Exactly solvable models and knot theory

Xie Chen

D. Ya. Petrina

Yasuhiro Akutsu, Miki Wadati

Presented is a general method to construct representations of the braid group, a basic object in the knot theory, from the Boltzmann weights of the exactly solvable models in statistical mechanics at criticality. The method is applied to a class of N -state vertex models ( N =2, 3 and 4) to have explicit braid group representations. Furthermore, the Markov traces are introduced and a sequence of new link polynomials, topological invariants for knots and links, is constructed.

X. M. Yang, L. Jin, Z. Song

Knot theory provides a powerful tool for understanding topological matters in biology, chemistry, and physics. Here knot theory is introduced to describe topological phases in a quantum spin system. Exactly solvable models with long-range interactions are investigated, and Majorana modes of the quantum spin system are mapped into different knots and links. The topological properties of ground states of the spin system are visualized and characterized using crossing and linking numbers, which capture the geometric topologies of knots and links. The interactivity of energy bands is highlighted. In gapped phases, eigenstate curves are tangled and braided around each other, forming links. In gapless phases, the tangled eigenstate curves may form knots. Our findings provide an alternative understanding of phases in the quantum spin system, and provide insights into one-dimension topological phases of matter.

Malte Henkel, Michel Pleimling