Krein Signature in Non-Hermitian Systems

• Krein signature framework is a rigorous method to classify stability, symmetry breaking, and exceptional points in non-Hermitian systems using biorthogonal extensions of conserved quantities.
• It employs generalized Brillouin zones that overcome conventional Bloch theory limitations, addressing the non-Hermitian skin effect and PT-symmetry transitions.
• The framework reveals mechanisms behind non-Bloch van Hove singularities and dimensional effects on PT-breaking thresholds, offering insights into spectral criticality.

A Krein signature framework provides a rigorous methodology for classifying and diagnosing stability, symmetry breaking, and exceptional points (EPs) in non-Hermitian systems, especially in the context of parity-time ($\mathcal{PT}$) symmetric non-Hermitian Hamiltonians. The central idea is to extend classical notions of energy signature, spectral stability, and conserved quantities to settings where the non-Hermitian skin effect (NHSE) and boundary sensitivity render conventional Bloch theory inapplicable. Such frameworks are indispensable for analyzing non-Bloch $\mathcal{PT}$-symmetry breaking transitions, the structure of generalized Brillouin zones (GBZ), and the emergence of non-Bloch EPs and novel spectral singularities.

1. Non-Hermitian Systems and $\mathcal{PT}$ Symmetry

Non-Hermitian (NH) Hamiltonian systems, characterized by the presence of gain and loss or nonreciprocal couplings, display spectral properties and phase transitions distinct from Hermitian counterparts. A non-Hermitian Hamiltonian or dynamical functional takes the general form $F = E + i\Gamma$, where $E$ is a real energy functional and $\Gamma$ encodes non-Hermitian contributions (e.g., gain/loss, asymmetric hopping). $\mathcal{PT}$-symmetry is implemented via a spatial exchange (parity, $P$) and antiunitary time-reversal $T$, such that for suitable variables, this symmetry maps $F$ to its complex conjugate.

In exchange-coupled spin systems, e.g., the two-spin model on the Bloch sphere, $\mathcal{PT}$0-symmetry is realized by exchanging spins and time-reversal (complex conjugation and $\mathcal{PT}$1). For parameters where $\mathcal{PT}$2 (with $\mathcal{PT}$3 exchange strength, $\mathcal{PT}$4 spin-torque), all eigenfrequencies are real and $\mathcal{PT}$5 is unbroken. At $\mathcal{PT}$6, eigenvalues coalesce at an exceptional point (EP), signifying symmetry breaking and the onset of complex conjugate eigenfrequencies for $\mathcal{PT}$7 (Komineas, 2022).

2. Generalized Brillouin Zone and Non-Bloch $\mathcal{PT}$8 Symmetry

In periodic NH systems, open boundary conditions induce the NHSE: bulk eigenstates localize at boundaries, violating the conventional bulk-boundary correspondence. The correct bulk spectral problem is recast in terms of a generalized Brillouin zone (GBZ), parameterized by complex $\mathcal{PT}$9 rather than real $\mathcal{PT}$0 via $\mathcal{PT}$1 (Xiao et al., 2020, Hu et al., 2022).

The non-Bloch Hamiltonian

$\mathcal{PT}$2

admits a spectrum determined by roots $\mathcal{PT}$3 of $\mathcal{PT}$4, with the physical GBZ curve defined by the "middle-pair" condition $\mathcal{PT}$5. The non-Bloch $\mathcal{PT}$6 symmetry is then

$\mathcal{PT}$7

When this holds, real spectra persist under open boundaries as long as $\mathcal{PT}$8 is unbroken (Hu et al., 2022, Longhi, 2019, Song et al., 2021).

3. Krein Signature and Exceptional Points in Non-Hermitian Systems

The Krein signature characterizes the local stability of spectral points and their ability to undergo symmetry breaking. In Hermitian or pseudo-Hermitian settings, Krein signature assigns a sign to an eigenmode (positive, negative, or indefinite) based on a conserved indefinite bilinear form (Krein–Pontryagin form). In NH systems, this approach persists via extension to biorthogonal frameworks or through geometric conserved quantities on phase space (e.g., integrals of motion $\mathcal{PT}$9 and $F = E + i\Gamma$0 on the Bloch sphere) (Komineas, 2022).

Exceptional points arise where eigenvalues and eigenvectors coalesce. In linearized two-spin systems, these appear when $F = E + i\Gamma$1. In non-Bloch lattice systems, EPs—specifically non-Bloch EPs—occur where cusps form on the GBZ, as diagnosed by resultant and discriminant conditions on the characteristic polynomial:

$F = E + i\Gamma$2

with EPs signaled by degeneracy of saddle-point energies and corresponding double roots on GBZ (Hu et al., 2022).

4. Geometric and Algebraic Structure of PT Symmetry Breaking

The geometric mechanism for non-Bloch $F = E + i\Gamma$3 symmetry breaking is the appearance of singularities (cusps) on the GBZ. The explicit process involves:

• Parameterizing GBZ as $F = E + i\Gamma$4, $F = E + i\Gamma$5.
• $F = E + i\Gamma$6 becomes multivalued (cusp) at points where $F = E + i\Gamma$7.
• Algebraic criterion: a cusp (nontrivial multiple root) occurs where the set

$F = E + i\Gamma$8

vanish at $F = E + i\Gamma$9 (Hu et al., 2022).

At the PT threshold, the GBZ transitions from smooth to cusp-bearing, and the real OBC spectrum gives way to complex conjugate branches. This is equivalent to a Krein collision in the extended, symmetry-adapted phase space.

5. Dynamical Signatures, Integrals of Motion, and Bistability

In nonlinear non-Hermitian spin systems, conserved quantities provide a phase space foliation, allowing precise tracking of stability islands, limit cycles, and bistabilities. For instance, for the two-spin system:

• $E$0 (with $E$1).
• $E$2 ($E$3 angular coordinate on projected circles; $E$4 set by $E$5).

The phase space is thereby foliated into invariant curves whose structure determines presence and basins of periodic orbits versus fixed points. At the EP ($E$6), these invariant manifolds coalesce, yielding abrupt loss of synchronization and the disappearance of periodic attractors (Komineas, 2022).

Bistability arises for $E$7, with coexisting attractors (limit cycles and fixed points), governed by the level sets of $E$8. As $E$9, these manifolds collide—an explicitly geometric Krein collision (Komineas, 2022).

6. Dimensionality and Universal PT-Breaking Thresholds

While in 1D non-Bloch systems the $\Gamma$0-breaking threshold is finite and set by the GBZ geometry, in two and higher dimensions a universal size-driven vanishing of the threshold occurs. The key scaling is the non-Hermiticity–size product $\Gamma$1, with $\Gamma$2 NH parameter and $\Gamma$3 linear size. As $\Gamma$4, $\Gamma$5 due to the scaling of boundary-localized mode overlaps (a consequence of the NHSE in higher dimensions). By contrast, in 1D, the threshold remains nonzero even for $\Gamma$6, reflecting a fundamental "dimensional surprise" in non-Bloch Krein analysis (Song et al., 2021).

7. Spectral Singularities: Non-Bloch van Hove Mechanism

At the non-Bloch PT threshold (i.e., when a cusp and saddle coincide on GBZ), the density of states diverges as a non-Bloch van Hove singularity. The singularity exponent is set by the order $\Gamma$7 of the merging saddle points:

$\Gamma$8

yielding novel, tunable spectral features absent in Hermitian or conventional Bloch settings. These non-Bloch van Hove singularities are observable in linear response and demonstrate the Krein-geometric underpinnings of non-Hermitian criticality (Hu et al., 2022).

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References:

• Non-Hermitian dynamics and phase space integrals: (Komineas, 2022)
• Non-Bloch PT symmetry and GBZ: (Xiao et al., 2020, Hu et al., 2022, Longhi, 2019)
• Dimensionality and universal threshold: (Song et al., 2021)