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).
Many years ago Baxter introduced an inhomogeneous two-dimensional classical spin model, now called the
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) 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.
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.
The relationship between the eigenspectrum of Ising and
XY quantum chains is well known. Although the Ising model has a
Z(2) symmetry and the
XY model a
U(1) symmetry, both models are described in terms of free-fermionic quasiparticles. The fermionic quasienergies are obtained by means of a Jordan-Wigner transformation. On the other hand, there exists in the literature a huge family of
Z(N) quantum chains whose eigenspectra, for
N>2, are given in terms of free parafermions, and they are not derived from the standard Jordan-Wigner transformation. The first members of this family are the
Z(N) free-parafermionic Baxter quantum chains. In this paper, we introduce a family of
XY models that, beyond two-body, also have
N-multispin interactions. Similar to the standard
XY model, they have a
U(1) symmetry and are also solved by the Jordan-Wigner transformation. We show that with appropriate choices of the
N-multispin couplings, the eigenspectra of these
XY models are given in terms of combinations of
Z(N) free-parafermionic quasienergies. In particular, all the eigenenergies of the
Z(N) free-parafermionic models are also present in the related free-fermionic
XY models. The correspondence is established via the identification of the characteristic polynomial, which fixes the eigenspectrum. In the
Z(N) free-parafermionic models, the quasienergies obey an exclusion circle principle that is not present in the related
N-multispin
XY models.
The relationship between the eigenspectrum of Ising and XY quantum chains is well known. Although the Ising model has a Z(2) symmetry and the XY model a U(1) symmetry, both models are described in terms of free-fermionic quasiparticles. The fermionic quasienergies are obtained by means of a Jordan-Wigner transformation. On the other hand, there exists in the literature a huge family of Z(N) quantum chains whose eigenspectra, for N>2, are given in terms of free parafermions, and they are not derived from the standard Jordan-Wigner transformation. The first members of this family are the Z(N) free-parafermionic Baxter quantum chains. In this paper, we introduce a family of XY models that, beyond two-body, also have N-multispin interactions. Similar to the standard XY model, they have a U(1) symmetry and are also solved by the Jordan-Wigner transformation. We show that with appropriate choices of the N-multispin couplings, the eigenspectra of these XY models are given in terms of combinations of Z(N) free-parafermionic quasienergies. In particular, all the eigenenergies of the Z(N) free-parafermionic models are also present in the related free-fermionic XY models. The correspondence is established via the identification of the characteristic polynomial, which fixes the eigenspectrum. In the Z(N) free-parafermionic models, the quasienergies obey an exclusion circle principle that is not present in the related N-multispin XY models.
Abstract A family of Z n -symmetric non-Hermitian models of Baxter was shown by Fendley to be exactly solvable via a parafermionic generalization of the Clifford algebra. We study these models with spatially random couplings, and obtain several exact results on thermodynamic singularities as the distributions of couplings are varied. We find that these singularities, independent of n, are identical to those in the random transverse-field Ising chain; correspondingly the models host infinite-randomness critical points. Similarities in structure to exact methods for random Ising models, a strong-disorder renormalization group, and generalizations to other models with free spectra, are discussed.