Anyon-Induced Criticality and Dynamical Stability in Non-Hermitian Many-Body Systems

Abstract

We show that anyonic statistics fundamentally reshapes non-Hermitian many-body physics by intrinsically breaking pseudo-Hermiticity, leading to a unique real-complex spectral transition with characteristically dense states in Im$E$. This anyon-induced transition occurs even when bosonic and pseudofermionic counterparts remain entirely real, revealing a form of non-Hermitian criticality driven purely by exchange statistics. The resulting spectrum exhibits enhanced gaps in Im$E$ that dynamically isolate dominant eigenstates, producing anomalously stable short-time quench dynamics for anyons. Our results identify anyonic statistics as an intrinsic mechanism for generating unconventional non-Hermitian critical behavior usually associated with highly non-local systems.

Figures (12)

• Figure 1: FIG. 1. Schematic illustration of representative interaction processes underlying the anyon-induced real-complex spectral transition (AIST) in coupled Abelian anyonic systems. (a) Processes without particle exchange: (a1) a scattering-bound process that hybridizes extended and bound two-particle configurations, (a2) a bound-bound process that renormalizes the bound-state sector, and (a3) a scattering-scattering process that reshapes the scattering continuum. These channels are determined by the composition of scattering and bound states and do not explicitly depend on the statistical angle $\theta$. (b) Process with particle exchange, in which interchain hopping acquires an occupation-dependent Peierls phase $f_j(\theta)$. This exchange-enabled statistical channel introduces a $\theta$-dependent asymmetry, breaks pseudo-Hermiticity, and can drive the emergence of AIST.
• Figure 2: Clean (Top Row) and Hybrid (Bottom Row) anyon-induced real-complex transition (AIST) for two particles. Anyons, unlike bosons ($\theta=0$) and pseudofermions ($\theta=\pi$), generically undergo a real-complex transition already in the scattering subspace at small $J_p$. (a) Number of eigenstates with nonzero imaginary energies ($\mathrm{Im}E>10^{-7}$ in numerics) as a function of $J_p$, with $U=16,\ \mu=4$. Compared with bosons ($\theta=0$) and pseudofermions ($\theta=\pi$), anyons acquire complex eigenenergies at an exponentially smaller $J_p$. (a1) to (a4) show representative spectra colored by the sublattice polarization $P=(N_A-N_B)/(N_A+N_B)$, with $N_\sigma=\sum_x\langle \hat{n}_{x,\sigma}\rangle$. The mixture of scattering and bound states lead to the mixed many-body CSE that generates complex eigenenergies under a weak $J_p$ [(a1) and (a2)], which are dominated by those from AIST in both their numbers and magnitudes [(a3) and (a4)].(b) and (b1) to (b4) show the same quantities as (a) and (a1) to (a4), but with different interaction and on-site potential $U=4,\ \mu=0.2$). Scattering and bound states are now separated in real energy. The resultant AIST and many-body CSE are qualitatively similar, except that the scatter many-body CSE overwhelms the AIST under a relatively strong $J_p$ in (b4). (c) and (d) Maximum imaginary eigenenergy, Max(Im$E$), for $U=16$ and $\mu=4$ as in (a). In (c), AIST is clearly recognized by zero and nonzero Max(Im$E$) with $\theta=0,\pi$ and generic $\theta$ as $L$ increases, respectively. Its critical nature can be identified in (d) by the threshold of $J_p$ to induce nonzero Max(Im$E$), which approaches zero exponentially as the system size $L$ increases. (e) and (f) the same as (c) and (d), but with $U=4$ and $\mu=0$ as in (b). The scattering many-body CSE dominates the complex eigenvalues, leading similar Max(Im$E$) for arbitrary $\theta$ values in (e). In (f), the threshold of $J_p$ also decreases exponentially as $L$ increases. Other parameters are $J=e^{\alpha}=1/\sqrt{2}$ and $L=10$, unless otherwise specified.
• Figure 3: Qualitatively distinct imaginary density of states and spectral transitions between anyonic and non-anyonic systems. Unlike bosons and pseudofermions, generic anyons exhibit broadly complex spectra, indicating pseudo-Hermiticity breaking via AIST. (a) and (b) imaginary spectrum (blue) versus $J_p$ for bosons and pseudofermions, respectively. Discontinuous jumps of eigenenergies from real to complex values signal the breaking of pseudo-Hermiticity for eigenstates in the many-body CSE. A transition of Max(Im$E$) (dark blue) is clearly seen in either case, where two states cross each other in Im$E$ (vertical dashed line). Top panels show the first derivative of Max(Im$E$), which jumps discontinuously at the transition point. (c) and (d) the same as (a) and (b), but for anyons with different $\theta$ values. Eigenenergies generally take complex values, indicating the invalidation of pseudo-Hermiticity for the Hamiltonian corresponding to AIST. The imaginary-energy spacing becomes much smaller, yet the transition of Max(Im$E$) still persists, as can be seen from the discontinuous change of its derivative. The transition is also marked by the jump of the edge correlation $C_{AL,B1}$ of the state with maximum Im$E$ (red dots). The imaginary energy distribution and its response to coupling differ markedly between anyonic and non-anyonic systems. Other parameters are $U = 16$, $\mu = 4$, and, $L = 10, J=e^{\alpha}=1/\sqrt{2}$. black stars and hexagons mark the parameters before and after quench is Fig. \ref{['fig:Quench']}.
• Figure 4: Dynamical stability induced by anyonic statistics. (a) and (b) boundary correlation $C_{AL,B1}(t)$ of the finial state and its overlap with the initial state $P(t)$, respectively. The pre- and post-quenched parameters are $(J_{\mathrm{ini}}, J_{\mathrm{{final}}})=(0.12, 0.124)$ for $\theta=0$, $(0.035, 0.045)$ for $0.4\pi$, $(0.04, 0.06)$ for $0.8\pi$, and $(0.08, 0.09)$ for $\pi$, as marked by black stars and hexagons in Fig. \ref{['anyonPT']}. In the short-time regime (approximately with $t<10^3$), bosons and pseudofermions show strong fluctuations in the demonstrated quantities, while anyons are dynamically stable after quenching. Beyond this regime, anyons exhibit strong oscillations in $C_{AL,B1}(t)$ as their maximal Im$E$ are two-fold degenerate (see supplemental Sec. IV). Other parameters are $N=2$, $U=16$, $\mu=4$, $L=10$, and $J=e^{\alpha}=1/\sqrt{2}$.
• Figure S1: (a) Imaginary energy levels around the crossing point for $\theta=0.4\pi$. (b) The same as (a) zoom in around the crossing point. (c), (d) the same as that of (a),(b) but for $\theta=0.8\pi$. We observe two states are nearly degenerate. Parameters are $U=16,\mu=4,L=10$, $J e^{\alpha} = 0.5$,$J e^{-\alpha} = 1$.