Generating function for Hermitian and non-Hermitian models

Abstract

It is well known that Hermitian and non-Hermitian models exhibit distinct physics and require different theoretical tools. In this work, we propose a unified generating-function framework for both classes with generic boundary conditions and local impurities. Within this framework, any finite lattice model can be mapped to a generating function of the form G(z)=P(z)/Q(z), where Q(z) and P(z) denote the bulk recurrence relation and boundary terms or impurities, respectively. The problem of solving for eigenstates reduces to a simple criterion based on the cancellation of zeros of Q(z) and P(z). Applying this method to the Hatano-Nelson (HN) model, we show how boundary conditions and impurities determine the location of the zeros, thereby demonstrating the boundary sensitivity of non-Hermitian systems. We further investigate topological edge states in the non-Hermitian Su-Schrieffer-Heeger (SSH) model and identify its topological phase transition. Inspired by generating-function techniques widely used in discrete mathematics, particularly in the study of the Fibonacci sequence, our results establish a direct connection between non-Hermitian physics and recurrence relations, providing a new perspective for analyzing non-Hermitian systems and exploring their connections with discrete mathematical structures.

Figures (4)

• Figure 1: (a) Diagram illustrating the recurrence relation of the Fibonacci sequence in terms of a non-Hermitian model. (b) Representation of the unified approach based on generating function for one-dimensional hopping models $\mathcal{G}(z)=P(z)/Q(z)$. (c) The criterion of cancellation of zeros, where admissible eigenvalues are selected by pairing the zeros, and then we get the wavefunction.
• Figure 2: Distribution of the zeros of $P(z)$ excluding $z=0$ in the complex plane with different boundary settings for the HN model. (a) OBC, (b) PBC, (c) OBC with an impurity, (d) PBC with an impurity. Blue dots mark zeros of $P(z)$, grey circles mark zeros of $Q(z)$. Arrows indicate the cancellation of zeros of $Q(z)$ with zeros of $P(z)$, which correspond to the admissible states. The red dashed circle represents the GBZ condition $|z|=\sqrt{t_\text{L}/t_\text{R}}$.
• Figure 3: Density plots of $\log |P(z)|$ for the impurity states of the HN model with different $V$. The black circles mark $z_1$ and $z_2$. (d) The corresponding wavefunctions localized at the left boundary, extended on the left hand side, and localized at the impurity. Parameters: (a) $V=0.2498$, (b) $V=0.4000$, and (c) $V=0.5711$ with $N=20$, $t_\text{L}=1$, $t_\text{R}=0.6$.
• Figure 4: Density plots of $\log |P_\text{A}(z)|$ for representative states of the non-Hermitian SSH model with different $\lambda=\sqrt{t_\text{L} t_\text{R}/t'_\text{L} t'_\text{R}}$, where the black circles indicate $z_1$ and $z_2$. The corresponding wavefunctions are shown alongside. (a),(b) Topological edge state with $\lambda=0.9$. (c),(d) Bulk state with $\lambda=1.2$ exhibiting skin effect. Parameters are chosen as $N=20$, $t_\text{L}=1.25\lambda$, $t_\text{R}=\lambda/1.25$, $t'_\text{L}=1$, and $t'_\text{R}=1$, keeping $\sqrt{t_\text{L} t'_\text{L}/t_\text{R} t'_\text{R}}=1.25$ fixed.

Theorems & Definitions (3)

• Theorem 1
• Corollary 1
• Corollary 2