Topological phases in the non-Hermitian Su-Schrieffer-Heeger model (1709.03788v2)

Abstract: We address the conditions required for a $\mathbb{Z}$ topological classification in the most general form of the non-Hermitian Su-Schrieffer-Heeger (SSH) model. Any chirally-symmetric SSH model will possess a "conjugated-pseudo-Hermiticity" which we show is responsible for a quantized "complex" Berry phase. Consequently, we provide the first example where the complex Berry phase of a band is used as a quantized invariant to predict the existence of gapless edge modes in a non-Hermitian model. The chirally-broken, $PT$-symmetric model is studied; we suggest an explanation for why the topological invariant is a global property of the Hamiltonian. A geometrical picture is provided by examining eigenvector evolution on the Bloch sphere. We justify our analysis numerically and discuss relevant applications.

• The paper introduces the cBerry phase as a novel invariant that predicts gapless edge modes in non-Hermitian SSH systems.
• Analytical and numerical methods reveal that chiral symmetry and pseudo-anti-Hermiticity ensure robust topological transitions without inversion symmetry.
• Bloch sphere mapping and simulations validate the emergence of nontrivial topological phases, informing advances in photonic and plasmonic applications.

Topological Phases in the Non-Hermitian Su-Schrieffer-Heeger Model

The Su-Schrieffer-Heeger (SSH) model represents a canonical framework for understanding topological insulators in one dimension, particularly known for hosting robust edge states due to its topological nature. Lieu’s paper expands on this traditional paradigm by probing the non-Hermitian generalization of the SSH model, a direction of significant interest due to the relevance of non-Hermitian systems in describing physical scenarios with gain or loss, such as optical systems. The paper is tethered in the pursuit of identifying topological phases within non-Hermitian models and discerning their classification compared to Hermitian counterparts.

Central to Lieu’s exploration is the introduction and examination of the "complex" Berry phase (cBerry phase), which emerges as a fundamental invariant dictating the topological characteristics of non-Hermitian systems. This serves to be the first scholarly work that utilizes the cBerry phase to predict the emergence of gapless edge modes within a non-Hermitian framework. What makes the cBerry phase particularly intriguing is its association with a phenomenon termed “conjugated-pseudo-Hermiticity,” arising from chiral symmetry alone, an insight that hitherto was not fully appreciated in the literature.

Contrasting with Hermitian models, where the topological invariant is often directly associated with band structures, in non-Hermitian settings, especially those with broken chiral symmetry but preserved $PT$-symmetry, the invariant manifests as a global property of the system. This distinction is crucial and emerges from the pseudo-anti-Hermitian nature, which guarantees the edge modes' gaplessness in real energy, albeit not in imaginary energy.

The paper performs a careful analytical exploration across two model cases: a generally chiral but otherwise unrestricted SSH model, and a chirally-broken, $PT$-symmetric version. For the former, the quantization of the cBerry phase confirms the emergence of topological transitions independent of inversion symmetry—a stark contrast to its real-valued Berry phase counterpart, which fails to achieve quantization absent inversion symmetry. This implies that non-Hermitian mechanics uniquely preserve topological integrity in a broader set of conditions than Hermitian ones, provided the presence of chiral symmetry.

By employing a geometrical representation, the paper bolsters its findings with a Bloch sphere mapping, which visually and analytically encapsulates the winding properties of eigenvectors. This technique elucidates the $\mathbb{Z}$ classification by demonstrating how under non-Hermiticity, the chiral symmetry ensures that eigenstates wrap nontrivially around the Bloch sphere, from where the Berry phase quantization emerges naturally away from band crossings.

Furthermore, by exploring numerical simulations, the paper validates the theoretical predictions concerning gapless edge states, thereby substantiating the role of cBerry phase as an authentic topological invariant in non-Hermitian settings. Noteworthy is the successful demonstration of edge state existence in scenarios breaking inversion symmetry, corroborating analytical precepts.

The implications of this research are multifaceted: it provides a substantive advancement in the classification framework for non-Hermitian topological phases, suggesting chirality and pseudo-anti-Hermiticity as pivotal symmetries in such settings. This also introduces broader application potential, particularly in photonic and plasmonic systems where non-Hermitian effects are consequential. Future inquiries may focus on generalizing these findings to higher-dimensional systems and exploring equivalent phenomena in discrete-time quantum systems and other analogous models.

Lieu’s work offers a robust expansion of our theoretical toolkit, promising utility in designing next-generation optical devices and fostering deeper insights into topological quantum matter in non-Hermitian contexts. By securing a rigorous conceptual ground, it provides a springboard for further exploration into the intricacies of non-Hermitian topological classifications and applications.