Entangled multi-knot lattice model of anyon current

TL;DR

This paper introduces an entangled multi-knot lattice model to simulate and analyze the topological and statistical properties of anyons, linking knot theory with quantum many-body physics and quantum Hall phenomena.

Contribution

It presents a novel knot lattice framework capturing anyon statistics, fusion rules, and phase transitions, connecting topological invariants with quantum states and potential quantum computing.

Findings

Anyon statistics are characterized by topological linking numbers.

Fusion rules are demonstrated through braiding on the knot lattice.

Quantum phase transitions are quantified by changes in linking numbers.

Abstract

We proposed an entangled multi-knot lattice model to explore the exotic statistics of anyon. This knot lattice model bears abelian and non-abelian anyons as well as integral and fractional filling states that is similar to quantum Hall system. The fusion rules of anyon are explicitly demonstrated by braiding on crossing states of the multi-knot lattice. The statistical character of anyon is quantified by topological linking number of multi-knot link. Long-range coupling interaction is a fundamental character of this knot lattice model. The short range coupling models, such as Ising model, fermion paring model, Kitaev honeycomb lattice model and so on, appears as the short range coupling case of the knot lattice model. We introduced link lattice pattern as geometric representation of the eigenstate of quantum many body model to explore the topological nature of quantum eigen-states. For example, a convection flow loop is introduced into the well-know BCS fermions pairing model to show the Pseudo-gap state in unconventional super-conducting state. The integral and fractional filling numbers in quantum Hall system is directly quantized by topological linking number. The quantum phase transition between different quantum states in quantum spin model is also directly quantified by the change of topological linking number, which revealed topological character of phase transition. This multi-knot lattice has a promising physical implementation by circularized photons in optical firbre network. It may also provide another different path to topological quantum computation.