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 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: Arjun Aggarwal - Introduction to quantum spin systems: tensor products, local Hamiltonians, supports and Lieb-Robinson bounds. This talk is a recap of the material covered in the first 3 weeks of the Winter quarter. We discuss the mathematical formalization of quantum spin systems. We define tensor products and relate them to formalization of systems of multiple spins. We also introduce Hamiltonians and discuss the quantum Ising model as an illustrative example. Time permitting, we state and give some intuition for the Lieb-Robinson bound. Notes
April 14th - 12-2PM in LOW 102: Miles Mai - Quantum Belief Propagation. We define the Gibbs state and state/prove the Quantum Belief Propagation theorem, which bounds how the Gibbs state responds to local perturbations. It shows that a local perturbation affects a Gibbs state by multiplying its density matrix by terms supported close to the perturbation. As usual, we prove this by exploiting Lieb-Robinson. Miles' notes, Hastings' original paper, and a more recent mathematically minded one by Capel-Moscolari-Teufel-Wessel
AGD Seminar, April 15th - 3:30-4:30 in DEN 111: Zhiqian (Simon) Du, UC Davis - Absence of Hall conductance in frustration-free fermionic systems. We give a new proof that translation-invariant free Fermion frustration-free models have zero Hall conductance. The central argument is that in frustration-free models, edge modes can’t merge into the ground state band. We also give an example of a gapped frustration-free model that is gapless on the half-plane.
April 21st - 12-2PM in LOW 102: Alexis Drouot - From decay of correlations to robustness of Gibbs states. First, I will use the quantum belief propagation principle to show that if a Gibbs state satisfies decay of correlations then it responds locally to perturbations (i.e. local defects do not affects far-away measurements). Then I will sketch a proof that Gibbs state in 1D systems do satisfy exponential decay of correlations. Notes. I am basing this talk on works of Capel-Moscolari-Teufel-Wessel and Kimura-Kuwahara.
April 28th - 12-2PM in LOW 102: Samiullah Khan Badal Khan - Lindbladians: nature's generators for preparing the Gibbs state I. We introduce Quantum channels, which are Completely Positive Trace Preserving (CPTP) maps, as the generalization of all transformations of quantum states and discuss their corresponding Kraus form. We then move on to studying how a state evolves when subjected to a family of maps that are natural for a system that is Markovian, which leads us to defining the Lindbladian as the generator for these channels. We will prove some properties of Lindbladians and then study the Davies generator, which describes the thermalization towards the Gibbs state for a quantum system weakly coupled to a heat bath, and begin sketching how one can prepare the Gibbs state using these ideas. Based on the works by Davies (I) and Davies II, and recent works by Anthony Chen, Kastoryano, and Gilyén. Notes
May 5th - 12-2PM in LOW 102: Dante Tjowasi - Lindbladians: nature's generators for preparing the Gibbs state II. We will continue our discussions on Lindbladians by introducing some tools to analyze it. In particular, we will discuss the quantum detailed balance condition, and how to analyze the “mixing time” of Lindbladians satisfying such conditions, analogous to similar results on classical Markov semigroups. I will try to sketch the proof of fast mixing time for the Lindbladians under high temperature conditions, showing an efficient algorithm to prepare/sample from a Gibbs state in high temperatures. Draws on work by Chen-Kastoryano-Gilyen and Rouze-Franca-Alhambra.
May 12th - 12-2PM in LOW 102: Dante Tjowasi - Lindbladians: nature's generators for preparing the Gibbs state III.
May 19th - 12-2PM in LOW 102: Matrix product states: an introduction
May 26th - 12-2PM in LOW 102: Low-entropy states have accurate MPS representations
June 2nd - 12-2PM in LOW 102:
AGD Seminar, June 3rd - 3:30-4:30 in DEN 111: Anthony Chi-Fang Chen, UC Berkeley
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