Mathematical Physics Seminar
Monte Carlo simulations on discrete space-time lattices have been successful as a non-perturbative numerical approach to particle physics. However, many important models such as supersymmetric systems, chiral gauge theories, and finite density QCD have complex or negative Boltzmann weights, and Monte Carlo simulations cannot be simply applied to such models. This is a strong motivation to seek deterministic ways rather than Monte Carlo schemes, and tensor network is a powerful tool to break the situation. We discuss a coarse-graining algorithm for tensor networks and its applications to particle physics. As a concrete example, we consider an application to the two-dimensional $\phi^4$ theory. Since the continuous d.o.f. do not get along with tensor networks, which are labeled by discrete indices, we introduce a new method to extract the tensor indices. To assess the validity of the new method, a continuum limit value of the critical coupling of the $\phi^4$ theory is calculated. After the basic case, an application to a supersymmetric theory is discussed. We put our focus on the two-dimensional Wess-Zumino model, which contains a one-component real scalar field and a two-component Majorana spinor field. To deal with the scalar field, the new method described above is employed. We show that the tensor renormalization group reproduces the analytical result of the Witten index.