Flat band and Bulk-Boundary correspondence in a non-Hermitian trimerized lattice model with generic boundary conditions

Abstract: We consider a Su-Schrieffer-Heeger(SSH)-type trimer model with next-nearest-neighbor(NNN) interaction and balanced loss-gain(BLG) to study the combined effect of lattice symmetries, topology, non-hermiticity and general boundary conditions(GBC)on the existence of flat band and the nature of Bulk-Boundary correspondence(BBC). We derive the necessary and sufficient conditions for the existence of an entirely real spectrum under the periodic boundary condition(PBC). The exact expressions for the compact localized states(CLS) and energy eigenvalues corresponding to flat bands are derived analytically under the PBC. We establish topological phase transitions(TPT) for PT-symmetry and pseudo-chiral symmetry through the computation of the Zak phase and sub-lattice Zak phase, respectively. The Hamiltonian under the open boundary condition(OBC) is studied numerically, and edge states are observed in the topologically non-trivial phase, thereby establishing the non-hermitian BBC. The CLS exists in both bulk and the boundary for systems having only pseudo-chiral symmetry, and an additional PT-symmetry destroys the CLS at the boundary. We generalize a known formalism to study the same Hamiltonian under GBC, and derive analytic expressions for the energy and eigenstates for a class of boundary conditions in parametric ranges which admit flat band under the PBC. The edge states for these boundary conditions, including the OBC, are obtained analytically in the topologically non-trivial phase, thereby establishing BBC. The non-hermitian skin effect(NHSE) is seen in the model with reciprocal bulk interaction and strongly non-reciprocal boundary terms. The winding number based on spectral topology is computed analytically.

• The paper derives necessary and sufficient conditions for a real periodic-boundary spectrum, including a geometric characterization based on intersecting hyperboloids in coupling-parameter space.
• It demonstrates that next-nearest-neighbor hopping is essential for a nontrivial zero-energy flat band in the trimerized chain, with real dispersive partners when the coupling separation exceeds the effective gain-loss scale.
• It analytically solves selected general boundary conditions to obtain edge-state energies and wave functions, while showing that asymmetric boundaries can generate spectral winding and the non-Hermitian skin effect despite a reciprocal bulk.

This paper analyzes a one-dimensional SSH-type trimerized tight-binding chain augmented with next-nearest-neighbor (NNN) hopping and balanced loss-gain (BLG) onsite imaginary potentials, studied under periodic, open, and general boundary conditions (2608.17428). The central results are threefold: analytic necessary and sufficient conditions for an entirely real spectrum under PBC; the identification of a non-trivial zero-energy flat band whose existence requires NNN interactions (in contrast with the standard SSH dimer); and an exact treatment of the model under general boundary conditions (GBC), from which edge-state energies, wave functions, and the non-Hermitian skin effect (NHSE) are derived analytically.

Model and boundary conditions

The bulk Hamiltonian on $N = 3m$ sites comprises three pieces: the standard trimerized SSH chain $H_{NN}$ with intra-cell couplings $\delta_1, \delta_2$ and inter-cell coupling $\delta_3$; a Hermitian NNN term $H_{NNN}$ with amplitudes $t_1, t_2, t_3$ connecting non-adjacent sublattices; and a BLG term with imaginary onsite potentials $i\gamma_1, i\gamma_2, -i(\gamma_1+\gamma_2)$ on the $a, b, c$ sublattices. The boundary Hamiltonian $H_B$ contains six couplings $(t_{il}, t_{ir}, \delta_{3l}, \delta_{3r})$ and encodes a hierarchy of boundary conditions: OBC ($H_{NN}$0), hermitian GBC (HGBC) with $H_{NN}$1, twisted (TBC), PBC/APBC as special TBC cases, and anti-Hermitian boundary conditions (AHBC) with $H_{NN}$2. The solvability criterion developed later requires $H_{NN}$3 or $H_{NN}$4, which all boundary conditions except the AHBC satisfy.

Reality of the spectrum under PBC

The Bloch Hamiltonian $H_{NN}$5 is a $H_{NN}$6 non-Hermitian matrix. Requiring all coefficients of its characteristic polynomial to be real yields two necessary conditions, which exist only when the BLG terms are nonzero. Under these constraints the characteristic polynomial is a depressed cubic $H_{NN}$7, with $H_{NN}$8 expressible through the combination $H_{NN}$9 and the vectors $\delta_1, \delta_2$0. The authors give the reality condition a geometric form: the necessary conditions describe the intersection of two hyperboloids in the $\delta_1, \delta_2$1-space (or $\delta_1, \delta_2$2-space) centered at $\delta_1, \delta_2$3. Sufficient conditions follow from the inequality $\delta_1, \delta_2$4 plus a bound on $\delta_1, \delta_2$5; a simpler sufficient condition is derived using $\delta_1, \delta_2$6, which in the scaling limit $\delta_1, \delta_2$7 forces either $\delta_1, \delta_2$8 or $\delta_1, \delta_2$9. The authors concede that the full eight-parameter problem is not exhaustively analyzed; only physically motivated limits are treated.

Flat band

Setting $\delta_3$0 for all $\delta_3$1 yields a zero-energy flat band with dispersive partners $\delta_3$2, real whenever $\delta_3$3 (equality defining the exceptional surface). Two structural results stand out. First, a non-trivial flat band does not exist without NNN interactions: with $\delta_3$4 the flat-band conditions force $\delta_3$5, i.e. disconnected trimers — whereas in the SSH dimer the NNN coupling destroys the flat band. The trimer thus occupies an intermediate regime where NNN hopping is essential. Second, although the flat-band equations appear independent of $\delta_3$6, this is misleading: the constraints are solved subject to the reality conditions, whose solutions depend explicitly on the BLG strengths. Flat-band solutions split into Type-I ($\delta_3$7) and Type-II ($\delta_3$8, $\delta_3$9, $H_{NNN}$0), with explicit closed-form solutions and their reality domains in each case; notably, Type-II solutions exist only when the two $H_{NNN}$1 have specific sign structures (same sign for one case, opposite signs for others).

Topological phase transitions

The paper identifies three $H_{NNN}$2-symmetric regimes (with parity operators $H_{NNN}$3 exchanging sublattice pairs, or rotated variants $H_{NNN}$4) and three pseudo-chiral regimes $H_{NNN}$5. For pseudo-chiral-symmetric limits, the compact localized states (CLS) are obtained analytically — each spanning only five sites over two or three unit cells — along with exact eigenvalues $H_{NNN}$6, $H_{NNN}$7. Because the conventional Zak phase is inadequate in this limit, the authors employ the sub-lattice Zak phase, computed analytically via biorthogonal projected eigenstates. It takes values $H_{NNN}$8 or $H_{NNN}$9 and jumps precisely at the hopping-amplitude equality (e.g., $t_1, t_2, t_3$0 for $t_1, t_2, t_3$1), marking the topological phase transition. For $t_1, t_2, t_3$2-symmetric regimes, the Zak phase is computed numerically from right eigenstates; the loss-gain strength $t_1, t_2, t_3$3 does not affect its value. Edge states under OBC appear in the gap between lower and middle bands when the lower-band Zak phase is non-trivial, and in the upper gap when the summed lower-plus-middle Zak phase is non-trivial, establishing non-Hermitian bulk-boundary correspondence (BBC). A noteworthy observation under OBC in the $t_1, t_2, t_3$4 pseudo-chiral limit: the flat-band pair becomes purely imaginary and localizes at an edge even when the sub-lattice Zak phase is trivial — attributed to the interplay of the imaginary potential with the flat band, and flagged as distinct from the other two cases.

A further structural result concerns the coexistence of localized states: with pseudo-chiral symmetry alone, CLS exist both in the bulk and at the boundary; imposing an additional $t_1, t_2, t_3$5 symmetry destroys the boundary CLS. The paper also exploits a duality $t_1, t_2, t_3$6 mapping between lattice configurations, which transfers all analytic results between paired parameter regimes.

Exact solution under general boundary conditions

Using the ansatz $t_1, t_2, t_3$7, the bulk equations reduce to a generalized Bloch problem $t_1, t_2, t_3$8. For fixed $t_1, t_2, t_3$9, $i\gamma_1, i\gamma_2, -i(\gamma_1+\gamma_2)$0 satisfies a palindromic quartic reducible to a bi-quadratic form; the boundary determinant $i\gamma_1, i\gamma_2, -i(\gamma_1+\gamma_2)$1 then yields a trigonometric equation $i\gamma_1, i\gamma_2, -i(\gamma_1+\gamma_2)$2, solved exactly in several limits via Chebyshev polynomials of the second kind. The paper is careful to note that the $i\gamma_1, i\gamma_2, -i(\gamma_1+\gamma_2)$3 limits are singular for this quartic and must be treated from the outset.

Under OBC the boundary condition reduces to $i\gamma_1, i\gamma_2, -i(\gamma_1+\gamma_2)$4 with $i\gamma_1, i\gamma_2, -i(\gamma_1+\gamma_2)$5; for $i\gamma_1, i\gamma_2, -i(\gamma_1+\gamma_2)$6 one root $i\gamma_1, i\gamma_2, -i(\gamma_1+\gamma_2)$7 becomes complex, signaling an edge state with energy $i\gamma_1, i\gamma_2, -i(\gamma_1+\gamma_2)$8 and closed-form wave functions. In the thermodynamic limit $i\gamma_1, i\gamma_2, -i(\gamma_1+\gamma_2)$9 — purely imaginary for the non-Hermitian model, and zero in the Hermitian limit. This analytic result confirms the numerical BBC established earlier.

For the solvable class $a, b, c$0 (encompassing PBC, APBC, and strongly non-reciprocal boundary terms), the spectrum forms closed loops in the complex plane whose number changes from two to one at a transition point; the spectral winding number around a suitably chosen reference energy equals $a, b, c$1. The complex $a, b, c$2 implies all eigenstates localize at one boundary — the NHSE — arising here from a reciprocal bulk with strongly asymmetric boundary terms ($a, b, c$3, $a, b, c$4 in the numerical demonstration). The coexistence of spectral topology and NHSE establishes non-Hermitian BBC in this regime, and the results persist when the BLG terms vanish, the only difference being real versus purely imaginary edge-state energies. Under AHBC ($a, b, c$5), a pair of edge states exists when $a, b, c$6.

Limitations and open questions

Several caveats are stated in the paper. The reality analysis of the full eight-dimensional parameter space is not exhaustive; the sufficient condition fails when $a, b, c$7 and $a, b, c$8 are parallel or nearly aligned, and only limiting cases are solved analytically. The $a, b, c$9-symmetric regimes resist complete analytic treatment — Zak phases and OBC spectra there are obtained numerically, and closed-form edge-state solutions are available only in pseudo-chiral limits. The exact GBC solutions cover a restricted class of boundary conditions satisfying $H_B$0 or $H_B$1; the AHBC, while admitting edge states, falls outside the exact solvability scheme. Finally, the mechanism by which the imaginary onsite potential drives flat-band edge localization with a trivial sub-lattice Zak phase in the $H_B$2 case is offered only as a qualitative explanation, and the paper leaves open whether this coexistence of trivial topology and boundary-localized flat-band states is generic.

Conclusion

The paper provides an analytically controlled account of how lattice symmetry, topology, non-Hermiticity, and boundary conditions interact in a trimerized SSH chain. Its principal contributions are the hyperboloid-geometric characterization of spectral reality, the demonstration that NNN coupling is necessary for non-trivial flat bands in the trimer (opposite to the dimer case), the sub-lattice Zak phase as the correct topological invariant in pseudo-chiral regimes with analytic BBC, and an exact GBC formalism yielding closed-form edge states, spectral winding, and NHSE driven by boundary asymmetry alone. The results extend, as special cases, to the Hermitian trimer chain under GBC, which had not been solved previously.