A system of confined charged electrons interacting via the long-range Coulomb force can form a Wigner crystal due to their mutual repulsion. This happens when 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 a quantum one. At T = 0, the system is governed by quantum mechanics, and hence the spatial density peaks associated with crystalline charge localization are sharpened for a lower average density. Conversely, in the classical limit of high temperatures, the crystalline spatial density peaks are suppressed (recovered) at a lower (higher) average density. In this paper, we study those two limits separately using an exact diagonalization of small one-dimensional (1D) systems containing few (\<10) electrons and propose an approximate method to connect them into a unified effective phase diagram for Wigner few-electron crystallization. The result is a qualitative quantum-classical crossover phase diagram of an effective 1D Wigner crystal. We show that although such a 1D system is at best an effective crystal with no true long-range order (and thus no real phase transition), the spatial density peaks associated with the quasicrystallization should be experimentally observable in a few-electron 1D system. We find that the effective crystalline structure slowly disappears with both the crossover average density and crossover temperature for crystallization decreasing with increasing particle number, consistent with the absence of any true long-range 1D order. Thus, an effective few-electron 1D Wigner crystal may be construed either as existing at all densities (manifesting short-range order) or as nonexisting at all densities (not manifesting any long-range order). Within one unified description, we show through exact theoretical calculations how a small 1D system interacting through the long-range Coulomb interaction could manifest effective Wigner solid behavior both in classical and quantum regimes. In fact, one peculiar aspect of the effective finite-size nature of 1D Wigner crystallization we find is that even a short-range interaction would lead to a finite-size 1D crystal, except that the crystalline order vanishes much faster with increasing system size in the short-range interacting system compared with the long-range interacting one.

}, issn = {2469-9950}, doi = {10.1103/PhysRevB.101.125113}, author = {DinhDuy Vu and Das Sarma, Sankar} }