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arXiv:1402.0061 · Journal of Physics A Mathematical and Theoretical

Parafermions in the τ2model

Helen Au-Yang, Jacques H. H. Perk

Published Jul 18, 2014• 27 citations
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Abstract

It has been shown recently by Baxter that the $τ_2(t_q)$ model with open boundary conditions can be solved by the "parafermionic" method of Fendley. In Baxter's paper there are several conjectures, which were formulated based on numerical short-chain calculations. Here we present the proof of two of them.

DOI
10.1088/1751-8113/47/31/315002

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Baxter operator and Baxter equation for $q$-Toda and Toda$_2$ chains

O. Babelon, Karol Kajetan Kozlowski, Vincent Pasquier

•May 22, 2018•5 citations
It has been shown recently by Baxter that the τ2(tq)τ_2(t_q)τ2​(tq​) model with open boundary conditions can be solved by the "parafermionic" method of Fendley. In Baxter's paper there are several conjectures, which were formulated based on numerical short-chain calculations. Here we present the proof of two of them.
Many years ago Baxter introduced an inhomogeneous two-dimensional classical spin model, now called the τ2(t)τ_2(t)τ2​(t) model with free boundary conditions, and he specialized the resulting quantum spin-chain Hamiltonian in a special limit to a simple clock Hamiltonian. Recently, Fendley showed that this clock Hamiltonian can be expressed in terms of free "parafermions." Baxter followed this up by showing that this construction generalizes to the more general τ2(t)τ_2(t)τ2​(t) model, provided some conjectures hold. In this paper, we will compare the different notations and approaches enabling us to express the Hamiltonians in terms of projection operators as introduced by Fendley. By examining the properties of the raising operators, we are then able to prove the last unproven conjecture in Baxter's paper left in our previous paper. Thus the eigenvectors can all be written in terms of these raising operators.
Certain spin chains, such as the quantum Ising chain, have free fermion spectra which can be expressed as the sum of decoupled two-level fermionic systems. Free parafermions are a generalisation of this idea to \mathbb{Z}_N ℤ N -symmetric clock models. In 1989 Baxter discovered a non-Hermitian but \mathcal{PT} 𝒫 𝒯 -symmetric model directly generalising the Ising chain, which was later described by Fendley as a free parafermion spectrum. By extending the model’s magnetic field parameter to the complex plane, it is shown that a series of exceptional points emerges, where the quasienergies defining the free spectrum become degenerate. An analytic expression for the locations of these points is derived, and various numerical investigations are performed. These exceptional points also exist in the Ising chain with a complex transverse field. Although the model is not in general \mathcal{PT} 𝒫 𝒯 -symmetric at these exceptional points, their proximity can have a profound impact on the model on the \mathcal{PT} 𝒫 𝒯 -symmetric real line. Furthermore, in certain cases an exceptional point may appear on the real line (with negative field).
Baxter operator and Baxter equation for qqq-Toda and Toda2_22​ chains