arXiv subjects

Daniel Massatt

Publications and source records attributed to Daniel Massatt.

At least 19 recordsLinked to original sources

arXiv2026

Relaxation of Incommensurate Structures via Quantum Models

Accurately modeling structural relaxation in incommensurate systems is intrinsically challenging due to the absence of global translational symmetry. In this work, we develop a variational quantum framework for structural relaxation in incommensurate Schr\"{o}dinger models, where displacement fields are formulated on the configuration space and the electronic Hamiltonian is represented in reciprocal space. This yields well-defined relaxed energy, local density of states, and forces through thermodynamic limits. We propose an anisotropic scattering-channel approximation, and prove exponential convergence of the approximate equilibria. Numerical experiments are performed to support the analysis and show that the model captures domain-wall formation and its impact on the electronic spectrum.

physics.comp-ph↗

arXiv2026

Momentum Space Algorithm for Electronic Structure of Double-Incommensurate Trilayer Graphene

Numerical algorithms for computing electronic structure of incommensurate 2D materials using ab initio models is critical for predicting material properties and guiding experiment. For bilayers, momentum space and continuum models have been introduced to approximate observables of ab initio tight-binding models using a momenta description despite the lack of periodicity in the tight-binding model required for Bloch theory. A similar structure has been introduced for double-incommensurate trilayers using a continuum model, where the three lattices are all mutually incommensurate. However, this description leads to a four-dimensional lattice space, and numerical convergence of the density of states was observed to have poor convergence. In this work, we introduce a momentum space framework for double incommensurate trilayer graphene, and introduce an efficient truncation scheme of the four-dimensional lattice to drastically improve convergence of the density of states and momentum local density of states (a parallel object to classical band structure). We implement this algorithm on an ab initio model of twisted trilayer graphene and validate convergence estimates. We further verify numerically that the momentum space algorithm, inherently higher order than the continuum model as it is an exact transformation of the tight-binding model, captures altered band behavior near the flat bands at magic angles.

math.NA↗

arXiv2026

Macroscopic approximation of tight-binding models near spectral degeneracies and validity for wave packet propagation

This paper concerns the derivation and validity of macroscopic descriptions of wave packets supported in the vicinity of degenerate points $(K,E)$ in the dispersion relation of tight-binding models accounting for macroscopic variations. We show that such wave packets are well approximated over long times by macroscopic models with varying orders of accuracy. Our main applications are in the analysis of single- and multilayer graphene tight-binding Hamiltonians modeling macroscopic variations such as those generated by shear or twist. Numerical simulations illustrate the theoretical findings.

math-ph↗

arXiv2025

A high-order regularized delta-Chebyshev method for computing spectral densities

We introduce a numerical method for computing spectral densities, and apply it to the evaluation of the local density of states (LDOS) of sparse Hamiltonians derived from tight-binding models. The approach, which we call the high-order delta-Chebyshev method, can be viewed as a variant of the popular regularized Chebyshev kernel polynomial method (KPM), but it uses a high-order accurate approximation of the $\delta$-function to achieve rapid convergence to the thermodynamic limit for smooth spectral densities. The costly computational steps are identical to those for KPM, with high-order accuracy achieved by an inexpensive post-processing procedure. We apply the algorithm to tight-binding models of graphene and twisted bilayer graphene, demonstrating high-order convergence to the LDOS at non-singular points.

physics.comp-ph↗

arXiv2025

Numerical computation of the density of states of aperiodic multiscale Schr\"odinger operators

Computing the electronic structure of incommensurate materials is a central challenge in condensed matter physics, requiring efficient ways to approximate spectral quantities such as the density of states (DoS). In this paper, we numerically investigate two distinct approaches for approximating the DoS of incommensurate Hamiltonians for small values of the incommensurability parameters $\epsilon$ (e.g., small twist angle, or small lattice mismatch): the first employs a momentum-space decomposition, and the second exploits a semiclassical expansion with respect to $\epsilon$. In particular, we compare these two methods using a 1D toy model. We check their consistency by comparing the asymptotic expansion terms of the DoS, and it is shown that, for full DoS, the two methods exhibit good agreement in the small $\epsilon$ limit, while discrepancies arise for less small $\epsilon$, which indicates the importance of higher-order corrections in the semiclassical method for such regimes. We find these discrepancies to be caused by oscillations in the DoS at the semiclassical analogues of Van Hove singularities, which can be explained qualitatively, and quantitatively for $\epsilon$ small enough, by a semiclassical approach.

math-ph↗

arXiv2025

An Atomic Cluster Expansion Potential for Twisted Multilayer Graphene

Twisted multilayer graphene, characterized by its moir\'e patterns arising from inter-layer rotational misalignment, serves as a rich platform for exploring quantum phenomena. Machine learning interatomic potentials (MLIPs) are a promising approach to model such systems. Our work develops a method to generate training and test datasets for fitting MLIPs that capture all possible misalignments but remain small-scale to facilitate efficient data generation and parameter estimation. To achieve this, we generate configurations with periodic boundary conditions suitable for DFT calculations, and then introduce an internal twist and shift within those supercell structures. Using this technique, supplemented with an active learning workflow, we fit an Atomic Cluster Expansion potential for simulating twisted multilayer graphene and test it for accuracy and robustness on a range of simulation tasks.

physics.comp-ph↗

arXiv2024

Construction and Accuracy of Electronic Continuum Models of Incommensurate Bilayer 2D Materials

Single-particle continuum models such as the popular Bistritzer-MacDonald model have become powerful tools for predicting electronic phenomena of incommensurate 2D materials and the development of many-body models aimed to model unconventional superconductivity and correlated insulators. In this work, we introduce a procedure to construct continuum models of arbitrary accuracy relative to tight-binding models for moir\'{e} incommensurate bilayers. This is done by recognizing the continuum model as arising from Taylor expansions of a high accuracy momentum space approximation of the tight-binding model. We apply our procedure in full detail to two models of twisted bilayer graphene and demonstrate both admit the Bistritzer-MacDonald model as the leading order continuum model, while higher order expansions reveal qualitative spectral differences.

math-ph↗

arXiv2023

Defect bound states in the continuum of bilayer electronic materials without symmetry protection

We analyze a class of bound defect states in the continuum electronic spectrum of bilayer materials, which emerge independent of symmetry protection or additional degrees of freedom. Taking graphene as a prototypical example, our comparative analysis of AA- and AB-stacked bilayer graphene demonstrates that these states originate from the intrinsic algebraic structure of the tight-binding Hamiltonian when trigonal warping is neglected rather than any underlying symmetry. Inclusion of trigonal warping and higher-order hoppings broaden the bound states into long-lived resonances. This discovery provides a pathway to previously unexplored approaches in defect and band-structure engineering. We conclude with a proposed protocol for observing these states in scanning tunneling microscopy experiments.

cond-mat.mes-hall↗

arXiv2022

Mathematical models of topologically protected transport in twisted bilayer graphene

Twisted bilayer graphene gives rise to large moir\'{e} patterns that form a triangular network upon mechanical relaxation. If gating is included, each triangular region has gapped electronic Dirac points that behave as bulk topological insulators with topological indices depending on valley index and the type of stacking. Since each triangle has two oppositely charged valleys, they remain topologically trivial. In this work, we address several questions related to the edge currents of this system by analysis and computation of continuum PDE models. Firstly, we derive the bulk invariants corresponding to a single valley, and then apply a bulk-interface correspondence to quantify asymmetric transport along the interface. Secondly, we introduce a valley-coupled continuum model to show how valleys are approximately decoupled in the presence of small perturbations using a multiscale expansion, and how valleys couple for larger defects. Thirdly, we present a method to prove for a large class of continuum (pseudo-)differential models that a quantized asymmetric current is preserved through a junction such as a triangular network vertex. We support all of these arguments with numerical simulations using spectral methods to compute relevant currents and wavepacket propagation.

math-ph↗

arXiv2022

Convergence of the Planewave Approximations for Quantum Incommensurate Systems

Incommensurate structures arise from stacking single layers of low-dimensional materials on top of one another with misalignment such as an in-plane twist in orientation. While these structures are of significant physical interest, they pose many theoretical challenges due to the loss of periodicity. In this paper, we characterize the density of states of Schr\"{o}dinger operators in the weak sense for the incommensurate system and develop novel numerical methods to approximate them. In particular, we (i) justify the thermodynamic limit of the density of states in the real space formulation; and (ii) propose efficient numerical schemes to evaluate the density of states based on planewave approximations and reciprocal space sampling. We present both rigorous analysis and numerical simulations to support the reliability and efficiency of our numerical algorithms.

math.NA↗

arXiv2021

Electronic Observables for Relaxed Bilayer 2D Heterostructures in Momentum Space

Momentum space transformations for incommensurate 2D electronic structure calculations are fundamental for reducing computational cost and for representing the data in a more physically motivating format, as exemplified in the Bistritzer-MacDonald model. However, these transformations can be difficult to implement in more complex systems such as when mechanical relaxation patterns are present. In this work, we aim for two objectives. Firstly, we strive to simplify the understanding and implementation of this transformation by rigorously writing the transformations between the four relevant spaces, which we denote real space, configuration space, momentum space, and reciprocal space. This provides a straight-forward algorithm for writing the complex momentum space model from the original real space model. Secondly, we implement this for twisted bilayer graphene with mechanical relaxation affects included. We also analyze the convergence rates of the approximations, and show the tight-binding coupling range increases for smaller relative twists between layers, demonstrating that the 3-nearest neighbor coupling of the Bistritzer-MacDonald model is insufficient when mechanical relaxation is included for very small angles. We quantify this and verify with numerical simulation.

math.NA↗

arXiv2021

Multiscale Invariants of Floquet Topological Insulators

This paper analyzes Floquet topological insulators resulting from the time-harmonic irradiation of electromagnetic waves on two dimensional materials such as graphene. We analyze the bulk and edge topologies of approximations to the evolution of the light-matter interaction. Topologically protected interface states are created by spatial modulations of the drive polarization across an interface. In the high-frequency modulation regime, we obtain a sequence of topologies that apply to different time scales. Bulk-difference invariants are computed in detail and a bulk-interface correspondence is shown to apply. We also analyze a high-frequency high-amplitude modulation resulting in a large-gap effective topology topologically that remains valid only for moderately long times.

math-ph↗

arXiv2020

Twisted Trilayer Graphene: a Precisely Tunable Platform for Correlated Electrons

We introduce twisted trilayer graphene (tTLG) with two independent twist angles as an ideal system for the precise tuning of the electronic interlayer coupling to maximize the effect of correlated behaviors. As established by experiment and theory in the related twisted bilayer graphene system, van Hove singularities (VHS) in the density of states can be used as a proxy of the tendency for correlated behaviors. To explore the evolution of VHS in the twist-angle phase space of tTLG, we present a general low-energy electronic structure model for any pair of twist angles. We show that the basis of the model has infinite dimensions even at a finite energy cutoff and that no Brillouin zone exists even in the continuum limit. Using this model, we demonstrate that the tTLG system exhibits a wide range of magic angles at which VHS merge and the density of states has a sharp peak at the charge-neutrality point through two distinct mechanisms: the incommensurate perturbation of twisted bilayer graphene's flat bands or the equal hybridization between two bilayer moiré superlattices.

cond-mat.mes-hall↗

arXiv2020

Modeling and Computation of Kubo Conductivity for 2D Incommensurate Bilayers

This paper presents a unified approach to the modeling and computation of the Kubo conductivity of incommensurate bilayer heterostructures at finite temperature. Firstly, we derive an expression for the large-body limit of Kubo-Greenwood conductivity in terms of an integral of the conductivity function with respect to a current-current correlation measure. We then observe that the incommensurate structure can be exploited to decompose the current-current correlation measure into local contribution and deduce an approximation scheme which is exponentially convergent in terms of domain size. Secondly, we analyze the cost of computing local conductivities via Chebyshev approximation. Our main finding is that if the inverse temperature $β$ is sufficiently small compared to the inverse relaxation time $η$, namely $β\lesssim η^{-1/2}$, then the dominant computational cost is $\mathcal{O}\bigl(η^{-3/2}\bigr)$ inner products for a suitably truncated Chebyshev series, which significantly improves on the $\mathcal{O}\bigl(η^{-2}\bigr)$ inner products required by a naive Chebyshev approximation. Thirdly, we propose a rational approximation scheme for the low temperature regime $η^{-1/2} \lesssim β$, where the cost of the polynomial method increases up to $\mathcal{O}\bigl(β^2\bigr),$ but the rational scheme scales much more mildly with respect to $β$.

math.NA↗

arXiv2020

Duality between atomic configurations and Bloch states in twistronic materials

The relative orientation (twist) of successive layers of stacked two-dimensional (2D) materials creates variations in the interlayer atomic registry. The variations often form a super lattice, called a moiré pattern, which can alter electronic properties. In this work we introduce a classification of the single-particle electronic structures that can occur in twisted stacks of 2D layers by characterizing them as "moiré molecules" or "moiré crystals". The molecules generate localized electronic states and moiré flat bands, while the crystals are sometimes unconventional and produce electronic banding in the configuration basis. The underpinning of this classification is the duality between interlayer configuration and monolayer Bloch momentum in moiré Hamiltonians. We apply this understanding to diagrams of local electron density in untwisted geometries to produce intuitive and quantitative predictions of twistronic properties. We provide a conceptual introduction to this framework through a one-dimensional model, and then apply it to 2D twisted bilayers of the semi-metal graphene, and of MoS$_2$, a representative material of the transition metal dichalcogenide (TMDC) family of semiconductors. This level of thorough understanding of twistronic phenomena is vital in the search for new material platforms for localized moiré electrons.

cond-mat.mes-hall↗

arXiv2019

Efficient Computation of Kubo Conductivity for Incommensurate 2D Heterostructures

Here we introduce a numerical method for computing conductivity via the Kubo Formula for incommensurate 2D bilayer heterostructures using a tight-binding framework. We begin with deriving the momentum space formulation and Kubo Formula from the real space tight-binding model using the appropriate Bloch transformation operator. We further discuss the resulting algorithm along with its convergence rate and computation cost in terms of parameters such as relaxation time and temperature. In particular, we show that for low frequencies, low temperature, and long relaxation times conductivity can be computed very efficiently using momentum space for a wide class of materials. We then demonstrate our method by computing conductivity for twisted bilayer graphene (tBLG) for small twist angles.

cond-mat.mes-hall↗

arXiv2019

Energy minimization of 2D incommensurate heterostructures

We derive and analyze a novel approach for modeling and computing the mechanical relaxation of incommensurate 2D heterostructures. Our approach parametrizes the relaxation pattern by the compact local configuration space rather than real space, thus bypassing the need for the standard supercell approximation and giving a true aperiodic atomistic configuration. Our model extends the computationally accessible regime of weakly coupled bilayers with similar orientations or lattice spacing, for example materials with a small relative twist where the widely studied large-scale moire patterns arise. Our model also makes possible the simulation of multi-layers for which no interlayer empirical atomistic potential exists, such as those composed of MoS2 layers, and more generally makes possible the simulation of the relaxation of multi-layer heterostructures for which a planar moire pattern does not exist.

physics.comp-ph↗

arXiv2019

Angle-Dependent {\it Ab initio} Low-Energy Hamiltonians for a Relaxed Twisted Bilayer Graphene Heterostructure

We present efficient angle-dependent low-energy Hamiltonians to describe the properties of the twisted bilayer graphene (tBLG) heterostructure, based on {\it ab initio} calculations of mechanical relxation and electronic structure. The angle-dependent relaxed atomic geometry is determined by continuum elasticity theory, which induces both in-plane and out-of-plane deformations in the stacked graphene layers. The electronic properties corresponding to the deformed geometry are derived from a Wannier transformation to local interactions obtained from Density Functional Theory calculations. With these {\it ab initio} tight-binding Hamiltonians of the relaxed heterostructure, the low-energy effective theories are derived from the projections near Dirac cones at K valleys. For twist angles ranging from 0.7$^\circ$ to 4$^\circ$, we extract both the intra-layer pseudo-gauge fields and the inter-layer coupling terms in the low-energy Hamiltonians, which extend the conventional low-energy continuum models. We further include the momentum dependent inter-layer scattering terms which give rise to the particle-hole asymmetric features of the electronic structure. Our model Hamiltonians can serve as a starting point for formulating physically meaningful, accurate interacting electron theories.

cond-mat.mes-hall↗