One-Particle Density Matrix Framework for Mode-Shell Correspondence: Characterizing Topology in Amorphous Higher-Order Topological Insulators

Technical Summary: A One-Particle Density Matrix Framework for Mode-Shell Correspondence in Amorphous Higher-Order Topological Insulators

Problem Statement
The characterization of topological phases, particularly higher-order topological insulators (HOTIs), traditionally relies on Hamiltonians and momentum-space invariants (e.g., Chern numbers, Wilson loops). However, topology is fundamentally a property of the quantum state itself, not merely the Hamiltonian that generates it. This poses challenges for systems lacking translational invariance, such as amorphous or disordered matter, where momentum-space definitions fail. Furthermore, distinguishing between "intrinsic" HOTIs (where bulk topology enforces boundary modes regardless of termination) and "extrinsic" HOTIs (where modes depend on specific boundary terminations) remains difficult in disordered settings. Existing methods often require an underlying Hamiltonian, limiting their applicability to interacting states or complex disordered geometries where a simple Hamiltonian description may be obscured or unavailable.

Methodology
The authors propose a framework based entirely on the one-particle density matrix ($\rho$), which encodes all single-particle correlations of a many-body Gaussian state. The core of their approach is a reformulation of the mode-shell correspondence, originally defined for single-particle Hamiltonians, directly in terms of $\rho$.

1. Density Matrix Formalism: The study utilizes the Bogoliubov–de Gennes (BdG) one-particle density matrix, $\rho$, which is a projector onto occupied bands for gapped Gaussian states. The framework assumes $\rho$ is exponentially localized in real space.
2. Mode Index ($I_{\text{mode}}$): Defined by restricting $\rho$ to a spatial region $A$ containing a boundary (e.g., a chain end or corner). The index counts chiral zero modes by summing contributions from eigenvalues $\lambda$ of the restricted matrix $\rho_A$ that are maximally entangled ($\lambda \approx 1/2$) across the boundary of $A$.
   $$I_{\text{mode}} = 4\text{Tr}((\rho_A - \rho_A^2)\theta C)$$
   where $C$ is the chiral symmetry operator and $\theta$ is a smooth cutoff function.
3. Shell Index ($I_{\text{shell}}$): To isolate bulk topology from trivial edge correlations, the authors introduce a "shell" region $A_{\text{shell}} \subset A$ surrounding the boundary modes but separated from the physical edge by a distance larger than the correlation length. The shell index is derived by rewriting the mode index as a commutator localized on this shell:
   $$I_{\text{shell}} = -2\text{Tr}(C\rho_A[\rho_A, \theta])$$
   In one dimension, this reduces to the local chiral marker, recovering the winding number in the translation-invariant limit.
4. Application to Disordered Systems: The authors apply this formalism to the Benalcazar-Bernevig-Hughes (BBH) model, a $C_4$-symmetric HOTI. They generate amorphous structures by perturbing a square lattice with Gaussian noise while enforcing $C_4$ rotational symmetry through a four-fold replication and rotation scheme. The indices are computed directly in real space without reference to momentum space.

Key Contributions and Results

• Hamiltonian-Free Characterization: The paper establishes that the mode-shell correspondence can be derived solely from the one-particle density matrix. This allows for the characterization of topological phases directly from the quantum state, applicable to both non-interacting and interacting systems (provided the occupation spectrum remains gapped).
• Intrinsic vs. Extrinsic Diagnosis: By applying the framework to the 2D BBH model, the authors demonstrate that the shell index provides a diagnostic for the nature of the HOTI. They show that an odd mode index leads to half-integer contributions to the shell index at each symmetry-related boundary intersection (e.g., corners). This fractional quantization implies the phase is intrinsic, as it cannot be trivialized by changing edge terminations. Conversely, an even mode index yields integer contributions, which may correspond to extrinsic phases.
• Robustness in Amorphous Matter: The framework is successfully applied to $C_4$-symmetric amorphous structures. While structural disorder increases the localization length of edge modes and introduces non-zero local bulk contributions to the shell index density, the total shell index remains quantized (approaching 1.0 in their numerical examples). The authors show that bulk contributions cancel out upon averaging over large distances, confirming that the mode-shell correspondence remains a valid diagnostic in the absence of translational invariance.
• Generalization to Interacting States: The authors note that if an interacting state has a gapped one-particle density matrix, it can be adiabatically flattened into a projector. The mode-shell correspondence applied to this flattened matrix defines the topology of the interacting state, extending the utility of the method beyond Gaussian states.

Significance and Claims
The paper claims to provide a "practical and diverse route" to characterize higher-order topology directly from the quantum state, bypassing the need for an underlying Hamiltonian. Its primary significance lies in:

1. Universality: The ability to define topological invariants for systems without translational symmetry (amorphous/disordered) and for interacting systems with gapped spectra.
2. State-Centric View: Reinforcing that topology is a property of the state itself, encoded in the one-particle density matrix.
3. Diagnostic Power: Offering a concrete method to distinguish intrinsic higher-order phases from extrinsic ones via the fractional quantization of the shell index, a distinction that is difficult to make in disordered systems using traditional indicators.

The authors conclude that while the shell index is not a bulk invariant in higher dimensions (unlike in 1D where it equals the winding number), it serves as a robust boundary diagnostic that generalizes the mode-shell correspondence to complex, disordered, and interacting topological phases.