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arXiv:1606.06319

Parafermions in the tau-2 model II

Helen Au-Yang, Jacques H. H. Perk

Published Jun 20, 2016• 10 citations

Abstract

Many years ago Baxter introduced an inhomogeneous two-dimensional classical spin model, now called the $τ_2(t)$ 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 $τ_2(t)$ 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.

DOI: 10.48550/arxiv.1606.06319

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•Nov 21, 2012•2 citations

Many years ago Baxter introduced an inhomogeneous two-dimensional classical spin model, now called the

τ_2(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

τ_2(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.

The spectrum of the quantum Ising chain can be found by expressing the spins in terms of free fermions. An analogous transformation exists for clock chains with symmetry, but is of less use because the resulting parafermionic operators remain interacting. Nonetheless, Baxter showed that a certain non-Hermitian (but PT -symmetric) clock Hamiltonian is 'free', in the sense that the entire spectrum is found in terms of independent energy levels, with the striking feature that there are n possibilities for occupying each level. Here I show this directly explicitly finding shift operators obeying a generalization of the Clifford algebra. I also find higher Hamiltonians that commute with Baxter's and prove their spectrum comes from the same set of energy levels. This thus provides an explicit notion of a 'free parafermion'. A byproduct is an elegant method for the solution of the Ising/Kitaev chain with spatially varying couplings.

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.

Abstract In this article we outline the historical development and key results obtained to date for free parafermionic spin chains. The concept of free parafermions provides a natural N-state generalization of free fermions, which have long underpinned the exact solution and application of widely studied quantum spin chains and their classical counterparts. In particular, we discuss the Baxter-Fendley free parafermionic Z(N) spin chain, which is a relatively simple non-Hermitian generalization of the Ising model.

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).

We give explicit formulas for the eigenvectors of the transfer matrix of the Baxter-Bazhanov-Stroganov (BBS) model (N-state spin model) with fixed-spin boundary conditions. We obtain these formulas from the formulas for the eigenvectors of the periodic BBS model by a limit procedure. The latter formulas were derived in the framework of Sklyanin’s method of separation of variables. In the case of fixed-spin boundaries, we solve the corresponding T-Q Baxter equations for the functions of separated variables explicitly. As a particular case, we obtain the eigenvectors of the Hamiltonian of the Ising-like ℤN quantum chain model.