Non-Hermitian XY Model Insights

• The non-Hermitian XY model is a family of spin chains extended with complex fields or couplings that preserves Jordan–Wigner solvability.
• It exhibits unique features such as exceptional points, parity-sector switching, and anti-linear symmetries that redefine phase transitions.
• Analytical techniques like Bogoliubov diagonalization reveal critical behavior, topological transitions, and fidelity zero phenomena in various formulations.

The non-Hermitian XY model denotes a family of non-Hermitian extensions of the one-dimensional anisotropic spin-$\tfrac12$ XY chain in which the transverse field, the exchange anisotropy, or both are continued to complex values, or else arise as effective generators of conditioned open-system dynamics. In these formulations, many of the algebraic advantages of the Hermitian XY chain survive—most notably Jordan–Wigner fermionization and Bogoliubov diagonalization—but the notions of symmetry, gap closing, and phase transition are reorganized around anti-linear symmetries, parity-sector switching, exceptional points, and, in some formulations, biorthogonal observables and fidelity zeros (Zhang et al., 2012, Gu et al., 9 Dec 2025).

1. Hamiltonian forms and exact-solvable structure

Several distinct Hamiltonian realizations are referred to as non-Hermitian XY models. A standard complex-field form is the anisotropic XY chain in a complex transverse magnetic field,

$$H = - J \sum_{\langle i, j \rangle} \left( \frac{1+\gamma}{2} \sigma_{i}^{x} \sigma_{j}^{x} + \frac{1-\gamma}{2} \sigma_{i}^{y} \sigma_{j}^{y} \right) - h \sum_{i} \sigma_{i}^{z},$$

with $h=h_r+i h_i$ or $h=g e^{i\theta}$. A second major class uses complex anisotropy rather than a complex field, for example

$$H=J\sum_{j=1}^{N}\left( \frac{1+i\gamma }{2}\sigma _{j}^{x}\sigma _{j+1}^{x}+\frac{1-i\gamma }{2}\sigma _{j}^{y}\sigma _{j+1}^{y}+\lambda \sigma _{j}^{z}\right).$$

A third route is dynamical: conditioning a decay process on no jump produces an effective non-Hermitian Hamiltonian

$$H_{\text{eff}}=H-\frac{i\gamma}{4}\sum_n(\sigma_n^z+1).$$

These realizations are not equivalent term by term, but all instantiate non-Hermitian deformations of XY spin physics (Gu et al., 9 Dec 2025, Zhang et al., 2012, Lee et al., 2014).

Variant	Representative non-Hermitian ingredient	Representative consequence
Complex transverse field	$h\in\mathbb C$	fidelity zeros and parity-sector switching
Complex anisotropy	$(1\pm i\gamma)/2$ couplings or $\gamma\in\mathbb C$	$\mathcal{RT}$ symmetry, EPs, topological transitions
No-jump effective model	$$H = - J \sum_{\langle i, j \rangle} \left( \frac{1+\gamma}{2} \sigma_{i}^{x} \sigma_{j}^{x} + \frac{1-\gamma}{2} \sigma_{i}^{y} \sigma_{j}^{y} \right) - h \sum_{i} \sigma_{i}^{z},$$0	steady-state selection and decay-conditioned criticality

The exact-solvable backbone is the Jordan–Wigner transformation,

$$H = - J \sum_{\langle i, j \rangle} \left( \frac{1+\gamma}{2} \sigma_{i}^{x} \sigma_{j}^{x} + \frac{1-\gamma}{2} \sigma_{i}^{y} \sigma_{j}^{y} \right) - h \sum_{i} \sigma_{i}^{z},$$1

followed by Fourier and Bogoliubov transformations. In the complex-field chain this yields

$$H = - J \sum_{\langle i, j \rangle} \left( \frac{1+\gamma}{2} \sigma_{i}^{x} \sigma_{j}^{x} + \frac{1-\gamma}{2} \sigma_{i}^{y} \sigma_{j}^{y} \right) - h \sum_{i} \sigma_{i}^{z},$$2

while in the no-jump model the quasiparticle dispersion becomes

$$H = - J \sum_{\langle i, j \rangle} \left( \frac{1+\gamma}{2} \sigma_{i}^{x} \sigma_{j}^{x} + \frac{1-\gamma}{2} \sigma_{i}^{y} \sigma_{j}^{y} \right) - h \sum_{i} \sigma_{i}^{z},$$3

In the intrinsic $$H = - J \sum_{\langle i, j \rangle} \left( \frac{1+\gamma}{2} \sigma_{i}^{x} \sigma_{j}^{x} + \frac{1-\gamma}{2} \sigma_{i}^{y} \sigma_{j}^{y} \right) - h \sum_{i} \sigma_{i}^{z},$$4-symmetric anisotropy model one obtains

$$H = - J \sum_{\langle i, j \rangle} \left( \frac{1+\gamma}{2} \sigma_{i}^{x} \sigma_{j}^{x} + \frac{1-\gamma}{2} \sigma_{i}^{y} \sigma_{j}^{y} \right) - h \sum_{i} \sigma_{i}^{z},$$5

The persistence of free-fermion solvability is one of the central reasons the non-Hermitian XY family has become a benchmark setting (Gu et al., 9 Dec 2025, Lee et al., 2014, Zhang et al., 2012).

2. Symmetry classes, sector structure, and non-Hermitian symmetry breaking

A central distinction between Hermitian and non-Hermitian XY chains is that the relevant symmetry may be parity, $$H = - J \sum_{\langle i, j \rangle} \left( \frac{1+\gamma}{2} \sigma_{i}^{x} \sigma_{j}^{x} + \frac{1-\gamma}{2} \sigma_{i}^{y} \sigma_{j}^{y} \right) - h \sum_{i} \sigma_{i}^{z},$$6, $$H = - J \sum_{\langle i, j \rangle} \left( \frac{1+\gamma}{2} \sigma_{i}^{x} \sigma_{j}^{x} + \frac{1-\gamma}{2} \sigma_{i}^{y} \sigma_{j}^{y} \right) - h \sum_{i} \sigma_{i}^{z},$$7, or an emergent symmetry visible only in selected sectors or modes. In the complex-field XY chain, the relevant $$H = - J \sum_{\langle i, j \rangle} \left( \frac{1+\gamma}{2} \sigma_{i}^{x} \sigma_{j}^{x} + \frac{1-\gamma}{2} \sigma_{i}^{y} \sigma_{j}^{y} \right) - h \sum_{i} \sigma_{i}^{z},$$8 structure is fermion-number parity after Jordan–Wigner mapping, i.e. the even and odd fermion sectors. The complex field does not merely preserve a parity sector and produce a conventional Hermitian degeneracy; rather, the ground state oscillates between parity sectors as the field is varied in the complex plane. This sector switching is the paper’s operational notion of non-Hermitian symmetry breaking for the XY chain (Gu et al., 9 Dec 2025).

In the complex-anisotropy chain, the key anti-linear symmetry is intrinsic rotation-time reversal. The Hamiltonian satisfies

$$H = - J \sum_{\langle i, j \rangle} \left( \frac{1+\gamma}{2} \sigma_{i}^{x} \sigma_{j}^{x} + \frac{1-\gamma}{2} \sigma_{i}^{y} \sigma_{j}^{y} \right) - h \sum_{i} \sigma_{i}^{z},$$9

with

$h=h_r+i h_i$0

while $h=h_r+i h_i$1 and $h=h_r+i h_i$2 separately do not commute with $h=h_r+i h_i$3. In the unbroken $h=h_r+i h_i$4-symmetric region the spectrum is entirely real; in the broken region eigenvalues occur in complex-conjugate pairs and eigenstates cease to be $h=h_r+i h_i$5-symmetric. The thermodynamic exceptional boundary is

$h=h_r+i h_i$6

This made the model an early many-body example of a non-$h=h_r+i h_i$7-symmetric Hamiltonian with a real spectrum controlled by an anti-linear symmetry (Zhang et al., 2012).

Open-boundary complex-anisotropy formulations introduce a related but distinct statement: in the $h=h_r+i h_i$8-parametrization, the Hamiltonian is $h=h_r+i h_i$9-symmetric when $h=g e^{i\theta}$0, so the pure imaginary $h=g e^{i\theta}$1-axis is a broken-$h=g e^{i\theta}$2 line containing exceptional points. For finite systems with $h=g e^{i\theta}$3 a multiple of $h=g e^{i\theta}$4, there are four EP values on that line (Henry et al., 6 Jul 2025).

An RG-based continuum extension adds clock anisotropy to the XY field theory,

$h=g e^{i\theta}$5

and shows that breaking $h=g e^{i\theta}$6 symmetry qualitatively changes the RG topology. In the broken regime there is no fixed-point line analogous to the BKT-like $h=g e^{i\theta}$7-symmetric case; instead, fixed points collide as $h=g e^{i\theta}$8, producing walking or pseudocriticality (Naichuk et al., 2024).

3. Phase structure and critical behavior

The phase structure of the non-Hermitian XY model is formulation-dependent. In the no-jump decay realization, the one-dimensional chain undergoes a transition from short-range order to quasi-long-range order at

$h=g e^{i\theta}$9

with exponential correlations below the threshold and power-law correlations above it. For $$H=J\sum_{j=1}^{N}\left( \frac{1+i\gamma }{2}\sigma _{j}^{x}\sigma _{j+1}^{x}+\frac{1-i\gamma }{2}\sigma _{j}^{y}\sigma _{j+1}^{y}+\lambda \sigma _{j}^{z}\right).$$0, the correlation-length exponent is $$H=J\sum_{j=1}^{N}\left( \frac{1+i\gamma }{2}\sigma _{j}^{x}\sigma _{j+1}^{x}+\frac{1-i\gamma }{2}\sigma _{j}^{y}\sigma _{j+1}^{y}+\lambda \sigma _{j}^{z}\right).$$1; for $$H=J\sum_{j=1}^{N}\left( \frac{1+i\gamma }{2}\sigma _{j}^{x}\sigma _{j+1}^{x}+\frac{1-i\gamma }{2}\sigma _{j}^{y}\sigma _{j+1}^{y}+\lambda \sigma _{j}^{z}\right).$$2, it is $$H=J\sum_{j=1}^{N}\left( \frac{1+i\gamma }{2}\sigma _{j}^{x}\sigma _{j+1}^{x}+\frac{1-i\gamma }{2}\sigma _{j}^{y}\sigma _{j+1}^{y}+\lambda \sigma _{j}^{z}\right).$$3. The ordered phase has a frustrated pattern: the chain splits into two alternating sublattices, correlations vanish for odd distances, and the $$H=J\sum_{j=1}^{N}\left( \frac{1+i\gamma }{2}\sigma _{j}^{x}\sigma _{j+1}^{x}+\frac{1-i\gamma }{2}\sigma _{j}^{y}\sigma _{j+1}^{y}+\lambda \sigma _{j}^{z}\right).$$4- and $$H=J\sum_{j=1}^{N}\left( \frac{1+i\gamma }{2}\sigma _{j}^{x}\sigma _{j+1}^{x}+\frac{1-i\gamma }{2}\sigma _{j}^{y}\sigma _{j+1}^{y}+\lambda \sigma _{j}^{z}\right).$$5-correlators display distinct antiferromagnetic or mixed frustrated behavior (Lee et al., 2014).

A different complex-field formulation with global gain/loss in the transverse field produces a three-region phase diagram organized by the non-Hermitian gap $$H=J\sum_{j=1}^{N}\left( \frac{1+i\gamma }{2}\sigma _{j}^{x}\sigma _{j+1}^{x}+\frac{1-i\gamma }{2}\sigma _{j}^{y}\sigma _{j+1}^{y}+\lambda \sigma _{j}^{z}\right).$$6. The exact gap-closing condition is

$$H=J\sum_{j=1}^{N}\left( \frac{1+i\gamma }{2}\sigma _{j}^{x}\sigma _{j+1}^{x}+\frac{1-i\gamma }{2}\sigma _{j}^{y}\sigma _{j+1}^{y}+\lambda \sigma _{j}^{z}\right).$$7

which defines an elliptical exceptional ring. Inside the ring lies a ferromagnetic phase with real gap and long-range order; for $$H=J\sum_{j=1}^{N}\left( \frac{1+i\gamma }{2}\sigma _{j}^{x}\sigma _{j+1}^{x}+\frac{1-i\gamma }{2}\sigma _{j}^{y}\sigma _{j+1}^{y}+\lambda \sigma _{j}^{z}\right).$$8 there is a paramagnetic phase with complex gap and exponential correlations; between them lies the critical transition zone, where the gap is purely imaginary and the long-range correlation function decays polynomially,

$$H=J\sum_{j=1}^{N}\left( \frac{1+i\gamma }{2}\sigma _{j}^{x}\sigma _{j+1}^{x}+\frac{1-i\gamma }{2}\sigma _{j}^{y}\sigma _{j+1}^{y}+\lambda \sigma _{j}^{z}\right).$$9

In this model the Hermitian Ising critical point expands into an extended critical-like region (Liu et al., 2020).

A biorthogonal treatment of the anisotropic XY chain in a complex transverse field reaches a different conclusion: the real parts of the correlation function and entanglement entropy reproduce Hermitian XY critical scaling. The phase diagram contains a ferromagnetic region inside

$$H_{\text{eff}}=H-\frac{i\gamma}{4}\sum_n(\sigma_n^z+1).$$0

a paramagnetic region for

$$H_{\text{eff}}=H-\frac{i\gamma}{4}\sum_n(\sigma_n^z+1).$$1

and a Luttinger-liquid region in the remaining domain

$$H_{\text{eff}}=H-\frac{i\gamma}{4}\sum_n(\sigma_n^z+1).$$2

In the LL phase,

$$H_{\text{eff}}=H-\frac{i\gamma}{4}\sum_n(\sigma_n^z+1).$$3

The paper attributes the FM phase to $$H_{\text{eff}}=H-\frac{i\gamma}{4}\sum_n(\sigma_n^z+1).$$4 symmetry breaking and the LL phase to an emergent $$H_{\text{eff}}=H-\frac{i\gamma}{4}\sum_n(\sigma_n^z+1).$$5 symmetry at the critical mode (Wang et al., 15 Nov 2025).

A further non-Hermitian variant, defined on an odd-length ring with symmetric non-collinear coupling, exhibits ring frustration. In its kink phase the true ground state is not the quasiparticle vacuum but a one-mode occupied state, and the low-energy sector becomes gapless with kink-like excitations. The winding number is $$H_{\text{eff}}=H-\frac{i\gamma}{4}\sum_n(\sigma_n^z+1).$$6 in the kink phase and $$H_{\text{eff}}=H-\frac{i\gamma}{4}\sum_n(\sigma_n^z+1).$$7 in the paramagnetic phase (Bi et al., 2020).

These results point in different directions. This suggests that the phrase “non-Hermitian XY model” does not designate a single universality scenario, but a class of exactly solvable and near-exactly solvable deformations whose phase diagrams depend on the non-Hermitian mechanism and on the definition of the physically relevant state.

4. Exceptional points, topology, and Lee–Yang-type structures

Exceptional points organize much of the spectral geometry of non-Hermitian XY chains. For the open chain with complex anisotropy parameter $$H_{\text{eff}}=H-\frac{i\gamma}{4}\sum_n(\sigma_n^z+1).$$8, quasi-energy degeneracies occur when the quasi-momentum equation develops a repeated root. The EP values satisfy

$$H_{\text{eff}}=H-\frac{i\gamma}{4}\sum_n(\sigma_n^z+1).$$9

For finite $h\in\mathbb C$0 there are $h\in\mathbb C$1 EP locations arranged on two concentric rings in the complex $h\in\mathbb C$2-plane. In the thermodynamic limit these rings converge to the unit circle $h\in\mathbb C$3, which is simultaneously the boundary between topological sectors with winding numbers $h\in\mathbb C$4 and $h\in\mathbb C$5 (Henry et al., 6 Jul 2025).

The open-chain solution has since been sharpened algebraically. In the Chebyshev-polynomial formulation, EPs are repeated roots of the same boundary polynomial in the quasi-energy variable $h\in\mathbb C$6. At EPs the quasi-Hamiltonian becomes defective, Jordan blocks appear, and generalized eigenvectors can be generated by differentiating the eigenvector branch with respect to $h\in\mathbb C$7. This construction gives the correct many-body state counting at defectiveness and makes the branch-point character of EPs explicit: encircling an EP in the complex anisotropy plane permutes eigenenergies and eigenstates (Li et al., 26 May 2026).

A distinct but closely related program concerns Lee–Yang theory. In the complex-field XY model, the overlap of ground states is treated as the ground-state analogue of a partition function, and its zeros are fidelity zeros. These occur only in the ordered phase. For $h\in\mathbb C$8 and $h\in\mathbb C$9, the zeros populate the complex-$(1\pm i\gamma)/2$0 plane and accumulate near the real critical point at $(1\pm i\gamma)/2$1. Writing $(1\pm i\gamma)/2$2 with fugacity variable $(1\pm i\gamma)/2$3, the zeros lie on the unit circle in $(1\pm i\gamma)/2$4: for $(1\pm i\gamma)/2$5 they are uniformly distributed along the circle, whereas for $(1\pm i\gamma)/2$6 they condense into two arcs separated by a gap, identified as the analogue of a Yang–Lee edge singularity. The leading zero, defined as the one with largest real part and smallest imaginary part, obeys

$(1\pm i\gamma)/2$7

and for $(1\pm i\gamma)/2$8 to $(1\pm i\gamma)/2$9 at $\gamma\in\mathbb C$0 yields

$\gamma\in\mathbb C$1

In this formulation, non-Hermitian symmetry breaking is visible not as a single level crossing but as a sequence of parity-sector changes traced by fidelity zeros (Gu et al., 9 Dec 2025).

Geometric diagnostics lead to the same boundary structure from another direction. In a generalized non-Hermitian XY model with field-dependent intrinsic $\gamma\in\mathbb C$2 symmetry, the exceptional boundary is exactly the locus where the Berry curvature diverges. Near that boundary the ground-state energy derivative scales as $\gamma\in\mathbb C$3, while the Berry curvature scales as $\gamma\in\mathbb C$4 (Zhang et al., 2013). Topological analyses of non-Hermitian XY extensions further connect zero quasienergy, quasienergy degeneracy, and winding number, with topological phases characterized by $\gamma\in\mathbb C$5 and trivial phases by $\gamma\in\mathbb C$6 (Liu et al., 23 Jan 2025).

5. Correlations, entanglement, metrology, and the problem of observables

Non-Hermitian XY chains have also become a testing ground for diagnostic observables. In the $\gamma\in\mathbb C$7KSEA extension,

$\gamma\in\mathbb C$8

nearest-neighbor logarithmic negativity and its derivative detect both the exceptional point

$\gamma\in\mathbb C$9

and the quantum critical point

$\mathcal{RT}$0

In the $\mathcal{RT}$1 regime there is also a vanishing-entanglement surface at

$\mathcal{RT}$2

Under quenches, the Loschmidt rate function detects some dynamical critical lines but can produce false positives, whereas the second moment of entanglement fluctuations is reported as a more robust diagnostic (Agarwal et al., 2023).

Factorization ideas from Hermitian spin chains persist in modified form. For the $\mathcal{RT}$3-symmetric $\mathcal{RT}$4 model, the Hermitian factorization surface analytically continued by $\mathcal{RT}$5 reproduces the non-Hermitian threshold

$\mathcal{RT}$6

which marks the transition between broken and unbroken $\mathcal{RT}$7 symmetry. At the same time, nearest-neighbor entanglement remains nonzero at the exceptional point, so the Hermitian correspondence is spectral rather than separability-based (Lakkaraju et al., 2021).

More recent entanglement analyses of the non-Hermitian XY chain with complex anisotropy and transverse field compare self-normalized and biorthogonal reduced density matrices. Concurrence, logarithmic negativity, mutual information, and quantum coherence all detect the $\mathcal{RT}$8 transition, but the standard biorthogonal concurrence can fail because the outer $\mathcal{RT}$9 truncation suppresses the signal near EPs. The introduced unconstrained concurrence,

$$H = - J \sum_{\langle i, j \rangle} \left( \frac{1+\gamma}{2} \sigma_{i}^{x} \sigma_{j}^{x} + \frac{1-\gamma}{2} \sigma_{i}^{y} \sigma_{j}^{y} \right) - h \sum_{i} \sigma_{i}^{z},$$00

restores the phase boundary (Zhang et al., 30 Sep 2025).

Metrological diagnostics add yet another layer. In the non-Hermitian $$H = - J \sum_{\langle i, j \rangle} \left( \frac{1+\gamma}{2} \sigma_{i}^{x} \sigma_{j}^{x} + \frac{1-\gamma}{2} \sigma_{i}^{y} \sigma_{j}^{y} \right) - h \sum_{i} \sigma_{i}^{z},$$01KSEA model, the quantum Fisher information for magnetic-field estimation scales as $$H = - J \sum_{\langle i, j \rangle} \left( \frac{1+\gamma}{2} \sigma_{i}^{x} \sigma_{j}^{x} + \frac{1-\gamma}{2} \sigma_{i}^{y} \sigma_{j}^{y} \right) - h \sum_{i} \sigma_{i}^{z},$$02 at both the quantum critical point and the exceptional point in the thermodynamic limit, and can reach $$H = - J \sum_{\langle i, j \rangle} \left( \frac{1+\gamma}{2} \sigma_{i}^{x} \sigma_{j}^{x} + \frac{1-\gamma}{2} \sigma_{i}^{y} \sigma_{j}^{y} \right) - h \sum_{i} \sigma_{i}^{z},$$03 for moderate sizes when $$H = - J \sum_{\langle i, j \rangle} \left( \frac{1+\gamma}{2} \sigma_{i}^{x} \sigma_{j}^{x} + \frac{1-\gamma}{2} \sigma_{i}^{y} \sigma_{j}^{y} \right) - h \sum_{i} \sigma_{i}^{z},$$04, a regime described as super-Heisenberg scaling (Agarwal et al., 31 Mar 2025).

The main conceptual complication is that non-Hermitian expectation values are not unique. A systematic comparison of right-right and left-right prescriptions in two non-Hermitian XY models concludes that the critical properties, including the phase diagram, depend on both the formalism used and the state considered. The same work argues in favor of standard quantum mechanics, i.e.

$$H = - J \sum_{\langle i, j \rangle} \left( \frac{1+\gamma}{2} \sigma_{i}^{x} \sigma_{j}^{x} + \frac{1-\gamma}{2} \sigma_{i}^{y} \sigma_{j}^{y} \right) - h \sum_{i} \sigma_{i}^{z},$$05

over biorthogonal expectation values for genuine open-system realizations, while also emphasizing that the relevant state—minimal-real-energy state or steady state—depends on the experimental preparation (Luo et al., 5 Jun 2026).

6. Physical realizations and broader developments

The non-Hermitian XY model is not purely formal. In the no-jump realization, three-level atoms provide a direct implementation: $$H = - J \sum_{\langle i, j \rangle} \left( \frac{1+\gamma}{2} \sigma_{i}^{x} \sigma_{j}^{x} + \frac{1-\gamma}{2} \sigma_{i}^{y} \sigma_{j}^{y} \right) - h \sum_{i} \sigma_{i}^{z},$$06 is pumped to an auxiliary decay channel, and absence of fluorescence heralds successful conditioning on no spontaneous-emission event. The effective Hamiltonian is then non-Hermitian, and the predicted transition from short-range order to quasi-long-range order is proposed for trapped ions, cavity QED, and atoms in optical lattices. For realistic parameters, the authors estimate that even $$H = - J \sum_{\langle i, j \rangle} \left( \frac{1+\gamma}{2} \sigma_{i}^{x} \sigma_{j}^{x} + \frac{1-\gamma}{2} \sigma_{i}^{y} \sigma_{j}^{y} \right) - h \sum_{i} \sigma_{i}^{z},$$07 atoms are sufficient (Lee et al., 2014).

Open-system derivations also appear in spin models with KSEA interactions. There the non-Hermitian XY-type Hamiltonian emerges as an effective no-jump Hamiltonian from a GKLS master equation for a Hermitian $$H = - J \sum_{\langle i, j \rangle} \left( \frac{1+\gamma}{2} \sigma_{i}^{x} \sigma_{j}^{x} + \frac{1-\gamma}{2} \sigma_{i}^{y} \sigma_{j}^{y} \right) - h \sum_{i} \sigma_{i}^{z},$$08+KSEA chain coupled to local and non-local reservoirs. This situates part of the non-Hermitian XY literature within the broader framework of monitored quantum trajectories rather than static phenomenological complex couplings (Agarwal et al., 2023).

Spectral-statistical studies have pushed the model toward many-body quantum chaos. Adding a random longitudinal $$H = - J \sum_{\langle i, j \rangle} \left( \frac{1+\gamma}{2} \sigma_{i}^{x} \sigma_{j}^{x} + \frac{1-\gamma}{2} \sigma_{i}^{y} \sigma_{j}^{y} \right) - h \sum_{i} \sigma_{i}^{z},$$09-field to an intrinsic-$$H = - J \sum_{\langle i, j \rangle} \left( \frac{1+\gamma}{2} \sigma_{i}^{x} \sigma_{j}^{x} + \frac{1-\gamma}{2} \sigma_{i}^{y} \sigma_{j}^{y} \right) - h \sum_{i} \sigma_{i}^{z},$$10 non-Hermitian XY chain drives a crossover from Poisson-like statistics toward GinUE-like complex spacing-ratio statistics. An additional imaginary $$H = - J \sum_{\langle i, j \rangle} \left( \frac{1+\gamma}{2} \sigma_{i}^{x} \sigma_{j}^{x} + \frac{1-\gamma}{2} \sigma_{i}^{y} \sigma_{j}^{y} \right) - h \sum_{i} \sigma_{i}^{z},$$11-field strengthens the complex-eigenvalue structure and improves agreement with 2D-Poisson-to-GinUE crossover behavior (Sarkar et al., 2023).

Non-Hermitian XY physics has also entered quantum thermodynamics. A two-qubit anisotropic XY model with staggered imaginary magnetic field,

$$H = - J \sum_{\langle i, j \rangle} \left( \frac{1+\gamma}{2} \sigma_{i}^{x} \sigma_{j}^{x} + \frac{1-\gamma}{2} \sigma_{i}^{y} \sigma_{j}^{y} \right) - h \sum_{i} \sigma_{i}^{z},$$12

naturally splits into a field-dependent working pair and field-independent idle levels. In the $$H = - J \sum_{\langle i, j \rangle} \left( \frac{1+\gamma}{2} \sigma_{i}^{x} \sigma_{j}^{x} + \frac{1-\gamma}{2} \sigma_{i}^{y} \sigma_{j}^{y} \right) - h \sum_{i} \sigma_{i}^{z},$$13-unbroken regime $$H = - J \sum_{\langle i, j \rangle} \left( \frac{1+\gamma}{2} \sigma_{i}^{x} \sigma_{j}^{x} + \frac{1-\gamma}{2} \sigma_{i}^{y} \sigma_{j}^{y} \right) - h \sum_{i} \sigma_{i}^{z},$$14, tuning $$H = - J \sum_{\langle i, j \rangle} \left( \frac{1+\gamma}{2} \sigma_{i}^{x} \sigma_{j}^{x} + \frac{1-\gamma}{2} \sigma_{i}^{y} \sigma_{j}^{y} \right) - h \sum_{i} \sigma_{i}^{z},$$15 compresses the idle-level gap and changes the operation mode of a quantum Otto cycle from dissipative behavior to positive-work heat-engine operation (Tusun et al., 21 Jun 2026).

Taken together, these developments show that the non-Hermitian XY model is less a single Hamiltonian than a structured family of solvable spin chains. Its recurrent themes are exact fermionization, anti-linear symmetries, exceptional geometry, and the nontrivial dependence of criticality on how states and observables are defined.