Classical field simulation of vortex lattice melting in a two-dimensional fast rotating Bose gas

Abstract

We present a classical field simulation study of the thermal melting of a two-dimensional vortex lattice in a rotating Bose gas, focusing on the role of finite-size effects on the melting temperature. This work constitutes a numerical continuation of the recent experimental investigation reported in [Physical Review Letters 133, 143401 (2024)], which addressed the thermal melting of a vortex lattice in a quasi-two-dimensional Bose gas. Using the stochastic projected Gross-Pitaevskii equation in a harmonic plus quartic trap, we simulate the finite-temperature equilibrium state and extract vortex configurations from density snapshots. Clear signatures of the two-step Kosterlitz--Thouless--Halperin--Nelson--Young melting scenario are identified. Our simulations enable a detailed characterization of the crystalline, hexatic, and liquid phases through correlation functions quantifying the translational and orientational order and through defect statistics. Finite-size effects are shown to play a crucial role at lower rotation frequencies, affecting the proliferation of lattice defects.

Figures (6)

• Figure 1: Convergence of the SPGPE simulation. Time evolution of (a) the coherent atom number $\mathcal{N}_C(t)$, (b) the angular momentum per particle $\mathcal{L}_z(t)/\mathcal{N}_C(t)$, and (c) the energy per particle $\mathcal{E}(t)/\mathcal{N}_C(t)$ for $\Omega=0.99\omega_r$, $\mu=2.38\hbar\omega_r$, and $k_BT=3\hbar\omega_r$. In panel (c), the dashed red and dash-dotted yellow curves show the interaction and single-particle energy contributions, respectively. The steady-state coherent atom number is $\mathcal{N}_{\mathcal{C}}=9913\pm130$, with an estimated incoherent fraction $\mathcal{N}_{\mathcal{I}}\simeq356$, see text for details.
• Figure 2: Example of finite-temperature vortex lattice. (a) Density profile. (b) Delaunay triangulation; the dashed blue circle indicates the Thomas-Fermi radius. (c) Pair correlation function $g(r)$ and orientational correlation function $G_6(r)$, averaged over $12\times10$ samples. The solid blue vertical line denotes the expected nearest neighbor spacing $a_v=\sqrt{2/\sqrt{3}n_v}$, where $n_v=M\Omega/\pi\hbar$ is the vortex density. The red dot indicates $|G_6(r=0)|$. Parameters: $\Omega=0.99\,\omega_r$, $\mu=2.38\,\hbar\omega_r$, and $k_BT=2.5\,\hbar\omega_r$. The dashed curves overlaid with the data are fits by damped cosine functions to extract the correlation lengths, see text for details.
• Figure 3: (a-c) Examples of density profiles and (d-f) corresponding vortex lattices at $\Omega=0.99\,\omega_r^{-1}$ for increasing temperatures $k_BT/\hbar\omega_r=\{0.1,1.4,2.5\}$ (left to right), in thermal equilibrium. The dashed blue circle indicates the Thomas-Fermi radius. In the vortex lattices, each site is labelled by its number of neighbors: blue disks for 6, red squares for 7 and pink diamonds for 5. The sequence illustrates the progression from crystalline to hexatic and liquid regimes (see Fig. \ref{['fig:4']}).
• Figure 4: (a) Correlation lengths $\ell_P$ (blue circles) and $\ell_G$ (green squares) as a function of temperature for $\Omega=0.99\,\omega_r$, computed within a disk of radius $R=0.9\times R_{\rm TF}$. The horizontal black dashed line indicates $R$. (b) Probability of having sites with 6 (blue circles), 5 (pink diamonds) and 7 (red squares) neighbors within a disk of radius $0.7\times R_{\rm TF}$. In both panels, the vertical dashed lines mark the estimated transition temperatures, and the shaded areas the uncertainties, see text for details. The error bars correspond to statistical uncertainties.
• Figure 5: Vortex lattice equilibrium phases as a function of the rotation frequency and temperature for $N=10^4$. Blue circles (orange squares) indicate the estimated crystal-hexatic $T_{s/h}$ (hexatic-liquid $T_{h/l}$) transition temperatures obtained from the simulations. The solid lines connect the points as a guide to the eye. The yellow dotted curve is an upper bound for the melting temperature, see Eq. eq:upperbound. Dashed curves indicate the estimated uncertainty on the transition temperatures (see text for details).