Competing compressible and incompressible phases in rotating atomic Bose gases at filling factor $\nu =2$

We study the groundstates of weakly interacting atomic Bose gases under conditions of rapid rotation. We present the results of large-scale exact diagonalization studies on a periodic geometry (a torus) which allows studies of compressible states with broken translational symmetry. Focusing on filling factor $\nu =2$, we show a competition between the triangular vortex lattice, a quantum smectic state, and the incompressible $k=4$ Read-Rezayi state. We discuss the corrections arising from finite size effects, and the likely behavior for large system sizes. The Read-Rezayi state is stabilized by a moderate amount of additional dipolar interactions.

Figure 1

Energy spectrum at $\nu =2$ as a function of $\sqrt{K_x^2+K_y^2}$, with torus aspect ratio chosen to be close to the value for which the groundstate energy is a local minimum. (a) $N_{\mathrm{v}}=8$, $N=16$, $a∕b=2$; (b) $N_{\mathrm{v}}=12$, $N=24$, $a∕b=4∕3$. The solid symbols mark the two lowest-energy states indicating the ordering wave vector of the smectic groundstate. The shaded symbols mark a band of excitations discussed in the text. [Note that the components $K_x,K_y$ are expressed in units of $2\pi ∕a$ and $2\pi ∕b$.]

Figure 2

Energy spectrum at $\nu =2$ as a function of $\sqrt{K_x^2+K_y^2}$, with torus aspect ratio chosen to be consistent with a triangular vortex lattice. (a) $N_{\mathrm{v}}=8$, $N=16$, $a∕b=\sqrt{3}$; (b) $N_{\mathrm{v}}=12$, $N=24$, $a∕b=2∕\sqrt{3}$. The solid symbols mark the wave vectors of a triangular vortex lattice. These states are not well separated from higher energy states, showing a weak tendency to vortex lattice ordering. [Note that the components $K_x,K_y$ are expressed in units of $2\pi ∕a$ and $2\pi ∕b$.]

Figure 3

(Color online) Variation of the overlap (squared) of the groundstate for contact and dipolar interactions with the $k=4$ RR state, for (a) $N_{\mathrm{v}}=8$, $N=16$, $a∕b=1$ and (b) $N_{\mathrm{v}}=10$, $N=20$, $a∕b=1$, for $a_{\perp }∕a_{\Vert }⪡1$. We parameterize the relative size of dipolar and contact interactions by the ratio of the first two Haldane pseudopotentials $V_2∕V_0$, as defined in Refs. 13, 27. The $k=4$ RR state has five degenerate states on the torus located at the zone center and corners [2, 11]; for $\mathit{K}=(0,0)$ the RR groundstate is a doublet and we plot the overlap of the lowest two states within this subspace, so the maximum overlap squared is 2. [In (b), the discontinuity of the overlap for $\mathit{K}=(0,0)$ close to $V_2∕V_0=0.13$ is due to a level crossing with a state of different point-group symmetry.]