The Tensor Network (TN) scheme has been evolving conceptually and practically over the past fifteen years, and has produced theoretical and computational tools in various fields. In elementary particle physics, we recently succeeded in applying the Grassmann Tensor Renormalization Group (GTRG) algorithm, a TN scheme, to the analysis of the phase structure of one-flavor lattice Schwinger model (two-dimensional QED). The results show that the algorithm is free from the sign problem, and the computational cost is comparable to bosonic case, thanks to the direct manipulation of the Grassmann variables. This was the first successful application of a TN scheme to a Euclidean lattice gauge-theory including relativistic fermions.
Furthermore, we also performed analysis on the phase structure of the one-flavor lattice Schwinger model with the so-called θ-term. In spite of the complex action, we have succeeded in reproducing the phase structure predicted by analytical calculations. This provides evidence that the GTRG algorithm has the ability to treat complex actions. Our work has shown that the GTRG algorithm solves the notorious sign problem inherent in the Monte Carlo method by using the Schwinger model as a testing ground. Toward the final goal of investigating four-dimensional lattice QCD, which is the SU(3) non-Abelian gauge theory with relativistic fermions, we are currently studying higher dimensional models including non-Abelian gauge theories.