PT symmetry-enriched non-unitary criticality

TL;DR

This work uncovers a new class of symmetry-enriched non-unitary critical points in one-dimensional non-Hermitian free-fermion models with PT symmetry. By analyzing PT-symmetric extensions of the SSH chain (the $\alpha$-nH SSH family), the authors show all critical points are described by a non-unitary CFT with central charge $c=-2$, while a PT symmetry-enriched sector hosts robust topological edge modes whose degeneracies are encoded in the imaginary part of the bulk entanglement entropy. They develop a branch-cut entanglement framework to properly define $S_A$ in the presence of complex eigenvalues, linking edge-state counts to $\operatorname{Im}[S_A]$ and establishing a generalized Li–Haldane correspondence at non-Hermitian criticality. A key physical mechanism, generalized mass inversion, explains how edge states persist at criticality without requiring long-range hopping, including at interfaces between Hermitian and non-Hermitian regions. These results extend symmetry-enriched criticality to non-Hermitian systems and provide concrete, exactly solvable models and entanglement diagnostics with potential photonic realizations.

Abstract

The interplay between topology and quantum criticality gives rise to the notion of symmetry-enriched criticality, which has attracted considerable attention in recent years. However, its non-Hermitian counterpart remains largely unexplored. In this Letter, we show how parity-time (PT) symmetry enriches non-Hermitian critical points, giving rise to a topologically distinct non-unitary universality class. By analytically investigating non-Hermitian free fermion models with $PT$ symmetry, we uncover a new class of conformally invariant non-unitary critical points that host robust topological edge modes. Remarkably, the associated topological degeneracy is surprisingly encoded in the purely imaginary part of the entanglement entropy scaling-a feature absent in Hermitian systems. The underlying mechanism for the emergence of edge states at non-Hermitian criticality is traced to a generalized mass inversion that is absent in Hermitian systems.

Figures (12)

• Figure 1: (a) Phase diagram of the $\mathcal{PT}$-symmetric nH SSH models in the $(u,w-v)$ plane. The Hermitian critical point at $(0,0)$ splits into two non-Hermitian critical lines $u=\pm(w-v)$ that border the $\mathcal{PT}$-broken region. The dashed line $w=v$ separates winding sectors $\omega=0$ and $\omega=1$. (b) Schematic nH SSH chain: red/blue circles denote $\pm iu$; black/purple bonds denote Hermitian hoppings. (c) Schematic $\alpha=2$ generalization.
• Figure 2: Entanglement entropy $S_A$ as a function of the subsystem size $\ell_A$ for a total system length $L=10000$. Blue dots indicate the real part of $S_A$, red dots indicate the imaginary part, and the blue line shows the numerical fit. (a) $\alpha=1$, $(v,w,u)=(2,1,1-10^{-12})$ --- trivial QCP, $\operatorname{Re}[S_A]=-0.665\ln[\sin(\pi \ell_A/L)]-10.18$. (b) $\alpha=1$, $(v,w,u)=(1,2,1-10^{-12})$ --- topological QCP, $\operatorname{Re}[S_A]=-0.665\ln[\sin(\pi \ell_A/L)]-10.87$. (c) $\alpha=2$, $(v,w,u)=(1,2,1-10^{-12})$ --- higher winding number QCP, $\operatorname{Re}[S_A]=-0.665\ln[\sin(\pi \ell_A/L)]-11.35$. (d) Finite-size scaling of the ground-state energy in the $\alpha$-nH SSH model under PBC with $v_F=\sqrt{2}$. All results support that these QCPs are described by a non-unitary CFT with central charge $c=-2$.
• Figure 3: Entanglement spectrum of the $\alpha$–nH SSH model in the topological phase. The left panels show the real part $\operatorname{Re}[\nu]$ and the right panels show the imaginary part $\operatorname{Im}[\nu]$ of the eigenvalues. (a) For $\alpha = 1$, a single complex-conjugate pair is present. (b) For $\alpha = 2$, two complex-conjugate pairs appear. In both cases, the number of complex-conjugate pairs matches the number of edge modes.
• Figure 4: (a) Disorder-averaged entanglement entropy $S_A$ at its topological QCP ($v(x) = 1 + \delta(x)$, $w = 2$, $u(x) = 1 - \delta(x) - 10^{-10}$; $L=1000$; $\delta(x) \in [-0.999, 0.999]$). Averaged over $1000$ disorder realizations; error bars denote ±1 s.e.m. across realizations. Blue dots represent the real part of $S_A$, red dots represent the imaginary part. (b) Interface schematic between a trivial gapped region (left) and the nH SSH chain at its topological QCP (right). Shown are the mass profile $m(x)$ (black), the spectral gap $\Delta(x) = \sqrt{m(x)^2-u(x)^2}$ (blue), and the bound-state density $|\psi(x)|^2$ (red) localized at the interface.
• Figure S1: Comparison of branch choices for entanglement in the non-Hermitian SSH chain at its topological QCP ($L=10^4$; $(v,w,u)=(1,2,1-10^{-12})$). (a) Entanglement spectrum $\{\nu\}$ at $\ell_A=100$ (left: $\operatorname{Re}[\nu]$; right: $\operatorname{Im}[\nu]$). (b) $S_A(\ell_A)$ evaluated with the principal branch. (c) $S_A(\ell_A)$ on the physically consistent branch, exhibiting the expected $c=-2$ scaling. Blue (red) markers denote $\operatorname{Re}[S_A]$ ($\operatorname{Im}[S_A]$).