Bipolar Non-Hermitian Skin Effect

• Bipolar NHSE is a boundary-localization phenomenon in non-Hermitian systems where eigenstates accumulate on opposite edges depending on energy, band, or symmetry.
• It arises from nonreciprocal couplings and balanced gain/loss that create twisted spectral loops characterized by point-gap winding numbers.
• Experimental realizations in photonic, topolectrical, and cold atom systems enable controllable directional transport and robust non-Hermitian localization.

The bipolar non-Hermitian skin effect (NHSE) is a boundary-localization phenomenon in non-Hermitian lattice systems whereby eigenstates with different energies or quantum numbers accumulate at opposite boundaries. Unlike the conventional NHSE, where all states localize unidirectionally due to asymmetric hopping or gain/loss, the bipolar NHSE features a reversal of localization direction as a function of energy, band, spin sector, or symmetry label. This effect fundamentally reflects nontrivial point-gap topology of the bulk complex energy spectrum, often induced by long-range asymmetric couplings, balanced gain and loss, or multi-component degrees of freedom. Bipolar NHSE appears in diverse systems including one- and two-dimensional non-Hermitian lattices, synthetic coupled chains, spin-orbit coupled Rashba models, and photonic crystals with engineered loss and gain.

1. Fundamental Mechanisms and Model Architectures

The essential mechanism of the bipolar NHSE arises from a non-Hermitian lattice Hamiltonian in which nonreciprocal (asymmetric) couplings extend beyond nearest neighbors or are combined with balanced gain and loss. The generic one-dimensional model is given by

$$H = \sum_{j=1}^{L-r_d} \sum_{s=1}^{r_d} \Bigl[(t_s - \gamma_s) c^\dagger_{j+s} c_j + (t_s + \gamma_s) c^\dagger_j c_{j+s}\Bigr],$$

where $t_s$ (Hermitian) and $\gamma_s$ (non-Hermitian) control hopping to the $s$th neighbor (Zeng, 2022). Under periodic boundary conditions (PBC), the Bloch dispersion becomes

$$E(k) = 2\sum_{s=1}^{r_d}\left[t_s\cos(sk) + i\gamma_s\sin(sk)\right],$$

producing spectra with multiple self-intersections ("twisted loops").

Distinct implementation examples include:

• One-dimensional chains with unidirectional long-range hoppings (Rafi-Ul-Islam et al., 2023).
• Two coupled nonreciprocal chains exhibiting interchain competition ("concurrent bipolar skin effect", CBSE) (Li et al., 4 Aug 2025).
• Ladders with balanced gain/loss realizing energy (Im $E$)–dependent skin polarization (Jiang et al., 2024).
• Spinful Rashba chains with nonreciprocal spin-dependent hopping protected by symmetry (Tozar, 2 Dec 2025).
• Two-dimensional photonic kagome crystals with engineered gain/loss on different sublattices (Yang et al., 19 Jan 2026).

By adjusting the structure or symmetry, the system can force part of the spectrum to localize to one edge (for "right-pumping" states) and another part to the opposite edge ("left-pumping" states), giving rise to the bipolar skin regime.

2. Topological Invariants and Bulk-Spectrum Analysis

The key bulk diagnostic for the bipolar NHSE is the point-gap winding number of the PBC complex energy spectrum,

$$W(E_B) = \frac{1}{2\pi i}\int_{-\pi}^\pi dk\, \partial_k \arg[E(k) - E_B]$$

for a reference energy $E_B$ (Zeng, 2022, Rafi-Ul-Islam et al., 2023). For multi-band models, the analogous invariant is computed by tracking the net winding of $\det[H(k) - E_B]$ (Jiang et al., 2024, Yang et al., 19 Jan 2026): $$V(E_B) = \frac{1}{2\pi i}\int dk\, \partial_k \ln\det[H(k) - E_B].$$ Bipolar NHSE is characterized by a spectrum forming multiple loops in the complex plane, each with opposite signs of $W$. State localization direction is thus set by the local winding: $W=+1$ (left localization), $W=-1$ (right), or, for spinful systems, by a $\mathbb{Z}_2$ spin-sector invariant (Tozar, 2 Dec 2025).

Intersections (self-intersections or "X points") of the twisted loops in $E(k)$ mark skin-effect edges where the winding index flips, which are associated with real-energy states immune to skin localization—a diagnostic signature differentiating the bipolar effect from conventional unipolar NHSE (Zeng, 2022, Rafi-Ul-Islam et al., 2023).

3. Localization Properties and Generalized Brillouin Zone

The open boundary condition (OBC) eigenstates in non-Hermitian systems often deviate from Bloch waves, necessitating the generalized Brillouin zone (GBZ) analysis. For a characteristic polynomial $P_E(\beta)$, the GBZ selects $\{\beta\}$ (complex exponentials) with $|\beta|$ set by boundary matching. The localization exponent for a given eigenstate is

$\kappa(E) = \ln|\beta_*|,$

with $\kappa>0$ ($<0$) corresponding to right (left) boundary pileup (Rafi-Ul-Islam et al., 2023, Jiang et al., 2024).

In the bipolar NHSE regime, the sign of $\kappa$ correlates with the winding of the local loop in the PBC spectrum. States near Bloch-like contact points ($|\beta|=1$) remain extended. In multi-component or spinful cases, one tracks $\kappa$ separately for each sector, enabling, for instance, simultaneous left- and right-localized modes ("bipolar" accumulation) or, in higher dimensions, multipolar corner localization (Yang et al., 19 Jan 2026).

4. Symmetry-Protected and Disordered Bipolar Skin Effects

In systems with internal symmetries such as time-reversal, spin-orbit coupling, or $\mathbb{Z}_2$ protection, the bipolar NHSE can be protected against moderate disorder (Tozar, 2 Dec 2025). In a non-Hermitian Rashba chain, the effective Hamiltonian ensures opposite-sign Lyapunov exponents for spin sectors under clean conditions; introducing disorder, one observes a regime where spin-up and spin-down mode profiles remain split between boundaries (quantified by a biorthogonal spin-separation index).

The phase diagram then displays a disorder-robust topological bipolar skin regime, a collapse to a trivial skin phase where localization persists without spin separation, and finally an Anderson localized phase where skin effects vanish. This hierarchical destruction under disorder is a distinguishing marker of symmetry-protected bipolar NHSE (Tozar, 2 Dec 2025).

5. Dimensional Extensions: Multipolar NHSE and Higher-Order Topology

The bipolar skin effect generalizes naturally to higher dimensions and multi-band models, giving rise to phenomena such as quadripolar NHSE (states accumulating at four corners) and the interplay of skin effects with higher-order topological modes. For instance, in a non-Hermitian 2D SSH ladder composed of stacked 1D bipolar chains, the combination of independent winding indices in both spatial directions yields four types of corner-localized skin states (Jiang et al., 2024).

In photonic kagome crystals with balanced gain and loss, the emergence of the bipolar skin effect occurs in tandem with higher-order topological corner modes. Non-Hermitian skin modes drive a breakdown of conventional (Hermitian) bulk-boundary correspondence, as point-gap winding, rather than Hermitian polarization, predicts the true accumulation pattern of states under OBC (Yang et al., 19 Jan 2026).

6. Experimental Realizations and Engineering Guidelines

Bipolar NHSE has concrete proposals and signatures in several synthetic platforms:

• Photonic waveguides exploiting engineered gain/loss or synthetic gauge fields (Rafi-Ul-Islam et al., 2023, Yang et al., 19 Jan 2026).
• topolectrical circuits incorporating non-reciprocal amplifiers, negative impedance converters, and tailored capacitive links (Jiang et al., 2024).
• Cold atom lattices with Floquet-engineered long-range tunneling (Rafi-Ul-Islam et al., 2023).
• Metamaterials with designed asymmetric electromagnetic interactions.

For engineering purposes, one selects:

• Nonreciprocal parameters (e.g., $\gamma_1, \gamma_2$ or balanced $\pm i\gamma$ on sublattices) to tailor twisted PBC loops.
• System size and interchain coupling to tune between concurrent and conventional NHSE (Li et al., 4 Aug 2025). The critical system size for CBSE scales as $N_c\sim 1/g$ for interchain coupling $g$.
• In electric circuits, tuning resistor values modulates gain/loss and hence the entry into partial, bipolar, or quadripolar NHSE regimes (Jiang et al., 2024).

Experimental signatures include imaging the spatial mode profile (photonic or acoustic systems), measurement of skin-mode accumulation at both ends/corners, and observation of suppression of localization at real-energy crossings.

7. Implications for Non-Hermitian Topology and Open Quantum Systems

The bipolar NHSE reveals new aspects of non-Hermitian topology, including a tunable interplay between spectral winding, localization, symmetry protection, and finite-size effects. The breakdown of Hermitian bulk-boundary correspondence underscores the necessity of point-gap topology as the genuine bulk predictor for edge/corner phenomena.

The ability to create, annihilate, or move skin-effect edges in the complex plane, and to realize protected regimes robust to moderate disorder, provides a platform for controllable directional transport, robust localization, and applications in wave manipulation, lasing, and synthetic quantum materials.

The concept extends far beyond single-band one-dimensional chains, enabling the study of multi-band, multi-component, and higher-dimensional non-Hermitian systems where intricate spectral and spatial localization patterns—such as quadripolar and symmetry-protected skin effects—can be realized and manipulated (Zeng, 2022, Rafi-Ul-Islam et al., 2023, Li et al., 4 Aug 2025, Jiang et al., 2024, Tozar, 2 Dec 2025, Yang et al., 19 Jan 2026).