Non-Bloch PT Symmetry in Lattice Systems

• Non-Bloch PT symmetry is defined by the existence of a purely real spectrum in non-Hermitian lattice systems under open boundary conditions, enforced by a generalized PT operation on the GBZ.
• The non-Hermitian skin effect causes exponential localization of bulk eigenstates at system boundaries, necessitating a β-based spectral analysis instead of standard Bloch theory.
• Non-Bloch PT symmetry breaking is marked by exceptional points and geometric singularities on the GBZ, which trigger significant changes in physical observables such as state norm and density of states.

Non-Bloch PT symmetry is a regime of parity-time ($\mathcal{PT}$) symmetry that arises in non-Hermitian lattice systems subject to the non-Hermitian skin effect (NHSE). In such systems, spectral and dynamical properties under open boundary conditions (OBC) fundamentally differ from those predicted by standard Bloch theory due to extreme boundary sensitivity. Non-Bloch PT symmetry is characterized by the existence of a purely real spectrum under OBC, enforced by a generalized or “non-Bloch” PT symmetry on a boundary-adapted spectral manifold known as the generalized Brillouin zone (GBZ), even when the spectrum under periodic boundary conditions (PBC) is entirely complex. The breakdown of this symmetry occurs via a non-Bloch PT transition, typically marked by the formation of exceptional points or geometric singularities in the GBZ, and is accompanied by qualitative changes in physical observables, including exponential growth or decay of state norms and novel van Hove singularities.

1. Conventional vs. Non-Bloch PT Symmetry

Conventional (Bloch) PT symmetry applies to non-Hermitian Hamiltonians $H$ where an antiunitary operator $A$ (often $P K$, with $P$ the parity/inversion operator and $K$ complex conjugation) satisfies $A H A^{-1} = H$. If this condition holds for each momentum $k$ in the Brillouin zone, the Bloch Hamiltonian $H(k)$ can exhibit entirely real spectra for certain parameter regions, with PT symmetry breaking (and the emergence of complex eigenvalues) occurring via band coalescence at exceptional points in the $k$-plane.

In contrast, non-Bloch PT symmetry emerges in systems with NHSE, where the bulk eigenstates localize at the system’s boundaries under OBC and are not captured by plane waves $H$0. Instead, the spectral analysis must be performed using an ansatz $H$1 with $H$2, and the effective Hamiltonian is recast as $H$3, parameterized on a GBZ that is generically non-circular and encodes the decay/amplification profile enforced by OBC. Non-Bloch PT symmetry is defined by the existence of a PT symmetry for $H$4; i.e., that $H$5 along the GBZ. This ensures that, for a range of system and non-Hermiticity parameters, the OBC spectrum is entirely real, despite the generic presence of complex PBC eigenvalues and absence of bulk-boundary correspondence (Longhi, 2019, Xiao et al., 2020, Hu et al., 2022).

2. The Non-Hermitian Skin Effect and Generalized Brillouin Zone

The NHSE refers to the accumulation of all bulk eigenstates at one (or more) boundaries of the system when non-reciprocal hopping or gain/loss is present, invalidating the conventional (PBC-based) band theory. In the NHSE regime, the ansatz $H$6 replaces $H$7, and the characteristic equation $H$8 must be solved with $H$9 restricted to the GBZ. The GBZ is explicitly determined by the requirement that, for each eigenenergy $A$0, the moduli of relevant pairs of roots of the characteristic equation are equal—a constraint that enforces the correct exponential localization profile matching OBC boundary conditions (Xiao et al., 2020, Hu et al., 2022, Song et al., 2021).

For one-dimensional models, the GBZ is typically a loop $A$1 in the complex plane, where $A$2 depends on the degree of non-Hermiticity and detailed hopping structure (Longhi, 2019, Hu et al., 2022).

3. Criterion and Mechanism for Non-Bloch PT Symmetry Breaking

Non-Bloch PT symmetry breaking refers to the transition from a regime where the OBC spectrum remains entirely real (PT-unbroken phase) to one where eigenvalues acquire complex conjugate pairs (PT-broken phase), with the threshold determined by parameters such as the non-Hermiticity $A$3 and system geometry.

For a broad class of one-dimensional systems, the GBZ approach yields exact analytic conditions for this transition. For example, in Floquet quantum walks,

$A$4

with the PT phase unbroken for $A$5 and broken for $A$6 (Longhi, 2019).

A geometric interpretation has identified that non-Bloch PT symmetry breaking in 1D occurs by the formation of cusps on the GBZ, corresponding to a coalescence of saddle points in $A$7:

$A$8

The algebraic condition for the PT-breaking threshold is obtained from the resultant of the characteristic polynomial and its $A$9-derivative, ensuring saddle-point coalescence (Hu et al., 2022).

In higher dimensions ($P K$0), the PT-breaking threshold acquires universal system-size dependence: the critical non-Hermiticity scales as $P K$1 as $P K$2, as a consequence of the scaling of NHSE-induced state overlaps, distinguishing higher-dimensional non-Bloch PT symmetry from both 1D non-Bloch and Bloch PT symmetry (Song et al., 2021).

4. Non-Bloch Exceptional Points and Non-Bloch van Hove Singularities

Non-Bloch exceptional points (EPs) materialize at the intersection between PT-exact and broken non-Bloch PT phases, operationalized on the GBZ rather than the conventional BZ. They are identified by the coalescence of roots of the characteristic equation for $P K$3:

$P K$4

with the resulting condition positionally off the unit circle. For example, in the photonic quantum walk model, EPs occur at $P K$5 (Xiao et al., 2020).

Non-Bloch van Hove singularities arise as divergent density-of-states (DOS) on the GBZ at saddle points where $P K$6. The singularity’s order depends on the local expansion shape of $P K$7, giving rise to distinctive scaling exponents not present in Hermitian (Bloch) analogues (Hu et al., 2022).

5. Physical Consequences and Experimental Signatures

Non-Bloch PT symmetry and its breaking fundamentally alter the relationship between observable spectra under different boundary conditions. Systems with non-Bloch PT symmetry can exhibit entirely real spectra under OBC while being generically complex under PBC. This boundary sensitivity enables several physical manifestations:

• In the PT-unbroken phase, observables such as the site-resolved amplitude or wavepacket norm remain bounded; in the PT-broken phase, exponential growth or decay is observed. The transition can be tracked using the Lyapunov exponent $P K$8, with

$P K$9

(Longhi, 2019).

• Divergent Green’s functions and non-Bloch van Hove singularities occur near the PT-breaking threshold, yielding extreme frequency and boundary sensitivity in response functions, directly observable in topo-electrical circuits, photonics, and cold-atom experiments (Hu et al., 2022, Song et al., 2021).
• The product $P$0 (non-Hermiticity × system size) acts as a universal indicator for PT symmetry breaking in $P$1, controlling the breakdown of perturbative approaches (Song et al., 2021).

6. Experimental Realizations and Platforms

Non-Bloch PT symmetry has been realized and observed in several experimental settings:

• Discrete-time quantum walks of photons in integrated photonics or fiber-loop architectures, where gain and loss are introduced in a controlled Floquet sequence (Longhi, 2019, Xiao et al., 2020).
• Topo-electrical circuits, with non-reciprocal couplings implemented by controlled impedance networks and OBC spectra extracted from impedance or transfer-function measurements (Song et al., 2021, Hu et al., 2022).
• Arrays of coupled microring resonators in photonic lattices, which allow direct engineering of site- or link-dependent non-Hermiticity and the measurement of transmission spectra (Song et al., 2021).
• Ultracold atomic systems in dissipative optical lattices with site-selective loss or Raman-induced NHSE and spatially resolved band-mapping (Song et al., 2021).

7. Theoretical Significance and Distinction from Conventional PT Symmetry

Non-Bloch PT symmetry establishes a new bulk-boundary paradigm for non-Hermitian physics. In conventional Hermitian and even standard non-Hermitian PT-symmetric systems, the Bloch framework suffices: spectral reality or PT-breaking is determined by closures of band gaps or Bloch exceptional points. In contrast, non-Bloch PT symmetry operates only when boundary condition–induced NHSE is strong, with a real OBC spectrum surviving up to a nonzero threshold in 1D, but vanishing in higher dimensions as system size increases (Song et al., 2021).

The non-Bloch program thus reveals a boundary-condition-driven phenomenon with no direct Hermitian analogue. Its breaking is set by geometric nonanalyticalities—cusps or saddle-point collisions—on the GBZ, and its observable consequences span steady-state, dynamical, and linear-response measurements, with unique spectral and spatial features not captured by Bloch theory (Hu et al., 2022, Song et al., 2021).