Topological bands exhibit obstruction to exponentially localized and symmetric Wannier functions, challenging the standard paradigm of representing projected interactions in terms of local orbitals with finite range. To faithfully capture the form factors and quantum geometry of topological bands we introduce a singular-value decomposition of the band-projected density form factors, enabling a geometry-based truncation scheme of the Hilbert space, exposing an intrinsic hierarchy on band-projected interactions that is determined by the underlying wavefunctions. This decomposition is most naturally described in terms of Cell Natural Orbitals (CNOs), as the eigenstates of the unit-cell reduced one-particle density matrix, whose occupation provide a measure of the minimal orbital complexity required to faithfully represent the band wavefunctions overlaps. The CNO decomposition identifies systematically the minimal number of local orbitals needed to reproduce short-ranged interactions while resolving the hierarchy of interaction strengths across CNO channels. Applied to magic-angle twisted bilayer graphene in the chiral limit, we find that the dominant CNO is centered at the AA site, resembling the
f-fermion of the heavy-fermion model. The subdominant CNO channels carry progressively weaker interaction matrix elements, allowing them to be treated at the static mean-field level, while the dominant channel requires a dynamical self-energy. The formalism illustrates how variations of charge density within the unit cell generate momentum dependence in the CNO envelope function and, consequently, dispersion in the single-particle spectral function. More broadly, our results establish CNOs as a geometry-informed bridge between band topology and real-space correlations, providing a systematic framework for analyzing interactions and emergent phases in quantum materials.
Typical Wannier-function downfolding starts with a mean-field or density functional set of bands to construct the Wannier functions. Here, we carry out a controlled approach, using density matrix renormalization group computed natural orbital bands, to downfold the three-band Hubbard model to an effective single-band model. A sharp drop-off in the natural orbital occupancy at the edge of the first band provides a clear justification for a single-band model. Constructing Wannier functions from the first band, we compute all possible two-particle terms and retain those with significant magnitude. The resulting single-band model includes two-site density-assisted hopping terms with ${t}_{n}\ensuremath{\sim}0.6t$. These terms lead to a reduction of the ratio
U/teff, and are important in capturing the doping-asymmetric carrier mobility, as well as in enhancing the pairing in a single-band model for the hole-doped cuprates.
In a previous work [Phys. Rev. B 77, 085122 (2008)], a procedure for constructing low-energy models of electrons in solids was proposed. The procedure starts with dividing the Hilbert space into two subspaces: the low-energy part (``
d space'') and the rest of the space (``
r space''). The low-energy model is constructed for the
d space by eliminating the degrees of freedom of the
r space. The thus derived model contains the strength of electron correlation expressed by a partially screened Coulomb interaction, calculated in the constrained random-phase approximation (cRPA), where screening channels within the
d space,
Pd, are subtracted. One conceptual problem of this established downfolding method is that for entangled bands it is not clear how to cut out the
d space and how to distinguish
Pd from the total polarization. Here, we propose a simple procedure to overcome this difficulty. The
d space is defined to be an isolated set of bands generated from a set of maximally localized Wannier basis, which consequently defines
Pd. The
r subspace is constructed as the complementary space orthogonal to the
d subspace, resulting in two sets of completely disentangled bands. Using these disentangled bands, the effective parameters of the
d space are uniquely determined by the cRPA method. The method is successfully applied to
3d transition metals.
The last several decades have seen significant advances in the theoretical modeling of materials within the fields of solid-state physics and materials science, but many methods commonly applied to this problem struggle to capture strong electron correlation accurately. Recent widespread interest in quantum materials -- where strong correlation plays a crucial role in the quantum effects governing their behavior -- further highlights the need for theoretical methods capable of rigorously treating such correlation. Here, we present a periodic generalization of variational two-electron reduced density matrix (2-RDM) theory, a bootstrapping-type method that minimizes the ground-state energy as a functional of the 2-RDM without relying on the wavefunction. The 2-RDM is computed directly by semidefinite programming with
N-representability conditions, ensuring accurate treatment of strongly correlated electronic systems. By exploiting translational symmetry, we significantly reduce computational scaling, enabling applications to realistic materials-scale systems. Additionally, we introduce an alternative to conventional energy band structures: natural-orbital occupation-number bands, which, being independent of mean-field assumptions, offer deeper insights into electron correlation effects. We demonstrate the effectiveness of this approach by applying the theory to hydrogen chains, molybdenum disulfide, and nickel oxide, showing that natural-orbital occupation bands correctly capture electronic character in regimes where density functional theory fails. This work represents a major step toward accurately describing the electronic structure of quantum materials using reduced density matrices rather than wavefunctions.
Mott insulators, in which strong Coulomb interactions fully localize electrons on single atomic sites, play host to an incredibly rich and exciting array of strongly correlated physics. One can naturally extend this concept to cluster Mott insulators, wherein electrons localize not on single atoms but across clusters of atoms, forming "molecules in solids". The resulting localized degrees of freedom incorporate the full spectrum of electronic degrees of freedom, spin, orbital, and charge. These serve as the building blocks for cluster Mott insulators, and understanding them is an important first step toward understanding the many-body physics that emerges in candidate cluster Mott insulators. Here, we focus on elementary building blocks, neglecting some of the complexity present in real materials which can often obfuscate the underlying principles at play. Through an extensive set of exact theoretical calculations on clusters of varying geometry, number of orbitals, and number of electrons, we uncover some of the basic organizing principles of cluster Mott phases, particularly when interactions dominate and negate a simple single-particle picture. These include, for example, the identification of an additional "cluster Hund's rule", of cluster ground states that are best understood from a purely interacting perspective, and of several localized degrees of freedom which are protected by an unusual combination of discrete spatial or orbital symmetries. Finally, we discuss the impact of adding additional terms, relevant to material candidates, on the phase diagrams presented throughout, as well as the potential next steps in the journey to building a more complete picture of cluster Mott insulators.
Effective descriptions of quantum materials rely on a separation of energy scales that identifies their low-energy degrees of freedom. When kinetic energy, lattice potential, and interactions compete, this hierarchy breaks down, and both the microscopic state and its effective description are unknown. Continuum neural quantum states (NQS) can find the ground state directly without projecting onto prescribed bands, orbitals, or spins, but their microscopic wavefunctions are difficult to interpret in terms of effective variables. Here, we introduce a wavefunction readout that maps the continuum NQS onto site-resolved charge and spin statistics, allowing effective degrees of freedom to emerge from the microscopic solution. Focusing on moiré transition metal dichalcogenides at filling
ν=3, the NQS reveals a low-spin ground state below the ferromagnetic crystal of spin-polarized Wigner molecules. Although the charge density retains triangular molecular motifs, our readout reveals significant charge fluctuations through bound nearest-neighbor doublon--hole pairs. Each molecular charge state has an inert spin-singlet core and rim electrons that carry the spin. Charge and spin fluctuations are therefore intertwined, with antiferromagnetic spin-correlations between sites, motivating an effective description by a spinful two-orbital Hubbard model. More generally, our results demonstrate a route from continuum wavefunctions to effective lattice models when competing scales preclude a controlled reduction.