Concurrent Bipolar Skin Effects (CBSE)

• CBSE is a phenomenon in non-Hermitian systems where a single eigenstate simultaneously localizes at opposite boundaries via sector-resolved accumulation.
• It leverages point-gap topology and generalized Brillouin zone techniques to diagnose finite-size effects and effective nonreciprocity across different sectors.
• CBSE insights inform experimental designs in photonics and spinful systems, highlighting the interplay between spectral topology and boundary localization.

Concurrent Bipolar Skin Effects (CBSE) denote a non-Hermitian boundary-accumulation regime in which opposite skin tendencies coexist concurrently rather than collapsing into a single global skin direction. In the most explicit usage, CBSE refers to a finite-size two-chain system in which a single open-boundary eigenstate is simultaneously localized at opposite boundaries on the two chains (Li et al., 4 Aug 2025). Closely related literatures use different names for nearby phenomena, including bipolar Floquet NHSE, reciprocal skin effect, helical spin skin effect, and symmetry-protected $\mathbb{Z}_2$ skin effect, but they share the core motif that distinct sectors—chains, momentum channels, quasienergy branches, spin sectors, or Kramers partners—accumulate at opposite boundaries under the same control parameters (Lin et al., 2024, Hofmann et al., 2019, Okuma et al., 2019).

1. Terminology and defining scope

The term CBSE is explicitly introduced for two weakly coupled non-reciprocal chains, where a state with eigenenergy in a $W=0$ region is “simultaneously localized at opposite boundaries of the two chains” (Li et al., 4 Aug 2025). In that formulation, “concurrent” means that the opposite localizations occur within the same eigenstate, and “bipolar” means that the two chain components accumulate at opposite ends.

Other papers study closely related but not identical phenomena under different labels. In the Floquet silicon-photonics literature, the “bipolar NHSE” denotes a phase in which “about half of the eigenstates” localize toward the left boundary and the other half toward the right at the same parameter point; the coexistence is therefore spectrally partitioned across quasienergy branches rather than chain-resolved within one state (Lin et al., 2024). In reciprocal two-dimensional systems, the “reciprocal skin effect” refers to the fact that modes at $+k_y$ and $-k_y$ localize on opposite transverse edges under the same open boundary condition (Hofmann et al., 2019). In symmetry-protected settings, a non-Hermitian $\mathbb{Z}_2$ skin effect produces Kramers-partner accumulation at opposite boundaries (Okuma et al., 2019). In gauge-field-induced spinful models, the “helical spin skin effect” means that spin-up and spin-down sectors accumulate on opposite edges or opposite corners, with hybrid-order and second-order variants (Li et al., 25 Apr 2025).

Taken together, these usages suggest that CBSE is best understood as a family of concurrent opposite-boundary skin phenomena whose sector label may be chain, band, momentum, spin, chirality, or symmetry partner. The terminology is therefore not universal, but the organizing structure is.

2. Spectral topology and non-Bloch structure

The basic topological language is point-gap topology. For a one-dimensional non-Hermitian Bloch Hamiltonian, the point-gap winding is

$$W(E)=\int_{0}^{2\pi}\frac{dk}{2\pi i}\frac{d}{dk}\log\det\!\left(H(k)-E\right),$$

and the central claim of the point-gap framework is that the skin effect originates from this intrinsic non-Hermitian topology rather than from boundary asymmetry alone (Okuma et al., 2019). The same spectral logic appears in Floquet form as

$$W(E)=\frac{1}{2\pi i}\oint_{\mathrm{BZ}}\partial_k \ln \det\!\left[H_F(k)-E\right]\, dk,$$

where the winding of complex quasienergy loops determines skin direction and its reversal (Lin et al., 2024). In photonic higher-order systems, the corresponding point-gap invariant is written as

$$W(k_x,f_{0})=\oint_{0}^{2\pi}\frac{d k_y}{2\pi i}\frac{\partial}{\partial k_y}\log \det [H(k_x,k_y)-f_{0}],$$

so the sign of spectral winding along a projected momentum cut fixes which boundary receives the skin modes (Yang et al., 19 Jan 2026).

A complementary non-Bloch description is given by generalized Brillouin-zone theory. In the explicit CBSE two-chain model, the periodic factor $e^{ik}$ is replaced by $\beta$, producing

$W=0$0

with characteristic equation $W=0$1 and GBZ condition $W=0$2 (Li et al., 4 Aug 2025). Localization is then controlled by whether the relevant $W=0$3-roots lie inside or outside the unit circle. In long-range unidirectional models, the same logic yields twisted spectral loops with opposite windings and corresponding GBZ sectors inside and outside $W=0$4, which directly produces opposite-edge localization channels (Rafi-Ul-Islam et al., 2023).

A real-space reformulation is proposed in the generalized NHSE framework,

$W=0$5

with eigenfunction

$W=0$6

There the localization factor is governed by $W=0$7, where $W=0$8 (Wei et al., 15 May 2025). This suggests a natural real-space language for CBSE: if different sectors carry opposite effective $W=0$9, the same inhomogeneous non-Hermitian profile can drive them to opposite boundaries.

3. Canonical finite-size CBSE in two coupled non-reciprocal chains

The most explicit CBSE model consists of two non-reciprocal chains $+k_y$0 and $+k_y$1 coupled by an onsite inter-chain amplitude $+k_y$2,

$+k_y$3

with $+k_y$4, $+k_y$5, and representative choices $+k_y$6, $+k_y$7, $+k_y$8 (Li et al., 4 Aug 2025). Under PBC the spectrum is determined by

$+k_y$9

where

$-k_y$0

$-k_y$1

Varying $-k_y$2 relative to $-k_y$3 produces nested, tangent, or intersecting complex loops.

The operational CBSE diagnostics are chain-resolved density imbalances

$-k_y$4

with $-k_y$5, together with

$-k_y$6

Extended states satisfy $-k_y$7 and $-k_y$8. CBSE is identified by $-k_y$9 and $\mathbb{Z}_2$0, meaning opposite chain polarizations within the same state. Unipolar NHSE instead has finite $\mathbb{Z}_2$1 and $\mathbb{Z}_2$2 (Li et al., 4 Aug 2025).

The key claim is that CBSE occupies a nominally trivial $\mathbb{Z}_2$3 region of the PBC point-gap spectrum, but only at finite size. For nested loops with $\mathbb{Z}_2$4, small systems show CBSE on the central OBC loop, whereas increasing $\mathbb{Z}_2$5 causes the OBC spectrum to expand into the $\mathbb{Z}_2$6 region and the CBSE sector disappears at about $\mathbb{Z}_2$7 for $\mathbb{Z}_2$8. For tangent loops with $\mathbb{Z}_2$9, the $$W(E)=\int_{0}^{2\pi}\frac{dk}{2\pi i}\frac{d}{dk}\log\det\!\left(H(k)-E\right),$$0 window shrinks much more slowly, with

$$W(E)=\int_{0}^{2\pi}\frac{dk}{2\pi i}\frac{d}{dk}\log\det\!\left(H(k)-E\right),$$1

so CBSE vanishes only asymptotically. For intersecting loops with $$W(E)=\int_{0}^{2\pi}\frac{dk}{2\pi i}\frac{d}{dk}\log\det\!\left(H(k)-E\right),$$2, finite-size OBC states first exhibit CBSE in the central $$W(E)=\int_{0}^{2\pi}\frac{dk}{2\pi i}\frac{d}{dk}\log\det\!\left(H(k)-E\right),$$3 region, then coexist with conventional bipolar NHSE when $$W(E)=\int_{0}^{2\pi}\frac{dk}{2\pi i}\frac{d}{dk}\log\det\!\left(H(k)-E\right),$$4 sectors appear, and finally lose the $$W(E)=\int_{0}^{2\pi}\frac{dk}{2\pi i}\frac{d}{dk}\log\det\!\left(H(k)-E\right),$$5 sector entirely in the thermodynamic limit. The fitted boundaries are

$$W(E)=\int_{0}^{2\pi}\frac{dk}{2\pi i}\frac{d}{dk}\log\det\!\left(H(k)-E\right),$$6

The result is a strict separation between CBSE and conventional bipolar NHSE: the former is a finite-size chain-resolved compromise in a $$W(E)=\int_{0}^{2\pi}\frac{dk}{2\pi i}\frac{d}{dk}\log\det\!\left(H(k)-E\right),$$7 sector, while the latter is an asymptotic $$W(E)=\int_{0}^{2\pi}\frac{dk}{2\pi i}\frac{d}{dk}\log\det\!\left(H(k)-E\right),$$8 partition into left- and right-skin states (Li et al., 4 Aug 2025).

4. Bipolar and concurrent variants beyond the canonical model

A direct experimental analogue appears in Floquet silicon photonics. There, a three-sublattice driven lattice with loss on sublattice $$W(E)=\int_{0}^{2\pi}\frac{dk}{2\pi i}\frac{d}{dk}\log\det\!\left(H(k)-E\right),$$9,

$$W(E)=\frac{1}{2\pi i}\oint_{\mathrm{BZ}}\partial_k \ln \det\!\left[H_F(k)-E\right]\, dk,$$0

and gauge-modulated couplings

$$W(E)=\frac{1}{2\pi i}\oint_{\mathrm{BZ}}\partial_k \ln \det\!\left[H_F(k)-E\right]\, dk,$$1

undergoes a topological transition from left-unipolar NHSE for $$W(E)=\frac{1}{2\pi i}\oint_{\mathrm{BZ}}\partial_k \ln \det\!\left[H_F(k)-E\right]\, dk,$$2, to bipolar NHSE for $$W(E)=\frac{1}{2\pi i}\oint_{\mathrm{BZ}}\partial_k \ln \det\!\left[H_F(k)-E\right]\, dk,$$3, to right-unipolar NHSE for $$W(E)=\frac{1}{2\pi i}\oint_{\mathrm{BZ}}\partial_k \ln \det\!\left[H_F(k)-E\right]\, dk,$$4 (Lin et al., 2024). In the bipolar phase, “about half of the eigenstates” localize at the left boundary and the other half at the right, with the transition traced to a change from isolated loops with the same winding to twisted or linked loops with opposite windings. Experimentally, the bipolar phase is observed at $$W(E)=\frac{1}{2\pi i}\oint_{\mathrm{BZ}}\partial_k \ln \det\!\left[H_F(k)-E\right]\, dk,$$5, where light injected near the center evolves toward both ends.

A distinct one-dimensional route uses long-range unidirectional hopping. In the generalized Hatano–Nelson chain

$$W(E)=\frac{1}{2\pi i}\oint_{\mathrm{BZ}}\partial_k \ln \det\!\left[H_F(k)-E\right]\, dk,$$6

the extra $$W(E)=\frac{1}{2\pi i}\oint_{\mathrm{BZ}}\partial_k \ln \det\!\left[H_F(k)-E\right]\, dk,$$7 term produces self-intersecting twisted loops with opposite winding signs. The corresponding OBC characteristic equation,

$$W(E)=\frac{1}{2\pi i}\oint_{\mathrm{BZ}}\partial_k \ln \det\!\left[H_F(k)-E\right]\, dk,$$8

has GBZ branches inside and outside the unit circle, so some eigenstates localize at the left edge and others at the right edge within the same open chain (Rafi-Ul-Islam et al., 2023). A related two-subchain model with nonconservative next-nearest-neighbor couplings,

$$W(E)=\frac{1}{2\pi i}\oint_{\mathrm{BZ}}\partial_k \ln \det\!\left[H_F(k)-E\right]\, dk,$$9

supports identical, opposite, and twisted winding phases, which map respectively to unipolar or bipolar NHSE, with opposite subchain-resolved transmission directions in the bipolar regime (Kong et al., 2024).

Spinful and higher-order variants generalize the same structure. A gauge-field-induced helical spin skin effect is realized in a bilayer reciprocal-dissipative lattice,

$$W(k_x,f_{0})=\oint_{0}^{2\pi}\frac{d k_y}{2\pi i}\frac{\partial}{\partial k_y}\log \det [H(k_x,k_y)-f_{0}],$$0

where gauge fields plus reciprocal dissipative couplings produce spin-resolved opposite-edge accumulation without on-site gain/loss or explicit asymmetric hopping (Li et al., 25 Apr 2025). In $$W(k_x,f_{0})=\oint_{0}^{2\pi}\frac{d k_y}{2\pi i}\frac{\partial}{\partial k_y}\log \det [H(k_x,k_y)-f_{0}],$$1, bulk modes show first-order opposite-edge skin accumulation while edge modes collapse to opposite corners, giving a hybrid-order concurrent structure; at $$W(k_x,f_{0})=\oint_{0}^{2\pi}\frac{d k_y}{2\pi i}\frac{\partial}{\partial k_y}\log \det [H(k_x,k_y)-f_{0}],$$2, only the second-order helical spin skin effect remains. In a photonic kagome crystal with balanced gain and loss, the point-gap winding similarly produces momentum-resolved opposite-edge localization in ribbon geometry and corner-resolved bipolar NHSE in full OBC, with bulk-state groups $$W(k_x,f_{0})=\oint_{0}^{2\pi}\frac{d k_y}{2\pi i}\frac{\partial}{\partial k_y}\log \det [H(k_x,k_y)-f_{0}],$$3 and $$W(k_x,f_{0})=\oint_{0}^{2\pi}\frac{d k_y}{2\pi i}\frac{\partial}{\partial k_y}\log \det [H(k_x,k_y)-f_{0}],$$4 accumulating at lower-left and upper-right corners, respectively (Yang et al., 19 Jan 2026). A related topological construction, the $$W(k_x,f_{0})=\oint_{0}^{2\pi}\frac{d k_y}{2\pi i}\frac{\partial}{\partial k_y}\log \det [H(k_x,k_y)-f_{0}],$$5 bi-directional skin-effect model

$$W(k_x,f_{0})=\oint_{0}^{2\pi}\frac{d k_y}{2\pi i}\frac{\partial}{\partial k_y}\log \det [H(k_x,k_y)-f_{0}],$$6

places one chirality on the top surface and the other on the bottom, which is close to CBSE in the sense of concurrent opposite-surface accumulation of distinct topological sectors (Ma et al., 2020).

5. Reciprocity, symmetry, geometry, and disorder

Concurrent opposite-boundary accumulation does not require microscopic nonreciprocity in the narrow Hatano–Nelson sense. In the reciprocal skin effect, a full two-dimensional reciprocal non-Hermitian model yields an effective one-dimensional nonreciprocal problem on each fixed-$$W(k_x,f_{0})=\oint_{0}^{2\pi}\frac{d k_y}{2\pi i}\frac{\partial}{\partial k_y}\log \det [H(k_x,k_y)-f_{0}],$$7 slice. The inverse decay length is

$$W(k_x,f_{0})=\oint_{0}^{2\pi}\frac{d k_y}{2\pi i}\frac{\partial}{\partial k_y}\log \det [H(k_x,k_y)-f_{0}],$$8

so

$$W(k_x,f_{0})=\oint_{0}^{2\pi}\frac{d k_y}{2\pi i}\frac{\partial}{\partial k_y}\log \det [H(k_x,k_y)-f_{0}],$$9

Hence $e^{ik}$0 and $e^{ik}$1 sectors localize on opposite transverse edges under the same OBC geometry (Hofmann et al., 2019). This is not chain-resolved CBSE in the strict finite-size two-chain sense, but it is a momentum-resolved bipolar precursor in a globally reciprocal system.

Geometry can also substitute for microscopic asymmetry. In a reciprocal two-dimensional photonic crystal for $e^{ik}$2 polarization, nonzero order-2 exceptional-point winding, spectral-area formation, and projected point-gap topology yield skin accumulation only for selected oblique interfaces. The guiding design sequence is

$e^{ik}$3

and the working criterion is

$e^{ik}$4

The realized effect is geometry-selected and unipolar rather than CBSE proper, but it demonstrates that reciprocal higher-dimensional systems can encode skin localization into boundary orientation and symmetry mismatch (Fang et al., 2022).

Symmetry protection can stabilize a genuinely bipolar spin-resolved skin phase. In the disordered non-Hermitian Rashba chain

$e^{ik}$5

with

$e^{ik}$6

$e^{ik}$7

spin-up modes localize at one boundary and spin-down modes at the other in a $e^{ik}$8 topological bipolar skin phase protected by

$e^{ik}$9

The disorder-driven sequence is

$\beta$0

with representative thresholds $\beta$1 and $\beta$2 at $\beta$3 (Tozar, 2 Dec 2025). This establishes that concurrent opposite-edge localization can be disorder-robust, while also showing that its topological protection can fail before skin accumulation itself disappears.

6. Diagnostics, limitations, and open directions

Across the literature, CBSE and related bipolar skin phenomena are diagnosed by a combination of spectral, spatial, and transport observables. The canonical two-chain CBSE model uses the chain-resolved imbalance pair $\beta$4 and explicit $\beta$5 assignment of each OBC eigenvalue to the surrounding PBC point-gap sector (Li et al., 4 Aug 2025). Floquet and photonic works use PBC–OBC loop collapse, field profiles, and time-domain migration toward one or both boundaries (Lin et al., 2024, Yang et al., 19 Jan 2026). Reciprocal and circuit realizations rely on momentum-resolved eigenspectra, Fourier reconstruction, or direct impedance-matrix diagonalization (Hofmann et al., 2019, Li et al., 25 Apr 2025). Spin-protected bipolar phases add biorthogonal observables such as

$\beta$6

the spin-separation index $\beta$7, and Lyapunov exponent $\beta$8, thereby separating topological bipolar skin phases from trivial skin and Anderson-localized regimes (Tozar, 2 Dec 2025).

Several limitations recur. First, the term CBSE is not universal: many authors instead describe the same structural idea as bipolar NHSE, reciprocal skin effect, helical spin skin effect, or $\beta$9 skin effect. Second, many demonstrations are sector-resolved rather than fully spectrum-wide. The Floquet bipolar phase is established primarily in band set I under the chosen excitation protocol, not as a statement that every quasienergy sector is equally bipolar (Lin et al., 2024). The reciprocal skin effect is momentum-partitioned, with special reciprocal momenta $W=0$00 remaining delocalized (Hofmann et al., 2019). The reciprocal photonic geometry-dependent skin effect is interface-selective and best described as unipolar rather than CBSE (Fang et al., 2022). Third, the explicit CBSE model of two weakly coupled chains is finite-size and unstable: as $W=0$01 grows, its $W=0$02 CBSE region is expelled into $W=0$03 sectors and crosses over to unipolar or conventional bipolar NHSE (Li et al., 4 Aug 2025).

Several adjacent directions indicate how the subject may broaden. Interaction-induced higher-order NHSE shows that doublon sectors can acquire corner skin accumulation even when the single-particle system has no such effect; this suggests that interactions can generate new sector-selective skin channels, although the work does not realize CBSE directly (Ling et al., 12 Jan 2025). Möbius-boundary ladders exhibit “concurrent skin-scale-free localization,” where one chain shows NHSE and the other scale-free localization, again indicating that concurrent but nonidentical boundary accumulations are possible in coupled non-Hermitian subsystems (Long et al., 8 Jul 2025). The generalized NHSE framework classifies such mode-dependent opposite localization as relative skin effect rather than global skin effect, which suggests a broader real-space taxonomy in which CBSE is one member of a larger family of sector-resolved non-Hermitian boundary accumulations (Wei et al., 15 May 2025).

In this broader view, CBSE is not a single universal phase but a structured class of non-Hermitian phenomena in which opposite-boundary localization survives concurrently across internal sectors. Its precise realization may be chain-resolved, branch-resolved, momentum-resolved, spin-resolved, symmetry-protected, geometry-selected, or higher-order; what unifies these cases is the coexistence of opposite skin channels within one open system, and the fact that their origin is most naturally described through point-gap topology, non-Bloch spectral selection, and sector-dependent effective nonreciprocity.