Quantum State Evolution and Berry Potentials at Exceptional Points and Quantum Phase Transitions

TL;DR

The paper addresses how quantum states evolve near exceptional points and quantum phase transition critical points, where Berry potentials often appear singular. It introduces a parameter-induced emergent dimension with an evolution generator K and shows that Berry potentials arise as projections of K in an adiabatic gauge, with the local curvature vanishing and implying gauge freedom. Through analytically tractable examples of an EP model and a diabolic point in the SSH chain, it demonstrates that the apparent singularities in K can be removed by alternative gauges, preserving information across critical points. The work connects these singularities to fidelity susceptibility divergences at degeneracies, linking EPs/DPs to QPTs, and offers a Black Hole–like geometric interpretation where coordinate singularities do not reflect physical information loss. Overall, the framework unifies EPs, DPs, and QPTs under a gauge-invariant, emergent-dimension approach with implications for detecting and understanding critical phenomena in quantum systems.

Abstract

The behavior of quantum states at exceptional points and at critical points associated with quantum phase transitions is intriguing yet puzzling. In this study, we present an alternative method for obtaining the Berry potentials using the evolution generator along the parameter induced dimension and demonstrate that they are singular at these critical points. Although these singularities may appear to indicate a breakdown in quantum state evolution, we show that the information carried by quantum states evolving across these critical points is not destroyed. Specifically, when the evolution generator of the full Hilbert space bundle is taken into account, the quantum states remain insensitive to the critical points. In physical terms, it's similar to the classical image of an object smoothly passing through a black hole's event horizon. Further similarities between exceptional points and quantum phase transitions are explored in this work.

Figures (2)

• Figure 1: Illustration of cells numbered $n - 1$, $n$, and $n + 1$. The red balls represent the A sites, and the blue balls represent the B sites within each cell. The coupling between A and B sites within the same cell is denoted by $v$, while the coupling between neighboring cells is denoted by $w$.
• Figure 2: A schematic diagram of a QPT shown in an energy-level diagram. The horizontal axis represents the parameter $q$, and the vertical axis denotes the energy. The critical point of the QPT occurs at $q = q_0$, where the ground state and the first excited state exchange roles (i.e., an energy-level crossing). The same parameter $q_0$ also corresponds to the singularity of the $q$-evolution generator.