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Exceptional point rings and $PT$-symmetry in the non-Hermitian XY model

Robert A. Henry, D. C. Liu, Murray T. Batchelor

TL;DR

This work extends the XY spin chain to a non-Hermitian setting by complexifying the anisotropy parameter $\lambda$, revealing two concentric rings of exceptional points in the complex $\\lambda$-plane where two quasi-energies and their eigenvectors coalesce. In the infinite-size limit these rings converge to the unit circle $\\Re(\\lambda)^2 + \\Im(\\lambda)^2 = 1$, which coincides with the topological phase boundary between winding numbers $w=1$ and $w=-1$, establishing a direct link between EP geometry and bulk topology. The study also identifies a line of broken $\\mathcal{PT}$ symmetry along the pure imaginary axis, containing four EPs when the system size is a multiple of 4, highlighting rich symmetry and spectral features in non-Hermitian many-body systems. The approach relies on the free-fermion structure and quasi-momentum formalism to map the spectrum, compute EP conditions, and relate them to the underlying topological invariants, with potential implications for experimental observation and broader classes of non-Hermitian quantum models.

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 $λ$ 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 two of the eigenvalues and their corresponding eigenvectors coalesce. The distinct $λ$ 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 $PT$ symmetry along the pure imaginary $λ$-axis. For finite systems, there are four EP values on this broken $PT$-symmetric line if the system size is a multiple of 4.

Exceptional point rings and $PT$-symmetry in the non-Hermitian XY model

TL;DR

This work extends the XY spin chain to a non-Hermitian setting by complexifying the anisotropy parameter , revealing two concentric rings of exceptional points in the complex -plane where two quasi-energies and their eigenvectors coalesce. In the infinite-size limit these rings converge to the unit circle , which coincides with the topological phase boundary between winding numbers and , establishing a direct link between EP geometry and bulk topology. The study also identifies a line of broken symmetry along the pure imaginary axis, containing four EPs when the system size is a multiple of 4, highlighting rich symmetry and spectral features in non-Hermitian many-body systems. The approach relies on the free-fermion structure and quasi-momentum formalism to map the spectrum, compute EP conditions, and relate them to the underlying topological invariants, with potential implications for experimental observation and broader classes of non-Hermitian quantum models.

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 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 two of the eigenvalues and their corresponding eigenvectors coalesce. The distinct 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 symmetry along the pure imaginary -axis. For finite systems, there are four EP values on this broken -symmetric line if the system size is a multiple of 4.

Paper Structure

This paper contains 8 sections, 25 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The non-Hermitian XY model
  3. Free fermion structure
  4. Quasi-momenta
  5. Quasi-energy degeneracies and EPs
  6. Infinite size limit
  7. $\mathcal{PT}$-symmetric points
  8. Concluding remarks

Figures (3)

  • Figure 1: Quasi-energies (left) and corresponding energy spectra (right) for the non-Hermitian XY model with $L=4$, and different values of the complex parameter $\lambda$. The energies (right) show the 16 values obtained via eq. (\ref{['eq:ff']}) from the four quasi-energies (blue circles), compared to the values obtained from exact diagonalisation of the full Hamiltonian (red crosses).
  • Figure 2: (left) Solutions of the quasi-energy degeneracy condition (\ref{['eq:kj_ep4']}) in the complex $k$-plane for increasing values of $L$. Each solution corresponds to an EP (right), where the absolute distance $\Delta\epsilon_{12}$ between the smallest two quasi-energies is shown for comparison. The green and blue circles label degeneracies and thus EPs appearing in the two different rings.
  • Figure 3: The topological phase diagram of the non-Hermitian XY model (\ref{['eq:1']}). The red line is a guide for the eye as the unit circle separating the regions with distinct topological phases $w=1$ and $w=-1$. In the infinite size limit there is a ring of EPs on the unit circle.