Yang-Lee edge singularity and quantum criticality in non-Hermitian PXP model

TL;DR

This work addresses quantum criticality in a non-Hermitian detuned PXP model by constructing a complete phase diagram that spans real- and complex-energy regimes. It develops biorthogonal entanglement entropy and Loschmidt-echo diagnostics to establish Ising universality ($c_{\rm eff}=\tfrac{1}{2}$) at the real-energy transition and identifies a confinement–deconfinement crossover in the PT-symmetric region. In the complex-energy regime, it reveals two PT-transition types and locates the Yang-Lee edge singularity (YLES) with finite-size scaling that yields $\beta=12/5$ and $c_{\rm eff}=2/5$, in agreement with non-unitary CFT predictions. The results are supported by an experimentally feasible proposal using Rydberg-atom arrays to observe YLES via self-normal and associated-biorthogonal Loschmidt echoes, highlighting a practical path to study non-Hermitian critical phenomena. The combination of kinetic constraints and non-Hermitian couplings offers a robust platform for exploring unconventional phase transitions and singularities in quantum many-body systems.

Abstract

We present a comprehensive theoretical framework for quantum criticality in the non-Hermitian detuned PXP model, and establish the complete phase diagram, which had remained elusive in previous studies. Starting from a numerically identified phase transition point, we construct an exact second-order phase transition boundary through a similarity transformation in the real-energy regime. By introducing the biorthogonal entanglement entropy and biorthogonal Loschmidt echo, we demonstrate from both equilibrium and nonequilibrium perspectives that this transition belongs to the Ising universality class. Using the correlation function, we further distinguish between confined and deconfined phases within the $\mathcal{PT}$-symmetric region. In the complex-energy regime, we identify both a full $\mathcal{PT}$ transition and a first-excited-state $\mathcal{PT}$ transition, respectively. Moreover, we identify the location of the Yang-Lee edge singularity (YLES) using both the associated-biorthogonal and self-normal Loschmidt echoes, and extract the corresponding critical exponent, which agrees with the predictions of non-unitary conformal field theory. Finally, we propose an experimental scheme to observe the YLES in Rydberg atomic arrays, which offers a promising route to exploring non-Hermitian critical phenomena and singularities in future experimental settings.

Figures (11)

• Figure 1: Quantum phase diagram of the non-Hermitian detuned PXP model [Eq. (\ref{['equ:nHdPXP']})], corresponding to the $\alpha=\pi/2$ slice of the parent Hamiltonian [Eq. (\ref{['eq:Ising']})]. The labels "deconfined" and "confined" refer to the ground-state phases. $\mathcal{PT}$ and BR$_f$ denote the $\mathcal{PT}$-symmetric and $\mathcal{PT}$-broken regions of the full many-body spectrum, respectively. Two distinct broken regimes are labeled BR$_{f1}$ and BR$_{f2}$. BR$_1$ corresponds to $\mathcal{PT}$ breaking in the first excited state, while the ground state remains $\mathcal{PT}$-symmetric. The red pentagram marks the EP at which all eigenvalues coalesce to zero. The yellow line indicates the second-order transition line and coincides with the YLES.
• Figure 2: Entanglement entropy for $g=0.1$. (a) The biorthogonal entanglement entropy with respect to $m$ for systems from $N = 16$ to $N = 24$. (b) The finite-size scaling of the biorthogonal entanglement entropy at peaks shown in (a), which are fitted by using Eq. (\ref{['eq:central charge']}). The central charge is identified as $c = 0.483\pm0.019$ from the biorthogonal entanglement entropy.
• Figure 3: Extraction of the critical exponent. (a) The short-time average rate function for $g=0.1$ with respect to $m$ with $\delta m = 0.01$. (b) The biorthogonal Loschmidt echo at the peak position $m^{(N)}$ of $t$ as a function of time. (c) Finite-size scaling of $(1-\mathcal{L}_{\rm min})$ obtained from panel (b) as a function of lattice sizes $N$. The correlation-length critical exponent $\nu=1.028\pm0.031$ is obtained from the fitting curve. (d) The scaling behavior of $\Delta_E$ versus $N$ around the critical point. The dynamical exponent $z$ obtained from the fitting is $z = 0.986\pm0.034$.
• Figure 4: The time evolution of $\mathcal{G}(l,t)$ as a function of $l$ and $t$ for $N = 20$: (a) $m = -5$, $g = 0.1$; (b) $m = 5$, $g = 0.1$.
• Figure 5: Energy spectrum as a function of $m$ for $g=1.5$. (a) Imaginary parts of the energy spectrum. The blue circles denote the ground state in the subspace of momentum $k=0$, the red circles represent the ground state for $k=\pi$, and the pink circles indicate the first excited state for $k=\pi$. Here, EP$_{\rm f}$ denotes the EP associated with the full $\mathcal{PT}$ transition, corresponding to the parameter $m_{c1}$, while EP$_{\rm 1}$ refers to the EP characterizing the first excited-state $\mathcal{PT}$ transition, corresponding to $m_{c2}$. (b) Imaginary parts of the energy spectrum. The blue circles denote the ground state for $k=0$, and the green circles represent the first excited state for $k=0$. The point $m_{c3}$ indicates a first-order phase transition. Inset shows the first derivative of the real part of the ground-state energy as a function of $m$. The point of discontinuity in the first derivative corresponds to $m_{c3}$ in panel (b). (c) Imaginary parts of the energy spectrum. The blue circles denote the ground state for $k=0$, and the green circles represent the first excited state for $k=0$. Here, EP$_{\rm f}$ denotes the EP associated with the full $\mathcal{PT}$ transition, corresponding to the parameter $m_{c4}$. Notably, this EP also marks the YLES.