Dimensional crossover of Bose-Einstein-condensation phenomena in quantum gases confined within slab geometries

We investigate systems of interacting bosonic particles confined within slablike boxes of size $L^2\times Z$ with $Z\ll L$, at their three-dimensional (3D) Bose-Einstein-condensation (BEC) transition temperature $T_c$, and below $T_c$ where they experience a quasi-two-dimensional (quasi-2D) Berezinskii-Kosterlitz-Thouless (BKT) transition (at $T_{\mathrm{BKT}}<T_c$ depending on the thickness $Z$). The low-temperature phase below $T_{\mathrm{BKT}}$ shows quasi-long-range order: the planar correlations decay algebraically as predicted by the 2D spin-wave theory. This dimensional crossover, from a 3D behavior for $T\gtrsim T_c$ to a quasi-2D critical behavior for $T\lesssim T_{\mathrm{BKT}}$, can be described by a transverse finite-size scaling limit in a slab geometry. Numerical evidence of the $3\mathrm{D}\to 2\mathrm{D}$ dimensional crossover is presented for the Bose-Hubbard model defined in anisotropic $L^2\times Z$ lattices with $Z\ll L$. We extend this scaling analysis to the case the slab geometry of the gas is effectively realized by a transverse (inhomogeneous) harmonic trap. Finally, we discuss off-equilibrium behaviors arising from slow time variations of the temperature across the BEC transition of gases confined within a slab geometry. We argue that the system develops an off-equilibrium transverse finite-size scaling under these time-dependent protocols.

Figure 1

Sketch of the $T\text{−}\mu$ (in units of the hopping parameter $t$) phase diagram of the 3D BH model in the hard-core $U\to \infty$ limit. The BEC phase is restricted to a finite region between $\mu =-6$ and 6. It is bounded by a BEC transition line $T_c\left(\mu \right)$, which satisfies $T_c\left(\mu \right)=T_c(-\mu )$ due to a particle-hole symmetry. Its maximum occurs at $\mu =0$, where [27, 29] $T_c(\mu =0)=2.01599\left(5\right)$; we also know that [26] $T_c(\mu \pm 4)=1.4820\left(2\right)$. At $T=0$, two further quantum phases exist: the vacuum phase ($\mu <-6$) and the incompressible $n=1$ Mott phase ($\mu >6$). Since we set $t=1$ for the hopping parameter of the BH model (1), $T$ and $\mu$ are expressed in units of $t$.

Figure 2

Sketch of the $L^2\times Z$ slab geometry with $L\gg Z$. The figure shows a slice of the lattice system on the $x\text{−}z$ plane (the $y$ direction runs orthogonal to the $x\text{−}z$ plane, along which the system extends analogously to the $x$ direction). The blobs indicate the lattice sites whose integer coordinates are limited by $1\le x,y\le L$ and $-(Z-1)/2\le z\le (Z-1)/2$. The one-particle correlation function $g\left(\mathit{x}\right)$ [cf. Eq. (4)] and the corresponding correlation length $\xi$ [cf. Eq. (6)] are defined along the $z=0$ plane.

Figure 3

Sketch of the phase diagram of the 2D BH model in the hard-core $U\to \infty$ limit. The normal and superfluid QLRO phases are separated by a finite-temperature BKT transition line, which satisfies $T_{\mathrm{BKT}}\left(\mu \right)=T_{\mathrm{BKT}}(-\mu )$ due to a particle-hole symmetry. Its maximum occurs at $\mu =0$, where [73] $T_{\mathrm{BKT}}(\mu =0)=0.6877\left(2\right)$. The superfluid QLRO phase is restricted to a finite region between $\mu =-4$ and 4, which is narrower than that of the 3D phase diagram (see Fig. 1). Since we set $t=1$ for the hopping parameter of the BH model, $T$ and $\mu$ are expressed in units of $t$.

Figure 4

The planar correlation length $\xi$ vs the temperature $T$ for the hard-core $U\to \infty$ BH model at zero chemical potential. We plot QMC data for $Z=5$ (top), $Z=9$ (middle), and $Z=13$ (bottom). Note that all figures show the same ranges of $T$ and $\xi$ values, to favor the comparison of the data for different thickness $Z$. Lengths are expressed in units of the lattice spacing, while $T$ is in units of the hopping parameter. The dashed vertical line indicates the BEC transition temperature $T_c$, the dotted vertical lines indicate our estimates of the $Z$-dependent BKT transition temperature. The statistical errors of the data are so small to be hardly visible.

Figure 5

$R_L\equiv \xi /L$ vs $Y$ for $Z=5$ (bottom) and $Z=9$ (top), and for several values of $L$ and $T$. The full line shows the spin-wave curve ${\mathcal{R}}_L\left(\mathcal{Y}\right)$ which is expected to be asymptotically approached for $L\to \infty$ within the QLRO phase; its end point corresponds to the BKT transition. In particular, for $Z=5$ the values of $T$ of the data shown in the bottom figure are (from right to left) $T=$ 1.1858, 1.3518, 1.5179, 1.6, 1.64, 1.65, 1.67, 1.6839, 1.85. The behavior of the data close to the BKT point suggests $T_{\mathrm{BKT}}(Z=5)\approx 1.65$. Analogously for $Z=9$ the data are for $T=1.8$, 1.81, 1.82, 1.83, 1.84,1.85, 1.9, 2; they suggest that $T_{\mathrm{BKT}}(Z=9)\approx 1.83$. The statistical errors of the data are so small to be hardly visible.

Figure 6

Data of $Y$ vs $T$ (in units of the hopping parameter), for $Z=9$ (bottom) and $Z=13$ (top) around the corresponding BKT temperatures. Analogous results have been obtained for $Z=5$. The dashed horizontal line indicates the BKT value $Y^{*}=0.6365...$. The dotted vertical lines indicate the interval corresponding to our best estimates of $T_{\mathrm{BKT}}$, i.e., $T_{\mathrm{BKT}}=1.829\left(1\right)$ for $Z=9$ and $T_{\mathrm{BKT}}=1.899\left(1\right)$ for $Z=13$, obtained by the matching procedure.

Figure 7

Finite-size dependence of $Y$ at the BKT transition. We plot data for $Z=5,9,13$ at their best estimates of $T_{\mathrm{BKT}}$, i.e., $T=1.645,1.829,1.899$, respectively, versus $L/\mathrm{\Lambda }\left(Z\right)$ with $\mathrm{\Lambda }\left(Z\right)={\mathrm{\Lambda }}_{XY}/\lambda \left(Z\right)=0.6,1.5,2.4$, respectively. The dashed line shows the $XY$ curve (A20). The inset shows data of $R_L$, with the corresponding $XY$ curve (dashed line) taken from Ref. [73].

Figure 8

Estimates of ${\delta }_{\mathrm{BKT}}\left(Z\right)\equiv 1-T_{\mathrm{BKT}}\left(Z\right)/T_c$ vs $Z^{-1/\nu }$ with $\nu =0.6717$ (the transverse size is reported in units of the lattice spacing). The data are compatible with the expected behavior ${\delta }_{\mathrm{BKT}}\left(Z\right)\sim Z^{-1/\nu }$. The inset shows the product $Z^{1/\nu }{\delta }_{\mathrm{BKT}}\left(Z\right)$ vs $Z^{-\omega }$ with $\omega =0.785$ (the dashed line is obtained by a linear fit), which is the expected behavior of the leading scaling corrections.

Figure 9

Scaling of $R_Z$ around $T_c$. Most data are taken for $L=120$, which is sufficiently large to provide a good approximation of the $L\to \infty$ limit in the range of $Z$ and $X$ values considered, within about 1%. We plot the data versus $X\equiv Z^{1/\nu }\delta$ with $\nu =0.6717$ and $\delta \equiv 1-T/T_c$. The inset shows the data of $R_Z$ at $T_c$ vs $Z^{-\omega }$ which is the expected behavior of the leading scaling corrections (the dashed line is obtained by a linear fit).

Figure 10

Scaling of the planar susceptibility $\chi$ around $T_c$. We plot $Z^{-1+\eta }\chi$ vs $X\equiv Z^{1/\nu }\delta$. The data support the asymptotic TFSS $\chi \approx Z^{1-\eta }\mathcal{C}\left(X\right)$.