The topological basis expression of four-qubit XXZ spin chain with twist boundary condition
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The topological basis expression of four-qubit XXZ spin chain with twist boundary condition Guijiao Du · Kang Xue · Chengcheng Zhou · Chunfang Sun · Gangcheng Wang
Received: 29 September 2012 / Accepted: 30 December 2012 © Springer Science+Business Media New York 2013
Abstract We investigate the XXZ model’s characteristic with the twisted boundary condition and the topological basis expression. Owing to twist boundary condition, the ground state energy will changing back and forth between E 13 and E 15 by modulate the parameter φ. By using TLA generators, the XXZ model’s Hamiltonian can be constructed. All the eigenstates can be expressed by topological basis, and the whole of eigenstates’ entanglement are maximally entangle states (Q(|φi ) = 1). Keywords
Topological basis · Quantum entanglement · Knot theory
1 Introduction The topological quantum field theory (TQFT) is one of the most intriguing features of quantum theory for it is related to quantum computing through anyons. The 2D braid behavior under the exchange of anyons is related to the ν = 5/2 state Fractional Quantum Hall Effect (FQHT) [1]. The topological basis plays the central role in
G. Du · K. Xue (B) · C. Sun · G. Wang School of Physics, Northeast Normal University, Changchun 130024, People’s Republic of China e-mail: [email protected] G. Du e-mail: [email protected] G. Wang e-mail: [email protected] C. Zhou School of Science, Changchun University of Science and Technology, Changchun 130022, People’s Republic of China e-mail: [email protected]
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TQFT and it can be denoted in terms of graphic technique [2]. To date, base on the application of braid relation in anyon theory, Ref. [2] nest Temperly-Lieb algebra (TLA) into 4D Yang-Baxter Equation (YBE) and reduce it to 2D YBE. In ν = 5/2 FQHE, quasiparticles are called Ising anyons which satisfy non-Abelian fractional statistics. the anyons obeys the following fusion rules, 1 1 × = 0 + 1, 2 2
1 1 × 1 = , 1 × 1 = 0, 2 2 1 1 0 × 0 = 0, 0 × = , 0 × 1 = 1. 2 2 There are two fusion ways for two 21 anyons. When four 21 anyous fuse together, we can divide the four 21 anyous into two pairs. Both pairs either fuse 0 or to 1. Then the result of anyons fuse together is 0. by previous theory, we have know two topological basis states |e1 , |e2 . The orthogonal basis states read [2–6],
where d describes a single loop . In the middle fusion chains (called conformal block), the internal edges are obey the fusion rules at each trivalent vertex. Motivated by the application of topological basis, we reconstruct the eigenstates by topological basis. Temperley-Lieb algebra (TLA) has croped up in a wide variety of mathematical and physical contexts. TLA was discovered by Temperley and Lieb who use it to research the single bond transfer matrices for Ising model. Later by Jones’ new polynomial invariant of knots and links [7,8], he is the first one who give the detailed research of TLA mathematical structure [9–11]. Then Baxter has further elaborated presented the relevance of this algebra to
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