Extended imaginary gauge transformation in a general nonreciprocal lattice

TL;DR

This work expands the imaginary gauge transformation framework to general nonreciprocal lattices with complex spectra by identifying η_I-pseudo-Hermiticity as the underlying mechanism. It proves that the generalized Brillouin zone is a circle with radius determined by hopping asymmetries, enabling analytic access to continuum bands, skin-mode localization lengths, and topological diagnostics even in PT-broken phases and with long-range hopping. The authors extend IGT to 2D systems under a path-independence condition, derive a necessary and sufficient criterion for long-range hopping, and apply the theory to the non-Hermitian SSH3 model and the 2D Hatano-Nelson model to establish bulk-boundary correspondences and corner-skin phenomena. These results offer a unified, analytically tractable route to understanding NHSE, GBZ topology, and boundary phenomena across real and complex spectra with potential experimental realizations in photonic, mechanical, and quantum-simulation platforms.

Abstract

Imaginary gauge transformation (IGT) provides a clear understanding of the non-Hermitian skin effect by transforming the non-Hermitian Hamiltonians with real spectra into Hermitian ones. In this paper, we extend this approach to the complex spectrum regime in a general nonreciprocal lattice model. We unveil the validity of IGT hinges on a class of pseudo-Hermitian symmetry. The generalized Brillouin zone of Hamiltonians respect such pseudo-Hermiticity is demonstrated to be a circle, which enables easy access to the continuum bands, localization length of skin modes, and relevant topological numbers. Furthermore, we investigate the applicability of IGT and the underlying pseudo-Hermiticity beyond nearest-neighbor hopping, offering a graphical interpretation. Our theoretical framework is applied to establish bulk-boundary correspondence in the nonreciprocal trimer Su-Schrieffer-Heeger model and to analyze the localization behaviors of skin modes in the two-dimensional Hatano-Nelson model.

Figures (6)

• Figure 1: (a) The logical relationship between the main concepts in this paper. The application of IGT in previous paper is limited to the shadowed area, which stands for the NN hopping models with entirely real spectra. With the help of $\eta_{\mathtt{I}}$-pseudo-Hermiticity, we extend it to the complex spectra regime and beyond NN hopping. We further demonstrate that it can also be extended to complex hopping models since the GBZ of the system is circular. (b) The generic nonreciprocal model with NN hoppings.
• Figure 2: (a) The energy spectra of SSH3 model in the PT-exact phase, the PT-broken phase, and with complex hoppings. The system size is $N=40$. Apart from the continuum band spectrum composed of the majority of eigenvalues, some isolated discrete energy levels exist. (b) The GBZs correspond to the energy spectra in (a), which are obtained by numerically solving the characteristic equations (discrete points) and analytically calculating the assisted GBZs (solid lines) AGBZ_2020. The shape of each GBZ is shown to be a circle and the radius is only relevant to the modulus of hopping strength. The corresponding squared modulus of continuum band eigenstates in the PT-exact phase (c), the PT-broken phase (d), and with complex hoppings (e); the red lines represent the theoretical exponential envelopes. Inset plots are the results on a logarithmic scale.
• Figure 3: The GBZ of HN model with long-range hopping between the $n$-th and $(n+3)$-th unit cells. (a) The schematic diagram of the model. (b) The GBZ of such a system. The red line represents the GBZ of a system that satisfies the path-independent condition, and the yellow dashed line represents the GBZ of a system that violates the path-independent condition. The parameters are taken as $N=40, t_\mathtt{R} = 0.35, t_\mathtt{L}=0.25$. The long-range hopping strength are taken as $t^\prime_\mathtt{R}=0.1$ and $t^\prime_\mathtt{L}=t^\prime_\mathtt{R} (t_\mathtt{L}/t_\mathtt{R})^3$ to fulfill the path-independent condition. An additional 0.014 is added on $t^\prime_\mathtt{L}$ to display the violated case.
• Figure 4: Numerical result of the topological number obtained from the NS Zak's phase and corresponding energy spectrum. The parameters are taken as $N=40, t_\mathtt{L_1}=2.025, t_\mathtt{R_1}=0.4, t_\mathtt{L_2}=-0.4, t_\mathtt{R_2}=0.9$ and $t_\mathtt{L_3}=t_\mathtt{R_3}=t_3$, such that the theoretical transition point locates at $t_3=0.6$ and $t_3 = 0.9$. The real and imaginary parts of the energy spectrum under different $t_3$ are plotted in (b) and (c). The discrete energy levels are marked by the red line. The emergence of discrete levels agrees with the change of topological number displayed in (a).
• Figure 5: The phase diagrams in both (a) non-Hermitian SSH3 and (b) Hermitian SSH3 model. The red lines display the boundary, and the numbers of edge states at the left edge are labeled in the plot. The blue dashed line in (a) displays the EPs.