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A system of confined charged electrons interacting via the long-range Coulombforce can form a Wigner crystal due to their mutual repulsion. This happenswhen the potential energy of the system dominates over its kinetic energy,i.e., at low temperatures for a classical system and at low densities for aquantum one. At $T=0$, the system is governed by quantum mechanics, and hence,the spatial density peaks associated with crystalline charge localization aresharpened for a lower average density. Conversely, in the classical limit ofhigh temperatures, the crystalline spatial density peaks are suppressed(recovered) at a lower (higher) average density. In this paper, we study thosetwo limits separately using an exact diagonalization of small one-dimensional(1D) systems containing few ($<10$) electrons and propose an approximate methodto connect them into a unified effective phase diagram for Wigner few-electroncrystallization. The result is a qualitative quantum-classical crossover phasediagram of an effective 1D Wigner crystal. We show that the spatial densitypeaks associated with the quasi-crystallization should be experimentallyobservable in a few-electron 1D system. We find that the effective crystallinestructure slowly disappears with both the crossover average density andcrossover temperature for crystallization decreasing with increasing particlenumber, consistent with the absence of any true long-range 1D order. In fact,one peculiar aspect of the effective finite-size nature of 1D Wignercrystallization we find is that even a short-range interaction would lead to afinite-size 1D crystal, except that the crystalline order vanishes much fasterwith increasing system size in the short-range interacting system compared withthe long-range interacting one.

One-dimensional few-electron effective Wigner crystal in quantum and classical regimes