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.
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.
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.
Abstract The XY spin chain is a paradigmatic example of a model solved by free fermions, in which the energy eigenspectrum is built from combinations of quasi-energies. In this article, we show that by extending the XY model’s anisotropy parameter
λ λ to complex values, it is possible for two of the quasi-energies to become degenerate. In the non-Hermitian XY model, these quasi-energy degeneracies give rise to exceptional points (EPs) where eigenvalues and their corresponding eigenvectors coalesce. The distinct
λ λ values at which EPs appear form concentric rings in the complex plane which are shown in the infinite system size limit to converge to the unit circle coinciding with the boundary between distinct topological phases. The non-Hermitian model is also seen to possess a line of broken
PT P T symmetry along the pure imaginary
λ λ -axis. For finite systems, there are four EP values on this broken
PT P T -symmetric line if the system size is a multiple of 4.
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.
We consider the non-Hermitian XY spin chain with open boundary conditions when the anisotropy parameter is extended to complex values. By analyzing the quasi-Hamiltonian matrix, we demonstrate that the free-fermion structure of the quasi-energy spectrum coincides with that of the Hermitian model and construct the corresponding biorthogonal fermionic basis away from exceptional points (EPs). We make use of an explicit Chebyshev-polynomial representation of the open-boundary eigenvectors in which the quasi-energy
ε is the natural spectral variable. This quasi-energy polynomial form is particularly useful at EPs, because EPs correspond to repeated roots of the same boundary polynomial, making the construction of generalized eigenvectors by
ε-differentiation transparent. At EPs, where the quasi-Hamiltonian becomes defective, we derive the Jordan normal form and construct the associated generalized eigenvectors, which yields the correct counting of independent many-body eigenstates. We further show that EPs act as branch points in the complex anisotropy plane, leading to the characteristic permutation of eigenenergies and eigenstates upon encirclement. The branch-cut structure of the biorthogonal eigenstates provides direct evidence for the exchange of eigenstates when an EP is encircled. These results provide an analytically controlled many-body platform for studying EP physics and non-Hermitian topology beyond momentum-space descriptions.