8.582 Selected Topics in Condensed Matter Physics

8.582 Selected Topics in Condensed Matter Physics

by Xiao-Gang Wen

2026 Spring, 2:30 PM to 4:00 PM on Wednesday and Friday in Room 56-154, MIT

Course site: https://canvas.mit.edu/courses/37294

Artificial intelligence is based on neural networks. A well-trained neural network develops an intricate internal structure that enables remarkable capabilities. However, describing and characterizing this internal structure remains a difficult and largely open challenge.

A similar problem arises in many-body physics. Many-body systems can be described using tensor networks. Under renormalization via coarse graining, a tensor network flows toward a fixed point. The fixed-point tensor network possesses rich internal structures that characterize gapped and gapless phases, emergent symmetries, emergent gauge interactions, as well as emergent Fermi and fractional statistics, etc. A central question is how to uncover and understand the intricate structures hidden in these fixed-point tensor networks.

The first part of this course introduces the tensor network approach to many-body systems. We will discuss numerical methods for driving the renormalization of tensor networks, and then explain how to extract the algebraic structures encoded in fixed-point tensor networks. Using the tensor network formalism, we will study many-body entanglement, topological order, symmetry-protected topological order, generalized symmetries, and related topics.

1) Discrete path integral (tensor network) for many-body systems.
2) Tensor network renormalization for gapped phases and gapless phase transitions
3) Algebra structures (Frobenius algebra, pentagon equation) in fixed-point tensor network
4) Fixed-point tensor network and exactly soluble model
5) Chain, cochain, cycle, cocycle – realization and classification of SPT phases
6) Quantum circuit: short-range entanglement and long-range entanglement
7) Topologically ordered phases
8) Holographic picture of generalized symmetry as topological order in one higher dimension.
9) Emergent generalized symmetry determine critical points.
10) … …

There will be weekly homework, due on Friday (65% of grade).
There will also be a term paper (35% of grade).

Office Hour: Wednesday 4:00 – 5:00 pm, Room 6c-317 (Xiao-Gang Wen).

The lecture notes will be updated frequently. Please download often.