[Paper Review] Quantum metric of non-Hermitian Su-Schrieffer-Heeger systems

Chao Chen Ye, W. L. Vleeshouwers+3 more|arXiv (Cornell University)|May 28, 2023

Quantum Mechanics and Non-Hermitian Physics51 references4 citations

TL;DR

This paper introduces a quantum geometric framework for non-Hermitian Su-Schrieffer-Heeger (SSH) models, demonstrating that the full topological phase diagram is captured only when the quantum metric tensor incorporates both left and right eigenvectors. It reveals that non-Hermiticity induces pseudo-Riemannian and complex quantum geometries, with topological phase transitions mathematically mirroring lightlike paths in general relativity and leading to a dimensional reduction in the quantum geometry, enabling zero-excitation responses to specific perturbations.

ABSTRACT

Topological insulators have been studied intensively over the last decades. Earlier research focused on Hermitian Hamiltonians, but recently, peculiar and interesting properties were found by introducing non-Hermiticity. In this work, we apply a quantum geometric approach to various Hermitian and non-Hermitian versions of the Su-Schrieffer-Heeger (SSH) model. We find that this method allows one to correctly identify different topological phases and topological phase transitions for all SSH models, but only when using the metric tensor containing both left and right eigenvectors. Whereas the quantum geometry of Hermitian systems is Riemannian, introducing non-Hermiticity leads to pseudo-Riemannian and complex geometries, thus significantly generalizing from the quantum geometries studied thus far. One remarkable example of this is the mathematical agreement between topological phase transition curves and lightlike paths in general relativity, suggesting a possibility of simulating space-times in non-Hermitian systems. We find that the metric in non-Hermitian phases degenerates in such a way that it effectively reduces the dimensionality of the quantum geometry by one. This implies that within linear response theory, one can perturb the system by a particular change of parameters while maintaining a zero excitation rate.

Motivation & Objective

• To extend quantum geometric analysis to non-Hermitian topological systems, particularly the SSH model, beyond the standard Hermitian framework.
• To identify the correct formulation of the quantum metric tensor (QMT) that fully captures topological phases and transitions in non-Hermitian systems.
• To explore how non-Hermiticity transforms quantum geometry from Riemannian to pseudo-Riemannian or complex structures.
• To establish a geometric correspondence between topological phase transitions and lightlike paths in general relativity.
• To demonstrate that the metric degeneracy in non-Hermitian phases reduces the effective dimensionality of quantum geometry, enabling zero excitation rates under specific perturbations.

Proposed method

• The study employs the full biorthogonal quantum geometric tensor (QGT), defined using both left and right eigenvectors of the non-Hermitian Hamiltonian, to compute the quantum metric tensor (QMT).
• The QMT is computed for various SSH models: Hermitian, pseudo-Hermitian, and non-reciprocal (NH-SSH-NR), and complex-hopping (NH-SSH-C) variants.
• The QMT is decomposed into components: $ g_{ ext{LR}} $, $ g_{ ext{LL}} $, and $ g_{ ext{RR}} $, to analyze the geometric structure and detect singularities signaling topological transitions.
• The parameter space is mapped using 3D visualizations of real and imaginary parts of the QMT components to identify phase boundaries and divergences.
• The geometric structure is analyzed in relation to general relativity, identifying when $ ds^2 = 0 $, corresponding to lightlike paths that align with topological phase transition curves.
• Finite-size effects are considered, and the role of phase terms in complex hopping amplitudes is analyzed to show their non-gaugable influence on the QMT.

Experimental results

Research questions

• RQ1 How does the quantum metric tensor in non-Hermitian systems differ from its Hermitian counterpart in capturing topological phases?
• RQ2 Can the full topological phase diagram of non-Hermitian SSH models be reconstructed using only the left or right eigenvector components of the QMT?
• RQ3 What geometric structure emerges in non-Hermitian systems—Riemannian, pseudo-Riemannian, or complex—when the full biorthogonal QGT is used?
• RQ4 Do topological phase transitions in non-Hermitian systems correspond to lightlike paths in the parameter space, as in general relativity?
• RQ5 How does metric degeneracy in non-Hermitian phases lead to a reduction in the effective dimensionality of quantum geometry and enable zero-excitation responses?

Key findings

• The full topological phase diagram of non-Hermitian SSH models is only correctly identified when the quantum metric tensor includes contributions from both left and right eigenvectors.
• Non-Hermiticity transforms the quantum geometry from Riemannian (Hermitian case) to pseudo-Riemannian or complex geometry, significantly generalizing the framework of quantum geometry.
• Topological phase transition curves in parameter space coincide mathematically with lightlike paths ($ ds^2 = 0 $) in the metric, suggesting a geometric duality with general relativity.
• The metric tensor degenerates in non-Hermitian phases, reducing the effective dimensionality of the quantum geometry by one, which enables a zero excitation rate under specific periodic perturbations.
• The $ LR $-component of the QMT contains the complete topological information, while $ LL $ and $ RR $ components each carry half, with divergences in opposite directions ($ +\infty $ and $ -\infty $), confirming the necessity of biorthogonal quantum mechanics.
• Complex phase terms in hopping amplitudes cannot be gauged away and lead to a complex-valued QMT, providing additional physical information not accessible through Hamiltonian analysis alone.