Topologically protected negative entanglement

TL;DR

This work shows that non-Hermitian topology can induce strongly negative entanglement in free-fermion systems via non-orthogonal edge states, especially when flat bands maximize state overlap. By employing a biorthogonal framework and analyzing both a 2-band flat-band edge-state model and a 4-band exceptional-crossing model, the authors identify non-Hermitian critical skin compression (nHCSC) as the mechanism behind dramatic negative scaling, including a $S_A\sim -\frac{1}{2}(BL_y)^2\log L$ form in gapless cases. They further show that negative entanglement extends to the second Rényi entropy $S_A^{(2)}$ and propose SWAP-operator-based measurements, while noting that EPs are not strictly required for negativity and that PT-symmetric cases yield strictly real entropies. The results reveal a novel interplay between topology, criticality, and non-Hermitian localization with potential experimental access in engineered quantum systems. Overall, the paper highlights topology as a control knob for probability non-conserving negative entanglement and expands the landscape of entanglement scaling beyond area- and volume-law paradigms.

Abstract

The entanglement entropy encodes fundamental characteristics of quantum many-body systems, and is particularly subtle in non-Hermitian settings where eigenstates generically become non-orthogonal. In this work, we find that negative biorthogonal entanglement generically arises from topologically protected non-orthogonal edge states in free fermion systems, especially for flat-band edge states. Departing from previous literature which associated negative entanglement with exceptional gapless points, we show that robustly negative entanglement can still occur in gapped systems. Gapless 2D flat-band edge states, however, exhibit novel $S_A\sim -\frac{1}{2}L_y^2\log L$ entanglement behavior which scales quadratically with the transverse dimension $L_y$, independent of system parameters. This dramatically negative scaling can be traced to a new mechanism known as non-Hermitian critical skin compression (nHCSC), where topological and skin localization in one direction produces a hierarchy of extensively many probability non-conserving entanglement eigenstates across a cut in another direction. Our discovery sheds light on new avenues where topology interplays with criticality and non-Hermitian localization, unrelated to traditional notions of topological entanglement entropy. This topologically protected negative entanglement also manifests in the second Rényi entropy, which can be measured through SWAP operator expectation values.

Figures (10)

• Figure 1: Robustly negative entanglement entropy from the gapped flat-band edge states of our 2-component Hamiltonian (Eq \ref{['Hssh']}), for parameters $B=1, t=0.8, a_0=1$ (with $b_0=1.2\ne 1$ to open up the gap). (a) For small cylinder length $L_y=5$ and $y$-direction OBCs, the energy spectrum $E(k)$ of the topological edge states $\ket{\psi_{e_1}},\ket{\psi_{e_2}}$ (bolded) exhibits a small but visible gap, but their overlap factor $\eta_\text{topo}(k)$ already approaches unity. (b) Upon increasing $L_y$ to 25, two edge bands with an exponentially small gap is observed within the non-trivial regime prescribed by Eq \ref{['topoCondition']}, with $\eta_\text{topo}(k)\approx 1$ extremely closely. (c) With $y$-PBCs, the midgap flat band disappears and $\eta(k)$ deviates markedly from unity, even though the gap is still small. (d) For $y$-OBCs but not $y$-PBCs, the overlap $\eta_\text{topo}(k)$ saturates very close to unity once $L_y\sim 10^1$. (e) The entanglement entropy scaling behavior $S_A$ for different $L_y$. Notably, as $L_y$ increases, $S_A$ decreases with $\log L$ more rapidly as $S_A\sim -(\kappa L_y+\xi)\log L$, with $\kappa\approx0.6633$, $\xi\approx -4.1817$ according to obtained from numerical fitting (black). It also saturates at $S_{min}\sim -L_y$ when $L\gtrapprox L_y$.
• Figure 2: Very robust quadratic scaling of the negative entanglement entropy $S_A$ arising from the gapless flat-band edge states of our 2-component Hamiltonian (Eq \ref{['Hssh']}) with parameters $b_0=1, t=0.5, a_0=2$. (a) Even for small $L_y=3, B=2$, two nearly flat gapless edge bands (bolded), with dispersion $E_{e_1,e_2}(k)\sim k^{BL_y}$ and overlap $\eta_\text{topo}(k)\approx 1$, emerge around $k=0$ under $y$-OBCs. (b) As $k$ approaches $0$, the skin effect ($r(k)\rightarrow 0$) strongly compresses the topological edge mode onto site $1+$, leading to nearly perfect localization. (c) For this gapless case, occupancy eigenvalues $p_i$ (Eq \ref{['pii']}) of $\bar{P}$ dramatically exceed the $[0,1]$ interval due to non-Hermitian critical skin compression (nHCSC), with $p_1,p_2,p_3,...$ (red, green, blue...) exhibiting a hierarchy of power-law dependencies with $L$. (d) The negative entanglement scaling is accurately approximated by $S_A\approx -\frac{1}{2}(BL_y)^2\log L$ (Eq \ref{['SLz2']}, black) across different $B,L_y$ combinations for sufficiently large cylinder circumference $L$. (e) The coefficient of $\log L$ in the numerical $S_A$, extracted through the gradient of the $dS_A/d(\log L)$ plots (shades of blue), agrees well with $-\frac{1}{2}(BL_y)^2$ (Eq \ref{['SLz2']}) when $L\gtrapprox 10^2$. At smaller $L$, the dependence $\propto (BL_y)^2$ still holds for smaller $BL_y$, albeit with a smaller coefficient.
• Figure 3: Negative entanglement in the 4-band exceptional topological crossing model (Eq \ref{['4bandHk']}) under $y$-OBCs with $L_y=3$. (a) In the topologically non-trivial but Hermitian case ($\alpha=0, M=1.2, \delta=0$), the overlap $\eta_\text{topo}(k)$ [Eq \ref{['Eta']}] of the topological edge states (blue) vanishes rigorously. (b) In the topologically trivial (line-gapped along Re$(E)=0$), non-Hermitian case ($\alpha=0.5\pi,M=3,\delta=2$), $\eta(k)$ of the closest bulk states (light blue) still vanishes essentially. (c) For the non-trivial Chern case ($\alpha=0, M=3, \delta=2$), perfect overlap i.e. $\eta_\text{topo}(k)=1$ is reached where topological edge modes (red) cross. (d) The free fermion entanglement entropy $S_A$ (considering only the real part) for cases (a,b) respectively increases and saturates with system circumference $L$ as expected, but that from the topological exceptional crossing (c) exhibits a new $-\frac{1}{3}\log L$ scaling. The entanglement subregion is taken to be the half-cylinder with width $L/2$.
• Figure 4: Parameter and boundary dependence of EPs in our 4-band model given by Eq. \ref{['H1General']}. (a) For $\alpha=0$ and OBCs in the $y$ direction, an EP occurs at $k=0$. (b) With PBC in the $y$ direction, no EP is observed, implying that the EP arises due to boundary localization. (c) For $\alpha=\pi/2$ with OBCs in the $y$ direction, no EP exists either. Other parameters: $L_y=6, M=3, Z=0.44, \lambda=\delta=1$.
• Figure 5: Absence of (a) divergent truncated projector $\bar{P}$ matrix elements and (b) eigenvalues outside of $[0,1]$ for an EP with square-root singularities. (c) Consequently, the real part of the entanglement entropy $S_A$ does not exhibit any negativity. Parameters and model are the same as in Fig. \ref{['FigS1']}(a), with $x$-direction size $L=50$ and $L_y=6$.