arXiv subjects

Paul Cazeaux

Publications and source records attributed to Paul Cazeaux.

16 recordsLinked to original sources

arXiv2026

Linear-Scaling Tensor Train Sketching

We introduce the TTStack sketch, a structured random projection tailored to the tensor train (TT) format that unifies existing TT-adapted sketching operators. By varying two integer parameters $P$ and $R$, TTStack interpolates between the Khatri-Rao sketch ($R=1$) and the Gaussian TT sketch ($P=1$). We prove that TTStack satisfies an oblivious subspace embedding (OSE) property with parameters $R = \mathcal{O}(d(r+\log 1/\delta))$ and $P = \mathcal{O}(\varepsilon^{-2})$, and an oblivious subspace injection (OSI) property under the condition $R = \mathcal{O}(d)$ and $P = \mathcal{O}(\varepsilon^{-2}(r + \log r/\delta))$. Both guarantees depend only linearly on the tensor order $d$ and on the subspace dimension $r$, in contrast to prior constructions that suffer from exponential scaling in $d$. As direct consequences, we derive quasi-optimal error bounds for the QB factorization and randomized TT rounding. The theoretical results are supported by numerical experiments on synthetic tensors, Hadamard products, and a quantum chemistry application.

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↗

arXiv2023

Mathematical models of topologically protected transport in twisted bilayer graphene

Twisted bilayer graphene gives rise to large moiré 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↗

arXiv2023

Relaxation and domain wall structure of bilayer moire systems

Moire patterns result from setting a 2D material such as graphene on another 2D material with a small twist angle or from the lattice mismatch of 2D heterostructures. We present a continuum model for the elastic energy of these bilayer moire structures that includes an intralayer elastic energy and an interlayer misfit energy that is minimized at two stackings (disregistries). We show by theory and computation that the displacement field that minimizes the global elastic energy subject to a global boundary constraint gives large alternating regions of one of the two energy-minimizing stackings separated by domain walls. We derive a model for the domain wall structure from the continuum bilayer energy and give a rigorous asymptotic estimate for the structure. We also give an improved estimate for the L2-norm of the gradient on the moire unit cell for twisted bilayers that scales at most inversely linearly with the twist angle, a result which is consistent with the formation of one-dimensional domain walls with a fixed width around triangular domains at very small twist angles.

math-ph↗

arXiv2022

Non-Abelian topological defects and strain mapping in 2D moiré materials

We present a general method to analyze the topological nature of the domain boundary connectivity that appeared in relaxed moiré superlattice patterns at the interface of 2-dimensional (2D) van der Waals (vdW) materials. At large enough moiré lengths, all moiré systems relax into commensurated 2D domains separated by networks of dislocation lines. The nodes of the 2D dislocation line network can be considered as vortex-like topological defects. We find that a simple analogy to common topological systems with an $S^1$ order parameter, such as a superconductor or planar ferromagnet, cannot correctly capture the topological nature of these defects. For example, in twisted bilayer graphene, the order parameter space for the relaxed moiré system is homotopy equivalent to a punctured torus. Here, the nodes of the 2D dislocation network can be characterized as elements of the fundamental group of the punctured torus, the free group on two generators, endowing these network nodes with non-Abelian properties. Extending this analysis to consider moiré patterns generated from any relative strain, we find that antivortices occur in the presence of anisotropic heterostrain, such as shear or anisotropic expansion, while arrays of vortices appear under twist or isotropic expansion between vdW materials. Experimentally, utilizing the dark field imaging capability of transmission electron microscopy (TEM), we demonstrate the existence of vortex and antivortex pair formation in a moiré system, caused by competition between different types of heterostrains in the vdW interfaces. We also present a methodology for mapping the underlying heterostrain of a moiré structure from experimental TEM data, which provides a quantitative relation between the various components of heterostrain and vortex-antivortex density in moiré systems.

cond-mat.mes-hall↗

arXiv2021

Randomized algorithms for rounding in the Tensor-Train format

The Tensor-Train (TT) format is a highly compact low-rank representation for high-dimensional tensors. TT is particularly useful when representing approximations to the solutions of certain types of parametrized partial differential equations. For many of these problems, computing the solution explicitly would require an infeasible amount of memory and computational time. While the TT format makes these problems tractable, iterative techniques for solving the PDEs must be adapted to perform arithmetic while maintaining the implicit structure. The fundamental operation used to maintain feasible memory and computational time is called rounding, which truncates the internal ranks of a tensor already in TT format. We propose several randomized algorithms for this task that are generalizations of randomized low-rank matrix approximation algorithms and provide significant reduction in computation compared to deterministic TT-rounding algorithms. Randomization is particularly effective in the case of rounding a sum of TT-tensors (where we observe 20x speedup), which is the bottleneck computation in the adaptation of GMRES to vectors in TT format. We present the randomized algorithms and compare their empirical accuracy and computational time with deterministic alternatives.

math.NA↗

arXiv2020

Moiré of Moiré: Modeling Mechanical Relaxation in Incommensurate Trilayer van der Waals Heterostructures

The incommensurate stacking of multi-layered two-dimensional materials is a challenging problem from a theoretical perspective and an intriguing avenue for manipulating their physical properties. Here we present a multi-scale model to obtain the mechanical relaxation pattern of twisted trilayer van der Waals (vdW) heterostructures with two independent twist angles, a generally incommensurate system without a supercell description. We adopt the configuration space as a natural description of such incommensurate layered materials, based on the local environment of atomic positions, bypassing the need for commensurate approximations. To obtain the relaxation pattern, we perform energy minimization with respect to the relaxation displacement vectors. We use a continuum model in combination with the Generalized Stacking Fault energy to describe the interlayer coupling, obtained from first-principles calculations based on Density Functional Theory. We show that the relaxation patterns of twisted trilayer graphene and $\mathrm{WSe_2}$ are "moiré of moiré", as a result of the incommensurate coupling two bilayer moiré patterns. We also show that, in contrast to the symmetry-preserving in-plane relaxation in twisted bilayers, trilayer relaxation can break the two-fold rotational symmetry about the xy-plane when the two twist angles are equal.

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↗

arXiv2018

Atomic and electronic reconstruction at van der Waals interface in twisted bilayer graphene

Control of the interlayer twist angle in two-dimensional (2D) van der Waals (vdW) heterostructures enables one to engineer a quasiperiodic moiré superlattice of tunable length scale. In twisted bilayer graphene (TBG), the simple moiré superlattice band description suggests that the electronic band width can be tuned to be comparable to the vdW interlayer interaction at a 'magic angle', exhibiting strongly correlated behavior. However, the vdW interlayer interaction can also cause significant structural reconstruction at the interface by favoring interlayer commensurability, which competes with the intralayer lattice distortion. Here we report the atomic scale reconstruction in TBG and its effect on the electronic structure. We find a gradual transition from incommensurate moiré structure to an array of commensurate domain structures as we decrease the twist angle across the characteristic crossover angle, $θ_c$ ~1°. In the twist regime smaller than $θ_c$ where the atomic and electronic reconstruction become significant, a simple moiré band description breaks down. Upon applying a transverse electric field, we observe electronic transport along the network of one-dimensional (1D) topological channels that surround the alternating triangular gapped domains, providing a new pathway to engineer the system with continuous tunability.

cond-mat.mtrl-sci↗

arXiv2018

Relaxation and Domain Formation in Incommensurate 2D Heterostructures

We introduce configuration space as a natural representation for calculating the mechanical relaxation patterns of incommensurate two-dimensional (2D) bilayers, bypassing supercell approximations to encompass aperiodic relaxation patterns. The approach can be applied to a wide variety of 2D materials through the use of a continuum model in combination with a generalized stacking fault energy for interlayer interactions. We present computational results for small-angle twisted bilayer graphene and molybdenum disulfide (MoS$_2$), a representative material of the transition metal dichalcogenide (TMDC) family of 2D semiconductors. We calculate accurate relaxations for MoS$_2$ even at small twist-angle values, enabled by the fact that our approach does not rely on empirical atomistic potentials for interlayer coupling. The results demonstrate the efficiency of the configuration space method by computing relaxations with minimal computational cost for twist angles down to $0.05^\circ$, which is smaller than what can be explored by any available real space techniques. We also outline a general explanation of domain formation in 2D bilayers with nearly-aligned lattices, taking advantage of the relationship between real space and configuration space.

cond-mat.mes-hall↗

arXiv2018

Quantum plasmons with optical-range frequencies in doped few-layer graphene

Although plasmon modes exist in doped graphene, the limited range of doping achieved by gating restricts the plasmon frequencies to a range that does not include visible and infrared. Here we show, through the use of first-principles calculations, that the high levels of doping achieved by lithium intercalation in bilayer and trilayer graphene shift the plasmon frequencies into the visible range. To obtain physically meaningful results, we introduce a correction of the effect of plasmon interaction across the vacuum separating periodic images of the doped graphene layers, consisting of transparent boundary conditions in the direction perpendicular to the layers; this represents a significant improvement over the Exact Coulomb cutoff technique employed in earlier works. The resulting plasmon modes are due to local field efffects and the non-local response of the material to external electromagnetic fields, requiring a fully quantum mechanical treatment. We describe the features of these quantum plasmons, including the dispersion relation, losses and field localization. Our findings point to a strategy for fine-tuning the plasmon frequencies in graphene and other two dimensional materials.

cond-mat.mes-hall↗

arXiv2016

Generalized Kubo Formulas for the Transport Properties of Incommensurate 2D Atomic Heterostructures

We give an exact formulation for the transport coefficients of incommensurate two-dimensional atomic multilayer systems in the tight-binding approximation. This formulation is based upon the C* algebra framework introduced by Bellissard and collaborators to study aperiodic solids (disordered crystals, quasicrystals, and amorphous materials), notably in the presence of magnetic fields (quantum Hall effect). We also present numerical approximations and test our methods on a one-dimensional incommensurate bilayer system.

math-ph↗

arXiv2016

Twistronics: Manipulating the Electronic Properties of Two-dimensional Layered Structures through their Twist Angle

The ability in experiments to control the relative twist angle between successive layers in two-dimensional (2D) materials offers a new approach to manipulating their electronic properties; we refer to this approach as "twistronics". A major challenge to theory is that, for arbitrary twist angles, the resulting structure involves incommensurate (aperiodic) 2D lattices. Here, we present a general method for the calculation of the electronic density of states of aperiodic 2D layered materials, using parameter-free hamiltonians derived from ab initio density-functional theory. We use graphene, a semimetal, and MoS$_2$, a representative of the transition metal dichalcogenide (TMDC) family of 2D semiconductors, to illustrate the application of our method, which enables fast and efficient simulation of multi-layered stacks in the presence of local disorder and external fields. We comment on the interesting features of their Density of States (DoS) as a function of twist-angle and local configuration and on how these features can be experimentally observed.

cond-mat.mes-hall↗

arXiv2016

Projective multiscale time-integration for electrostatic particle-in-cell methods

The simulation of problems in kinetic plasma physics are often challenging due to strongly coupled phenomena across multiple scales. In this work, we propose a wavelet-based coarse-grained numerical scheme, based on the framework of Equation-Free Projective Integration, for a kinetic plasma system modeled by the Vlasov-Poisson equations. A kinetic particle-in-cell (PIC) code is used to simulate the meso scale dynamics for short time intervals. This allows the extrapolation over long time-steps of the behavior of a coarse wavelet-based discretization of the system. To validate the approach and the underlying concepts, we perform two 1D1V numerical experiments: nonlinear propagation and steepening of an ion wave, and the expansion of a plasma slab in vacuum. The direct comparisons to resolved PIC simulations show good agreement. We show that the speedup of the projective integration scheme over the full particle scheme scales linearly with the system size, demonstrating efficiency while taking into account fully kinetic, non-Maxwellian effects. This suggests that the approach is potentially interesting for kinetic plasma problems with a large separation of scales, especially in higher dimensions.

physics.comp-ph↗

arXiv2016

Analysis of rippling in incommensurate one-dimensional coupled chains

Graphene and other recently developed 2D materials exhibit exceptionally strong in-plane stiffness. Relaxation of few-layer structures, either free-standing or on slightly mismatched substrates occurs mostly through out-of-plane bending and the creation of large-scale ripples. In this work, we present a novel double chain model, where we allow relaxation to occur by bending of the incommensurate coupled system of chains. As we will see, this model can be seen as a new application of the well-known Frenkel-Kontorova model for a one-dimensional atomic chain lying in a periodic potential. We focus in particular on modeling and analyzing ripples occurring in ground state configurations, as well as their numerical simulation.

math.NA↗

arXiv2013

Application of Hierarchical Matrix Techniques To The Homogenization of Composite Materials

In this paper, we study numerical homogenization methods based on integral equations. Our work is motivated by materials such as concrete, modeled as composites structured as randomly distributed inclusions imbedded in a matrix. We investigate two integral reformulations of the corrector problem to be solved, namely the equivalent inclusion method based on the Lippmann-Schwinger equation, and a method based on boundary integral equations. The fully populated matrices obtained by the discretization of the integral operators are successfully dealt with using the H-matrix format.

math.AP↗