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arXiv:2608.09932

Cell Natural Orbitals in Interacting Topological Bands

Nishchhal Verma, Harshitra Mahalingam, Daniel Muñoz‐Segovia, Raquel Queiroz

Published Aug 10, 2026• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2608.09932

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Dynamics and Phase Transitions in Interacting Electron Systems

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This dissertation studies interacting electron models for quantum matter from two perspectives with different approaches respectively. The first part of this text focuses on rigorously proving Lieb-Robinson bounds for continuous space integer quantum Hall systems perturbed by short range two-body interactions. A Lieb-Robinson bound is an operator norm inequality that implies a finite rate of information propagation in a quantum system, a low energy analogue to the relativistic limit given by the speed of light. Many results of this nature have been obtained for lattice systems, but over the last several years, progress has been made in identifying classes of continuous space models that admit propagation estimates. In this thesis, we prove Lieb-Robinson bounds for two-dimensional continuum fermions in a constant, perpendicular magnetic field using the strategy of Gebert et al. \cite{gebert:2020}, and use this result to prove strong continuity of the Heisenberg dynamics in the limit where inter-particle interactions occur in infinite volume. We then shift to a different framework, namely that of lattice-localized frames, to represent a class of interacting quantum Hall systems in continuous space, as introduced by Bachmann and De Nittis, \cite{bachmann2025lieb}. With this framework, we define a family of parametrized Hamiltonians and construct a continuum analogue of the Hastings-Wen quasi-adiabatic evolution, i.e. dynamics in parameter space that satisfy Lieb-Robinson bounds themselves. Moreover, we then directly apply this to prove quantization of Hall conductance for these models. The new results of these sections are based on works in preparation, \cite{hingorani2026lieb,hingorani2026quantization}. The second portion of this dissertation focuses on two specific materials and phases: magic angle twisted bilayer graphene (MATBG) and superconducting lanthanum nickel gallium-2 (LaNiGa$_2$). MATBG has garnered a tremendous amount of attention since works such as \cite{MB} and \cite{tarnopolsky2019origin}, which first suggested the existence of flat bands in the electronic band structure of twisted bilayer graphene: bilayer graphene with a relative twist between the layers. Astonishingly, this phenomenon occurs for specific twist angles dubbed magic angles. This quenching of kinetic energy sets the stage for electron-electron interactions to dominate the behavior of the material. The onset of Mott insulating states at several carrier densities as seen through peaks in inverse compressibility \cite{expts-b} drew our interest. In collaboration with Oitmaa and Singh, \cite{hingorani2022onset}, we study the SU(4) Fermi-Hubbard model as a candidate effective model for describing the low energy behavior of MATBG, where we uncover the onset of charge incompressibility and Mott gaps, indicating the existence of the insulating phase.Additionally, we investigate the superconducting pairing mechanism of LaNiGa$_2$, which has been highlighted as a potential time-reversal symmetry breaking (TRSB) material below its critical temperature due to experimental evidence for internal magnetization, \cite{Ghosh2020}, \cite{Hillier2012}, \cite{Sundar2024}. In our recent work, \cite{tsnb-1mxc}, we report the existence of a Hebel-Slichter (HS) peak in the nuclear-spin relaxation rate, an effect attributed to the divergence in the electronic density of states as the superconducting gap opens. Furthermore, we study models with triplet and singlet spin pairing respectively to ascertain a mechanism that reproduces a HS peak. In doing so, we find that two distinct superconducting gaps and TRSB may not coexist, which raises questions about whether LaNiGa$_2$ really does break time reversal symmetry or whether a different model is required to fully characterize this material in its superconducting phase.

•Jun 25, 2026•0 citations
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 fff-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.
Understanding correlated quantum matter starts with an accurate model of the single-particle states that interact at low energies: their dispersion, band geometry, orbital content and charge density. In many cases, notably the topological bands of moire materials, it is not straightforward to find a real-space description with a few local orbitals that accomplishes this task. Here we provide a systematic way to identify the local degrees of freedom that best capture the band geometry and charge density of any chosen set of bands. We use the unit-cell one-particle reduced density matrix (UC-1pRDM), obtained by restricting the projector onto the target bands to a single unit cell. Its eigenstates, which we call cell natural orbitals (CNOs), form a local, symmetric basis uniquely determined by the Bloch wavefunctions and the choice of real-space partition. Their eigenvalues measure the occupation of each CNO in the target bands, quantifying entanglement across unit-cell boundaries and the importance of multi-orbital character. A set of CNOs that maximizes total spectral weight and reproduces the target band symmetries provides optimal trial states for Wannierization. We exemplify this by constructing a lattice model for twisted bilayer WSe2_22​ that tracks the orbital content across twist angles.
Mott physics in topological bands has recently attracted considerable attention, particularly in the context of twisted bilayer graphene (TBG). However, the essential ingredients for stabilizing this physics remain unclear. Here, we demonstrate a quantum Mott semimetal phase as the ground state within a one-dimensional spinful Hubbard model featuring only one orbital per unit cell, protected by inversion and particle-hole symmetries. We start from a two-orbital model where a localized fff orbital on the A sublattice hybridizes with a delocalized ccc orbital on the B sublattice. Projecting the fff-orbital Hubbard UUU onto the active flat band yields a lattice model with Wannier orbitals centered on the B sublattice. Similar to TBG, a momentum-space scale k∗k_*k∗​ emerges, setting the interaction range in the projected model to 1/k∗1/k_*1/k∗​. While the ground state is ferromagnetic with only the Hubbard UUU, introducing an inter-site antiferromagnetic spin coupling JJJ stabilizes a Mott semimetal∗^*∗ phase with a central charge c=3c=3c=3. Using exact diagonalization (ED) and density matrix renormalization group (DMRG) methods, we show that this phase hosts a spinful Dirac fermion coexisting with a neutral spin mode -- analogous to the fractionalized Fermi liquid (FL∗^*∗) phase in higher dimensions. Furthermore, breaking particle-hole (PH) symmetry via dispersion transforms the Mott semimetal into a Mott insulator, which is separated from a distinct Mott insulating phase by a continuous transition with a polarization jump of 1/21/21/2. Our work provides the first unbiased evidence of a Mott semimetal ground state and demonstrates that this 1D model captures some essential aspects of TBG physics, despite lacking a Wannier obstruction.
Motivated by the higher Chern bands of twisted graphene multilayers, we consider flat bands with arbitrary Chern number C with ideal quantum geometry. While C > 1 bands differ from Landau levels, we show that these bands host exact fractional Chern insulator (FCI) ground states for short-range interactions. We show how to decompose ideal higher Chern bands into separate ideal bands with Chern number 1 that are intertwined through translation and rotation symmetry. The decomposed bands admit an SU ( C ) action that combines real space and momentum space translations. Remarkably, they also allow for analytic construction of exact many-body ground states, such as generalized quantum Hall ferromagnets and FCIs, including flavor-singlet Halperin states and Laughlin ferromagnets in the limit of short-range interactions. In this limit, the SU ( C ) action is promoted to a symmetry on the ground-state subspace. While flavor singlet states are translation symmetric, the flavor ferromagnets correspond to translation broken states and admit charged skyrmion excitations corresponding to a spatially varying density wave pattern. We confirm our analytic predictions with numerical simulations of ideal bands of twisted chiral multilayers of graphene, and discuss consequences for experimentally accessible systems such as monolayer graphene twisted relative to a Bernal bilayer. Published by the American Physical Society 2023
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/teffU/{t}_{\mathrm{eff}}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.
The rich phenomenology of twisted bilayer graphene (TBG) near the magic angle is believed to arise from electron correlations in topological flat bands.An unbiased approach to this problem is highly desirable, but also particularly challenging, given the multiple electron flavors, the topological obstruction to defining tight-binding models, and the long-ranged Coulomb interactions.While numerical simulations of realistic models have thus far been confined to zero temperature, typically excluding some spin or valley species, analytic progress has relied on fixed point models away from the realistic limit.Here, we present unbiased Monte Carlo simulations of realistic models of magic-angle TBG at charge neutrality.We establish the absence of a sign problem for this model in a momentumspace approach and describe a computationally tractable formulation that applies even on breaking chiral symmetry and including band dispersion.Our results include (i) the emergence of an insulating Kramers intervalley coherent ground state in competition with a correlated semimetal phase, (ii) detailed temperature evolution of order parameters and electronic spectral functions that reveal a "pseudogap" regime, in which gap features are established at a higher temperature than the onset of order, and (iii) predictions for electronic tunneling spectra and their evolution with temperature.Our results pave the way towards uncovering the physics of magic-angle graphene through exact simulations of over a hundred electrons across a wide temperature range.
This dissertation studies interacting electron models for quantum matter from two perspectives with different approaches respectively. The first part of this text focuses on rigorously proving Lieb-Robinson bounds for continuous space integer quantum Hall systems perturbed by short range two-body interactions. A Lieb-Robinson bound is an operator norm inequality that implies a finite rate of information propagation in a quantum system, a low energy analogue to the relativistic limit given by the speed of light. Many results of this nature have been obtained for lattice systems, but over the last several years, progress has been made in identifying classes of continuous space models that admit propagation estimates. In this thesis, we prove Lieb-Robinson bounds for two-dimensional continuum fermions in a constant, perpendicular magnetic field using the strategy of Gebert et al. \cite{gebert:2020}, and use this result to prove strong continuity of the Heisenberg dynamics in the limit where inter-particle interactions occur in infinite volume. We then shift to a different framework, namely that of lattice-localized frames, to represent a class of interacting quantum Hall systems in continuous space, as introduced by Bachmann and De Nittis, \cite{bachmann2025lieb}. With this framework, we define a family of parametrized Hamiltonians and construct a continuum analogue of the Hastings-Wen quasi-adiabatic evolution, i.e. dynamics in parameter space that satisfy Lieb-Robinson bounds themselves. Moreover, we then directly apply this to prove quantization of Hall conductance for these models. The new results of these sections are based on works in preparation, \cite{hingorani2026lieb,hingorani2026quantization}. The second portion of this dissertation focuses on two specific materials and phases: magic angle twisted bilayer graphene (MATBG) and superconducting lanthanum nickel gallium-2 (LaNiGa2_22​). MATBG has garnered a tremendous amount of attention since works such as \cite{MB} and \cite{tarnopolsky2019origin}, which first suggested the existence of flat bands in the electronic band structure of twisted bilayer graphene: bilayer graphene with a relative twist between the layers. Astonishingly, this phenomenon occurs for specific twist angles dubbed magic angles. This quenching of kinetic energy sets the stage for electron-electron interactions to dominate the behavior of the material. The onset of Mott insulating states at several carrier densities as seen through peaks in inverse compressibility \cite{expts-b} drew our interest. In collaboration with Oitmaa and Singh, \cite{hingorani2022onset}, we study the SU(4) Fermi-Hubbard model as a candidate effective model for describing the low energy behavior of MATBG, where we uncover the onset of charge incompressibility and Mott gaps, indicating the existence of the insulating phase.Additionally, we investigate the superconducting pairing mechanism of LaNiGa2_22​, which has been highlighted as a potential time-reversal symmetry breaking (TRSB) material below its critical temperature due to experimental evidence for internal magnetization, \cite{Ghosh2020}, \cite{Hillier2012}, \cite{Sundar2024}. In our recent work, \cite{tsnb-1mxc}, we report the existence of a Hebel-Slichter (HS) peak in the nuclear-spin relaxation rate, an effect attributed to the divergence in the electronic density of states as the superconducting gap opens. Furthermore, we study models with triplet and singlet spin pairing respectively to ascertain a mechanism that reproduces a HS peak. In doing so, we find that two distinct superconducting gaps and TRSB may not coexist, which raises questions about whether LaNiGa2_22​ really does break time reversal symmetry or whether a different model is required to fully characterize this material in its superconducting phase.