Activating non-Hermitian skin modes by parity-time symmetry breaking

The study addresses how parity-time (PT) symmetry—known to ensure real eigenenergies for PT-unbroken states by balancing gain and loss—can be leveraged not only for stability but also as a design principle to engineer and control non-Hermitian skin effects (NHSE) in higher dimensions. Prior work established that NHSE requires complex spectral loops under periodic boundary conditions and thus PT-symmetry breaking. However, how PT symmetry can be selectively broken or restored in different sectors (bulk vs. boundary) to manipulate NHSE remains largely unexplored. This work proposes a systematic framework where PT symmetry (and in 3D, CP symmetry) is selectively broken or recovered in bulk or boundary subspaces to activate specific NHSE channels, enabling bulk, surface, hinge, or corner-localized dynamics and directional toggling of NHSE without chiral pumping.

The paper situates itself within non-Hermitian physics where PT symmetry guarantees real spectra and has been widely explored in optical, acoustic, circuit, and atomic platforms. It builds on the understanding that NHSE arises with spectral winding under PBC and is tied to PT breaking, and references non-Bloch PT symmetry which can yield real OBC spectra even when PBC spectra are complex. The authors highlight gaps in systematically using PT symmetry to manipulate NHSE, especially in higher dimensions and selectively in bulk vs. boundary sectors, and connect to hybrid skin-topological phenomena and pseudo-Hermiticity frameworks.

• Models: The core 2D models are constructed from a non-Hermitian Su–Schrieffer–Heeger (nH-SSH) Hamiltonian extended with ky-dependent couplings. One representative 2D Hamiltonian Hy,2D(kx, ky) uses Pauli matrices on a pseudospin-1/2 sublattice basis with components h1 = t1 cos kx, h2 = u + v cos ky, h3 = v sin kx + iγ/2 + i t2 sin ky. For fixed ky this reduces to a 1D nH-SSH with generalized PT symmetry (K H K = H†) under x-OBC.
• Spectral and localization diagnostics: The non-Hermiticity is tuned to yield PT-broken and unbroken regimes. Fractal dimension FD = −ln[∑r |wr|2]/ln√N quantifies effective dimensionality: FD ≈ 0 (corner), 1 (edge), 2 (bulk). 1D projections x-FD and y-FD assess direction-selective localization.
• Boundary interpolation: To toggle NHSE directionally, the authors interpolate x-boundary conditions via H2D^β = H2D^OBC + e^{−β} Hx^{N.N.}, with β from 0 (x-PBC) to ∞ (x-OBC), keeping y-OBC fixed. This tunes PT breaking/restoration in selected sectors and switches NHSE between x and y directions, revealing intermediate corner-localized states.
• Selective boundary NHSE with topology: A second 2D model H2D = ∑a hx,y,z,0(kx, ky) σa satisfies PT with PT = K. With parameter condition u^2 + v^2 − 2uv > g + t2 (all positive), bulk remains PT-unbroken and free of NHSE, while x-edge topological states (obtained via Wilson loop/edge projectors P± = (1 ± σz)/2) obey an effective edge Hamiltonian Hedge = t1 cos kx + i(g + t2 sin kx) that is PT-asymmetric and exhibits spectral winding, thus undergoing y-NHSE under y-OBCs. An added anti-Hermitian term −i(t2 sin kx + g)σz defines H2D′, restoring reality to one branch of edge states, selectively turning off their NHSE.
• Dynamics: Time evolution |ψ(t)⟩ = e^{−iHt}|ψ(0)⟩ is simulated from a center-localized initial state on N × N lattices. Center-of-mass ⟨x⟩, ⟨y⟩ trajectories reveal bulk-then-boundary stages and non-monotonic flows when multiple NHSE channels compete.
• 3D generalizations: Constructed 3D Hamiltonians Hy/g,3D(k) by stacking the 2D models with a Hermitian SSH in kz: HSSH(kz) = (u + v cos kz) τx + v′ sin kz τy, with τ Pauli matrices. Effective surface Hamiltonians are derived via projectors P3D on z-surfaces, showing emergent non-Bloch CP symmetry on surfaces (due to factors of i) enabling CP-activated NHSE. Additional deformations (e.g., adding (t2 sin kz + δ) σx τz) selectively restore CP for certain hinge branches. A contrasting 3D design H3D = H2D(kx, ky) ⊗ τz + HSSH(kz) realizes bulk-selective NHSE, where generalized PT breaking in bulk drives accumulation to surfaces/hinges while conventional surface/hinge states remain extended.
• Non-Bloch PT framework: The Methods establish non-Bloch PT symmetry for asymmetric-hopping SSH (γ ≠ 0, g = 0) via complex momentum shifts k → k + iκ (κ = ln((u + γ/2)/(u − γ/2))) and similarity transforms that remove NHSE, guaranteeing real OBC spectra when |u| > |γ/2|. This is extended to the 2D/3D constructions and to CP symmetry on surfaces.
• Numerical procedures: Spectra under PBC and OBC (partial and full), computation of FD/x-FD/y-FD, spatial localization profiles (bulk/surface/hinge/corner), and time dynamics are presented for representative parameter sets consistent with figures. Codes and raw data are available upon request.

• Corner NHSE via generalized PT breaking: For the 2D model Hy,2D under x-OBC, ky-slices exhibit PT-broken (complex OBC spectra) for ky ∈ (0, π) and PT-unbroken (real OBC spectra) for ky ∈ [π, 0], with non-Bloch exceptional points at ky = 0, π. With full x,y-OBCs, states from PT-broken slices undergo y-NHSE in addition to x-NHSE, yielding corner-localized modes (FD ≈ 0) at Im(E) ≈ 0, while PT-unbroken slices produce edge-localized modes (FD = 1) with real energies.
• Directional toggling of NHSE: By tuning β in H2D^β from ∞ (x-OBC) to 0 (x-PBC) while keeping y-OBC, localization toggles from x-edge NHSE to y-edge NHSE. An intermediate regime (β ≈ 10) exhibits corner modes with FD ≈ 0.5 as a byproduct of PT activation. Average FD shows a trough near β ≈ 10, while x-FD decreases and y-FD increases nearly monotonically, evidencing mutual exclusion of x- and y-NHSE channels.
• Selective boundary NHSE (hybrid skin-topological effect): In the PT-symmetric bulk model H2D, bulk states remain PT-unbroken and have a purely real spectrum, showing no NHSE. However, topological x-edge states have effective Hedge = t1 cos kx + i(g + t2 sin kx) with spectral winding over ky, so under y-OBCs they experience y-NHSE and accumulate at opposite corners depending on Im(E)’s sign, while bulk remains extended. This is geometry-independent (intrinsic hybrid skin-topological effect).
• Turning NHSE off for a chosen edge branch: Adding −i(t2 sin kx + g)σz to form H2D′ renders one branch of edge eigenenergies real (H′edge = t1 cos kx), leaving those edge states localized on the edge without further y-NHSE, while the other branch remains PT-broken and collapses to corners.
• Anomalous non-monotonic dynamics: For H2D, a center-initialized state first exhibits bulk propagation with nearly constant ⟨x⟩, ⟨y⟩, then after t ≈ 30 boundary-induced corner accumulation with Im(E) > 0 dominates and ⟨x⟩, ⟨y⟩ → 1. For H2D′, bulk y-NHSE initially pushes the state to the top boundary (⟨y⟩ increases, ⟨x⟩ constant), and after reaching side boundaries (t ≈ 50), dynamics become dominated by PT-protected left edge states (Im(E) > 0), deactivating y-NHSE and causing flow from top to left edge with ⟨x⟩ → 1, ⟨y⟩ → Ny/2.
• 3D generalizations with surface/hinge selectivity and CP symmetry: In Hy/g,3D, z-surface effective Hamiltonians exhibit non-Bloch CP symmetry enabling CP-activated NHSE. CP breaking yields surface-to-corner pumping; toggling x-boundaries switches between x- and y-directed NHSE on surfaces. Selective CP restoration via added Hermitian terms removes NHSE for one hinge branch, leaving it hinge-localized while the other undergoes corner accumulation. In a contrasting 3D model H3D, generalized PT breaking produces bulk-selective NHSE where bulk states accumulate to surfaces (first order) or hinges (second order), while conventional surface (blue) and hinge (green) states remain NHSE-free and extended.
• Quantitative markers and conditions: FD discriminates dimensionality (FD ≈ 0 corner, 1 edge). Directional toggling shows FD trough at β ≈ 10. Bulk PT-unbroken condition stated as u^2 + v^2 − 2uv > g + t2. Non-Bloch PT-unbroken regime for SSH slices occurs when |u| > |γ/2|, yielding real OBC spectra even with complex PBC spectra.

The findings demonstrate that PT symmetry, by forbidding spectral winding when unbroken, can be strategically broken or restored to activate or suppress NHSE in chosen sectors (bulk vs. boundary) and directions (x vs. y), including intermediate corner localization. This framework extends naturally to 3D where CP symmetry on surfaces provides an additional handle, enabling surface- and hinge-selective NHSE and on-demand switching between first- and second-order NHSE. The results address the gap in systematically controlling NHSE via symmetry operations and yield novel state dynamics, including non-monotonic boundary flows without chiral pumping. The paradigm generalizes to other winding-forbidding symmetries (non-Bloch or generalized PT, CP, pseudo-Hermiticity) and provides design rules for devices exploiting NHSE. The geometry-based electrostatics analogy for achieving real OBC spectra with complex PBC spectra offers further avenues for NHSE activation/deactivation strategies. The proposed phenomena are compatible with a broad range of experimental platforms that support gain/loss and non-reciprocity.

This work establishes PT symmetry as a versatile design paradigm for selective activation and control of the non-Hermitian skin effect across bulk and boundary sectors in 2D and 3D systems. It introduces directional NHSE toggling, PT-mediated corner modes, and selective edge PT breaking/restoration that lead to hybrid skin-topological effects and anomalous, non-monotonic state dynamics. In 3D, interplay with CP symmetry enables surface- and hinge-selective NHSE and switching between first- and second-order bulk NHSE. The framework generalizes to other symmetries that preclude spectral winding and offers a systematic scheme for constructing models with coexisting, sector-dependent NHSE behaviors. Future directions include extending to non-Hermitian gapless phases, leveraging the electrostatics analogy for geometry-based control, and experimental realizations in photonic, acoustic, electronic circuit, and quantum platforms.

The study is theoretical and based on non-interacting lattice models analyzed via spectral properties, localization diagnostics (FD and its projections), and numerical time evolutions. While experimental platforms are proposed, no experimental validation is presented. Quantum implementations are discussed within effective non-Hermitian Hamiltonian descriptions that approximate short-time Lindbladian dynamics by neglecting quantum jumps, which may limit quantitative applicability in open quantum systems. Code and raw data are available on request, indicating reproducibility but not open-source distribution within the article.