The possibility of crystallization and magnetic ordering in an electron gas is examined in the determinantal approximation, using a variational method. A Mott transition is found at r s similar, equals 10 to a ferromagnetic crystalline state of the Wigner type, while an antiferromagnetic crystalline phase seems to be excluded. A simple way of including correlation effects is proposed, in which a correlation function is introduced into the interaction energy between electrons of antiparallel spin. The possibility then emerges of a crystalline antiferromagnetic phase over the large intermediate density range 10 less, equals r s less, equals 40.
The possibility of crystallization and magnetic ordering in an electron gas is examined in the determinantal approximation, using a variational method. A Mott transition is found at r s similar, equals 10 to a ferromagnetic crystalline state of the Wigner type, while an antiferromagnetic crystalline phase seems to be excluded. A simple way of including correlation effects is proposed, in which a correlation function is introduced into the interaction energy between electrons of antiparallel spin. The possibility then emerges of a crystalline antiferromagnetic phase over the large intermediate density range 10 less, equals r s less, equals 40.
We study finite-temperature properties of a Hubbard model including sites of a particle bath, which was proposed as a microscopic model to show itinerant ferromagnetism at finite electron density. We use direct numerical methods, such as exact diagonalization and random vector methods. The temperature dependence of quantities is surveyed in the full range of the temperature. We find that the specific heat has several peaks, which correspond to ordering processes in different energy scales. In particular, magnetic correlations develop at very low temperature. The system exhibits an itinerant ferromagnetic state or an antiferromagnetic state of the Mott insulator depending on the chemical potential of the particle bath and the Coulomb interaction. From a microscopic viewpoint, the competition between these two types of magnetic states causes a peculiar ordering process of local spin correlations. Some local ferromagnetic correlations are found to be robust, which indicates that the ferromagnetic correlation originates from the motion of itinerant electrons in a short-range cluster.
A discussion of the various properties of pair correlation functions of a magnetic electron gas is presented. The cusp conditions obeyed by the many-electron wave functions for a pair of electrons with parallel and antiparallel spins in close proximity are expressed as derivative conditions on the corresponding parallel-and antiparallel-spin pair correlation functions. In particular, it is shown that in a fully ferromagnetic gas the parallel spin correlations determine the correlation energy whereas in the paramagnetic case, this function is of less importance in comparison to the antiparallel-spin correlation function. In the high-momentum limit, it is found that the momentum distribution of electrons is dominated by antiparallel-spin correlations in the partially magnetic case whereas in the fully ferromagnetic case, it is dominated by parallel-spin correlations but is smaller by a factor of ${q}^{\ensuremath{-}2}$. A calculation of the parallel-spin correlation function for the ferromagnetic gas in perturbation theory is presented. A model for the exchange correlation energy of the system which takes into account the exact conditions given here is proposed. Corresponding results for the two-dimensional electron gas are also stated.
We construct a model for the value of the pair distribution function g ↑↓ (0) between antiparallel-spin electrons at contact in a two-dimensional electron gas with e 2 / r interactions, as a function of the coupling-strength parameter r s . The model involves an interpolation between the result of a low- r s expansion, including the second-order direct and exchange contributions to the energy in the paramagnetic state, and the result of a partial-wave phase-shift analysis near Wigner crystallization. The interpolation formula is in excellent agreement with many-body calculations based on the ladder approximation. We further show through an STLS self-consistent calculation that g ↑↓ (0) is essentially independent of the state of spin polarization of the electron gas.
Introduction In this chapter we study metal–insulator transitions which occur due to correlations between the charge carriers. This quantum phase transitions results from the competition between kinetic energy and Coulomb repulsion where the latter has the tendency to localise the electrons. The simplest model which describes this competition is the one-band Hubbard model with its two energy scales, the bandwidth W and the Coulomb repulsion U between electrons of opposite spins on the same site. For a fixed density of electrons, n = 1; as the ratio U/W increases, we expect to find a zero-temperature phase transition from a metallic state to an insulator where the electrons are frozen at the lattice sites. The method employed here to investigate this transition is the Gutzwiller approach (Gutzwiller, 1965). This is a variational method which played an important role in providing a framework for the ideas of Mott on the problem of the metal–insulator transition (Mott, 1974). On the metallic side of the transition, this approach describes a metal with renormalised Fermi liquid parameters, essentially with enhanced effective mass and uniform susceptibility. This is the so-called highly correlated electron gas (Brinkman and Rice, 1970) for which the metal oxide V 2 O 3 is the prototype system. In the insulating side, Gutzwiller's approximation yields a noninteracting system of localised electrons. This insulating phase is incompressible , i.e. has zero compressibility and infinite zero temperature susceptibility. The latter is not a consequence of interactions but simply due to the divergence of the Curie susceptibility, χ ∝ 1 /T , of non-interacting moments at T = 0. Then, in Gutzwiller's approach the transition is from a paramagnetic metal to a paramagnetic insulator. We will continue using the designation Mott transition to describe this type of phase transition where the insulating phase has no long-range magnetic order. In real systems, most frequently, localisation due to interactions is accompanied by the appearance of antiferromagnetic order.
Alexander Shashkin Institute of Solid State Physics, Chernogolovka, Moscow District 142432, Russia Sergey Kravchenko Physics Department, Northeastern University, Boston, Massachusetts 02115, U.S.A. Two-dimensional (2D) electron systems are realized when the electrons are free to move in a plane but their motion perpendicular to the plane is quantized in a confining potential well. Quantum phase transitions realized experimentally in such systems so far include metal-insulator transitions in perpendicular magnetic fields, metal-insulator transition in zero magnetic field, and a possible transition to a Wigner crystal. The first transition is governed by the externally controlled electron density or magnetic field, while the other two are governed by the electron density. At low electron densities in 2D systems, the strongly-interacting limit is reached because the kinetic energy is overwhelmed by the energy of electron-electron interactions. The interaction strength is characterized by the ratio between the Coulomb energy and the Fermi energy, r∗s = Eee/EF . Assuming that the effective electron mass is equal to the band mass, the interaction parameter r∗s in the single-valley case reduces to the Wigner-Seitz radius, rs = 1/(πns)1/2aB, and therefore increases as the electron density, ns, decreases (here aB is the Bohr radius in the semiconductor). Possible candidates for the ground state of the system include a Wigner crystal characterized by spatial and spin ordering [1], a ferromagnetic Fermi liquid with spontaneous spin ordering [2], a paramagnetic Fermi liquid [3], etc. In the strongly-interacting limit (rs 1), no analytical theory has been developed to date. According to numerical simulations [4], Wigner crystallization is expected in a very dilute regime, when rs reaches approximately 35. Refined numerical simulations [5] have predicted that prior to the crystallization, in the range of the interaction parameter 25 ≤ rs ≤ 35, the ground state of the system is a strongly correlated ferromagnetic Fermi liquid. At higher electron densities, rs ∼ 1, the electron liquid is expected to be paramagnetic, with the effective mass, m, and Lande´ g factor renormalized by interactions. Apart from the ferromagnetic Fermi liquid, other intermediate phases between the Wigner crystal and the paramagnetic Fermi liquid may also exist.
We use quantum Monte Carlo methods to calculate the zero-temperature phase diagram of the two-dimensional homogeneous electron gas. We find a transition from a paramagnetic fluid to an antiferromagnetic triangular Wigner crystal at density parameter r(s)=31(1) a.u. and a transition to a ferromagnetic crystal at r(s)=38(5) a.u. The fully spin-polarized fluid is never stable. We search for, but do not find, the ferromagnetic "hybrid" phase proposed by H. Falakshahi and X. Waintal [Phys. Rev. Lett. 94, 046801 (2005)10.1103/PhysRevLett.94.046801].