The XY Spin Chain and the Topological Basis Realization
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The XY Spin Chain and the Topological Basis Realization Feifei Wang1 · Yue Cao1 · Yizi Zhu1 · Chunfang Sun1 · Gangcheng Wang1 · Kang Xue1 Received: 23 April 2020 / Accepted: 18 August 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Temperley-Lieb (T-L) algebra plays an important role in quantum computation, quantum teleportation, knot theory, statistical physics and topological quantum field theory. At the same time, a large number of models are represented by the generators of T-L algebra. In this paper, it is shown that the XY model can be constructed from the linear combination of the T-L algebra generators.We construct three new topological basis states, then investigate the particular properties of the topological basis in this system. For instance the energy ground state of the system falls on one of the topological basis states. Keywords Topological basis states · XY model · Temperly-Lieb algebra
1 Introduction Like the famous Ising model [1–3] and Heisenberg models [4], the quantum Heisenberg XY model (XY model) [5, 6] is one of the many highly simplified models in statistical mechanics. It was intensively investigated by Lieb, Schultz, and Mattis in 1960 [7]. The XY model is a special case of the n-vector model (the n-vector model or O(n) model is a simple system of interacting spins on a crystalline lattice.). The classical XY model has been widely used as a model for phase transitions in materials with interacting spins. XY model exhibits the well known Kosterlitz-Thouless(KT) transition [8]. It also can be realized in cavity QED system [9] and in the quantum-Hall system [10]. Meanwhile, the XY model can be used to construct the swap gate in quantum computation. In the field of topological quantum computation, based on the υ = 25 fractional quantum Hall effect [11–14], the two-dimensional(2D) braid behavior under the exchange of anyons [15] has been investigated. Anyons can be divided into three types: 0, 12 , 1 which satisfy non-abelian fractional statistics. When two anyons fuse into a new particle, its quantum
Chunfang Sun
[email protected] Gangcheng Wang [email protected] 1
School of Physics, Northeast Normal University, Changchun 130024, China
International Journal of Theoretical Physics
number is obtained by the following laws: × 12 = 0 + 1, 12 × 1 = 12 , 1 × 1 = 0, 0 × 0 = 0, 0 × 12 = 12 , 0 × 1 = 1. 1 2
(1)
It is worth noting that there are two different fusion channels for two 1/2 anyons. When four 1/2 anyons fuse together to get 0, via dividing the four 1/2 anyons into two pairs, both pairs either fuse to 0 or to 1 then fuse the resulting anyons together to form 0. Thus a twodimensional space of such states can be obtained. In general, the orthogonal topological basis states read [11]
(2) where the parameter d corresponds to the values of a loop in topology. Lately, the research shown that orthogonal topological basis has important applications in many aspects [16, 17]. In ref [16], with the aid of topological basis states, the fourdimens
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