Scaling of Quantum Geometry Near the Non-Hermitian Topological Phase Transitions

TL;DR

The paper analyzes scaling of quantum geometry near non-Hermitian topological transitions in a long-range Kitaev chain by developing a complex Bogoliubov transformation and biorthonormal ground-state formalism. It demonstrates that the derivative of the geometric phase, the Wannier-state correlations, and the quantum geometric tensor reveal distinct universality classes controlled by the long-range exponent $\alpha$ and the critical momentum $k_0$, including across exceptional points. The results quantify universal exponents $\nu$ and $\gamma$ for various transitions and show spatially anomalous Wannier correlations, offering a framework for quantum geometry scaling in non-Hermitian topological systems. These findings highlight how non-Hermiticity and long-range couplings reshape critical geometry and topology, with potential implications for experimental platforms realizing non-Hermitian superconducting-like chains.

Abstract

The geometry of quantum states can be an indicator of criticality, yet it remains less explored under non-Hermitian topological conditions. In this work, we unveil diverse scalings of the quantum geometry over the ground state manifold close to different topological phase transitions in a non-Hermitian long-range extension of the Kitaev chain. The derivative of the geometric phase, as well as its scaling behavior, shows that systems with different long-range couplings can belong to distinct universality classes. Near certain criticalities, we further find that the Wannier state correlation function associated with extended Berry connection of the ground state exhibits spatially anomalous behaviors. Finally, we analyze the scaling of the quantum geometric tensor near phase transitions across exceptional points, shedding light on the emergence of novel universality classes.

Figures (5)

• Figure 1: (Color online) (a) Schematic representation of the long-range non-Hermitian Kitaev chain as expressed in Eq. \ref{['e1']}. (b) Phase diagram of the long-range non-Hermitian Kitaev chain in the PT-unbroken region calculated through the extended Zak phase (Eq. \ref{['wn']}) with $\Delta=J=1$ and $\delta=0.1$. Yellow and white lines represent analytical solutions $\mu=-2J \text{Li}_{\alpha}(\mp 1)$ corresponding to $k_0=\pi$ and $k_0=0$ criticalities, respectively. The black line at $\alpha=1$ represents the transition from the fractional to the topological phase occurring without gap closing.
• Figure 2: (Color online) (a) First-order derivative of the geometric phase. For $\alpha > 2$, around criticalities corresponding to both $k_0 = 0$ and $k = \pi$, $\frac{dG}{d\mu}$ exhibits non-analytic peaks. For $1 < \alpha < 2$, while a non-analytic peak still appears near the $k_0 = \pi$ criticality, $\frac{dG}{d\mu}$ fails to display such a non-analytic behavior near the $k_0 = 0$ criticality, as shown in the brown box. (b) Near the $k_0=0$ criticality in the range $\alpha > 2$, non-analytic peaks increase with system size. (c) Near the $k_0=0$ criticality in the range $1 < \alpha < 2$, $\frac{dG}{d\mu}$ does not exhibit non-analyticity. (d) The non-analytic peaks in (a) are associated with critical exponent $\nu \approx 1$, with parameters $J = \Delta = 1$, $\delta = 0.2$.
• Figure 3: (Color online) (a) Spatial behavior of the Wannier correlation function for $\alpha = 1.5$ and within the integer Zak phase. Here we choose $\mu=1.54$ which is near the $k_0 = \pi$ ($\mu_c=1.53$) criticality and $\mu=5.17$ which is near $k_0 = 0$ ($\mu_c=5.22$) criticality. (b) Scaling of fidelity susceptibility yielding a critical exponent, which is absent around $k_0=0$ for the range $1<\alpha<2$, and gives $\gamma = 2$ around other criticalities.
• Figure 4: (Color online) (a) Phase diagram for imbalanced pairings with parameter $J=1,\alpha=1.5,\mu=0.$ Topological phases ($W=\pm1$) for $\Delta^2-\delta^2>0$ and coalescing phases for $\Delta^2-\delta^2<0$. (b1) Scaling of ground state fidelity susceptibility $g_{(\Delta+\delta)(\Delta-\delta)}$ with critical exponent $\gamma = 1$ for the transition between $W=1$ and $W=-1$ around $\Delta=\delta=0$ (denoted by red arrow). (b2) Scaling of ground state fidelity susceptibility $g_{(\Delta+\delta)(\Delta-\delta)}$ with critical exponent $\gamma \approx -1$ for the transition between $W=1$ and coalescing phase (denoted by blue arrow).
• Figure 5: (Color online) Dynamical critical exponent for different ranges of the long-range decay parameter $\alpha$ and non-Hermitian parameter $\delta$. The blue and red colors correspond to criticalities $k_0=0$ and $\pi$ at $\mu_c=-2J\text{Li}_{\alpha}(\pm1)$ with $J=1$ respectively.