Elementary excitations of ultracold soft-core bosons across the superfluid-supersolid phase transition

We investigate the zero-temperature excitation spectrum of two-dimensional soft-core bosons for a wide range of parameters and across the phase transition from a superfluid to a supersolid state. Based on mean-field calculations and recent quantum Monte Carlo results, we demonstrate the applicability of the Bogoliubov-de Gennes equations, even at high interaction strengths where the system forms an insulating cluster crystal. Interestingly, our study reveals that the maximum energy of the longitudinal phonon band in the supersolid phase connects to the maxon energy of the superfluid at the phase transition.

Figure 1

Superfluid fraction of two-dimensional bosons as a function of the dimensionless interaction strength $\alpha =m\rho V_0{R}_0^4/{\hslash }^2$. The points show results of Monte Carlo simulations for 320 particles with a density of $R_0^2\rho =4.4$, whereas the continuous line is a guide for the eye. The superfluid fraction drops abruptly at $\alpha \approx 13.4$ (dotted line), marking a first-order superfluid-supersolid phase transition, in close agreement with the mean-field prediction of $\alpha =12.7$ (dashed line). Around $\alpha \approx 22$, the systems enter an insulating phase. The insets show PIMC snapshots for different $\alpha$, illustrating the particle density profile in the three different phases. We checked that the results do not change for a temperature range of 0.1–$0.75{\hslash }^2/m{R}_0^2$ and particle numbers 160, 320, 640.

Figure 2

Excitation spectrum in the superfluid phase according to Eq. (6) (line) compared to the PIMC data (circles) of Ref. [25].

Figure 3

Mean-field spectra (lines) at (a) $\alpha =16.93$ and (b) $\alpha =30.62$ obtained from Eqs. (5) along the three symmetry directions of the Brillouin zone [see inset of panel (b)]. The symbols represent the PIMC data of Ref. [25] for longitudinal excitations computed along the direction $\Gamma$–$M$–$\Gamma$ in the first two Brillouin zones. (c)–(e) Density fluctuations $\Delta {\rho }_{\mathbf{k}}^n\left(\mathbf{r}\right)={|u_{n,\mathbf{k}}\left(\mathbf{r}\right)-v_{n,\mathbf{k}}\left(\mathbf{r}\right)|}^2$ and (f)–(h) phase fluctuations $\Delta {\phi }_{\mathbf{k}}^n\left(\mathbf{r}\right)={|u_{n,\mathbf{k}}\left(\mathbf{r}\right)+v_{n,\mathbf{k}}\left(\mathbf{r}\right)|}^2$ computed at $\alpha =16.93$ and $\mathbf{k}=\overrightarrow{\Gamma K}/10$ (directed along the horizontal axis) for the (c), (f) lowest (solid line) band, (d), (g) middle (dashed line) band, and (e), (h) higher (dot-dashed line) gapless band.

Figure 4

(a) Linear dispersion ${v=\partial \omega /\partial k|}_{k=0}({\hslash }^2/m{R}_0)$ of the gapless modes and (b) their energy $\omega ({\hslash }^2/m{R}_0^2)$ at distinct momenta as a function of $\alpha$. In the superfluid phase ($\alpha <12.7$), panel (b) shows the maxon energy (line) and the excitation energies (circles) of the three gapless modes at the $M$ point (${\omega }_M$) and of the longitudinal and transverse phonon band at the $K$ point (${\omega }_K$) in the supersolid phase (see Fig. 3).