Intrinsic-to-extrinsic supersolid transition and fractionally modulated states in a lattice ultracold Bose gas with long-range interaction

We investigate the intrinsic-to-extrinsic supersolid (SS) transition in a lattice ultracold Bose gas with strong long-range interaction. When changing the depth of the periodic lattice potential, the transition is shown to manifest in the ground-state wave function and energy, the change of superfluid fraction $f_s$, and a roton instability. Near the transition in the extrinsic SS phase, due to the competition between the long-range interaction and the periodic potential, we show that there exist a variety of stable fractionally modulated states (FMSs) upon the change of the effective length of the long-range interaction. Consequence of the transition across different FMSs is discussed.

Figure 1

(a)–(f) Intrinsic-to-extrinsic SS transition manifested by the ground-state wave functions (black lines) of a long-range interacting lattice ultracold Bose gas. For long-range interaction, $\alpha =30$ and $r_c=0.75$ are fixed in all frames. Lattice potentials are schematically shown by red curves with the depth $V_0=2.0,1.55,1.45,1.0,0.5$, and $0.1$ respectively from (a) to (f). See text for the units of energy and length. Panels (g)–(i) show the ground-state energy $E$, the derivative $dE/d{V}_0$, and the superfluid fraction $f_s$ as a function of $V_0$ crossing the critical point at $V_c=1.478$.

Figure 2

(a) Roton instability occurred in the elementary excitations of a 1D homogeneous ultracold Bose gas $\left(V_0=0\right)$ when the long-range interaction strength $\alpha$ is above a critical value ${\alpha }_c\simeq 24.95$. Blockade radius is fixed at $r_c=0.75$. (b) Roton instability occurred in the elementary excitations of a lattice system when the lattice potential depth $V_0$ is below a critical value $V_c\simeq 1.478$. The interaction strength and range are fixed at $\alpha =30$ $\left(>{\alpha }_c\right)$ and $r_c=0.75$. In both frames, yellow zone corresponds to $\mathrm{Im}\left(\omega \right)\ne 0$ for the red curve.

Figure 3

(a) $\tilde{U}\left(2\pi \nu \right)/\alpha$ plotted as a function of $r_c/d$ for $\nu =1$ (blue), $4/5$ (green), $3/4$ (red), $2/3$ (cyan), $3/5$ (magenta), and $1/2$ (gold) respectively. For a given $r_c$, the ground state can be estimated corresponding to a particular $\nu$ which makes $\tilde{U}\left(2\pi \nu \right)/\alpha$ minimum (see text). (b) The calculated superfluid fraction $f_s$ as a function of $r_c/d$, with fixed $\alpha =30$ and $V_0=11$, showing transitions across various fractionally modulated states.

Figure 4

Fractionally modulated states revealed by the $k=0$ Bloch wave density distribution ${\left|{\phi }_{k=0}\right|}^2$ of the lowest band. The blockade radius $r_c/d=48/64$ (a), $57/64$ (b), $62/64$ (c), $68/64$ (d), $78/64$ (e), and $87/64$ (f) (black curves), which correspond respectively to $\nu =1,4/5,3/4,2/3,3/5$, and $1/2$ states (see Fig. 3). For comparison, the cyan curve in (a) is for $r_c/d=30/64$, which corresponds to $\nu =1$ as well, but has relatively minor modulation. Thin red curves represent schematically the fixed lattice potential. $V_0=11$ and $\alpha =30$ are fixed in all frames.

Figure 5

(a) Critical depth $V_c$ of the lattice potential as a function of $\alpha$ for three different $r_c$. The dots correspond to critical strength ${\alpha }_c$ for the SF-to-intrinsic SS transition when $V_0=0$. (b) Critical depth $V_c$ as a function of $r_c$ for three different $\alpha$. The dots correspond to critical length $r_c$ for the SF-to-intrinsic SS transition when $V_0=0$.