Les Houches Lectures Notes on Tensor Networks
Bram Vancraeynest-De Cuiper, Weronika Wiesiolek, Frank Verstraete
TL;DR
Les Houches Tensor Networks presents a coherent framework that uses MPS/PEPS and MPO algebras to classify and simulate strongly correlated quantum matter. The five lectures develop a geometry of variational manifolds, the calculus of MPS/DMRG/VUMPS, and the role of generalized (fusion‑category) symmetries in 1D and 2D, culminating in string‑net and topological order formalisms. A central theme is that global properties and phase structure are encoded locally in tensor networks via projective representations, fusion rules, and intertwiners, enabling a unified treatment of SPTs, topological order, and dualities through the tube algebra and module categories. The work highlights a powerful, holographic view where entanglement structure and generalized symmetries dictate phase classification, edge physics, and universal behavior, with practical computational tools for 1D and 2D systems and deep connections to TFT/CFT concepts.
Abstract
Tensor networks provide a powerful new framework for classifying and simulating correlated and topological phases of quantum matter. Their central premise is that strongly correlated matter can only be understood by studying the underlying entanglement structure and its associated (generalised) symmetries. In essence, tensor networks provide a compressed, holographic description of the complicated vacuum fluctuations in strongly correlated systems, and as such they break down the infamous many-body exponential wall. These lecture notes provide a concise overview of the most important conceptual, computational and mathematical aspects of this theory.