Symmetries, Conservation Laws and Entanglement in Non-Hermitian Fermionic Lattices

Abstract

Non-Hermitian quantum many-body systems feature steady-state entanglement transitions driven by the competition between unitary dynamics and dissipation. In this work, we reveal the fundamental role of conservation laws in shaping this competition. Focusing on translation-invariant non-interacting fermionic models with U(1) symmetry, we present a theoretical framework to understand the structure of the steady-state of these models and their entanglement content based on two ingredients: the nature of the spectrum of the non-Hermitian Hamiltonian and the constraints imposed on the steady-state single-particle occupation by the conserved quantities. These emerge from an interplay between Hamiltonian symmetries and initial state, due to the non-linearity of measurement back-action. For models with complex energy spectrum, we show that the steady state is obtained by filling single-particle right eigenstates with the largest imaginary part of the eigenvalue. As a result, one can have partially filled or fully filled bands in the steady-state, leading to an entanglement entropy undergoing a filling-driven transition between critical sub volume scaling and area-law, similar to ground-state problems. Conversely, when the spectrum is fully real, we provide evidence that local observables can be captured using a diagonal ensemble, and the entanglement entropy exhibits a volume-law scaling independently on the initial state, akin to unitary dynamics. We illustrate these principles in the Hatano-Nelson model with periodic boundary conditions and the non-Hermitian Su-Schrieffer-Heeger model, uncovering a rich interplay between the single-particle spectrum and conservation laws in determining the steady-state structure and the entanglement transitions. These conclusions are supported by exact analytical calculations and numerical calculations relying on the Faber polynomial method.

Figures (15)

• Figure 1: Sketch of the steady-state occupation of single-particle energy levels by class. For simplicity, we consider the case of a two-band single-particle Hamiltonian. The black balls represent the initial distribution. This should be interpreted as the expectation value of $\hat{n}_k$ for class B and class C.
• Figure 2: Scaling of steady-state entanglement entropy, depending on the spectrum of the non-hermitian Hamiltonian and the structure of the steady state. In the band occupation column, we illustrate the case of a two-band model for simplicity.
• Figure 3: Time evolution of the expectation value of $\left\langle \hat{n}_{k} \right\rangle$ distribution. In panel $a)$, the initial state corresponds to a charge density wave, $\left.\lvert \Psi_0 \right\rangle= \left.\lvert 101\cdots101 \right\rangle$, and so it is in class B (corresponding to an initial filling equal to $1/2$), while in panel $b)$ the initial state belongs to class C. Other parameters: $L=128$ and $\gamma=0.4J$.
• Figure 4: Time evolution of the entanglement entropy for different system sizes, starting from initial states in class B (panel $a)$) and class C (panel $b)$). The inset in both panel shows the steady-state entanglement entropy as a function of the total system size. The scaling behavior is found to be $S(+\infty) \propto a_2 \ln(L) +\mathcal{O}\left( 1\right)$, with $a_2 = 0.321 \pm 0.05$ in panel a and $a_2 = 0.3336 \pm 0.0005$ in panel $b)$. This is consistent with the analytical prediction $a_2 = 1/3$ for the thermodynamic limit. Other parameters: $\gamma = 0.4J$.
• Figure 5: Scheme of the non-Hermitian SSH model: Each unit cell is enclosed within a grey area. The intra-cell hopping is given by $-J-h/2$, and the inter-cell hopping is given by $-J+h/2$. The red sites (A sublattice) have a local gain term, $i\gamma$, while the blue sites (B sublattice) have local damping, $-i\gamma$.