NLM DIR Seminar Schedule

Abstract:

The spin chain interacting systems have applications in various biological systems. The simplest spin interaction model - Ising model has been used in study of network states in the neural population, the spread of covid-19, development of cancer models, and even in financial markets and social sciences. We show that a generalization of the Ising model - quantum anisotropic XY model spin chains can be solved in periodic and open boundary conditions with arbitrary site lengths. The Hamiltonian with spin operators are converted to fermionic raising and lowering operators followed by Jordan-Wigner transformation. We discern the fermion parity e^[iπN] = (−1)^N in the boundary term in the periodic chain carefully to derive the ground state and excited state energy and their degeneracies. The periodic case was diagonalized by the Bogoliubov transformation; while the open boundary case is solved by matrix analysis with recursive relations. We present the expression of ground state energy. The even number sites have no degeneracy; while the odd number site chains have 4-fold degeneracy. For the open chain, the odd number chain can be expressed as a sum of analytical terms; while the fermionic energies have to be solved graphically. The degeneracy in the open chains is 2-fold degenerate with the odd number chains and no degeneracy with the even ones. The analytical form of the energy spectrum could potentially facilitate the simulations of the interacting systems.