Full symmetry implementation in condensed matter and molecular physics—Modified group projector technique

Abstract

Symmetry is well established as one of the fundamental concepts in physics, accurately extracting relevant characteristics of the studied object, giving deep and transparent insight to its properties. In the solid state and molecular physics the most abundant application is reduction of the dimension of the eigenproblem of the Hamiltonian, with the resulting eigenvectors labeled by good quantum numbers, forming the so called symmetry adapted basis. Such a basis is the starting point for subsequent analysis of the physical properties of the system, performed usually by applying adequate perturbation technique. Standard procedure for finding a symmetry adapted basis involves Wigner operators, which are sums of the operators acting in the quantum state space (Hilbert space, most usually) over all elements of the symmetry group of the systems. However, both the dimension of the state space and the number of the symmetry transformations are infinite even in the simplest approximate models in crystal physics making obstacles for direct application of the standard Wigner projector technique, and its numerical implementation. On the other hand, there is a minimal part of the system, the full symmetry elementary cell (symcell), from which the whole system can be built by action of the full symmetry group elements on it. A clear heuristic idea, that symcell and full symmetry group, determine the properties of the entire system, is fully realized within modified group projector technique. Namely, when applying this technique, the full symmetry of the system is used to provide reduction of calculations to the symcell only, singling out its state space (of a finite dimension!) as the effective state space to be worked in. Physical observables, expressed through their irreducible tensor components, obtain their counterparts in this finite-dimensional space of a symcell. It remains to consider only the symmetry transformations which leave the symcell invariant. This is absolutely sufficient for the complete information on the properties of the system. In this report we review the modified group projector technique, giving detailed algorithm for its application, which is also a clear prescription for its numerical implementation. Generality of the method allows unified treatment of the most important dynamical models for ions (lattice dynamics within harmonic approximation), electrons (tight-binding method for the quantum states) and spin subsystem (spin waves in quasi classical approach), with specific details considered separately. Finally, relaxation techniques, optimization of the structure and of the spin ordering are analyzed, in order to provide the most efficient symmetry based procedures.