I am organizing a working group on the mathematics of quantum spin systems. If you are interested in joining, please contact me by email first to check that you have the prerequisite. I expect a strong background in linear algebra and analysis. (typically proof-based abilities on problems involving analysis, singular values and/or tensor products. Typically Math 318, 340 and 424 with excellent grades, equivalent classes, or personal experience).
If joining, you will be expected to give a 1+hour long presentation. Possible topics
Interest and availability form
Pizza (with vegetarian options) is provided during our regular meetings, Tuesdays 12-2PM, LOW 102.
Spring 2026
In the Spring 2026, we will discuss two classes of quantum states: matrix product states (MPS) and Gibbs (thermal) states:
MPS are representations of quantum states that arise in the density-matrix renormalization group algorithm, one of the main computational tools to compute ground states. We will see that low-entropy states admit a low-complexity MPS representation, and conversely.
Gibbs states describe the behavior of a physical system at a specific temperature. We will discuss the relation between decay of correlations and the impact of local perturbations on far-away measurements.
Spring 2026 schedule (tentative):
April 7th - 12-2PM in LOW 102: Introduction to quantum spin systems. Tensor products, locality, support and Lieb-Robinson bounds.
April 14th - 12-2PM in LOW 102: Gibbs states. Decay of correlations at high temperature.
April 21st - 12-2PM in LOW 102: Quantum propagation belief
April 28th - 12-2PM in LOW 102: Gibbs states preparation
May 5th - 12-2PM in LOW 102: Matrix product states
May 12th - 12-2PM in LOW 102: Density matrices, Schmidt decomposition, entanglement entropy
May 19th - 12-2PM in LOW 102: Low-entropy states have accurate MPS representations
May 26th - 12-2PM in LOW 102: Density matrix renormalization group algorithm
June 2nd - 12-2PM in LOW 102:
Winter 2026
In the Winter 2026, we introduced quantum spin systems. We discussed the locality of the dynamics (expressed through Lieb-Robinson bounds) and the decay of correlations for gapped ground states. We then introduced entropy of entanglement and proved the area law for ground states of gapped 1D quantum spin systems.
Schedule
January 8th - 12-2PM in PDL C401 - Alexis Drouot: An introduction to quantum spin systems. Notes
January 22nd - 12-2PM in PDL C401 - Miles Mai: Reduced States and Observables on Subsystems. Notes
January 28th - 3:30-4:30PM in DEN 111 - AGD seminar by Matthew Hastings: Lieb-Robinson bounds and applications to quantum spin systems: how dynamics control statics.
January 29th - 12-2PM in MOR 234 - Shiang-Bin Chiu: Emergent Speed Limits from Locality: Lieb-Robinson Bounds. References (Chapter 3.1, 3.2)
February 5th - 12-2PM in MOR 234 - Yusen Ye: Gapped ground states have exponentially small correlations. References (Chapter 6.1), Notes
February 12th - 12-2PM in MOR 234 - Alexis Drouot: Hastings' ground state factorization. Lemma 1 of Hastings' area law paper; see also this paper. Notes
February 19th - 12-2PM in MOR 234 - Amelie Martin: Classical and quantum entropy. Notes on Shannon entropy. Notes on entanglement entropy.
See also the videos below by Lieb, Topics in Quantum Entropy and Entanglement, given at the Princeton Summer school on "Quantum Information and Computation"
February 19th - 4:30-6PM in PDL C38 - Alexis Drouot: Up, down, and all around: A gentle introduction to quantum spins systems.
February 26th - 12-2PM in MOR 234 - Dante Tjowasi: Bounds on Entanglement Entropy of Approximately Low-Schmidt Rank Ground States. Notes
March 5th - 12-2PM in MOR 234 - Liam Bonds: Hasting's Area Law for One Dimensional Quantum Systems. Notes