Exceptional points in the Baxter-Fendley free parafermion model

Abstract

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}_NℤN-symmetric clock models. In 1989 Baxter discovered a non-Hermitian but \mathcal{PT}𝒫𝒯-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}𝒫𝒯-symmetric at these exceptional points, their proximity can have a profound impact on the model on the \mathcal{PT}𝒫𝒯-symmetric real line. Furthermore, in certain cases an exceptional point may appear on the real line (with negative field).

Features, Paradoxes and Amendments of Perturbative Non-Hermitian Quantum Mechanics

“…also a more recent review of the field in [41]). In essence, the latter authors revealed that in a number of specific toy models the conventional ansatz (45) may still serve as a productive constructive tool, yielding, at the small and real coupling constants, the real (i.e., energy-like) as well as complex (i.e., resonance-representing) lowlying spectra after standard resummation.…”

“…The reality of spectra of the Hamiltonians has independently been noticed in the context of quantum field theory [12]. This attracted the attention of the physics community to the applicability of expansions (45) in the non-Hermitian setting of Equation (44). The authors of the innovated studies of imaginary cubic anharmonic-oscillator Hamiltonians…”

“…Another, even more straightforward explanation of the existence of the emergent open-system instabilities becomes best visible when the system in question happens to lie close to a Kato's exceptional-point singularity [42][43][44]. In such a vicinity, indeed, the operator of metric Θ becomes singular and dominated by a projector [45]. Perturbations H 1 , which are small with respect to the correct physical Hilbert-space norm (in our present notation this means that ♯H 1 ♯ ≪ 1) may still be, simultaneously, very large with respect to the conventional open-system norm as defined in the standard regular limit of Θ → I (i.e., ∥H 1 ∥ ≫ 1).…”

Critical behavior of dirty free parafermionic chains

“…A Jordan-Wigner-like transformation [4], unlike in the TFIC, does not make this task at all trivial: the aforementioned parafermionic operators are highly nonlinear in these Jordan-Wigner parafermions. While a number of observables in the homogeneous (J j = J, h j = h for all j) version of the model, especially in the vicinity of h = J, have been calculated [5][6][7][8][9][10][11], the fact that the model is in principle exactly solvable for inhomogeneous couplings has not been exploited. This is the goal of the present paper.…”