Knotted Configurations in the Continuous Heisenberg Spin Chain with Lower Bound of the Energy

Abstract

When studying the classical Heisenberg spin models, we usually map the normalised unit vector, that represents the spin, on the tangent of a space curve. The total chirality of the curve (the spin configuration) i.e., the total momentum of the spin system, is a conserved quantity [1] and therefore self-crossing of the curve is not allowed. Using this fact and the non-contractibility of some closed loops in SO (3), we have proposed new topological spin configurations for the Heisenberg spin model [1]. Now we propose new topological spin configuration for the Heisenberg model, which represents a knot and has higher lower bound for the energy as the previously proposed configurations. The same analysis holds for a thin elastic rod that is bent to an open knot. PACS numbers:  75.10. Pq, 75.10. Hk, 02.40. Hw, 02.10. Kn