Non-stabilizerness as a Diagnostic of Criticality and Exceptional Points in Non-Hermitian Spin Chains

TL;DR

The paper develops stabilizer Rényi entropy, a measure of non-stabilizerness, as a diagnostic tool for non-Hermitian quantum matter, focusing on PT-symmetric spin chains. By computing M_2^{RR} from right-ground-state MPS using non-Hermitian DMRG, it analyzes the non-Hermitian transverse-field Ising model and the non-Hermitian XX model, finding model-specific signatures: M_2^{RR} peaks at conventional critical lines in Ising but vanishes at exceptional points, while in real-space XX it peaks at the exceptional line; momentum-space XX shows a minimum at exceptional points. Finite-size scaling shows that these features sharpen with system size, highlighting magic as a sensitive marker of criticality and symmetry breaking in non-Hermitian systems, complementing entanglement-based diagnostics. The results point toward broader applications in disordered, non-equilibrium, and experimental non-Hermitian quantum matter.

Abstract

We investigate non-stabilizerness, also known as ``magic,'' to understand criticality and exceptional points in non-Hermitian quantum many-body systems. Our focus is on parity-time ($\mathcal{PT}$) symmetric spin chains, specifically the non-Hermitian transverse-field Ising and XX models. We calculate stabilizer Rényi entropies in their ground states using non-Hermitian matrix product state methods. Our findings show that magic exhibits unique and model-specific signs of phase transitions. In the Ising chain, it peaks along the regular Hermitian-like critical line but disappears across exceptional points. In contrast, in the XX chain, it reaches its maximum at the exceptional line where $\mathcal{PT}$ symmetry is broken. Finite-size scaling reveals that these effects become more pronounced with larger systems, highlighting non-stabilizerness as a sensitive marker for both quantum criticality and non-Hermitian spectral degeneracies. We also investigate magic in momentum space for the XX model analytically and find that is reaches a minimum around exceptional points. Our results indicate that magic takes extremal values at the exceptional points and serves as a valuable tool for examining complexity, criticality, and symmetry breaking in non-Hermitian quantum matter.

Figures (7)

• Figure 1: Ground-state density plot of $M_2^{RR}(L)$. The dashed black and white lines indicate the theoretical critical lines $\gamma_{c1}$ and $\gamma_{c2}$ in the thermodynamic limit. Magic peaks along the Ising transition line and vanishes at the exceptional points across the phase diagram. System size is fixed to $L=16$.
• Figure 2: (a) Horizontal cuts through the phase diagram at several values of $\gamma$ in the $\mathcal{PT}$-symmetric regime, showing that $M_2^{RR}$ peaks at the Ising transition. (b) Vertical cuts along the non-Hermitian parameter $\gamma$ for different external magnetic fields, illustrating that magic reaches its maximum at the Ising transition and vanishes at the exceptional point $\gamma=1$.
• Figure 3: Scaling of $M_2^{RR}$ with system size. As the system size increases, the location of the transition becomes sharper and better resolved.
• Figure 4: Ground-state density plot of $M_2^{RR}(L)$. The dashed black lines indicate the theoretical exceptional points line. Magic peaks along the $\mathcal{PT}$ exceptional line. The system size is $L=16$.
• Figure 5: Horizontal cuts through the phase diagram at several values of $\delta$ showing that $M_2^{RR}$ peaks at the exceptional point.