Spin and exchange in mixed quantum states from first principles in Feynman path integral quantum statistical mechanics representation

Abstract

The exchange and space correlations in the system of indistinguishable electrons against the background of thermal fluctuations are studied from first principles. A complete set of mutually orthogonal spin-coordinate basis functions symmetric with respect to permutations in coordinate and spin variables is constructed by the action of Young symmetry operators. As an illustration of a general approach, Path Integral Monte Carlo (PIMC) computer simulation of mixed quantum states of four electrons in a model ellipsoidal nanoscopic trap with soft walls is performed. The equilibrium value of the spin is calculated as well as the mechanism of its formation depending on the temperature and shape of the trap in connection with the evolution of the spatial organization of electrons is analyzed. The Quantum-statistical calculation, executed with a fundamentally accurate description of the permutation symmetry of electrons and the spin variable, includes in explicit form the Coulomb and exchange correlations of all orders and the energy exchange with a thermal bath. The equilibrium values ​​of the spin in dependence of the shape of the potential well and temperature as well as statistical distributions over the eigenfunctions of the spin squared operator are obtained. A general, fundamentally accurate and numerically implementable on computers available today computers approach for describing the spin states of systems of different numbers of indistinguishable fermions with spin 1/2 in the representation of path integrals is formulated.The regularities in dependence of spin on the shape of the system and on the temperature are studied. The variations in the geometry of the trap in direction of elongated shapes are accompanied by the transfer of a part of the electron cloud first from the equator to the poles of the ellipsoidal potential well, and then in the opposite direction with the subsequent collapse of the ring-shaped form into a compact clot in the center and its decomposition into two fragments localized on the axis of rotation. The spin of the mixed quantum state of four electrons at room temperature varies depending on the geometry of the system in the opposite direction, compared with the system of two and three electrons, reaching its maximum at a spherical shape. Spin states of electron ensembles several nanometers in size are unstable to thermal perturbations already in the region of room temperatures. Compression leads to a stabilization of spin states. The calculation results indicate a strong dependence of the energy of collective mixed spin states of electrons on a detailed account of exchange and Coulomb high-orders spatial correlations.