R_{\theta }$ the one-particle density matrix is equal to the condensate density $n_0$.

Figure 5

Energy scales in Popov’s approach: $T∕\ln \mid {ϵ}_0∕\mu \mid \ll k_0^2∕2m\sim T\ll (k_0^{\prime }{)}^2∕2m\ll {ϵ}_0=1∕m{a}^2$.

Figure 6

(Color online) Schematic phase diagram of a uniform dilute Bose gas. ${\rho }_s$ is the superfluid density, normalized by the mass density of the gas $\rho =mn$, $T_{\mathrm{KT}}$ is the critical temperature of the Kosterlitz-Thouless transition, $T_{\mathrm{MF}}$ is the mean-field temperature, calculated perturbatively. The size of the transition critical region is defined by a parameter $1∕\ln \ln (1∕n{a}^2)$, where $a$ is characteristic $s$-wave scattering length. The region, dominated by phononic quasiparticle behavior, is of the width $1∕\ln (1∕n{a}^2)$.

Figure 7

Schematic phase diagram of a two-dimensional trapped weakly interacting dilute Bose gas. ${\rho }_0$ stands for the density of the condensate, ${\rho }_s$ is the superfluid density, ${\rho }_{\mathrm{MF}}$ is the mean-field density, which can be estimated perturbatively. $T_{\mathrm{BEC}}$ is the crossover temperature from the superfluid regime to the true Bose-Einstein condensation phase. $T_{\mathrm{BKT}}$ is the critical temperature of the Kosterlitz-Thouless transition. $T_{\mathrm{MF}}$ marks the critical region of BKT transition.