The University of Arizona

On the existence and stability of spectral gaps of quantum spin systems

On the existence and stability of spectral gaps of quantum spin systems

Series: Mathematical Physics and Probability Seminar
Location: Math 402
Presenter: Amanda Young, University of Arizona

Quantum spin systems are many-body systems in which particles are bound to the sites of a lattice. These models have recently received significant interest in part due to their promise for developing fault-tolerant quantum codes. Because the emergence of exotic particles such as anyons, specific interest has been given to two-dimensional models. The behavior of these systems is often dictated by the low-lying energy regime, including the existence or nonexistence of a spectral gap above the ground state energy. While the notion of a gapped ground state phase is easy to describe, proving that a system is gapped remains a notoriously difficult problem, especially for multi-dimensional models. We will discuss the martingale method for estimating spectral gaps, as well as recent progress in proving the existence of a gap for a class of multi-dimensional quantum spin systems known as the Product Vacua and Boundary State (PVBS) models. 
 
Once a spectral gap is known, a natural next step is to determine if the model remains gapped in the presence of perturbations. One situation where stability has been proved is for models with topologically ordered ground states. These are models for which the ground states cannot be distinguished by local operators. Previous works in this area have been for models with periodic boundary conditions and a unique infinite volume frustration-free ground state. In this talk, we will outline the stability results and discuss how they can be extended in several directions: including to quantum spin models with discrete symmetry breaking and lattice fermion models.

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