Survival of Hermitian Criticality in the Non-Hermitian Framework

TL;DR

This work shows that quantum phase transitions known from Hermitian systems persist in a non-Hermitian, one-dimensional XY model with a complex transverse field, by employing biorthogonal quantum mechanics to compute ground-state correlations and entanglement. The model is mapped to free fermions via the Jordan–Wigner transformation and Fourier analysis, yielding an exact $k$-space solution with ground-state eigenvectors $|g_k\rangle$ and $\langle\tilde{g}_k|$, and an EP-defined spectrum that reveals an emergent $U(1)$ symmetry underpinning the Luttinger liquid (LL) phase. Remarkably, the real parts of the spin-spin correlation function scale as $|\mathrm{Re}C_{ll+r}^x|\sim r^{-1/2}$ and the entanglement entropy scales as $\mathrm{Re}S_L\sim \tfrac{1}{3}\ln L$ in the LL region, just as in the Hermitian XY model, with the FM phase characterized by $Z_2$ symmetry breaking and a paramagnetic region by gapped behavior. A topological winding number around exceptional points further distinguishes the LL phase, highlighting nontrivial topology in a non-Hermitian setting. These results underscore the robustness of Hermitian criticality in open systems and open new pathways for observing quantum critical phenomena under dissipation.

Abstract

In this work, we investigate many-body phase transitions in a one-dimensional anisotropic XY model subject to a complex-valued transverse field. Within the biorthogonal framework, we calculate the ground-state correlation functions and entanglement entropy, confirming that their scaling behavior remains identical to that in the Hermitian XY model. The preservation of Hermitian phase transition features in the non-Hermitian setting is rooted in the persistence and emergence of symmetries and their breaking. Specifically, the ferromagnetic (FM) phase arises from the breaking of a $Z_2$ symmetry, while the Luttinger liquid (LL) phase is enabled by the emergence of a $U(1)$ symmetry together with the degeneracy of the real part of the energy spectrum. The nontrivial topology of the LL phase are characterized by the winding number around the exceptional point (EP). Given that non-Hermitian systems are inherently open, this research opens a new avenue for exploring conventional quantum phase transitions that are typically vulnerable to decoherence and environmental disruption in open quantum systems.

Figures (5)

• Figure 1: (Color online) The phase diagram of the non-Hermitian XY model as a function of $\mathrm{Re}\lambda$ and $\mathrm{Im}\lambda$ for a given $\gamma$. The blue elliptical interior ($(\mathrm{Re}\lambda)^2+(\mathrm{Im}\lambda)^2/\gamma^2<1$) corresponds to the FM phase. The green region ($|\mathrm{Re}\lambda| > 1$) represents the PM phase. The rest yellow region [$(\mathrm{Re}\lambda)^2+(\mathrm{Im}\lambda)^2/\gamma^2>1$ and $\mathrm{Re}\lambda < 1$] exhibits the glass phase. The purple circles represent the critical point corresponding to FM-PM phase transition, i.e., $\lambda=1$, in the Hermitian XY model.
• Figure 2: (Color online) (a1)-(b1) The longitudinal spin-spin correlation function $\mathrm{Re}C_{l,l+r}^x$ and the entanglement entropy $\mathrm{Re}S_L$ as a function of spin-spin distance $r$ and the subsystem size $L$ for $\gamma=1$ and $\lambda=\lambda_0e^{i\pi/3}$ with $\lambda_0=0.5$ (bright blue), $\lambda_0=1$ (blue), $\lambda_0=1.5$ (yellow), $\lambda_0=2$ (olive) and $\lambda_0=3$ (light green). The inset of (a1) shows the scaling law of the spin-spin correlation at $\lambda=1.5e^{i\pi/3}$. (a2)-(b2) The longitudinal spin-spin correlation function $\mathrm{Re}C_{l,l+r}^x$ and the entanglement entropy $\mathrm{Re}S_L$ as functions of $\lambda_0$ for $\gamma=1$ and $\lambda=\lambda_0e^{i\pi/3}$ (along the red arrow in Fig. \ref{['fig1']}). (a3)-(b3) The longitudinal spin-spin correlation function $\mathrm{Re}C_{l,l+r}^x$ and the entanglement entropy $\mathrm{Re}S_L$ as functions of $\lambda_0$ for $\gamma=1$ and $\lambda=\lambda_0+i(1-\lambda_0)$ (along the brown arrow in Fig. \ref{['fig1']}). The insets of (a3) and (b3) display the derivation of the short spin-spin correlation $\mathrm{Re}C_{l,l+2}^x$ and local entanglement $\mathrm{Re}S_2$, respectively.
• Figure 3: (Color online) The minimal values of the real and imaginary parts of the energy as the function of $\lambda_0$ with $\lambda=\lambda_0e^{i\pi/3}$ and $\gamma=1$.
• Figure 4: (Color online) The real (a) and imaginary (b) parts of the full energy across all mode as the function of $\lambda_0$. The parameters are $\lambda=\lambda_0e^{i\pi/3}$ and $\gamma=1$.
• Figure 5: (Color online) The winding number for different phases as the function of $\lambda_0$ for $\lambda=\lambda_0e^{i\pi/3}$ and $\gamma=1$.