Non-Hermitian SSH Chains

• Non-Hermitian SSH chains are one-dimensional dimerized lattices that extend the conventional SSH model by incorporating asymmetric hopping, gain/loss, or effective self-energies to generate complex spectra.
• They feature unique spectral characteristics such as point and line gaps, exceptional points, and non-Hermitian skin effects that dramatically alter edge, defect, and transport behaviors.
• Their analysis utilizes advanced topological invariants and generalized Brillouin zone techniques to address the breakdown of bulk-boundary correspondence in open-boundary systems.

Non-Hermitian Su–Schrieffer–Heeger (SSH) chains are one-dimensional dimerized lattices obtained by extending the bipartite SSH model to Hamiltonians with complex spectra. In these systems, non-Hermiticity is introduced by asymmetric hopping, alternating or modulated gain and loss, or effective self-energies generated by leads, substrates, or other reservoirs. The resulting bands can exhibit point gaps or line gaps, exceptional points, biorthogonal topology, non-Hermitian skin effects, and edge, defect, or boundary modes whose amplification, attenuation, or localization differs qualitatively from the Hermitian SSH limit (Edalatmanesh et al., 14 Jan 2025, Aquino et al., 2022, Ostahie et al., 2020). Because different realizations preserve different symmetries—chiral symmetry, $\mathcal{PT}$ symmetry, anti-$\mathcal{PT}$ symmetry, pseudo-Hermiticity, or inversion—the appropriate topological and dynamical description depends strongly on how non-Hermiticity is introduced (Slootman et al., 27 Apr 2025, Jangjan et al., 2024, Ye et al., 2023).

1. Canonical formulations and physical mechanisms

The standard SSH chain is a bipartite lattice with two sites $A,B$ per unit cell and alternating intracell and intercell hoppings. A common starting point is

$$H_{\mathrm{SSH}}=\sum_{n=1}^{N}\Big(t_1\,|n,A\rangle\langle n,B|+t_2\,|n,B\rangle\langle n+1,A|+\mathrm{h.c.}\Big),$$

for which the Hermitian limit is trivial when $|t_1|>|t_2|$ and topological when $|t_2|>|t_1|$; under open boundaries, the latter hosts the familiar pair of zero-energy edge modes (Edalatmanesh et al., 14 Jan 2025).

Non-Hermiticity enters this framework through several inequivalent mechanisms. In substrate- or lead-coupled chains, the SSH lattice acquires an effective self-energy that acts as an imaginary onsite term. In the substrate formulation,

$$H_{\mathrm{eff}}=H_{\mathrm{SSH}}-i\gamma\sum |m,n\rangle\langle m,n|,$$

so $\gamma>0$ describes loss or attenuation into the substrate and $\gamma<0$ describes gain; in lead-coupled chains the effective Hamiltonian acquires imaginary onsite terms at the two ends, again rendering the spectrum complex (Edalatmanesh et al., 14 Jan 2025, Ostahie et al., 2020). In other realizations, non-Hermiticity is intrinsic to the lattice couplings: asymmetric intracell hopping appears as $v\pm g$ in a minimal non-Hermitian SSH model, while staggered gain/loss appears as $\mathcal{PT}$0 on the two sublattices in $\mathcal{PT}$1-symmetric chains (Aquino et al., 2022, Slootman et al., 27 Apr 2025).

A large family of generalizations preserves the SSH dimerization motif while enlarging the internal structure. These include tetramerized chains with onsite pattern $\mathcal{PT}$2, periodically modulated hopping with an additional staggered imaginary potential, multipartite chains with $\mathcal{PT}$3 sublattice sites per unit cell, nonreciprocal ladders, and spinful chains dressed by SU(2) gauge matrices (Li et al., 2021, Mandal et al., 2024, Nehra et al., 2022, Zhou et al., 7 Feb 2025, Miao et al., 31 Mar 2025). The common feature is that the SSH connectivity survives, while the complex structure of the Hamiltonian alters spectral topology and boundary physics.

2. Complex spectra, gaps, and exceptional points

A defining distinction in non-Hermitian SSH chains is the type of complex spectral gap. In the asymmetric-hopping SSH chain with

$\mathcal{PT}$4

the winding number

$\mathcal{PT}$5

takes values $\mathcal{PT}$6. The region with $\mathcal{PT}$7 has a real line gap, the regions with $\mathcal{PT}$8 have a point gap, and the trivial regions with $\mathcal{PT}$9 may have either real or imaginary line gaps (Aquino et al., 2022). This half-integer structure has no Hermitian counterpart and is tied directly to exceptional-point encirclement.

Exceptional points organize much of the phase structure. In the same asymmetric-hopping model, the trajectory traced by the Bloch Hamiltonian in the $A,B$0 plane determines whether zero, one, or two exceptional points are encircled; encircling both gives $A,B$1, encircling only one gives $A,B$2, and encircling none gives $A,B$3 (Aquino et al., 2022). In substrate-coupled chains, a minimal two-site model with one lossy site has an exceptional point at $A,B$4, while in an extended SSH chain with one decoupled end site the numerically observed exceptional point is shifted to about $A,B$5, showing that boundary structure and chain connectivity renormalize the two-level condition (Edalatmanesh et al., 14 Jan 2025). In modulated four-sublattice chains, anti-$A,B$6 transitions also occur at exceptional points of edge states and can be separated from the conventional topological phase transition (Jangjan et al., 2024).

For chiral non-Hermitian SSH systems, the relevant bulk invariant is often written in terms of the off-diagonal blocks. In spinful SU(2)-gauged chains and in nonreciprocal ladders, the gauge-invariant winding number takes the form

$A,B$7

which remains well defined in the chiral non-Hermitian setting and diagnoses topological transitions together with gap closing or gap reconstruction (Miao et al., 31 Mar 2025, Zhou et al., 7 Feb 2025). In multipartite models, however, ordinary bandwise Zak phases can become ill defined because several bands may merge into composite loops that encircle one or multiple exceptional points. The appropriate invariant is then a composite nonadiabatic geometric phase attached to the full stitched loop rather than to a single band (Nehra et al., 2022).

3. Edge, defect, and boundary-localized states

The most immediate departure from the Hermitian SSH chain appears in the zero-energy sector. In substrate-coupled chains, uniform coupling shifts the entire spectrum vertically in the complex plane, but partial decoupling of one end produces much richer behavior. In the Hermitian-trivial regime $A,B$8, increasing $A,B$9 drives two states toward an exceptional point and then into one decaying and one amplifying zero-energy mode, both strongly localized near the decoupled end. In the Hermitian-topological regime $$H_{\mathrm{SSH}}=\sum_{n=1}^{N}\Big(t_1\,|n,A\rangle\langle n,B|+t_2\,|n,B\rangle\langle n+1,A|+\mathrm{h.c.}\Big),$$0, the edge mode attached to the substrate side is damped away while the opposite edge mode survives, yielding an asymmetric zero-energy “monomode.” A further parity effect appears when $$H_{\mathrm{SSH}}=\sum_{n=1}^{N}\Big(t_1\,|n,A\rangle\langle n,B|+t_2\,|n,B\rangle\langle n+1,A|+\mathrm{h.c.}\Big),$$1 terminal sites are decoupled: in Case 1, odd $$H_{\mathrm{SSH}}=\sum_{n=1}^{N}\Big(t_1\,|n,A\rangle\langle n,B|+t_2\,|n,B\rangle\langle n+1,A|+\mathrm{h.c.}\Big),$$2 induces a zero-energy boundary peak while even $$H_{\mathrm{SSH}}=\sum_{n=1}^{N}\Big(t_1\,|n,A\rangle\langle n,B|+t_2\,|n,B\rangle\langle n+1,A|+\mathrm{h.c.}\Big),$$3 suppresses it; in Case 2, the parity dependence reverses, while the monomode associated with the suspended segment remains robust regardless of parity (Edalatmanesh et al., 14 Jan 2025).

Interface and defect geometries introduce additional possibilities. In the complex SSH model with alternating gain and loss, the Hermitian topological mid-gap defect state becomes an $$H_{\mathrm{SSH}}=\sum_{n=1}^{N}\Big(t_1\,|n,A\rangle\langle n,B|+t_2\,|n,B\rangle\langle n+1,A|+\mathrm{h.c.}\Big),$$4-symmetric state with purely imaginary energy when it remains localized, but it can also disappear into the bulk continuum or undergo pairwise spontaneous $$H_{\mathrm{SSH}}=\sum_{n=1}^{N}\Big(t_1\,|n,A\rangle\langle n,B|+t_2\,|n,B\rangle\langle n+1,A|+\mathrm{h.c.}\Big),$$5-symmetry breaking. The resulting defect-state phase diagram contains regions with one, two, or three localized states, plus a critical line with none at all (Lang et al., 2018). This establishes that non-Hermiticity can destabilize the SSH zero mode not only by broadening it, but also by converting it into a symmetry-broken defect pair.

Other SSH extensions create new boundary states rather than merely modifying the conventional ones. Tetramerized chains with periodic imaginary onsite potentials extend the topologically nontrivial region, generate new edge states whose locality differs from the Hermitian SSH case, and can drive bulk states into a purely imaginary spectrum when the imaginary potentials are strong enough (Li et al., 2021). Cross-linked SSH chains, where several SSH segments meet at a single defect site, support defect-localized states in both trivial and topological regimes; after gain and loss are added, the topologically protected mid-gap state undergoes an abrupt transition from anti-localized to localized near the defect, and both localized and bulk states exhibit cascaded spatial symmetry breaking at exceptional points (Sivan et al., 2022).

4. Bulk-boundary correspondence and the non-Hermitian skin effect

In many non-Hermitian SSH chains, periodic-boundary and open-boundary spectra do not coincide. In the spinful SU(2)-gauged model, the periodic spectrum forms closed loops in the complex-energy plane, while the open-boundary spectrum deforms into open arcs; this mismatch is the characteristic signature of the non-Hermitian skin effect. The generalized Brillouin zone restores the correct open-boundary bulk description by replacing $$H_{\mathrm{SSH}}=\sum_{n=1}^{N}\Big(t_1\,|n,A\rangle\langle n,B|+t_2\,|n,B\rangle\langle n+1,A|+\mathrm{h.c.}\Big),$$6 with $$H_{\mathrm{SSH}}=\sum_{n=1}^{N}\Big(t_1\,|n,A\rangle\langle n,B|+t_2\,|n,B\rangle\langle n+1,A|+\mathrm{h.c.}\Big),$$7, solving

$$H_{\mathrm{SSH}}=\sum_{n=1}^{N}\Big(t_1\,|n,A\rangle\langle n,B|+t_2\,|n,B\rangle\langle n+1,A|+\mathrm{h.c.}\Big),$$8

and selecting the contour on which the relevant roots satisfy the modulus condition that defines the generalized Brillouin zone (Miao et al., 31 Mar 2025). An analogous reconstruction is required in the non-Hermitian SSH4 model, where the generalized Brillouin zone and the real-space winding number restore bulk-boundary correspondence that fails under naive Bloch theory (He et al., 2020).

The skin effect itself is not unique. The SU(2)-gauged SSH chain supports both unipolar and bipolar non-Hermitian skin effects, depending on the interplay of asymmetric hoppings and the gauge phases $$H_{\mathrm{SSH}}=\sum_{n=1}^{N}\Big(t_1\,|n,A\rangle\langle n,B|+t_2\,|n,B\rangle\langle n+1,A|+\mathrm{h.c.}\Big),$$9; twisted structures in the complex-energy loops are interpreted as signatures of the bipolar regime, in which different spectral branches localize at opposite edges (Miao et al., 31 Mar 2025). In a nonreciprocal SSH ladder built from two chains with opposite nonreciprocity, weak inter-leg coupling produces a critical skin effect rather than an ordinary one: the decay length $|t_1|>|t_2|$0 of the bulk skin modes varies with system size and is numerically verified to satisfy the scale-free law $|t_1|>|t_2|$1, while bulk-boundary correspondence is restored in the thermodynamic limit (Zhou et al., 7 Feb 2025). At the level of the elementary non-Hermitian building block, the unidirectional Hatano–Nelson chain exhibits a spectral winding transition $|t_1|>|t_2|$2 under binary disorder, and the nontrivial winding regime contains two completely delocalized states with diverging localization length (Ghosh et al., 12 Jan 2026).

A common misconception is that all non-Hermitian SSH chains display a skin effect. This is not so. The gain-and-loss $|t_1|>|t_2|$3-symmetric SSH chain studied thermodynamically has a line gap rather than a point gap and therefore does not display a non-Hermitian skin effect; its behavior under periodic and open boundaries differs mainly through the presence of topological edge modes (Slootman et al., 27 Apr 2025). Likewise, reciprocal chains with periodically modulated hopping and staggered imaginary onsite potentials have extended bulk states in generic parameter regimes, with localization of bulk states appearing only at the maximally dimerized limit $|t_1|>|t_2|$4 and with state-selective $|t_1|>|t_2|$5-breaking thresholds instead of conventional skin accumulation (Mandal et al., 2024).

5. Transport, quench dynamics, and thermodynamic response

Non-Hermitian SSH chains have been analyzed extensively as open transport systems. When a finite SSH chain is coupled to two semi-infinite leads, the effective Hamiltonian becomes non-Hermitian through imaginary onsite terms at the two ends. The resulting complex eigenvalues give the mid-gap edge states a finite lifetime, permit lead-induced mid-gap states in the non-topological regime at sufficiently strong coupling, and yield an exact zero-energy Landauer transmission formula in the topological regime,

$|t_1|>|t_2|$6

The transmission is non-monotonic in the chain-lead coupling and reaches a unitary maximum at a specific $|t_1|>|t_2|$7; notably, chiral disorder can enhance unitary conductance in the topological phase when the lead coupling is sufficiently strong (Ostahie et al., 2020).

Boundary engineering can also control transport in more local ways. In partially suspended substrate-coupled chains, zero-energy transmission peaks appear only for parity configurations that support a boundary state: in the $|t_1|>|t_2|$8 regime, strong coupling yields zero-energy transmission only for odd $|t_1|>|t_2|$9, while in the $|t_2|>|t_1|$0 regime the parity dependence reverses and even $|t_2|>|t_1|$1 enhances zero-energy transmission (Edalatmanesh et al., 14 Jan 2025). In quenched finite chains with an embedded non-Hermitian segment, transport from an initially localized topological edge state becomes reflection-asymmetric: for one range of the final dimerization $|t_2|>|t_1|$2, right reflection exceeds left reflection, while at larger $|t_2|>|t_1|$3 the asymmetry switches. The reported explanation is a partial reorganization of bulk states in both localization and energy near the critical or exceptional-point region (Ghosh et al., 2024).

Dynamical and thermodynamic phenomena go beyond stationary transport. In the SU(2)-gauged non-Hermitian SSH chain, non-Abelian gauge couplings enhance self-healing under a moving imaginary scattering potential: compared with the Abelian case, the long-time deviation is smaller and recovery of the original eigenstate is stronger, an effect attributed to skin modes with large imaginary eigenenergies and to gauge-induced tuning of spin-dependent nonreciprocity (Miao et al., 31 Mar 2025). In a different direction, the gain-and-loss $|t_2|>|t_1|$4-symmetric SSH chain supports an imaginary time crystal phase in the symmetry-broken regime. The key resonance condition is

$|t_2|>|t_1|$5

which allows the Green’s function to oscillate in imaginary time with Matsubara frequency. The effect is statistics-dependent: for bosons it requires a fully imaginary spectrum, while for fermions the choice $|t_2|>|t_1|$6 allows resonance earlier. Under open boundaries, topological edge states in the topological regime also exhibit such oscillatory behavior (Slootman et al., 27 Apr 2025).

6. Generalizations, diagnostics, and unresolved distinctions

A central lesson of the literature is that no single diagnostic captures all non-Hermitian SSH phases. In quantum-geometric treatments, the left-right biorthogonal quantum metric tensor is required to recover the full phase structure. In the pseudo-Hermitian SSH chain with real nonreciprocal hoppings, the metric singularities lie on the phase boundaries $|t_2|>|t_1|$7, while the left-left and right-right constructions contain only half of the topological information. In more general complex non-Hermitian SSH models, the quantum metric becomes pseudo-Riemannian or complex, may develop negative directions or null directions, and can exhibit dimensional reduction in the sense that one parameter direction carries zero metric weight and corresponds to zero excitation rate in linear response (Ye et al., 2023).

Other extensions show that the SSH framework remains useful even when the notion of a single two-band winding breaks down. In periodically driven non-Hermitian SSH chains, the bi-orthonormal geometric phase distinguishes trivial insulator, non-trivial insulator, and Möbius metallic phases, with the drive amplitude acting as a control parameter through the Bessel renormalization $|t_2|>|t_1|$8 in the high-frequency regime (Vyas et al., 2020). In multipartite non-Hermitian chains, several bands may merge into composite loops that encircle one or multiple exceptional points, and the appropriate invariant is a composite cyclic geometric phase rather than the ordinary band Zak phase (Nehra et al., 2022). In the non-Hermitian SSH3 model, non-Hermiticity itself creates a point-gap topology with winding $|t_2|>|t_1|$9, while the SSH4 model retains a more standard chiral topology but requires generalized Brillouin-zone or real-space formulations to recover bulk-boundary correspondence (He et al., 2020).

Quasiperiodic and extended models add a further layer of complexity. In non-Hermitian SSH chains with quasiperiodic onsite potentials, the localization-delocalization boundary becomes a mobility ring in the complex-energy plane, and increasing the mosaic period number produces multiple mobility rings (Li et al., 28 Jan 2026). In extended SSH chains with next-nearest-neighbor hopping and imaginary onsite energies, the transmission coefficient oscillates both with next-nearest-neighbor strength and with system size; balanced gain-loss and pure gain/loss lead to qualitatively different transport responses, including regimes with $$H_{\mathrm{eff}}=H_{\mathrm{SSH}}-i\gamma\sum |m,n\rangle\langle m,n|,$$0 in the $$H_{\mathrm{eff}}=H_{\mathrm{SSH}}-i\gamma\sum |m,n\rangle\langle m,n|,$$1-broken phase (Rahaman et al., 13 Jun 2026).

An important objective controversy concerns which invariant is physically decisive in a given model. The ordinary Berry phase can fail to capture anti-$$H_{\mathrm{eff}}=H_{\mathrm{SSH}}-i\gamma\sum |m,n\rangle\langle m,n|,$$2-protected edge states: in doubled-period non-Hermitian chains, edge states can persist even when the twisted-boundary-condition Berry phase is trivial, and parity-based invariants $$H_{\mathrm{eff}}=H_{\mathrm{SSH}}-i\gamma\sum |m,n\rangle\langle m,n|,$$3 and $$H_{\mathrm{eff}}=H_{\mathrm{SSH}}-i\gamma\sum |m,n\rangle\langle m,n|,$$4 are then used to characterize the edge spectrum (Jangjan et al., 2024). Likewise, ordinary single-band Zak phases fail in multipartite composite-loop phases, while left-right quantum geometry succeeds where left-left and right-right metrics do not (Nehra et al., 2022, Ye et al., 2023). This suggests that the non-Hermitian SSH family is best understood not as a single topological class, but as a set of symmetry- and gap-dependent problems in which point-gap, line-gap, parity, biorthogonal, and generalized-Brillouin-zone descriptions each have a sharply delimited domain of validity.