Exceptional flat bands in bipartite non-Hermitian lattices

Technical Summary: Exceptional Flat Bands in Bipartite Non-Hermitian Lattices

Problem Statement
Flat bands, characterized by quenched kinetic energy and macroscopic quantum degeneracy, are known to host exotic correlated and topological phases in Hermitian systems, such as unconventional superconductivity and fractional Chern insulators. While the formation mechanisms of flat bands in Hermitian bipartite lattices are well-established—relying on a mismatch in the number of degrees of freedom between sublattices—the behavior of these states in non-Hermitian (NH) crystals remains an open question. NH systems, relevant to open and driven systems, exhibit unique phenomena like exceptional points (EPs) and the non-Hermitian skin effect (NHSE). It is unclear whether the symmetry-based principles governing Hermitian flat bands extend to the NH regime, or if NH effects can generate new types of dispersionless states with no Hermitian analogue.

Methodology
The authors employ a general symmetry-based framework to construct non-Hermitian bipartite crystalline lattices (BCLs). The methodology proceeds as follows:

1. Hamiltonian Construction: Starting from a Hermitian chiral BCL Hamiltonian ($H^C_k$) with sublattice imbalance ($N_A \neq N_B$), the authors introduce a non-Hermitian deformation via a chiral operator $C$. The deformed Hamiltonian is defined as $H_k = (1 + \alpha C)H^C_k$, where $\alpha \in \mathbb{R}$ quantifies the non-Hermiticity (representing gain/loss or non-reciprocal couplings).
2. Generalization: The framework is extended to include asymmetric chemical potentials ($\mu_A \neq \mu_B$) and arbitrary intra-sublattice hopping ($B_k$), breaking chiral symmetry while maintaining the bipartite structure.
3. Spectral Analysis: The eigenvalues and eigenstates are derived using the singular value decomposition of the inter-sublattice hybridization matrix $S_k$. The authors analyze the behavior of the spectrum as $\alpha$ varies, specifically focusing on the regime where $|\alpha| \geq 1$ (the exceptional point regime).
4. Geometric and Topological Probes: The study evaluates the non-Hermitian quantum geometric tensor (QGT) to determine the quantum metric and Berry curvature, assessing how NH effects alter the geometric properties of the bands.
5. Model Applications: The theoretical framework is applied to specific models, including the Lieb lattice, a ten-band model of twisted bilayer graphene (TBG), and a tight-binding model for the compound $Ca_2Ta_2O_7$.

Key Contributions and Results

1. Extension of Hermitian Principles: The paper demonstrates that the Hermitian mechanism for flat-band formation—based on sublattice degeneracy mismatch ($n_a > N_B$)—extends directly to the NH regime. Whenever one sublattice hosts a momentum-independent eigenvalue with degeneracy exceeding that of its partner, flat bands arise regardless of the specific NH parameters (gain, loss, or complex couplings).
2. Discovery of Exceptional Flat Bands (EFBs): A novel class of states, termed Exceptional Flat Bands (EFBs), is identified. These arise when dispersive bands coalesce at exceptional points (EPs) where $|\alpha| = 1$. Unlike standard flat bands, EFBs:
   • Persist beyond the spectral singularities ($|\alpha| > 1$).
   • Exhibit biorthogonal eigenmodes that span both sublattices.
   • Possess energies and lifetimes that are tunable via sublattice asymmetry and non-reciprocal couplings.
   • Can emerge even in bipartite lattices with imbalanced but constant sublattice chemical potentials, a scenario with no closed-system analogue.
3. Spectral Collapse and Stability: In the presence of sublattice asymmetry, the authors show that dispersive states can collapse onto a dispersionless mode at energy $\bar{\mu}$ (the average chemical potential) when $|\alpha| > 1$. For specific momentum-dependent conditions, these states can become long-lived (infinite lifetime) as the imaginary part of the energy vanishes.
4. Geometric Signatures: While the NH deformation preserves the trivial topological character (vanishing Berry curvature) of the flat bands in chiral models, it induces distinct geometric signatures. The quantum metric exhibits a divergence at high-symmetry points (e.g., the M-point in the Lieb lattice) due to spectral collapse. This divergence is linked to the self-orthogonality of eigenstates at EPs and the blowup of the Petermann factor, suggesting enhanced energy absorption and sensitivity to perturbations.
5. Robustness and Versatility: The framework is shown to be robust against strong on-site asymmetry (as seen in the $Ca_2Ta_2O_7$ model), where the system can maintain real-valued spectra and unitary evolution despite NH deformation. The construction applies to systems with or without chiral symmetry, provided the anticommutation relations between the chiral operator and the momentum-independent terms are satisfied.

Significance and Claims
The paper claims to provide a general prescription for engineering flat bands in open quantum matter. By unifying Hermitian and NH flat-band constructions, the work reveals a new class of dispersionless states (EFBs) that are intrinsic to non-Hermitian physics. The authors assert that this framework is directly applicable to synthetic platforms where gain, loss, and non-reciprocity can be engineered, including:

• Photonic crystals with tailored gain-loss profiles.
• Ultracold atomic arrays with controlled dissipation.
• Classical metamaterials.

The significance lies in the potential to realize interaction-driven and topological phases far from equilibrium that have no Hermitian counterpart. The authors note that while their symmetry-based construction avoids the non-Hermitian skin effect (NHSE) by design, the framework offers a systematic route to explore correlated phenomena in NH systems, such as enhanced density of states and characteristic transport anomalies. The work does not claim to solve the many-body problem in these systems but establishes the single-particle foundation necessary for such phases to emerge.