NLM DIR Seminar Schedule

Abstract:

The spin-interacting models have wide applications in studying biological systems such as pattern generations, neural networks, and the spread of disease. In addition, the central charge of conformal field theory (CFT) could quantify the universality classes and give the magnitude of the Casimir effect. It has led to research on the categorization of cell membranes with multiple phases which have implications for cell trafficking and communications. To better understand the spin interaction systems, we revisit the well-known XX model, along with the energy spectrum and the ground state degeneracy. While imposing the translational invariance, we obtain the energy spectrum of the finite-length periodic chain via Jordan-Wigner transformation with suitable momentum mode choices. The finite open chain violates the translational symmetry and is solved by matrix analysis in addition to the Jordan-Wigner transformation. By investigating the long chain length asymptotics, we find different dominant correction terms for chains under open and periodic boundary conditions as well as for chains of even and odd number of sites. By comparing the asymptotic form of the ground state energy with the one from CFT, we confirm that the conformal central charge for the XX chain is c = 1 for the even chain lengths, albeit for open boundary conditions there exists an additional boundary energy term. For the odd number site chains, while the boundary energy for the open boundary remains the same, the system is not describable by CFT with the central charge c = 1.