Recurrence method in Non-Hermitian Systems

TL;DR

This work addresses the difficulty of obtaining open-boundary spectra in non-Hermitian systems by introducing a recurrence method based on the characteristic polynomial $D_N(E)=\det(E-H_N)$. By exploiting size-independent recurrences and expressing $D_N(E)$ in terms of roots $z_i(E)$, the authors separate bulk spectra (where two maximal $|z_i|$ coincide) from edge spectra (where the leading coefficient vanishes), eliminating the need for GBZ construction. They derive general bulk expressions for multi-band nearest-neighbor models (e.g., non-Hermitian SSH and Rice-Mele) and provide a targeted, efficient route to edge spectra, including size-parity effects and robustness to boundary perturbations. The method is extended to the non-Hermitian Hofstadter model, enabling tractable computation of complex bulk and edge spectra in large systems with non-reciprocity and magnetic fields. Overall, the recurrence approach offers a scalable, accurate alternative to diagonalization and non-Bloch GBZ methods, with direct insights into bulk-edge correspondences and topological edge modes in open non-Hermitian systems, and is accompanied by accessible code on GitHub.

Abstract

We propose a novel and systematic recurrence method for the energy spectra of non-Hermitian systems under open boundary conditions based on the recurrence relations of their characteristic polynomials. Our formalism exhibits better accuracy and performance on multi-band non-Hermitian systems than numerical diagonalization or the non-Bloch band theory. It also provides a targeted and efficient formulation for the non-Hermitian edge spectra. As demonstrations, we derive general expressions for both the bulk and edge spectra of multi-band non-Hermitian models with nearest-neighbor hopping and under open boundary conditions, such as the non-Hermitian Su-Schrieffer-Heeger and Rice-Mele models and the non-Hermitian Hofstadter butterfly - 2D lattice models in the presence of non-reciprocity and perpendicular magnetic fields, which is only made possible by the significantly lower complexity of the recurrence method. In addition, we use the recurrence method to study non-Hermitian edge physics, including the size-parity effect and the stability of the topological edge modes against boundary perturbations. Our recurrence method offers a novel and favorable formalism to the intriguing physics of non-Hermitian systems under open boundary conditions.

Figures (7)

• Figure 1: We compare the OBC (red curves and dots) and PBC (blue curves) spectra of the non-Hermitian Rice-Mele model in Eq. \ref{['eq:nH_Rice_Mele']} with two sublattices A and B and asymmetric hoppings $v_1\neq v_2^*$ or $w_1\neq w_2^*$, illustrated in (a). The OBC spectra are fully real when both $v_1v_2, w_1w_2>0$, irrespective of $V$: (b) $v_1=2$, $v_2=1$, $w_1=1.2$, $w_2=0.8$, and $V=0.5$; (c) $v_1=0.8$, $v_2=0.2$, $w_1=1.2$, $w_2=0.8$, and $V=0.5$; (d) $v_1=0.8$, $v_2=0.2$, $w_1=-1$, $w_2=1$, and $V=0.5$. However, when both $v_1v_2, w_1w_2<0$, a non-Bloch PT symmetry breaking may occur: (e-g) $v_1=-0.8$, $v_2=0.2$, $w_1=-1$, and $w_2=1$, thus the transition points are at $V_c=1.4$. The spectrum is complex for (d) and (e) $V=1.2$ at a critical point at (f) $V=1.4$ and becomes fully real for (g) $V=1.5$. Note that $|v_1v_2|>|w_1w_2|$ and thus no edge state exists in (b), while $|v_1v_2|<|w_1w_2|$ and two isolated edge states appear at $E=\pm V$ (red dots) in (c-g).
• Figure 2: The real [(a) and (c)] and imaginary [(b) and (d)] parts of the OBC spectra for non-Hermitian Rice-Mele models in Eq. \ref{['eq:nH_Rice_Mele']} on even-size [$2N=40$ in (a) and (b)] and odd-size [$2N+1=41$ in (c) and (d)] systems display a size-parity effect. The black lines represent the OBC bulk spectrum, and the red (blue) lines represent the left (right) localized edge modes. The model parameters $v_1=0.9+1.2\sin\theta$, $v_2=0.9-1.2\sin\theta$, $w_1=w_2=0.6$, and $V=0.2\sin\theta$ are variable with respect to $\theta$. For the $2N$-size models, a pair of edge modes at $E=\pm V$ appear when $0.56<\sin\theta<0.90$, consistent with condition $|v_1v_2|<|w_1w_2|$; for the $(2N+1)$-size models, an isolated edge mode at $E=V$ always exists irrespective of $\theta$, yet switches between the right and left edges as $\theta$ evolves.
• Figure 3: The spectra and localization properties of the edge modes for the boundary-perturbed non-Hermitian models in Eq. \ref{['eq:perturbed_nH_ssh']} show the stability of the topological edge modes. (a) As the perturbation $V_1$ varies on the left edge, the right edge mode (blue) stays unchanged, while the original left edge mode (red) develops an energy shift yet remains in the bulk gap. An additional edge mode emerges on the left boundary for $V_1>0.7$. (b) The wave functions of the respective edge modes show full localizations at $V_1=1.2$. $u_1=1$, $u_2=2$, and $\gamma=1$.
• Figure 4: We consider the non-Hermitian Hofstadter butterfly on a two-dimensional triangular lattice. The model, given by the Hamiltonian in Eq. \ref{['eq:nH_triangle_lattice_model']}, consists of unequal counterclockwise hoppings $t_1=\sqrt{(1-\delta)(1+\delta)}$ (blue arrows) and clockwise hoppings $t_2=\sqrt{(1+\delta)(1-\delta)}$ (red arrows). Also, we introduce a perpendicular magnetic field with $p/2q$ magnetic flux quantum per triangle plaquette. We set $\boldsymbol{x}$ and $\boldsymbol{y}$ as our lattice vectors.
• Figure 5: The spectra of the triangular lattice models in Eq. \ref{['eq:nH_triangle_lattice_model']} versus the applied perpendicular magnetic field ($p/q$ magnetic flux quantum per lattice plaquette) displays clear fractal and self-similar features resembling a (Hofstadter) butterfly, both in (a) Hermitian $\delta=0$ and (b-c) non-Hermitian $\delta=0.2$ scenarios. We consider a geometry with PBC in the $\boldsymbol{y}$ direction and OBC in the ${x}$ direction. The blue (b) and red (c) spectra are the real and imaginary parts, respectively. (d) The Landau fan structure near the band top (and bottom) exhibits a linear dependence on the magnetic field (red lines) consistent with $\sqrt{1-\delta^2}E_n\propto 6-3B(n+1/2), n=0, 1, 2, \cdots$ from the complex semiclassical theory Yang2024supp and the interpretation of a complex effective magnetic field for a finite $\delta=0.2$.

Theorems & Definitions (4)

• Theorem 1
• proof
• Theorem 2
• Theorem 3