Evidence for a Quantum Phase Transition to a Ferromagnetic State in a 2D Electron System.

Dongzi Liu

J.H Davies, A.R Long

The nature of the ground state of degenerate two dimensional (2D) fermions at zero magnetic field is an outstanding open problem, which has not been deciphered despite decades of research. In the absence of disorder, the ground state is believed to be determined by the interplay between the kinetic energy, E f, and the inter-particle interaction energy, Ec = e2/ica, where a = (nn)~1^ 2 is the typical inter-particle distance, n is the areal particle density, and k is the host dielectric constant. The relative importance of the two energy scales is characterized by rs = a/ao, with ao being the Bohr radius. For electrons in a single band, rs = Ec/ E f while for the (100) surface of silicon rs = Ec/2E p , due to the two-fold valley degeneracy. At very high densities (rs <£ 1) a 2D system approaches the paramagnetic limit of a non-interacting degenerate gas characterized by the Pauli susceptibility, xo • As the density is reduced, the growing ferromagnetic correlations lead to substantial enhancement of the spin susceptibility, The system is predicted to remain paramagnetic up to rs « 20, where numerical calculations [1] find a quantum phase transition to a ferromagnetic liquid phase [2]. At lower density, rs & 37 [3], the Coulomb correlations are predicted to lead to a second phase transition to an anti-ferromagnetic quantum Wigner crystal. The energy balance between the ferro and paramagnetic states is very subtle, therefore the density window where ferromagnetism can take place is small [1].

Systems Alexander Shashkin, Sergey Kravchenko

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.

Anders W. Sandvik, Marco Vekić

Xin Wan

Yuan-Heng Tseng, Y.-H. Tseng, Fu-Jiun Jiang

A universal (supervised) neural network (NN), which is only trained once on a one-dimensional lattice of 200 sites, is employed to study the phase transition of the two-dimensional (2D) 5-state ferromagnetic Potts model on the square lattice. In particular, the NN is obtained by using merely two artificially made configurations as the training set. Due to the elegant features of the considered NN, results associated with systems consisting of over 4000000 spins can be obtained with ease, and convincing evidence showing the investigated phase transition is weakly first order is reached. The outcomes demonstrated here can hardly be achieved with the standard NNs that are commonly used in the literature.