Unexpected turns of a humble theorem

Most of interacting quantum many-body systems are not exactly solvable, and many of them are still challenging for computational approaches. On the other hand, there are a few rigorous theorems which can be applied to a wide range of systems to limit the possible phases of matter to be realized. A theorem proved by Lieb, Schultz, and Mattis in 1961 apparently did not draw much attention (even by the authors themselves) initially, but has evolved over time to become one of the most important theorems for quantum many-body systems. Under certain conditions, the theorem forbids the system to be in a featureless trivial phase. Thus it gives a useful clue for finding nontrivial topological phases. In this talk, I will review the basics and historical evolution of Lieb-Schultz-Mattis theorem, including some of the recent developments.