Spontaneous Chern-Euler Duality Transitions

TL;DR

The paper shows a spontaneous Chern-Euler duality in non-Hermitian PT-symmetric three-band systems, where the topological invariant remains conserved in magnitude but changes from Euler to Chern character during PT-symmetry breaking. Through an interpolation $H_oldsymbol{\lambda}(oldsymbol{k})$, oblique projection operators, and the realification/complexification of Bloch bundles, it proves that a dual transition occurs when $|C_ u|=|oldsymbol{ ext Chi}|$, with real Euler bands transforming into complex Chern bands that share the same topological magnitude. The analysis unifies real and complex bundle descriptions via $ ext{GL}^+_2(\mathbb{R}) o SO(2) o GL_1(\mathbb{C})$ equivalences and demonstrates continuity of the evolving subspaces $V_oldsymbol{\lambda}(oldsymbol{k})$, supported by Wilson-loop and covariant-connection calculations. This topological duality connects symmetry classes in non-unitary settings and provides experimental paths to observe dual Chern and Euler characteristics through exceptional rings and interferometric probes of geometric phases.

Abstract

Topological phase transitions are typically characterized by abrupt changes in a quantized invariant. Here we report a contrasting paradigm in non-Hermitian parity-time symmetric systems, where the topological invariant remains conserved, but its nature transitions between the Chern number, characteristic of chiral transport in complex bands, and the Euler number, which characterizes the number of nodal points in pairs of real bands. The transition features qualitative changes in the non-Abelian geometric phases during spontaneous parity-time symmetry breaking, where different quantized components become mutually convertible. Our findings establish a novel topological duality principle governing transitions across symmetry classes and reveal unique non-unitary features intertwining topology, symmetry, and non-Abelian gauge structure.

Figures (6)

• Figure 1: Schematics of the transitions studied in this article. The boxes schematically indicate the spectra at the three types of models $H_{\rm Euler}$, $H_{\rm Chern}$, and $H_{\rm trivial}$ and during the transitions between them. The Chern-Euler duality transition between bands 1 and 2 (shown in red and blue) takes place without involvement of the third, real band (green). The non-dual transition between bands with different Chern number and Euler number forces connection between all three bands. In the Hermitian regime, the Euler topology stably protects the nodal points (NPs) between the bands 1 and 2.
• Figure 2: The spontaneous Chern-Euler duality transition. (a) Top: The off-diagonal Berry curvature $B_{12}$ at $\lambda=0$ (left) and the diagonal Berry curvature $B_{++}$ at $\lambda=0.55$ (right) over the two-dimensional BZ. Their integrals are equal to $\chi=2$ and $C_+ =2$, respectively. Bottom: In the symmetry-preserving regime, the paired eigenstates are described by a real rank-two vector bundle, whose frame bundle is a $\hbox{GL}^+_2(\mathbb R)$ bundle. After spontaneous symmetry breaking, the eigenstates are described by two complex rank-one vector bundles, whose frames form two $\hbox{GL}_1(\mathbb C)$ bundles. The $\hbox{GL}^+_2(\mathbb R)$ topology is continuously related to the $\hbox{GL}_1(\mathbb C)$ topology, leading to the conservation $|\chi|=|C_\pm|$. (b)-(e) The minimum of the local eigenvalue gaps $|\Delta\omega(\mathbf k)|_{\text{min}}\equiv\textrm{Min}_{i,j}|\omega_i(\mathbf k)-\omega_j(\mathbf k)|$ of $H_{\lambda}$ at $\lambda=0.1$ (b), $\lambda=0.44$ (c), and $\lambda=0.55$, reflecting the shapes of level crossings in the BZ. The corresponding complex spectra are shown in the right panel of each figure, with the three bands labeled by different colors.
• Figure 3: The non-dual transition from trivial bands to Chern bands. (a)-(d). The minimum eigenvalue gap $|\Delta\omega(\mathbf k)|_{\text{min}}=\textrm{Min}_{i,j}|\omega_i(\mathbf k)-\omega_j(\mathbf k)|$ and the spectra of $H_{\mu}$ on the complex plane at $\mu=0$ in (a), $\mu=0.41$ in (d), $\mu=0.55$ in (c), and $\mu=0.72$ in (d). All the three bands are connected together during the non-dual transition. (e) The real and imaginary parts of the spectra as a function of $k_y$ at fixed $k_x=\pi$ for different $\mu$. Triple level crossings, i.e., EP3s are generated during this non-dual transition.
• Figure S1: Partition of eigenvalues/bands and spontaneous symmetry breaking (SSB). (a) When the eigenvalues are on the real axis, we may distinguish them by their real parts, and there can be an energy gap between different connected components in the spectra along the real axis. When the eigenvalues can detach from the real axis, we will also have different connected components on the lower and upper half complex planes.They are distinguished by disspation gaps. (b) The SSB inside a BZ. Exceptional rings are the boundaries between symmetry-preserving regionsand SSB regions. They convert real gauge structures of the eigenstates into complex gauge structures. The protections of the Dirac points are erased after the gauge type shift.
• Figure S2: How to choose the enclosing disc $D$ on the complex plane of spectrum. The blue discs represent the approximate region of all eigenvalues $\omega_j$ for $\mathbf k,\lambda$ in a neighbourhood of the EP.