Melting of Trapped Few-Particle Systems

In small confined systems predictions for the melting point strongly depend on the choice of quantity and on the way it is computed, even yielding divergent and ambiguous results. We present a very simple quantity that allows us to control these problems—the variance of the block averaged interparticle distance fluctuations.

Figure 1

Configuration of a 2D trapped quantum system [Eq. (2)] of $N=8$ spin-polarized bosons at a temperature close to the ground state. (a),(b) The liquidlike ($\lambda =14$) and solidlike ($\lambda =30$) states, respectively. (c) The corresponding radial density profiles $\rho (r)$.

Figure 2

(a) Distance of an arbitrary pair of $N=4$ classical particles in 2D as a function of the MC step. From top to bottom: $T_1=0.02$ (solidlike), $T_2=0.06$ and $T_3=0.09$ (transition region), and $T_4=0.5$ (liquidlike). $a$ and $b=\sqrt{2}a$ denote the two possible interparticle distances in the ground state. (b) Temperature dependence of the mean block averaged IDF ${\overline{u}}_{\mathrm{rel}}$, for different block lengths $M={10}^3$, ${10}^4$, ${10}^5$, ${10}^6$ (right to left) [equivalent to computing $u_{\mathrm{rel}}$, Eq. (1), from multiple simulations of length $L=M$]. (c) The corresponding second moment ${\sigma }_{u_{\mathrm{rel}}}$, Eq. (3). (d) Specific heat $C_v$ and energy correlation time $k_{\mathrm{corr}}$. (e) Total energy autocorrelation function $C_E$, Eq. (4), for three temperatures.

Figure 3

Left: Typical behavior of the block averaged IDF vs block number $s$ for $N=8$ charged bosons in 2D (cf. Fig. 1) for different coupling strengths: ${\lambda }_1=14$, ${\lambda }_2=22$, ${\lambda }_3=26$, and ${\lambda }_4=30$. Each point is an average over a block of length $M=1000$. Right: Histograms show the probability $P$ of different values $u_{\mathrm{rel}}$ averaged over a total of 9,000 blocks. Results are from PIMC simulations of system (2).

Figure 4

Mean value ${\overline{u}}_{\mathrm{rel}}$ (top) and second moment ${\sigma }_{u_{\mathrm{rel}}}$ (bottom) of the block averaged IDF for different particle numbers $N$. Left: temperature dependence of a classical 3D system (classical melting, classical MC simulations). Right: 2D quantum system, dependence on the quantum coupling parameter $\lambda$ (quantum melting, PIMC results). In both cases the block length equals $M=1000$. Dashed lines locate the critical values of $T$ (or $\lambda$) and $u_{\mathrm{rel}}^{\mathrm{crit}}$.