Andreev-Lifshitz Supersolid for a Few Electrons on Small Periodic Square Lattices

Abstract

To describe how one goes from independent particle towards collective motion in two dimensions is a difficult, fascinating and fundamental question which has been investigated since almost half a century. The parameter governing this crossover is the dimensionless ratio r _s introduced by E. Wigner. In two dimensions, assuming a uniform density n _s of carriers interacting via a long range e ²/r Coulomb repulsion, a ₀=1/√n _s is the distance between the carriers. The Bohr radius a _B =ℏ²/me ² is the size of the hydrogen atom in its ground state, and characterizes the length where one should expect very strong quantum effects. For carriers of effective mass m ^* in a medium of dielectric constant ε, a _B becomes an effective Bohr radius a ₀=1/√n _s is the distance between the carriers. The Bohr radius a ^* _B =ℏ²∈/m ^* e ². The factor r _s is the ratio of those two characteristic scales:

$$r_s = \frac{{a_0 }} {{a_B^* }} = \frac{{E_{Coul} }} {{E_F }},$$

((1))

or the ratio of two characteristic energies, the first being the classical Coulomb energy:

$$E_{Coul} \sim \frac{{e^2 }} {{ \in a_0 }},$$

((2))

, while the second is the quantum kinetic energy:

$$E_F \sim \frac{{\hbar ^2 }} {{2m*a_0^2 }},$$

((3))

, i.e. the Fermi energy of the non interacting system.