Non-Hermitian XY Spin Chains

• Non-Hermitian XY spin chains are extensions of the 1D spin-½ XY model using complex anisotropy and magnetic fields to introduce non-Hermiticity while retaining exact solvability via Jordan–Wigner transformation.
• They feature exceptional points, biorthogonal eigenstates, and rich spectral topology that reveal transitions between symmetry-unbroken and broken regimes.
• Analytical techniques like Chebyshev polynomial representation and BdG formulations enable precise diagnostics of quantum criticality, entanglement, and phase transitions in these models.

Searching arXiv for the cited non-Hermitian XY spin-chain literature to ground the article and verify the referenced papers. arxiv_search(query="all: \"Exceptional point rings and PT-symmetry in the non-Hermitian XY model\" OR id:(Henry et al., 6 Jul 2025)", max_results=5, sort_by="relevance") arxiv_search(query="all: \"Exact Solution for Non-Hermitian Free Fermions: A Case Study of the XY Chain\" OR id:(Li et al., 26 May 2026)", max_results=5, sort_by="relevance") arxiv_search(query="all: \"Characterizing phase transitions and criticality in non-Hermitian extensions of the XY model\" OR id:(Liu et al., 23 Jan 2025)", max_results=5, sort_by="relevance") arxiv_search(query="all: \"Quantum critical properties of non-Hermitian XY models with magnetic field\" OR id:(Luo et al., 5 Jun 2026)", max_results=5, sort_by="relevance") arxiv_search(query="non-Hermitian XY spin chain exceptional point PT symmetry", max_results=10, sort_by="relevance") Non-Hermitian XY spin chains are non-Hermitian extensions of the one-dimensional spin-$\tfrac12$ XY model in which non-Hermiticity is introduced through complex anisotropy, complex magnetic fields, gain/loss terms, staggered imaginary fields, boundary fields, or non-collinear couplings. A recurring feature is that many of these models remain exactly solvable after Jordan–Wigner fermionization, so their many-body spectra can still be organized in terms of free-fermion quasienergies even when eigenvalues become complex, eigenvectors become biorthogonal, and exceptional points (EPs) appear. Across the literature, the subject connects spectral degeneracies, $\mathcal{PT}$- or $\mathcal{RT}$-type symmetry breaking, non-Hermitian topology, modified criticality, and correlation or entanglement diagnostics in a setting that is analytically controlled (Henry et al., 6 Jul 2025).

1. Model class and routes to non-Hermiticity

The Hermitian parent is the anisotropic XY chain in a magnetic field,

$$H= - \sum_{n}\left(\frac{1+\gamma}{2}\sigma^{x}_n\sigma^{x}_{n+1}+\frac{1-\gamma}{2}\sigma^{y}_n\sigma^{y}_{n+1}\right) + h \sum_{n}\sigma^{z}_n,$$

with variants differing by boundary conditions and field conventions. Non-Hermitian continuations arise by allowing $\gamma$ or $h$ to be complex, or by adding explicitly imaginary couplings. In the open-chain continuation analyzed in "Exceptional point rings and $PT$-symmetry in the non-Hermitian XY model" (Henry et al., 6 Jul 2025), the anisotropy parameter is analytically continued to complex values through the rescaled form

$$H_\lambda = -\sum_{j=1}^{L-1}\left(\sigma_j^x\sigma_{j+1}^x+\lambda\,\sigma_j^y\sigma_{j+1}^y\right), \qquad \lambda=\frac{1-\gamma}{1+\gamma}.$$

For real $\lambda$ the Hamiltonian is Hermitian; complex $\lambda$ produces a genuinely non-Hermitian XY chain.

Other constructions realize non-Hermiticity differently. A global complex transverse field yields

$\mathcal{PT}$0

with $\mathcal{PT}$1, and the imaginary term acts as gain/loss of the $\mathcal{PT}$2 state (Liu et al., 2020). A staggered imaginary transverse field gives the dimerized $\mathcal{PT}$3-symmetric chain

$\mathcal{PT}$4

added to alternating XY couplings, and the reality of the full spectrum is controlled by the competition between $\mathcal{PT}$5 and dimerization (Giorgi, 2010). Another spin-$\mathcal{PT}$6 XY model introduces an alternating imaginary field $\mathcal{PT}$7, producing a $\mathcal{PT}$8-symmetric two-site problem with an EP at $\mathcal{PT}$9 and a chain treated by a two-spin cluster mean-field approximation (Li et al., 2022).

Further variants include an intrinsic $\mathcal{RT}$0-symmetric anisotropic XY chain with complex conjugate $\mathcal{RT}$1 and $\mathcal{RT}$2 couplings,

$\mathcal{RT}$3

a ring-frustrated odd-site chain with an imaginary symmetric non-collinear interaction $\mathcal{RT}$4, and an XY-type chain with local loss and nearest-neighbor pair creation/annihilation,

$\mathcal{RT}$5

which the authors identify as an anisotropic XY Hamiltonian with $\mathcal{RT}$6 (Zhang et al., 2012, Bi et al., 2020, Zhang et al., 2022).

2. Exact free-fermion structure and solvability

A central structural fact is that many non-Hermitian XY chains remain exactly solvable by the same Jordan–Wigner/free-fermion mechanism as their Hermitian counterparts. In the open non-Hermitian XY chain with complex anisotropy, the many-body spectrum retains the free-fermion form

$\mathcal{RT}$7

and the quasienergies are determined from the eigenvalues of $\mathcal{RT}$8 or, equivalently, from the quantization condition

$\mathcal{RT}$9

This is the basis on which quasi-energy degeneracies can be lifted to many-body EPs (Henry et al., 6 Jul 2025).

"Exact Solution for Non-Hermitian Free Fermions: A Case Study of the XY Chain" (Li et al., 26 May 2026) makes this structure explicit at the quasi-Hamiltonian level. After Jordan–Wigner transformation, the open-chain Hamiltonian is written as

$$H= - \sum_{n}\left(\frac{1+\gamma}{2}\sigma^{x}_n\sigma^{x}_{n+1}+\frac{1-\gamma}{2}\sigma^{y}_n\sigma^{y}_{n+1}\right) + h \sum_{n}\sigma^{z}_n,$$0

with

$$H= - \sum_{n}\left(\frac{1+\gamma}{2}\sigma^{x}_n\sigma^{x}_{n+1}+\frac{1-\gamma}{2}\sigma^{y}_n\sigma^{y}_{n+1}\right) + h \sum_{n}\sigma^{z}_n,$$1

For complex $$H= - \sum_{n}\left(\frac{1+\gamma}{2}\sigma^{x}_n\sigma^{x}_{n+1}+\frac{1-\gamma}{2}\sigma^{y}_n\sigma^{y}_{n+1}\right) + h \sum_{n}\sigma^{z}_n,$$2, $$H= - \sum_{n}\left(\frac{1+\gamma}{2}\sigma^{x}_n\sigma^{x}_{n+1}+\frac{1-\gamma}{2}\sigma^{y}_n\sigma^{y}_{n+1}\right) + h \sum_{n}\sigma^{z}_n,$$3 is generally non-Hermitian but remains complex symmetric, $$H= - \sum_{n}\left(\frac{1+\gamma}{2}\sigma^{x}_n\sigma^{x}_{n+1}+\frac{1-\gamma}{2}\sigma^{y}_n\sigma^{y}_{n+1}\right) + h \sum_{n}\sigma^{z}_n,$$4. The determinant identity

$$H= - \sum_{n}\left(\frac{1+\gamma}{2}\sigma^{x}_n\sigma^{x}_{n+1}+\frac{1-\gamma}{2}\sigma^{y}_n\sigma^{y}_{n+1}\right) + h \sum_{n}\sigma^{z}_n,$$5

shows that the characteristic polynomial has exactly the same structure as in the Lieb–Schultz–Mattis solution, so the many-body spectrum still consists of independent $$H= - \sum_{n}\left(\frac{1+\gamma}{2}\sigma^{x}_n\sigma^{x}_{n+1}+\frac{1-\gamma}{2}\sigma^{y}_n\sigma^{y}_{n+1}\right) + h \sum_{n}\sigma^{z}_n,$$6 contributions.

Away from EPs, one can construct a full biorthogonal fermionic basis. In the notation of (Li et al., 26 May 2026), the left and right fermionic operators satisfy

$$H= - \sum_{n}\left(\frac{1+\gamma}{2}\sigma^{x}_n\sigma^{x}_{n+1}+\frac{1-\gamma}{2}\sigma^{y}_n\sigma^{y}_{n+1}\right) + h \sum_{n}\sigma^{z}_n,$$7

and the Hamiltonian becomes diagonal in that basis. The same paper also rewrites open-boundary eigenvectors in a Chebyshev-polynomial form, using

$$H= - \sum_{n}\left(\frac{1+\gamma}{2}\sigma^{x}_n\sigma^{x}_{n+1}+\frac{1-\gamma}{2}\sigma^{y}_n\sigma^{y}_{n+1}\right) + h \sum_{n}\sigma^{z}_n,$$8

so that the boundary conditions are encoded by polynomial equations in the spectral variable $$H= - \sum_{n}\left(\frac{1+\gamma}{2}\sigma^{x}_n\sigma^{x}_{n+1}+\frac{1-\gamma}{2}\sigma^{y}_n\sigma^{y}_{n+1}\right) + h \sum_{n}\sigma^{z}_n,$$9. This polynomial representation is especially useful at EPs because repeated roots of the boundary polynomial directly signal defective modes.

An analogous free-fermion or BdG structure appears in the periodic-field and quench problems. With a complex field and periodic boundary conditions, the fermionized Hamiltonian can be written in Bogoliubov–de Gennes form

$\gamma$0

and the topological phase diagram may then be read off from quasienergy degeneracies and winding numbers (Liu et al., 23 Jan 2025). In the post-selected quench problem with imaginary field,

$\gamma$1

each $\gamma$2 sector becomes a two-level non-Hermitian BdG problem with quasiparticle spectrum

$\gamma$3

allowing exact computation of time-dependent correlations and entanglement (Turkeshi et al., 2022).

3. Exceptional points, quasi-energy degeneracies, and spectral geometry

In non-Hermitian XY chains, EPs are generated when repeated roots occur in the quasiparticle quantization problem. In the complex-$\gamma$4 open chain, EPs are found by solving simultaneously

$\gamma$5

and

$\gamma$6

Eliminating $\gamma$7 yields a condition for special momenta $\gamma$8, and then

$\gamma$9

At these values, the paper verifies that the biorthogonal overlap vanishes,

$h$0

which is the hallmark of a Jordan-block singularity rather than an ordinary degeneracy (Henry et al., 6 Jul 2025).

For finite $h$1, those EPs form a highly structured set. The total number is

$h$2

arranged on two concentric rings in the complex $h$3-plane, one inside and one outside the unit circle. For $h$4, the explicit values are

$h$5

In the thermodynamic limit the EPs approach the $h$6th roots of unity. Using the large-$h$7 expansion

$h$8

the EP condition reduces to

$h$9

with solutions

$PT$0

excluding $PT$1. The two EP rings therefore collapse onto

$PT$2

as $PT$3 (Henry et al., 6 Jul 2025).

The Chebyshev-polynomial solution of (Li et al., 26 May 2026) clarifies why this happens. There, EPs correspond to repeated roots of the open-boundary polynomial in $PT$4, and generalized eigenvectors are obtained transparently by $PT$5-differentiation: $PT$6 The same work further shows that EPs are branch points in the complex anisotropy plane: encircling an EP permutes eigenenergies and eigenstates, and the branch-cut structure of biorthogonal overlaps provides direct evidence of the state exchange (Li et al., 26 May 2026).

Other non-Hermitian XY realizations exhibit different EP geometries. In the intrinsic $PT$7-symmetric anisotropic chain, the thermodynamic-limit boundary is the hyperbola

$PT$8

separating unbroken and broken $PT$9 regions (Zhang et al., 2012). In the model with a global complex transverse field, gap closing yields an elliptical exceptional ring

$$H_\lambda = -\sum_{j=1}^{L-1}\left(\sigma_j^x\sigma_{j+1}^x+\lambda\,\sigma_j^y\sigma_{j+1}^y\right), \qquad \lambda=\frac{1-\gamma}{1+\gamma}.$$0

which the authors identify as the locus of a second-order quantum phase transition (Liu et al., 2020). In the two-site lossy XY-type chain, the EP condition is

$$H_\lambda = -\sum_{j=1}^{L-1}\left(\sigma_j^x\sigma_{j+1}^x+\lambda\,\sigma_j^y\sigma_{j+1}^y\right), \qquad \lambda=\frac{1-\gamma}{1+\gamma}.$$1

or equivalently

$$H_\lambda = -\sum_{j=1}^{L-1}\left(\sigma_j^x\sigma_{j+1}^x+\lambda\,\sigma_j^y\sigma_{j+1}^y\right), \qquad \lambda=\frac{1-\gamma}{1+\gamma}.$$2

and the relevant eigenvalues coalesce there (Zhang et al., 2022).

4. Symmetry, topology, and phase structure

The symmetry content of non-Hermitian XY chains is model dependent. Some examples are explicitly $$H_\lambda = -\sum_{j=1}^{L-1}\left(\sigma_j^x\sigma_{j+1}^x+\lambda\,\sigma_j^y\sigma_{j+1}^y\right), \qquad \lambda=\frac{1-\gamma}{1+\gamma}.$$3-symmetric, while others are controlled by intrinsic $$H_\lambda = -\sum_{j=1}^{L-1}\left(\sigma_j^x\sigma_{j+1}^x+\lambda\,\sigma_j^y\sigma_{j+1}^y\right), \qquad \lambda=\frac{1-\gamma}{1+\gamma}.$$4 or $$H_\lambda = -\sum_{j=1}^{L-1}\left(\sigma_j^x\sigma_{j+1}^x+\lambda\,\sigma_j^y\sigma_{j+1}^y\right), \qquad \lambda=\frac{1-\gamma}{1+\gamma}.$$5 symmetry. In the open complex-$$H_\lambda = -\sum_{j=1}^{L-1}\left(\sigma_j^x\sigma_{j+1}^x+\lambda\,\sigma_j^y\sigma_{j+1}^y\right), \qquad \lambda=\frac{1-\gamma}{1+\gamma}.$$6 chain, parity reverses the lattice and time reversal complex conjugates the coefficients, implying

$$H_\lambda = -\sum_{j=1}^{L-1}\left(\sigma_j^x\sigma_{j+1}^x+\lambda\,\sigma_j^y\sigma_{j+1}^y\right), \qquad \lambda=\frac{1-\gamma}{1+\gamma}.$$7

The Hamiltonian is therefore $$H_\lambda = -\sum_{j=1}^{L-1}\left(\sigma_j^x\sigma_{j+1}^x+\lambda\,\sigma_j^y\sigma_{j+1}^y\right), \qquad \lambda=\frac{1-\gamma}{1+\gamma}.$$8-symmetric when $$H_\lambda = -\sum_{j=1}^{L-1}\left(\sigma_j^x\sigma_{j+1}^x+\lambda\,\sigma_j^y\sigma_{j+1}^y\right), \qquad \lambda=\frac{1-\gamma}{1+\gamma}.$$9, namely on the pure imaginary axis. Along that axis, the quasienergies and many-body energies appear in complex-conjugate pairs, so the line is a broken-$\lambda$0-symmetric line; for $\lambda$1, four EP values lie exactly on it (Henry et al., 6 Jul 2025).

By contrast, the anisotropic periodic model with couplings $\lambda$2 is not $\lambda$3-symmetric but satisfies

$\lambda$4

In the unbroken $\lambda$5 phase the spectrum is fully real; when the condition

$\lambda$6

is met for some mode, complex energies appear and the system enters the broken $\lambda$7 phase (Zhang et al., 2012). A later exact study of the imaginary-$\lambda$8 XY model uses the anti-unitary $\lambda$9 symmetry and locates the transition at

$\lambda$0

with $\lambda$1-symmetric and $\lambda$2-broken regimes distinguished by whether the dispersion is real or partly purely imaginary (Luo et al., 5 Jun 2026).

Topological classification is equally prominent. In the complex-$\lambda$3 open chain, the infinite-size EP locus coincides with the boundary between winding-number sectors $\lambda$4 and $\lambda$5, given by

$\lambda$6

The EP rings are therefore finite-size spectral precursors of a non-Hermitian topological phase boundary (Henry et al., 6 Jul 2025). In a broader treatment of non-Hermitian XY extensions, the winding number is defined from the BdG Hamiltonian as

$\lambda$7

and the topological/trivial boundaries become, for example,

$\lambda$8

for a complex-field Ising chain and

$\lambda$9

for an XY chain with purely imaginary field (Liu et al., 23 Jan 2025).

Phase diagrams can differ sharply between models and even between candidate states of the same model. In the complex-$\mathcal{PT}$00 periodic XY chain, the minimal-energy state has ferromagnetic, LL, and paramagnetic regions,

$\mathcal{PT}$01

$\mathcal{PT}$02

$\mathcal{PT}$03

whereas the steady state has only LL and PM phases (Luo et al., 5 Jun 2026). In the model with a global complex transverse field, the usual Hermitian Ising critical points expand into a critical transition zone (CTZ): inside the exceptional ring the phase is ferromagnetic with real gap, outside the ring but with $\mathcal{PT}$04 the CTZ has purely imaginary gap, and for $\mathcal{PT}$05 the phase is paramagnetic with complex gap (Liu et al., 2020).

5. Correlations, entanglement, and critical diagnostics

Static correlations, entanglement, and fidelity furnish the principal diagnostics of non-Hermitian XY criticality, but the results depend strongly on the chosen formalism. In exactly solved chains, spin correlators are typically expressed as Pfaffians or Toeplitz determinants of fermionic contractions. For the global complex-field model, the long-range correlation function

$\mathcal{PT}$06

distinguishes the three regions cleanly: in the ferromagnetic phase it tends to a nonzero constant; in the CTZ it decays algebraically,

$\mathcal{PT}$07

and in the paramagnetic phase it decays exponentially,

$\mathcal{PT}$08

The exceptional ring is second order because the ground-state energy density is continuous, its first derivatives are continuous, and second derivatives diverge at the exceptional point $\mathcal{PT}$09 (Liu et al., 2020).

In the exactly solved comparison of standard and biorthogonal formalisms, the magnetization and asymptotic correlators differ qualitatively between right-right and left-right expectation values. The right-right magnetization

$\mathcal{PT}$10

is real, while the left-right quantity is generically complex. The same work concludes that the critical properties, including the phase diagram, depend on both the formalism used and the state considered, and it provides arguments in favor of standard quantum mechanics for physically realizable open-system implementations (Luo et al., 5 Jun 2026). This is one of the main controversies in the subject: what should count as the analogue of a ground state, and which expectation values should be regarded as physically meaningful.

Biorthogonal entanglement is another major diagnostic. In non-Hermitian XY extensions with complex field or anisotropy, the biorthogonal fidelity susceptibility

$\mathcal{PT}$11

peaks near criticality and obeys the finite-size scaling form

$\mathcal{PT}$12

For two representative models the exponent is $\mathcal{PT}$13, while for the anisotropy-driven transition the fitted value is

$\mathcal{PT}$14

The same study finds that topological transitions are accompanied by entanglement transitions: in the topological phase the half-chain entropy is essentially size independent, whereas in the trivial phase it grows logarithmically with subsystem size; at a topological-topological transition the fitted central charge is

$\mathcal{PT}$15

The authors also introduce the long-time averaged Loschmidt echo as a dynamical phase-transition diagnostic (Liu et al., 23 Jan 2025).

Entanglement can also probe $\mathcal{PT}$16-symmetry breaking directly. In the two-site XY model with alternating imaginary field, the concurrence of the eigenstate depending only on the imaginary field is always equal to one in the $\mathcal{PT}$17-symmetric region, decreases in the broken region, and shows non-analytic behavior at the EP; the biorthogonal concurrence shows a different $\mathcal{PT}$18-dependence, illustrating basis sensitivity (Li et al., 2022). In the post-selected quench problem with imaginary field, the long-time entanglement entropy behaves as

$\mathcal{PT}$19

in the gapless decay-mode phase, while it rapidly saturates in the gapped phase; the same gapless regime exhibits correlation spreading outside the Lieb–Robinson cone, with collapses and revivals (Turkeshi et al., 2022).

6. Applications, finite-size phenomena, and related directions

Non-Hermitian XY chains have been used as analytically tractable platforms for metrology, state preparation, frustration physics, and spectral-chaos diagnostics. In steady-state metrology, the lossy XY-type chain with local field encoding on the first site yields explicit two-site quantum Fisher informations

$\mathcal{PT}$20

so the QFI for the field amplitude diverges at the EP while the QFI for the azimuthal angle reaches its maximum there,

$\mathcal{PT}$21

For longer chains, the QFI for $\mathcal{PT}$22 is enhanced with system size but saturates because the local perturbation induces only short-range correlations (Zhang et al., 2022).

Finite non-Hermitian XY chains can also use EPs as dynamical attractors for entangled-state preparation. In a chain with imaginary boundary fields,

$\mathcal{PT}$23

the one-magnon sector reduces to a non-Hermitian tight-binding problem. At $\mathcal{PT}$24, the EP occurs at

$\mathcal{PT}$25

and the coalescent state is the W-like state

$\mathcal{PT}$26

For $\mathcal{PT}$27, the EP yields a distant Bell state, and the same dynamical mechanism was extended numerically to a GHZ state in a non-Hermitian transverse-field Ising chain (Li et al., 2015).

Odd-site periodic geometry introduces another distinctive phenomenon. In the ring-frustrated non-Hermitian XY model with symmetric non-collinear coupling, the Jordan–Wigner mapping gives an imbalanced-pairing Kitaev chain with parity-dependent boundary term. In the real-spectrum regime with $\mathcal{PT}$28, the true spin ground state is not the Bogoliubov vacuum but

$\mathcal{PT}$29

a one-mode-occupied state. The excitation gap then vanishes in the thermodynamic limit, and the low-energy sector is a topological kink phase with winding number $\mathcal{PT}$30 (Bi et al., 2020).

The subject also connects to spectral-statistical questions. For the intrinsic $\mathcal{PT}$31-symmetric anisotropic XY model, adding a random $\mathcal{PT}$32-field or varying the $\mathcal{PT}$33-field produces a crossover in complex spacing-ratio statistics from Poisson-like behavior toward Ginibre-unitary behavior. The paper interprets this as a non-Hermitian integrability-to-chaos transition facilitated by random fields and $\mathcal{PT}$34-symmetry breaking (Sarkar et al., 2023).

A plausible implication of this body of work is that non-Hermitian XY spin chains occupy an unusual position among exactly solvable many-body models: they preserve enough of the free-fermion algebra to allow exact control of EPs, topology, and asymptotic correlations, while simultaneously exposing ambiguities that are absent in Hermitian systems, especially the choice of state and the choice of expectation-value formalism. That combination is why they recur as testbeds for non-Hermitian criticality, symmetry breaking, and many-body spectral singularities (Luo et al., 5 Jun 2026).