Dimensional crossover of nonrelativistic bosons

We investigate how confining a transverse spatial dimension influences the few- and many-body properties of nonrelativistic bosons with pointlike interactions. Our main focus is on the dimensional crossover from three to two dimensions, which is of relevance for ultracold-atom experiments. Using functional-renormalization-group equations and $T$-matrix calculations we study how the phase transition temperature changes as a function of the spatial extent of the transverse dimension and relate the three- and two-dimensional $s$-wave scattering lengths. The analysis reveals how the properties of the lower-dimensional system are inherited from the higher-dimensional one during renormalization-group evolution. We limit the discussion to confinements in a potential well with periodic boundary conditions and argue why this qualitatively captures the physics of other compactifications such as transverse harmonic confinement as in cold-atom experiments.

Figure 1

Schematics of the energy spectrum in the dimensional crossover from three to two dimensions. (a, b) Discrete eigenenergies for a noninteracting system in a 3D box potential which confines the system to the cuboid $(x,y,z)\in {[0,L_x]}^2\times [0,L]$. In (a) we have set $L_x/L=2$, whereas (b) displays the case of $L_x/L=3$. As we increase the ratio $L_x/L$, an effective 2D continuum of states (thin gray lines) fills the region between the discrete spectrum of excitations in the transverse $z$ direction (thick red lines). The lowest excitations appear above the zero-point energy, which is generally not 0. (c) Sketch of the more realistic scenario of a system which is interacting and confined by a smooth trapping potential. Although the concrete mode spectrum and level spacing in (c) differ from those in (a) and (b), we find the same qualitative behavior.

Figure 2

The crossover function $F\left(\tilde{L}\right)$ modifies the flow equations of the 3D system. For small $\tilde{L}=Lk$ it diverges like ${\tilde{L}}^{-1}$ and the 2D flow equations are recovered. For large $\tilde{L}$ it approaches a constant and the system behaves three-dimensionally. In the FRG approach we start at an ultraviolet scale $k_{\mathrm{\Lambda }}=\mathrm{\Lambda }$ with $L\mathrm{\Lambda }\gg 1$, so that the description of the microscopic theory is three-dimensional.

Figure 3

Superfluid fraction ${\rho }_0/n$ at $k\to 0$ for different values of $L$. Larger values of $L$ correspond to the curves farther to the right. We directly see that the critical temperature, here in units of the chemical potential $\mu$, is higher in more 3D-like systems. For all compactification lengths $L$ we recover the correct limit ${\rho }_0{/n|}_{T\to 0}=1$, where the whole gas is superfluid due to Galilean invariance. Curves correspond to a coupling strength $g_{3\mathrm{D}}\sqrt{\mu }=0.025$.

Figure 4

Dimensional crossover of the superfluid critical temperature $T_{\mathrm{c}}$ from two to three dimensions. For large $L$ the superfluid transition goes along with Bose-Einstein condensation. The transition temperature is substantially lower in a smaller system, i.e., when $L$ is smaller. For the crossover we fix the 3D coupling constant to the same value as in Fig. 3. In experiments with ultracold atoms, $g_{3\mathrm{D}}$ is usually controlled by a magnetic Feshbach resonance.

Figure 5

Critical temperature $T_{\mathrm{c}}/\mu$ in two dimensions as a function of $g_{2\mathrm{D}}$ over four orders of magnitude. For better comparability with Eq. (A2) we have multiplied the function by $g_{2\mathrm{D}}$. The blue circles and solid line constitute the FRG result for $t_{\mathrm{f}}=-10$ (as used in the text), whereas the dashed red line corresponds to the asymptotic form of Eq. (A2) for small $g_{2\mathrm{D}}$. For the dashed line we use the fit result from Table 1.

Figure 6

Dimensional crossover of the superfluid critical temperature $T_{\mathrm{c}}$ for a fixed value of $a_{2\mathrm{D}}$. We use the parameters described in Appendix pp1. One observes three characteristic regimes for the behavior of the critical temperature: (i) For small $0.1\le L\sqrt{\mu }\le 1$, the transition temperature increases. However, since we have chosen $\mathrm{\Lambda }/\sqrt{\mu }=10$ here, this region corresponds to $1\le L\mathrm{\Lambda }\le 10$ and thus violates the criterion $L\gg {\mathrm{\Lambda }}^{-1}$. Thus it is not of relevance for this discussion. (ii) For intermediate $L\sqrt{\mu }$, the critical temperature settles at $T_{\mathrm{c}}/\mu =1.32$ (dashed horizontal line), which is the transition temperature for the corresponding 2D gas with the same value of $a_{2\mathrm{D}}$. In this region, which corresponds to $L/a_{2\mathrm{D}}=(1-3)\times {10}^3$ in this particular case, the gas is truly two-dimensional. (iii) For larger $L$, we find an increase in the critical temperature. Note that the crossover scale $L\sqrt{\mu }\approx 3$ between the 2D and the quasi-2D regime is still much larger than $T^{-1/2}$ and ${\mu }^{-1/2}$. The slight dimple in $T_{\mathrm{c}}/\mu$ before the rise seems to be rooted in the same oscillatory crossover behavior as found for small $L$, and we do not attribute any deep physical meaning to it at this point.

Figure 7

Top: Characteristic scale dependence of $\eta$ and ${\eta }_z$ within the anisotropic derivative expansion. Whereas the planar anomalous dimension $\eta$ (blue curve) remains continuous, the transverse anomalous dimension ${\eta }_z$ has discontinuous jumps due to the successive integration of modes and the step function in the regulator, Eq. (B40). The function ${\eta }_z\left(k\right)$ follows the behavior of $\eta \left(k\right)$ on average for $k>\frac{2\pi }{L}$ and vanishes below $k=\frac{2\pi }{L}$ (shown as the dashed vertical line). The plot is shown for $T=0,L\sqrt{\mu }=15,g_{3\mathrm{D}}\sqrt{\mu }=0.025$. The qualitative behavior of $\eta$ and ${\eta }_z$ is independent of this choice. Bottom: Ratio $\xi =A_z/A$ at the lowest momentum $k_{\mathrm{f}}$ with parameters as in Fig. 4 for zero temperature (upper, blue curve) and at $T_{\mathrm{c}}$ (lower, red curve). Since $\xi$ is of order unity—and not exceptionally smallthe isotropic derivative expansion is sufficient to compute observables like the critical temperature.