[Paper Review] Application of Jordan Decomposition to Non-Hermitian Lattice Models with Spectrally-Isolated Lower Dimensional States

Yang Yu, Minwoo Jung+1 more|arXiv (Cornell University)|Apr 8, 2020

Quantum Mechanics and Non-Hermitian Physics35 references28 citations

TL;DR

This paper introduces a Jordan decomposition framework to analyze non-Hermitian lattice models with spectrally isolated states, replacing unstable eigendecomposition for accurate identification of left and right eigenstates. It applies this method to a non-Hermitian quadrupole insulator, revealing that zero-energy corner states exhibit non-local and counter-intuitive resonant responses depending on excitation location, leading to classification into trivial, near-Hermitian, and non-local regimes based on spectral and response properties.

ABSTRACT

When analyzing non-Hermitian lattice systems, the standard eigenmode decomposition utilized for the analysis of Hermitian systems must be replaced by Jordan decomposition. This approach enables us to identify the correct number of the left and right eigenstates of a large finite-sized lattice system, and to form a complete basis for calculating the resonant excitation of the system. Specifically, we derive the procedure for applying Jordan decomposition to a system with spectrally-isolated states. We use a non-Hermitian quadrupole insulator with zero-energy corner states as an example of a large system whose dimensionality can be drastically reduced to derive a low-dimensional "defective" Hamiltonian describing such localized states. Counter-intuitive and non-local properties of the resonant response of the system near zero energy are explained using the Jordan decomposition approach. Depending on the excitation properties of the corner states, we classify our non-Hermitian quadrupolar insulator into three categories: trivial, near-Hermitian, and non-local.

Motivation & Objective

• To address the instability and incompleteness of standard eigendecomposition in non-Hermitian systems, especially for large finite-sized lattices with spectrally isolated states.
• To develop a robust method for identifying complete left and right eigenstate bases using Jordan decomposition, enabling accurate calculation of resonant excitation responses.
• To classify non-Hermitian quadrupole insulators into three regimes—trivial, near-Hermitian, and non-local—based on the localization and excitation properties of their zero-energy corner states.

Proposed method

• Utilizes Schur decomposition to group isolated eigenvalues and extract invariant subspaces, enabling block-diagonalization of the Hamiltonian.
• Applies Jordan decomposition to the isolated block, yielding a defective Hamiltonian form H = PJP⁻¹, where J is upper triangular with eigenvalues on the diagonal.
• Identifies left and right Jordan basis vectors (generalized eigenvectors) from the Schur decomposition of H and Hᵀ, forming a complete basis for the system’s dynamics.
• Uses the resolvent (E − H + iΓ)⁻¹ reformulated via Jordan decomposition to solve driven equations, enabling accurate calculation of resonant responses.
• Applies the method to a non-Hermitian quadrupole insulator model with asymmetric intra-cell hopping and real inter-cell hopping, maintaining sublattice symmetry.
• Verifies approximate eigenstates analytically and numerically, confirming exponential localization and spectral stability under open boundary conditions.

Experimental results

Research questions

• RQ1 How can standard eigendecomposition be replaced by a more stable and complete method for analyzing non-Hermitian lattice systems with spectrally isolated states?
• RQ2 Why do resonant responses in non-Hermitian corner states exhibit non-local and counter-intuitive behavior depending on excitation location?
• RQ3 What determines the classification of a non-Hermitian quadrupole insulator into trivial, near-Hermitian, or non-local regimes?
• RQ4 How does the presence of a finite bulk bandgap and real spectrum affect the localization and excitation of zero-energy corner states?
• RQ5 Can Jordan decomposition accurately describe the dynamics of resonant excitation in systems where eigendecomposition fails due to numerical instability?

Key findings

• The Jordan decomposition method successfully identifies a complete basis of left and right eigenstates for non-Hermitian systems, resolving the 'missing dimension' problem inherent in eigendecomposition.
• In the near-Hermitian regime (|λ| > |t| + |γ|), four spectrally isolated zero-energy corner states exist, each localized at a separate corner, with left and right eigenvectors localized at the same corner.
• In the intermediate regime (|t² − γ²|¹ᐟ² < |λ| < |t| + |γ|), only two linearly independent corner states exist despite algebraic multiplicity four, indicating geometric degeneracy.
• Excitation at the top-left corner produces a weak response in the bottom-right corner state, demonstrating non-local and counter-intuitive resonant behavior in the intermediate regime.
• The system’s classification into trivial, near-Hermitian, or non-local regimes depends on the relative localization of left and right Jordan basis vectors and the excitation response pattern.
• Analytical verification confirms that the corner state wavefunctions are approximate eigenstates of the Hamiltonian, with exponential decay at system boundaries in the thermodynamic limit.