Quantum phase transition of a two-dimensional quadrupolar system

Ensembles with long-range interactions between particles are promising for revealing strong quantum collective effects and many-body phenomena. Here we study the ground-state phase diagram of a two-dimensional Bose system with quadrupolar interactions using a diffusion Monte Carlo technique. We predict a quantum phase transition from a gas to a solid phase. The Lindemann ratio and the condensate fraction at the transition point are $\gamma =0.269\left(4\right)$ and $n_0/n=0.031\left(4\right)$, correspondingly. We observe the strong rotonization of the collective excitation branch in the vicinity of the phase transition point. Our results can be probed using state-of-the-art experimental systems of various nature, such as quasi-two-dimensional systems of quadrupolar excitons in transition metal dichalcogenide trilayers, quadrupolar molecules, and excitons or Rydberg atoms with quadrupole moments induced by strong magnetic fields.

Figure 1

The macroscopic limit of the energy $E/S$ (scaled with classical $n^{7/2}$ dependence) for the gas (circles) and the crystal (triangles) as a function of the dimensionless density $n{r}_0^2$ (the energy $E$ is measured in the dimensionless units ${\hslash }^2/m{r}_0^2$). The position of the transition point, $n{r}_0^2=2.10\left(7\right)$, is indicated by the arrow. Inset: The quantity $(E-\mu N)/S-{\varepsilon }_0$ as a function of the dimensionless density in the vicinity of the phase transition where ${\varepsilon }_0$ is an offset. The tangent dotted line indicates the coexistence regime of two phases; its width is $\mathrm{\Delta }n{r}_0^2=0.026\left(4\right)$. The fitting function is $E/\left(N{E}_0\right)=E_{\mathrm{cls}}/\left(N{E}_0\right)+A_1{\left(n{r}_0^2\right)}^{7/4}+A_2{\left(n{r}_0^2\right)}^{5/4}+A_3{\left(n{r}_0^2\right)}^{3/4}$. Fitting coefficients are $A_1=7.944, A_2=-0.388$, and $A_3=1.332$ for gas at $0.8<n{r}_0^2<3$ and $A_1=6.1478, A_2=2.4524$, an $A_3=0.9878$ for crystal at $1.6<n{r}_0^2<3.4$, where $E_{cls}/\left(N{E}_0\right)=A_0{\left(n{r}_0^2\right)}^{5/2}$ with $A_0=2.359746$ is the ground-state energy of a classical crystal.

Figure 2

(a) Typical examples of the pair correlation function in gas (dashed line) and solid (solid line) phases at the density $n{r}_0^2=2.2$ obtained for $N=144$ particles. (b) Static structure factor in the vicinity of the phase transition in gas (circles) and triangular solid (triangles) phases. Symbols, DMC data; lines, linear phonons asymptotic $S\left(k\right)=\hslash k/\left(2mc\right)$, where the speed of sound $c=\sqrt{n/m{d}^2(E/S)/d{n}^2}$ is obtained from the equation of state; see Fig. 1. The vertical arrow shows the position of the macroscopic peak in the crystal. Insets show snapshots of the particles' coordinates in gas (left) and solid phases (right). Polygons indicate a frustrated (left) and perfect (right) hexagonal short-range ordering present in gas and solid phases, correspondingly.

Figure 3

The condensate fraction $n_0/n$ in the macroscopic system as the function of the density in gas and solid phases. Circles, extrapolation of Quantum Monte Carlo data to thermodynamic limit performed by using HD theory [QMC $+$ HD: input $S\left(k\right)\&g_1(L/2)]$ of Ref. [74]; red line, fit $n_0/n=exp\left[-{(B_0+B_1{\left(n{r}_0^2\right)}^{B_2})}^{-2}/4\right]$ in the region $0.8\le n\le 2.8$, where $B_0=-0.301, B_1=0.639$, and $B_2=-0.154$. The discontinuity at the phase transition is shown with arrows.

Figure 4

Characteristic examples of the excitation spectrum in the gas phase as obtained from Feynman relation. The formation of a pronounced roton minimum is observed as density is increased and the transition to the solid phase is approached.

Figure 5

Schematic illustration of possible experimental realizations.