Dipolar fermions in a multilayer geometry

We investigate the behavior of identical dipolar fermions with aligned dipole moments in two-dimensional multilayers at zero temperature. We consider density instabilities that are driven by the attractive part of the dipolar interaction and, for the case of bilayers, we elucidate the properties of the stripe phase recently predicted to exist in this interaction regime. When the number of layers is increased, we find that this “attractive” stripe phase exists for an increasingly larger range of dipole angles, and if the interlayer distance is sufficiently small, the stripe phase eventually spans the full range of angles, including the situation where the dipole moments are aligned perpendicular to the planes. However, in this regime, we expect the behavior to be strongly modified by the binding of dipoles between layers. In the limit of an infinite number of layers, we derive an analytic expression for the mean-field interlayer effects in the density-density response function and, using this result, we find that the stripe phase is replaced by a collapse of the dipolar system.

Figure 1

Schematic representation of the system geometry. A gas of dipolar fermions is confined to $N$ 2D layers labeled by an index $j=1,2,\cdots ,N$ and separated by a distance $d$. We assume that each layer has the same density of dipoles $n$. The dipole moment of each fermion is aligned by an electric field $\mathbf{E}=(E_x,0,E_z)$ in the $x-z$ plane ($\phi =0$ direction), at an angle $\theta$ with respect to the $\widehat{\mathbf{z}}$ direction.

Figure 2

Main panel: Phase boundary for the $\phi =0$ stripe phase in the two-layer geometry as a function of the tilt angle $\theta$ and the dimensionless interaction strength $U$ at fixed interlayer distance $k_{F}d=2$. Density modulations are in the direction $\phi =0$ of the dipole tilt (schematic figure). The instability to this phase is to the right of the plotted boundaries: The full STLS results [38] ([green] open triangles) are compared to the results obtained with exchange-only correlations ([blue] solid line), also referred to as the X-STLS approximation scheme. Within X-STLS, the $\phi =0$ stripe phase appears for $\theta >{\theta }_c\simeq 0.79$. At $\theta =\pi /2$, the gas is unstable towards mechanical collapse for $U\gtrsim 1.57$ ([red] diamond symbol and thick solid line), where the gas compressibility is infinite. The density modulations in the two layers have a phase shift $\eta$ (lower inset) equal to $2\theta$, i.e., the shift between the modulations $2\theta /q_c$ (schematic figure).

Figure 3

Schematic representation of a single dipole in the top layer interacting classically with an infinite bottom layer of dipoles arranged in a stripe modulated phase. As in Fig. 1, all dipoles are aligned by an electric field $\mathbf{E}$. In the left (right) panel, the single dipole is shifted by $\overline{x}$ ($\overline{y}$) with respect to one of the stripe crests for the $\phi =0$ ($\phi =\frac{\pi }{2}$) stripe phase.

Figure 4

Evolution of the $\phi =0$ stripe phase for the trilayer $N=3$ geometry when varying the dimensionless layer separation $d{k}_F$. Left panel: phase boundary in the $\theta$ versus $U$ plane; the instability to the stripe phase is on the right side of the plotted boundary. The boundary is gray at higher values of $U$ and smaller values of the tilt angle $\theta$, where interlayer pairing correlations cannot be neglected (see Sec. 6a). Right panel: Rescaled stripe wave vector $q_c/k_F$ at the phase boundaries as a function of the tilt angle $\theta$.

Figure 5

Phase boundaries (left panels) and rescaled stripe wave vector at the boundaries $q_c/k_F$ (right panels) for the $\phi =0$ stripe phase for different values of the number of layers $N$ and two fixed values of the layer distance: $d{k}_F=1.2$ (top panels) and $d{k}_F=2$ (bottom panels). The phase boundary for stripe formation at $N=2$ ([red] open circles) is gray when the system preferentially forms dimers. The shaded region indicates the crossover to a “bound-state regime” dominated by dipolar chains (here evaluated for the case of an infinite number of layers).

Figure 6

Critical value of the interlayer distance $d_c{k}_F$ at which ${\theta }_c=0$ (at $U\to \infty$) as a function of the inverse number of layers, $1/N$. In the limit $N\to \infty ,d_c{k}_F=1.47$. Numerical data are represented as (red) circles, while the dashed (blue) line is a linear fit to the data. Inset: maximum value of the stripe wave vector $q_c/k_F$ for $d=d_c$ as a function of $1/N$. Data are (orange) triangles, while the (turquoise) dashed line is a nonlinear fit giving $f(1/N)=2.83{(1/N)}^{0.49}$.

Figure 7

Left panel: Phase boundary of the collapsed region for an infinite number of layers $N\to \infty$ and different values of the rescaled layer density $d{k}_F$. The instability to collapse is signalled by an infinite compressibility of the dipolar gas, i.e., by a divergence of the static response function for ${\mathbf{q}}_c=0$. While for $d>d_c$, collapse occurs only for a tilt angle larger than a critical value ${\theta }_c$, for $d<d_c$, the collapsed phase spans the entire range of angles. The phase boundaries are plotted in gray when the system crosses over to the “bound state regime” dominated by infinitely long dipolar chains (see Sec. 6b). Right panel: We plot the asymptotic values of the tilt angle at the phase boundary for $U\to \infty , {\theta }_c$, as a function of the rescaled layer distance $d{k}_F$; we find that the critical distance for infinite layers is given by $d_c(N\to \infty )k_F=1.47$.

Figure 8

Left panel: Size in momentum space, $k_{0}d$, of dimers ($N=2$ [filled symbols]) and bound chains ($N\to \infty$ [empty symbols]) as a function of the interaction strength $U_d$ for $\theta =0$ ([red] circles) and $\theta =\pi /4$ ([blue] squares). Right panel: “bound-state” region for fixed $k_{F}d=1.2$, indicating where $k_0/k_F\gtrsim 1$ and interlayer correlations are important (see Fig. 5).

Figure 9

Classical configuration of an infinite chain of static dipoles (dipole in the layer $j^{*}$ positioned at $\mathbf{r}=0$): dipoles in adjacent top and bottom layers are shifted by a vector $\pm {\mathbf{r}}_{01}=\pm (x_0,0)$, respectively, in order to minimize the potential energy $v_{\mathrm{cl}}\left(x_0\right)$ (see text). Fluctuations around this classical configuration are introduced by allowing one dipole, say in layer $j^{*}$, to move freely to any in-plane position $\mathbf{r}$.

Figure 10

Two identical circles of radius $k_F$, where the centers are separated by a distance $q$ in $k$ space. The area of the shaded overlapping region corresponds to the integral in Eq. (A1), defining the noninteracting static structure factor in 2D.