Topological nature of edge states for one-dimensional systems without symmetry protection (2412.10526v2)

Abstract: We numerically verify and analytically prove a winding number invariant that correctly predicts the number of edge states in one-dimensional, nearest-neighbor (between unit cells), two-band models with any complex couplings and open boundaries. Our winding number uses analytical continuation of the wave-vector into the complex plane and involves two special points on the full Riemann surface band structure that correspond to bulk eigenvector degeneracies. Our winding number is invariant under unitary or similarity transforms. We emphasize that the topological criteria we propose here differ from what is traditionally defined as a topological or trivial phase in symmetry-protected classification studies. It is a broader invariant for our model that supports nonzero energy edge states and its transition does not coincide with the gap closing condition. When the relevant symmetries are applied, our invariant reduces to well-known Hermitian and non-Hermitian symmetry-protected topological invariants.

• The paper introduces a novel bulk-derived topological invariant that determines the existence of edge states without relying on symmetry constraints.
• It employs a two-band tight-binding model with complex, nearest-neighbor couplings to analytically and numerically validate the emergence of asymmetric edge modes.
• Its findings offer practical insights for experimental platforms such as photonic and topolectrical systems by quantifying edge state robustness in non-Hermitian setups.

Topological Edge States in 1D Systems Without Symmetry Protection

The paper "Topological nature of edge states for one-dimensional systems without symmetry protection" (2412.10526) presents a detailed theoretical and numerical analysis of edge states in generic 1D, two-band systems with complex, nearest-neighbor couplings. An outstanding aspect of this work is the establishment of a new topological invariant for the existence of edge states, derived purely from bulk quantities, applicable even in the absence of conventional symmetry protection.

Context and Motivation

Traditionally, topological edge states in one-dimensional systems are classified within the Altland-Zirnbauer (10-fold way) or the extended 38-fold way for non-Hermitian systems, relying heavily on underlying symmetries such as chiral, time-reversal, or particle-hole. In these frameworks, topological invariants—Berry phases, winding numbers—determine the presence or absence of edge-localized states, with robust bulk-edge correspondence. However, many realistic models either lack such symmetries due to parameter asymmetries or deliberate design, especially in engineered systems (e.g., photonic or circuit platforms).

This work addresses the existence and stability of edge modes in asymmetric 1D systems—Hermitian or non-Hermitian—where translational invariance in the bulk is assumed, but no further symmetry protection is imposed. Notably, robust edge-localized states are observed in such systems, motivating a generalized, symmetry-agnostic framework.

Model and Analytical Framework

The authors consider the canonical 1D two-band tight-binding model with arbitrary complex couplings:

H(z) = [ t_aa,0 + t_aa,-1/z + t_aa,1*z t_ab,0 + t_ab,-1/z + t_ab,1*z ] [ t_ba,0 + t_ba,-1/z + t_ba,1*z t_bb,0 + t_bb,-1/z + t_bb,1*z ]

with $z = e^{ik}$, and $k$ generally complex.

Unlike the canonical SSH model, the model allows fully asymmetric Hermitian or non-Hermitian couplings. The open boundary condition is enforced as vanishing amplitudes outside the finite chain. The authors employ the generalized Bloch theorem to construct solutions in terms of $p$ roots ($p=4$ for this model) of the characteristic equation $\det[H(z)-E\,I]=0$.

Edge states correspond to isolated eigenvalues away from the continuum formed by the bulk spectrum, with their spatial profile determined by the magnitude ordering of $|z_i|$. The key technical insight is to analyze the eigenvector structure via the ratio $M(z,E) = a(z)/b(z)$, which encodes the internal “polarization” of the Bloch wave. By mapping $M$ over the Brillouin zone or generalized Brillouin zone (GBZ), robust information about edge state existence is extracted.

Topological Invariant for Edge States without Symmetry

Eigenvector Degeneracy and the $M$-Riemann Sphere

A central result is the identification of bulk eigenvector degeneracy points in $M$, i.e., values where two $z_i$ solutions at a given $E$ yield the same $M$ (but distinct $z$), denoted $M_{\mathrm{deg}}$. These correspond geometrically to two sheets of the $(z,E)$ band structure Riemann surface touching in the $M$-Riemann mapping.

The authors analytically derive the condition for these points, which in general are two per system (for nearest-neighbor models). Importantly, they prove that the existence and number (0, 1, or 2) of edge states is topologically determined by the winding of mapped bulk states around these $M_{\mathrm{deg}}$ points.

Generalized Winding Number

The general invariant is constructed as an integer count of how many $M_{\mathrm{deg}}$ points are enclosed by the loop traced by $M(\mathcal{C}_{\mathrm{GBZ}})$ on the $M$-plane or $M$-Riemann sphere. For Hermitian models, this reduces (modulo 2) to a simplified form over the ordinary Brillouin zone; for generic non-Hermitian systems, the GBZ must be used, and nuances of branch points and contour orientation are critical.

The explicit formula:

W_j = (1 + (1/2πi) ∮ dM (d/dM) ln [ (M - M_deg,j)/(M - M_branch) ] ) mod 2

ensures generality and robustness under all $z$-independent unitary and similarity transformations of $H$.

Key Properties

• Independence from symmetry: The invariant is well-defined even when no chiral, sublattice, or other protecting symmetry exists.
• Reduction to known invariants: In the presence of relevant symmetries, the invariant collapses to well-known Berry or non-Hermitian winding invariants.
• Non-coincidence with gap closing: The emergence or annihilation of edge states, as determined by this invariant, does not necessarily coincide with the closure of the bulk energy gap, in contrast to the usual symmetry-protected paradigm.

Numerical Validation and Analytical Proof

Comprehensive numerical analyses are provided. Sweeping over model parameters (linear interpolation between random asymmetric and SSH-type models), the authors compute both:

• The topological invariant defined above, purely from bulk (GBZ) quantities.
• The direct analytical edge state criterion via examination of OBC eigenvalues and eigenvectors.

The phase diagrams reveal perfect correspondence, with the invariant precisely predicting the number of edge states in all cases, modulo minor finite-size effects.

Analytically, the construction is shown to be equivalent to tracking the crossing of $M_{\mathrm{deg}}$ points through the GBZ or BZ in parameter space. This formalizes edge state emergence as a topological transition in the bulk eigenvector structure, rooted in algebraic geometry of the band equations.

Implications and Outlook

Practical and Theoretical Impact

• Bulk Calculability: The invariant depends only on bulk quantities—no need for explicit calculation of edge states or Green's functions of a truncated system.
• Experimental Design: This provides a powerful characterization tool in photonic, circuit, or cold atom platforms where asymmetric couplings are accessible and symmetries may be absent or broken by design.
• Edge State Robustness: The structure of the invariant allows a quantitative measure of the “distance” to edge state annihilation or creation, and thus the robustness of edge-localized phenomena to perturbations.
• Generalization to Higher Models: The approach can be extended to models with longer-range coupling, continuous systems, or higher dimensions, provided a Riemann surface (or similar algebraic structure) can be constructed for the bulk band equations.

Open Questions and Future Research

Several promising directions arise:

• Physical realization: Systematic exploration in platforms with tunable asymmetry, particularly in tailored photonic or topolectrical lattice systems.
• Topological protection in the absence of symmetries: Further investigation into the stability of these states under disorder, interactions, or coupling to environments.
• Relation to non-Hermitian topology: A deeper synthesis with recent work on point-gap topology, braiding, and spectral flow in non-Hermitian systems.
• Algorithmic tools: Development of automated methods to compute these invariants for arbitrary systems, with relevance to materials search and device engineering.

Summary Table: Comparison with Conventional Topological Phases

	Symmetry-Protected (usual)	This Work: No Symmetry
Symmetry	Chiral/sub. lattice/…	None (only transl. invariance)
Protected state	Zero-energy edge states	Generic energy edge states
Invariant	Berry phase, SSH winding	$M$-Riemann winding
Derived from	Bulk Bloch eigenvectors, BZ	Bulk eigenvector map, GBZ
Gap closing at transition	Yes	Not necessarily
Dependence on boundaries	No	No
Applicability	SSH, Majorana chains, etc.	Generic 2-band 1D systems

Conclusion

This paper solidifies a foundational understanding of edge states in 1D systems with arbitrary couplings and no symmetry protection. By connecting edge state existence to the global algebraic topology of the bulk band structure, it bridges a major gap between conventional symmetry-based classification and the practical observation of robust edge phenomena in asymmetric systems. This framework is poised to become a reference in future studies of engineered quasi-1D systems and non-Hermitian topological matter.