Fidelity and quantum geometry approach to Dirac exceptional points in diamond nitrogen-vacancy centers

TL;DR

The paper investigates Dirac exceptional points (EPs) within a non-Hermitian NV-center system by using fidelity susceptibility in a biorthogonal Hilbert space. The authors show that the Dirac EP, residing entirely in the PT-unbroken phase with linear dispersion, induces a geometric singularity where the real part of the fidelity susceptibility diverges to $-\infty$, and this divergence is strongly anisotropic, occurring along the non-reciprocal coupling axis while remaining finite along the detuning axis. This anisotropy is traced to the defective Jordan structure at the EP, where leading-order eigenstate deformation is confined to a single generalized Jordan direction. The results establish fidelity-based diagnostics as a robust, direction-sensitive probe of non-Hermitian singularities and highlight the NV-center platform as a practical testbed for quantum control and sensing near Dirac EPs.

Abstract

Dirac exceptional points (EPs) represent a novel class of non-Hermitian singularities that, unlike conventional EPs, reside entirely within the parity-time unbroken phase and exhibit linear energy dispersion. Here, we theoretically investigate the quantum geometry of Dirac EPs realized in nitrogen-vacancy centers in diamond, utilizing fidelity susceptibility as a probe. We demonstrate that despite the absence of a symmetry-breaking phase transition, the Dirac EP induces a pronounced geometric singularity, confirming the validity of fidelity in characterizing non-Hermitian EPs. Specifically, the real part of the fidelity susceptibility diverges to negative infinity, which serves as a signature of non-Hermitian criticality. Crucially, however, we reveal that this divergence exhibits a distinct anisotropy, diverging along the non-reciprocal coupling direction while remaining finite along the detuning axis. This behavior stands in stark contrast to the omnidirectional divergence observed in conventional EPs. Our findings provide a comprehensive picture of the fidelity probe near the Dirac EP, highlighting the critical role of parameter directionality in exploiting Dirac EPs for quantum control and sensing applications.

Figures (3)

• Figure 1: The energy spectrum of the non-Hermitian Hamiltonian, Eq. \ref{['eq:Hamiltonian']}. (a) The real part of the eigenenergies, $\mathrm{Re}(E_n)$, revealing a Dirac cone structure centered at the Dirac EP $(q_1,q_2)=(0,1)$ within the PT-unbroken phase. (b) The imaginary part of the eigenenergies, $\mathrm{Im}(E_n)$. The system remains in the PT-unbroken phase characterized by purely real eigenvalues for small $q_2$, but transitions to the PT-broken phase exhibiting non-zero imaginary components when the non-reciprocal coupling exceeds a critical threshold bounded by conventional EPs.
• Figure 2: (a) Density plot of the real part of the fidelity susceptibility $|\mathrm{Re}(\chi_F)|$ in the $(q_1, q_2)$ plane. The red spot at $(0,1)$ corresponds to the Dirac EP, while the red curves indicate the exceptional lines formed by conventional EPs. (b) Behavior of $\mathrm{Re}(\chi_F)$ along the $q_2$ axis with $q_1=0$ fixed, demonstrating sharp divergences to negative infinity at both the Dirac EP and conventional EPs. (c) The real part of the fidelity $F_0$ between states at $q_2$ and $q_2+\delta q$ with fixed $q_1=0$. As the system straddles the conventional EP boundary, $\mathrm{Re}(F_0)$ approaches the universal limit of $1/2$.
• Figure 3: The real part of the fidelity susceptibility, $\mathrm{Re}(\chi_0)$, calculated along the radial direction as a function of the polar angle $\phi$ surrounding the Dirac EP. The curves correspond to different radial distances $r=0.1, 0.2, 0.3$. The susceptibility exhibits a strong anisotropic divergence, peaking at $\phi=\pi/2$ and $3\pi/2$ (approaching along the $k_2$ axis) while vanishing within numerical precision at $\phi=0$ and $\pi$ (approaching along the $k_1$ axis). This behavior indicates that the leading-order geometric response near the Dirac EP is highly directional in parameter space.