Exact Solution for Non-Hermitian Free Fermions: A Case Study of the XY Chain

• URL: http://arxiv.org/abs/2605.26813v1
• Date: Tue, 26 May 2026 10:30:22 GMT
• Title: Exact Solution for Non-Hermitian Free Fermions: A Case Study of the XY Chain
• Abstract summary: We consider the non-Hermitian boundary spin chain with open conditions when the anisotropy boundary is extended to complex values.<n>We show that exceptional points (EPs) act as branch points in the complex anisotropy plane, leading to the characteristic permutation of eigenenergies and eigenstates upon encirclement.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 $\varepsilon$ 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 $\varepsilon$-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.

Related papers

• Typical entanglement in anyon chains: Page curves beyond Lie group symmetries [41.99844472131922]
  We study bipartite entanglement statistics in one-dimensional anyon chains constrained by fusion rules of unitary pre-modular categories.<n>Surprisingly, despite the constrained Hilbert space structure, the large $L$ expansion has no universal $O(sqrtL)$ or $O(1)$ symmetry-type corrections.<n> Numerical simulations of the integrable and quantum-chaotic golden chain Hamiltonian show that chaotic mid-spectrum eigenstates match the Haar-random predictions.
  arXiv  Detail & Related papers  (2026-03-26T18:00:03Z)
• Symmetry-protected topology and deconfined solitons in a multi-link $\mathbb{Z}_2$ gauge theory [45.88028371034407]
  We study a $mathbbZ$ lattice gauge theory defined on a multi-graph with links that can be visualized as great circles of a spherical shell.<n>We show that this leads to state-dependent tunneling amplitudes underlying a phenomenon analogous to the Peierls instability.<n>By performining a detailed analysis based on matrix product states, we prove that charge deconfinement emerges as a consequence of charge-fractionalization.
  arXiv  Detail & Related papers  (2026-03-02T22:59:25Z)
• Geometric phase of exceptional point as quantum resonance in complex scaling method [0.0]
  We analyze the self-orthogonality in the vicinity of an EP, the Berry phase, and the Chern characteristic.<n>Our results provide a bridge between non-Hermitian spectral theory and the traditional theory of quantum resonances.
  arXiv  Detail & Related papers  (2025-12-31T00:23:18Z)
• Singularity Selector: Topological Chirality via Non-Abelian Loops around Exceptional Points [0.0]
  Chirality governs measurable asymmetries across nature, from enantiomer-selective drugs and left-handed fermions in particle physics to handed charge transport in Weyl semimetals.<n>We extend this universal concept to non-Hermitian systems by defining topological chirality, an invariant that emerges whenever an exceptional-points (EP) pair is present.
  arXiv  Detail & Related papers  (2025-12-21T16:01:16Z)
• Fragmented exceptional points and their bulk and edge realizations in lattice models [0.0]
  We show how fragmented exceptional points (FEPs) can be induced into the bulk and edge spectrum of two-dimensional and three-dimensional lattice models.<n>The free design of FEPs significantly opens up a new frontier for non-Hermitian physics.
  arXiv  Detail & Related papers  (2025-07-29T18:45:34Z)
• Topological crystals and soliton lattices in a Gross-Neveu model with Hilbert-space fragmentation [39.146761527401424]
  We explore the finite-density phase diagram of the single-flavour Gross-Neveu-Wilson (GNW) model.<n>We find a sequence of inhomogeneous ground states that arise through a real-space version of the mechanism of Hilbert-space fragmentation.
  arXiv  Detail & Related papers  (2025-06-23T14:19:35Z)
• Isospectrality and non-locality of generalized Dirac combs [41.94295877935867]
  We consider a generalization of Dirac's comb model, describing a non-relativistic particle moving in a periodic array of generalized point interactions.<n>We classify a large class of isospectral relations, determining which Hamiltonians are spectrally unique, and which are instead related by a unitary or anti-unitary transformation.
  arXiv  Detail & Related papers  (2025-05-23T13:58:50Z)
• Symmetries, Conservation Laws and Entanglement in Non-Hermitian Fermionic Lattices [37.69303106863453]
  Non-Hermitian quantum many-body systems feature steady-state entanglement transitions driven by unitary dynamics and dissipation.<n>We show that the steady state is obtained by filling single-particle right eigenstates with the largest imaginary part of the eigenvalue.<n>We illustrate these principles in the Hatano-Nelson model with periodic boundary conditions and the non-Hermitian Su-Schrieffer-Heeger model.
  arXiv  Detail & Related papers  (2025-04-11T14:06:05Z)
• Entanglement Hamiltonian and orthogonal polynomials [0.0]
  We study the entanglement Hamiltonian for free-fermion chains with a particular form of inhomogeneity.<n>We show that this deformation is interpreted as a local inverse temperature and can be obtained in the continuum limit.<n>Using this prediction, the properly rescaled eigenvalues of the commuting operator are found to provide a very good approximation of the entanglement spectrum and entropy.
  arXiv  Detail & Related papers  (2024-12-16T17:46:53Z)
• Tensor product random matrix theory [39.58317527488534]
  We introduce a real-time field theory approach to the evolution of correlated quantum systems. We describe the full range of such crossover dynamics, from initial product states to a maximum entropy ergodic state.
  arXiv  Detail & Related papers  (2024-04-16T21:40:57Z)
• Exceptional points and quantum phase transition in a fermionic extension of the Swanson oscillator [8.84834042985207]
  We propose a fermionic extension of a non-Hermitian quantum system consisting of a general representation of a quadratic Hamiltonian. The model admits a quantum phase transition - we discuss the two phases and also demonstrate that the ground-state entanglement entropy exhibits a discontinuous jump.
  arXiv  Detail & Related papers  (2024-01-30T17:20:34Z)
• Enhanced eigenvector sensitivity and algebraic classification of sublattice-symmetric exceptional points [3.8399157726466986]
  Exceptional points (EPs) are degeneracy of non-Hermitian Hamiltonians, at which the eigenvalues, along with their eigenvectors, coalesce. We show that odd-order EPs can exist and exhibit enhanced sensitivity in the behaviour of eigenvector-coalescence.
  arXiv  Detail & Related papers  (2022-11-15T19:01:05Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.