Anomalous entanglement scaling from eigenvector nonorthogonality in critical non-Hermitian free fermions

Technical Summary: Anomalous Entanglement Scaling from Eigenvector Nonorthogonality in Critical Non-Hermitian Free Fermions

Problem Statement
Entanglement entropy serves as a universal diagnostic for quantum phases and critical points. In one-dimensional (1D) Hermitian critical systems, the von Neumann entanglement entropy $S_1$ of a subsystem of length $\ell$ scales logarithmically as $S_1 \sim (c/3) \ln \ell$, where the coefficient is fixed by the central charge $c$ of the underlying conformal field theory (CFT). While non-Hermitian systems—arising in open quantum systems, gain-loss processes, and post-selected measurement trajectories—exhibit unique phenomena like exceptional points and the non-Hermitian skin effect, a generic understanding of entanglement scaling in their critical steady states has been lacking. Existing studies show diverse behaviors, including logarithmic scaling with negative central charges or parameter-dependent coefficients, but a unified mechanism explaining these anomalies in non-Hermitian free-fermion steady states was absent.

Methodology
The authors investigate the steady states of critical 1D non-Hermitian free-fermion chains, specifically focusing on the non-Hermitian Su–Schrieffer–Heeger (SSH) model. The steady state $|\Psi_S\rangle$ is defined as the long-time limit of non-unitary evolution, which selects the right eigenstate $|R_\alpha\rangle$ with the largest imaginary part of the energy ($\text{Im } E_\alpha$).

1. Model and Steady State Definition: The system is described by a quadratic Hamiltonian with asymmetric hopping (imaginary gauge fields). The steady state corresponds to a Slater determinant formed by filling all single-particle right eigenmodes with $\text{Im } \varepsilon_m > 0$.
2. Entanglement Calculation: The Rényi entanglement entropies $S_n$ are computed using the correlation matrix method. For non-Hermitian systems, the authors emphasize that the correlation matrix must be constructed from the occupied right eigenstates, which are generally non-orthogonal. This requires an orthonormalization step (via QR decomposition or Schur decomposition) to define the physical density matrix $\rho = |\Psi_S\rangle\langle\Psi_S|$.
3. Analytical Derivation: The authors perform a low-energy expansion of the correlation matrix near the "occupation boundary," defined by the momenta where $\text{Im } E(k) = 0$. They analyze the discontinuity of the projector across this boundary.
4. Numerical Verification: Large-scale lattice numerics ($L \le 65536$) are performed to compute entanglement entropies for various parameter sets and disorder strengths, comparing the fitted logarithmic coefficients against the analytical predictions.

Key Contributions and Results

• Anomalous Scaling and the "Imaginary" Dirac Point:
  The study reveals that while the entanglement entropy scales logarithmically ($S_n \sim \zeta_n \ln \ell$), the coefficient $\zeta_n$ is not universal in the Hermitian sense. Instead, $\zeta_n$ varies continuously with system parameters and forms a family that cannot be reproduced by a single central charge.
  The authors identify the origin of this anomaly as an "imaginary" Dirac point: a crossing in the imaginary part of the energy spectrum ($\text{Im } E(k) = 0$) where the real parts of the bands remain split. At this point, the system transitions between two Bloch states.

• Role of Eigenvector Nonorthogonality:
  In Hermitian critical points, the eigenvectors at the crossing are orthogonal, leading to a maximal discontinuity in the occupation projector. In the non-Hermitian case, the right eigenvectors at the imaginary Dirac point are non-orthogonal. This non-orthogonality is quantified by an overlap angle $\theta$, defined by the inner product of the occupied spinors.
  The non-orthogonality weakens the discontinuity of the occupation projector from unity to $\cos \theta$. This "weakened discontinuity" opens a gap in the entanglement spectrum and reduces the logarithmic coefficient $\zeta_n$.

• Closed-Form Coefficients:
  The authors derive a closed-form expression for the Rényi coefficients $\zeta_n(\theta)$ as a function of the overlap angle $\theta$.
  $$\zeta_n = \frac{2}{\pi^2} \int_{E_\theta}^\infty d\epsilon \frac{\cos \theta \tanh(\epsilon/2)}{\sqrt{\tanh^2(\epsilon/2) - \sin^2 \theta}} h_n\left(\frac{1}{1+e^{-\epsilon}}\right)$$
  where $E_\theta = 2 \text{arctanh}|\sin \theta|$. These analytical results show excellent quantitative agreement with lattice numerics across the entire gapless regime and for all Rényi indices $n$.

• Additivity and Generalization:
  The framework is generalized to arbitrary 1D critical non-Hermitian steady states. The total logarithmic coefficient is shown to be additive:
  $$\zeta_n = \sum_{i=1}^m \zeta_n(\theta_i) + N_F \frac{1 + 1/n}{12}$$
  where $m$ is the number of imaginary Dirac points (with overlap angles $\theta_i$) and $N_F$ is the number of single-band Fermi points (contributing the standard chiral fermion value). This formula also applies to non-Hermitian ground states defined by crossings of $\text{Re } E(k) = 0$.

• Robustness to Disorder:
  A significant finding is the robustness of this logarithmic scaling against weak real onsite disorder. Unlike Hermitian disordered free fermions, which typically flow to an Anderson localized (area-law) phase or a random-singlet fixed point, the non-Hermitian steady states retain their logarithmic scaling. Weak disorder renormalizes the effective overlap angle $\theta_{\text{eff}}$, thereby enhancing the entanglement coefficient, but does not destroy the scaling form. The authors attribute this to the absence of impurity backscattering in the complex energy spectrum, as each complex energy is supported by a single propagating channel.

Significance
The paper provides a generic mechanism for entanglement scaling in critical non-Hermitian free-fermion steady states. It establishes that the non-universality of the logarithmic coefficient is a bulk effect arising from the non-orthogonality of eigenvectors at imaginary Dirac points, rather than a boundary effect or a specific feature of a particular model. This distinguishes non-Hermitian criticality from both Hermitian criticality (fixed $c$) and other non-Hermitian scenarios (e.g., measurement-induced transitions). The results offer a unified classification for entanglement in these systems, linking the geometric properties of the eigenvector bundle (overlap angle) directly to the entanglement entropy, and demonstrate that this scaling is robust against weak disorder, contrasting sharply with the behavior of Hermitian disordered systems.