Cell Natural Orbitals in Quantum Materials

Technical Summary: Cell Natural Orbitals in Quantum Materials

Problem Statement
The accurate modeling of correlated quantum matter requires a minimal description of the relevant low-energy single-particle degrees of freedom. While momentum-space descriptions (Bloch wavefunctions) capture dispersion and symmetry, real-space models are essential for treating local electronic interactions. However, identifying a minimal set of local orbitals that faithfully reproduces the target band's dispersion, symmetry, and quantum geometry (specifically the intracell charge density and dipole matrix elements) is a non-trivial task. Existing methods, such as Maximally Localized Wannier Functions (MLWFs) or disentanglement techniques, often require pre-specifying the number of orbitals or rely on physically motivated initial projections, failing to optimize the minimal local Hilbert space dictated by the bands themselves. Furthermore, topological obstructions (e.g., in Chern bands) or obstructed atomic limits can prevent the existence of a symmetric, localized basis spanning the target manifold using standard approaches.

Methodology
The authors introduce a systematic construction of local orbitals, termed Cell Natural Orbitals (CNOs), derived solely from the projector onto a set of target Bloch bands. The core mathematical object is the Unit-Cell One-Particle Reduced Density Matrix (UC-1pRDM), denoted as $L$.

1. Construction of UC-1pRDM: Let $P$ be the projector onto the target Bloch bands and $\Pi$ be the projector restricting the Hilbert space to a single reference unit cell. The UC-1pRDM is defined as $L = \Pi P \Pi$. In the orbital basis, the matrix elements are given by $L_{\alpha\beta} = \langle c^\dagger_\beta c_\alpha \rangle$, calculated by summing over the target bands and including an embedding matrix $V_k$ that accounts for the real-space position of orbitals within the unit cell.
2. Diagonalization: The eigenstates of $L$, denoted $|\tau_a\rangle$, are the CNOs. Their corresponding eigenvalues, $\lambda_a$, represent the occupation of each CNO within the target bands.
3. Physical Interpretation:
   • Spectral Weight: $\lambda_a$ quantifies the importance of each local orbital. A single $\lambda_1 \approx 1$ implies the band is well-described by a single orbital; fractional occupations indicate a multi-orbital character.
   • Entanglement: The eigenvalues $\lambda_a$ are the singular values of the map between local orbitals and the band subspace. They define the entanglement spectrum across the unit-cell partition, with the von Neumann entropy $S_{cell}$ measuring the failure of the occupied subspace to factorize across the cell boundary.
   • Symmetry and Topology: By choosing a unit-cell partition centered at a specific Wyckoff position $q$, the CNOs transform as irreducible representations (irreps) of the site-symmetry group $G_q$. This allows CNOs to identify candidate Elementary Band Representations (EBRs).
   • Zeros: The envelope functions $s_{a,k}$ (rescaled Bloch coefficients) reveal zeros in the Brillouin zone. Topological obstructions (e.g., non-zero Chern numbers) enforce irremovable zeros, while obstructed atomic limits or incompatible Wyckoff choices lead to symmetry-enforced zeros that can be removed by shifting the partition center.

Key Contributions and Results

• Systematic Basis Selection: The paper provides a method to determine the optimal local orbital basis without prior assumptions about the number of orbitals or their specific atomic character. The CNOs are uniquely determined by the Bloch wavefunctions and the choice of real-space partition.
• Application to Obstructed Atomic Insulators: Using a monolayer 1H-TMD model, the authors demonstrate that choosing a unit-cell partition centered at the atomic site (1a Wyckoff position) yields fractional occupations and symmetry-enforced zeros in the envelope functions, reflecting the obstructed nature of the band. Shifting the partition to the Wannier center (1c Wyckoff position) results in a dominant CNO with $\lambda_1 \approx 1$ and no zeros, correctly identifying the minimal local description.
• Application to Topological Bands: In the spinless Bernevig-Hughes-Zhang (BHZ) model, the CNO occupations vary smoothly across the topological phase transition, but the envelope functions exhibit irremovable zeros in the topological phase ($C=1$), manifesting the Wannier obstruction in real space.
• Construction of Local Models for Twisted Bilayer WSe$_2$: The authors apply the method to twisted bilayer WSe$_2$ (tWSe$_2$) across a range of twist angles ($1^\circ$ to $5^\circ$).
  • They identify that the top two moiré bands require different local descriptions depending on the twist angle and the choice of partition (MM, MX, or XM stacking sites).
  • In the topological regime ($\theta > 1.51^\circ$), no single local orbital spans the bands due to the finite total Chern number.
  • By selecting subsets of CNOs (e.g., a multi-centered set combining MM, MX, and XM orbitals) that maximize total spectral weight and satisfy the completeness condition (non-zero determinant of the overlap matrix $S_k$), they construct tight-binding models.
  • Single-Shot Wannierization: These CNO subsets serve as trial states for a single-shot Wannierization procedure. The resulting tight-binding models reproduce not only the band dispersion but also the quantum geometry (Berry curvature) of the continuum model.
  • Trade-offs: The authors compare an MM-centered model (3 orbitals on one site) with a multi-centered model (orbitals on different sites). The multi-centered model yields more localized Wannier functions (shorter-range hoppings) but requires a larger cluster for symmetry-preserving many-body treatments (e.g., cluster DMFT), whereas the MM-centered model preserves $C_3$ symmetry more easily but has longer-range interactions.

Significance and Claims
The paper claims that CNOs offer a direct, wavefunction-derived measure of the multi-orbital character of target bands. By reversing the conventional approach of inferring orbitals from chemistry or symmetry eigenvalues, CNOs allow the local degrees of freedom to be inferred directly from the band manifold itself.

The significance lies in:

1. Optimality: CNOs provide the optimal trial states for Wannierization, minimizing the number of orbitals required to span the target bands while preserving quantum geometry.
2. Diagnostic Power: The occupation spectrum and envelope function zeros serve as diagnostics for topological obstructions and obstructed atomic limits, distinguishing between removable (symmetry) and irremovable (topological) zeros.
3. Basis for Interacting Models: Since CNOs diagonalize the unit-cell correlation matrix, they provide a natural basis for band-projected interactions. The occupation hierarchy $\lambda_a$ orders the interaction scales, offering a route to constructing multi-orbital Hubbard models and defining impurity spaces for quantum embedding methods (like DMFT) without arbitrary choices.
4. Generality: The method applies to both lattice and continuum models (via a real-space grid derived from the plane-wave basis) and is compatible with first-principles calculations, as it requires only the band projector as input.

The authors conclude that this perspective bridges first-principles wavefunctions and continuum models with the minimal local degrees of freedom required for effective theories of correlated and topological quantum materials.