Wigner Crystallization of Electrons in a One-Dimensional Lattice: A Condensation in the Space of States

We study the ground state of a system of spinless electrons interacting through a screened Coulomb potential in a lattice ring. By using analytical arguments, we show that, when the effective interaction compares with the kinetic energy, the system forms a Wigner crystal undergoing a first-order quantum phase transition. This transition is a condensation in the space of the states and belongs to the class of quantum phase transitions discussed in [M. Ostilli and C. Presilla, J. Phys. A 54, 055005 (2021).]. The transition takes place at a critical value ${r_s}_c$ of the usual dimensionless parameter $r_s$ (radius of the volume available to each electron divided by effective Bohr radius) for which we are able to provide rigorous lower and upper bounds. For large screening length these bounds can be expressed in a closed analytical form. Demanding Monte Carlo simulations allow to estimate ${r_s}_c\simeq 2.3\pm 0.2$ at lattice filling $3/10$ and screening length 10 lattice constants. This value is well within the rigorous bounds $0.7\le {r_s}_c\le 4.3$. Finally, we show that if screening is removed after the thermodynamic limit has been taken, ${r_s}_c$ tends to zero. In contrast, in a bare unscreened Coulomb potential, Wigner crystallization always takes place as a smooth crossover, not as a quantum phase transition.

Figure 1

Value of $g_{\text{cross}}(N,N_p)$, solution of Eq. (12), as a function of $N_p$ with filling $N_p/N=3/10$ and screening length $R=10a$. Four different condensed subspaces (different $\max V_{\mathrm{cond}}$) are considered [30]. Numerical MC data (symbols) are extrapolated to $N_p\to \infty$ by fitting $A+B/N_p$ (solid lines) to the points with largest $N_p$. The horizontal dashed lines are the rigorous bounds of Eqs. (26) and (27). Inset: derivative of the GS energy per particle versus $g$ for different values of $N_p$. The shaded column indicates the value of $g_c=2.76\pm 0.24$ extrapolated as explained above.