Non-Hermitian Skin Effect in Lattices

• Non-Hermitian skin effect is a phenomenon in nonreciprocal lattices where most bulk eigenstates localize at one boundary under open boundary conditions.
• Its characterization relies on asymmetric hopping, complex Berry phases, and winding numbers, revealing topological features distinct from Hermitian systems.
• Controllable switching between localized edge modes and extended topological states offers promising applications in topological lasers and unidirectional transport devices.

The non-Hermitian skin effect (NHSE) is a spectral and wavefunction phenomenon unique to non-Hermitian systems, in which a macroscopic fraction of eigenstates, including bulk states, accumulates at the boundaries under open boundary conditions (OBC). This effect arises in nonreciprocal tight-binding lattices, fundamental to many non-Hermitian topological insulators, and is underpinned by a complex interplay between nonreciprocity, point-gap topology, chiral symmetry, and the breakdown of the conventional bulk–boundary correspondence that governs Hermitian systems.

1. Structure and Origin in Nonreciprocal Lattices

The canonical setting for the NHSE in the cited work is a one-dimensional tight-binding chain with nonreciprocal nearest-neighbor hopping defined as: $$H = \sum_{n=1}^{N-1} \left[ t_n c_{n+1}^\dagger c_n + t_n' c_n^\dagger c_{n+1} \right]$$ where $t_n, t_n'$ are (possibly site-dependent) forward and backward hopping amplitudes, and $c_n$, $c_n^\dagger$ are site annihilation and creation operators. The system is non-Hermitian for $t_n' \neq t_n^*$. Nonreciprocity breaks the symmetry between forward and backward dynamics, introducing an effective "imaginary gauge field" and resulting in asymmetric amplification and decay along the chain.

Under OBC, this nonreciprocity leads all bulk eigenstates—not only topological edge modes—to localize near a boundary, a phenomenon absent in Hermitian models. The localization direction and depth depend on the magnitude and sign of the hopping asymmetry.

2. Band Theory and Topological Characterization

When translation invariance is assumed, the system's behavior can be analyzed under periodic boundary conditions (PBC) via its Bloch Hamiltonian. For a two-site (dimerized) unit cell: $$\mathcal{H}(k) = \begin{bmatrix} 0 & t_1 + t_2 e^{ik} \ t_1' + t_2' e^{-ik} & 0 \end{bmatrix}$$ with eigenvalues: $$E_\pm = \pm \sqrt{(t_1 + t_2 e^{ik}) (t_1' + t_2' e^{-ik})}$$ The Hamiltonian exhibits chiral symmetry, $\sigma_z \mathcal{H}(k) \sigma_z = -\mathcal{H}(k)$.

Non-Hermitian topology is characterized via complex winding invariants:

• The complex Berry phase (non-Hermitian generalization):

$$\nu_{\pm} = \frac{1}{\pi} \int dk\, \left\langle \psi_{\pm}^{L} \middle| i \partial_k \middle| \psi_{\pm}^{R} \right\rangle$$

where $|\psi^{R/L}\rangle$ are right/left eigenstates, and

• The spectral (energy) winding number:

$$\nu_E = \frac{1}{2\pi} \oint dk\, \partial_k \arg(E_+ - E_-)$$

These invariants remain well-defined even though the conventional bulk–boundary correspondence is not directly applicable: the bulk spectrum under OBC is not generally predictable from the PBC spectrum.

3. Behavior under Open Boundaries and Skin Depth

Open boundaries sharply reveal the NHSE. Unlike Hermitian topological insulators, where bulk states remain extended and only edge states localize, in non-Hermitian, nonreciprocal systems:

• Bulk states accumulate (‘pump’ or ‘pile up’) at one boundary.
• The skin depth $\xi_{\text{bulk}}$ (localization length) for bulk states decreases as nonreciprocity, typically parameterized by the magnitude of hopping asymmetry $|t_n' - t_n|$, increases. In the strong nonreciprocal regime, this localization becomes extremely sharp.

For example, in models with two-site unit cells and alternating hopping, tuning the backward hopping strengths $t_1', t_2'$ modulates the skin effect's strength and direction. Numerical studies confirm that for various parameter regimes, bulk states are strongly localized at one edge while topological zero-energy states can become unexpectedly extended.

4. Switching between Edge and Extended Topological States

The interplay of nonreciprocity and chiral symmetry enables a controllable switching of the spatial character of the topological zero-energy state. Specifically:

• In the Hermitian or weakly non-Hermitian regime, topological edge states are sharply localized at the boundary associated with a topological phase transition.
• As nonreciprocity increases, while skin modes become increasingly localized at one edge, the topological zero-energy state (protected by chiral symmetry: $C H C^{-1} = -H$ and for a two-site unit cell, $C=\sigma_z$) can become delocalized or extended. For odd lattice sizes, the zero-mode remains robust due to chiral symmetry, but its profile spreads throughout the lattice—a clear deviation from standard bulk-boundary correspondence.

The condition $\psi_0(n) = C\psi_0(n)$, which for two-site unit cells imposes $\psi_0(n) = 0$ for even $n$, constrains the spatial profile of the topological mode. Upon sufficient nonreciprocity (for example, by increasing $t_1'$ relative to $t_1$), the bulk states localize more tightly while the zero-mode broadens, eventually extending across the full system.

5. Variation of Skin Depth and Physical Implications

The skin depth demonstrates the following trends:

• For bulk skin modes: Skin depth decreases monotonically with increasing nonreciprocity—a stronger nonreciprocal hopping rapidly localizes modes against an edge.
• For topological zero-energy states: Skin depth can increase with nonreciprocity (unless the parameter regime flips the localization to the other edge). This observation, especially in systems with an odd number of sites or with weak disorder respecting chiral symmetry, implies a decoupling between topology and localization in non-Hermitian lattices.

Implications include:

• Breakdown of bulk-boundary correspondence: In Hermitian systems, bulk invariants accurately count and localize edge states. In non-Hermitian systems with strong nonreciprocity, the distinction between bulk and boundary states blurs.
• Controllable spatial profile of topological states: Tuning nonreciprocal parameters can switch topological edge states into extended modes, allowing for new device design possibilities in topological lasers and unidirectional transport elements.
• Robustness to disorder: Chiral-symmetry-protected zero modes may remain extended and resist localization even as bulk skin modes remain tightly bunched—significant for disorder-resilient device functionality.

6. Key Formulas

The following formulas encode the NHSE physics:

Concept	Formula / Description
Nonreciprocal tight-binding Hamiltonian	$$H = \sum_{n=1}^{N-1} [ t_n c_{n+1}^\dagger c_n + t_n' c_n^\dagger c_{n+1} ]$$
Bloch Hamiltonian (2-site unit cell, PBC)	$$\mathcal{H}(k) = \begin{bmatrix} 0 & t_1 + t_2 e^{ik} \ t_1' + t_2' e^{-ik} & 0 \end{bmatrix}$$
Eigenenergies	$$E_\pm = \pm \sqrt{ (t_1 + t_2 e^{ik}) (t_1' + t_2' e^{-ik}) }$$
Chiral symmetry	$\sigma_z \mathcal{H}(k) \sigma_z = -\mathcal{H}(k)$
Complex winding number (Berry phase)	$$\nu_{\pm} = \frac{1}{\pi} \int dk\, \langle \psi_{\pm}^{L}| i\partial_k | \psi_{\pm}^{R} \rangle$$
Energy winding number	$$\nu_E = \frac{1}{2\pi} \oint dk\, \partial_k \arg(E_+ - E_-)$$

7. Significance and Applications

The non-Hermitian skin effect in nonreciprocal tight-binding systems under OBC reveals unprecedented control over the localization of bulk and edge states via nonreciprocal hopping. The existence of controllably extended topological modes in the presence of strong NHSE breaks the usual expectations from Hermitian bulk–boundary correspondence, opening avenues for:

• Realization of topological devices with tunable localization properties (e.g., topological lasers or wave routers).
• Engineering robust, boundary-sensitive lasing and amplification schemes exploiting skin accumulation and manipulable zero modes.
• Quantitative understanding of how symmetry, disorder, and nonreciprocity interface to yield (or disrupt) topological protection in non-Hermitian matter.

These findings establish the NHSE as a key element in the theory and application of non-Hermitian topological insulators, driving an expanded exploration of bulk–boundary phenomena far beyond their Hermitian analogues.