Topological order from quantum loops and nets

Abstract

I define models of quantum loops and nets which have ground states with topological order. These make possible excited states comprised of deconfined anyons with non-abelian braiding. With the appropriate inner product, these quantum loop models are equivalent to net models whose topological weight involves the chromatic polynomial. A useful consequence is that the models have a quantum self-duality, making it possible to find a simple Hamiltonian preserving the topological order. For the square lattice, this Hamiltonian has only four-spin interactions

Link invariants, the chromatic polynomial and the Potts model

“…This also provides a relationship between certain string-net models and loop models, cf [12], [7]. Applying traces to these algebras, we express the SO(3) Kauffman polynomial in terms of the chromatic polynomial, see corollary 6.5.…”

“…Recently such connections found an important new application in condensed matter physics, in the study of topological states of matter, cf [14], [12], [7], [24]. In this paper we present a number of results at the intersection of combinatorics, quantum topology, and statistical mechanics which have applications both in mathematics, specifically to the properties of the chromatic polynomial of planar graphs and its relation to link invariants, and in physics (in the study of the Potts model and of quantum loop models).…”

“…In particular, the chromatic algebra discussed in this paper underlies the quantum loop models discussed in [8] and further developed in [12,7] (closely related models were introduced in [24]). Quantum loop models provide lattice spin systems whose low-energy excitations in the continuum limit are described by a topological quantum field theory [14].…”

Tutte chromatic identities from the Temperley–Lieb algebra

“…The origin of Beraha's definition of the numbers B n is his observation [2] that the zeros of the chromatic polynomial of large planar triangulations seem to accumulate near them. Tutte's estimate (1)(2)(3)(4)(5) gives a hint about this phenomenon for B 5 . Efforts have been made (see Kauffman and Saleur [14] and Saleur [20]) to explain Beraha's observation using quantum groups.…”

“…We show that the existence of this map is implied by the relation (1-2), and then (1-1) follows from applying the algebra traces to the homomorphism ‰ . The golden identity has an interesting application in physics in quantum loop models of "Fibonacci anyons", where it implies that these loop models should yield topological quantum field theories in the continuum limit (see Fendley [5], Fidkowski et al [8] and Walker [26] for more details).…”