Deciphering Root-of-Unity XXZ Quantum Spin Chains: Irreducible Representations of the Temperley–Lieb Algebra

Overview

When the number of qubits in a quantum system becomes extremely large, the system exhibits continuous, rather than discretized, behavior. These macroscopic systems are of interest at the intersection of quantum mechanics and statistical physics. One such canonical system is the spin-½ chain, which contains a chain of qubits each with spin upward or downward. Substantial work remains in identifying the smallest building-block spaces invariant under the Hamiltonian operators, a central question in the field. One promising direction lies in XXZ spin chains—a family of Heisenberg-Ising models—which depend on a complex parameter β. For generic β, Wenzl mapped out these building-block spaces; however, the special case where β = 2 cos(cπ) for rational c is significantly less tractable. In this work, I make foundational progress by solving this problem for β = 0, furnishing a complete set of building-block spaces—my key breakthrough being the novel construction of an exact sequence on representations of the non-semisimple Temperley–Lieb algebra. This case is the most difficult, deviating furthest from Wenzl’s generic-β regime. Moving forward, my research has the potential to improve the effectiveness of error detection and mitigation in quantum simulations, especially using the XXZ spin chain at β = 0 as a concrete benchmark.

Competition history

• CSEF 2026 Mathematical Sciences (Senior Division) · Entry S-14-07

Related projects

Closest projects by meaning, across every fair and year in the corpus.

Browse more like this

• All Mathematics projects
• CSEF Mathematics projects
• Every CSEF 2026 project