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Alexander B. Watson

Publications and source records attributed to Alexander B. Watson.

At least 19 recordsLinked to original sources
arXiv2026

Relaxation effects on Hartree-Fock ground states in twisted bilayer graphene at even integer fillings

A standard approach for studying magic angle twisted bilayer graphene (MATBG)'s correlated electronic phase diagram is to project the Coulomb interaction down to effective models only involving electrons in single-particle flat bands and some nearby remote bands. We provide a novel systematic derivation of a single-particle continuum model of MATBG's single-particle properties which incorporates structural relaxation while remaining in the Lagrangian frame. We project the Coulomb interactions down to electrons occupying the flat bands of this model and compute the Hartree-Fock many-body ground states at fillings $\nu = \pm 2$. We find that incorporating relaxation effects drives the model into a semi-metallic phase at $- 2$ because of particle-hole asymmetry in the relaxed model's single-particle dispersion and because the flat band wavefunctions become more concentrated leading to an enhanced Hartree potential. Our results corroborate recent ab initio density functional theory studies which also found semi-metallic phases at $-2$. We discuss potential explanations for why such phases have not been seen in experiments.

cond-mat.mes-hall↗
arXiv2026

WKB Spectral Asymptotics for a One-Dimensional Dirac Operator with a Slowly Varying Mass Profile

We study the semiclassical spectral theory of a one-dimensional Dirac operator describing waves at the interface between topologically distinct media. We derive a modified Bohr-Sommerfeld quantization condition for the squared operator via a systematic formal WKB construction producing approximate eigenpairs. Our result differs from the standard result by the half-integer shift depending on the pseudo-spin index which allows for recovering the topologically protected zero mode. We verify our result for the solvable P\"{o}schl--Teller potential and provide numerical computations confirming the convergence of eigenvalues and eigenfunctions to their WKB approximations.

math-ph↗
arXiv2026

Ergodic properties of one-dimensional incommensurate bilayer materials

We consider one-dimensional deterministic and random tight-binding Hamiltonians modeling electronic properties of twisted bilayer materials. When the twisted structure is incommensurate, we prove convergence of the density of states measure in the thermodynamic limit and Pastur's theorem on shift-invariance of the spectrum. Our results extend those of Massatt et al. and Cancès et al. in allowing for randomness. We provide numerical density of states computations for the operators we consider.

math-ph↗
arXiv2025

Formal justification of a continuum relaxation model for one-dimensional moiré materials

Mechanical relaxation in moiré materials is often modeled by a continuum model where linear elasticity is coupled to a stacking penalty known as the Generalized Stacking Fault Energy (GSFE). We review and compute minimizers of a one-dimensional version of this model, and then show how it can be formally derived from a natural atomistic model. Specifically, we show that the continuum model emerges in the limit $ε\downarrow 0$ and $δ\downarrow 0$ while holding the ratio $η:= \frac{ε^2}δ$ fixed, where $ε$ is the ratio of the monolayer lattice constant to the moiré lattice constant and $δ$ is the ratio of the typical stacking energy to the monolayer stiffness.

math-ph↗
arXiv2025

Interacting Twisted Bilayer Graphene with Systematic Modeling of Structural Relaxation

Twisted bilayer graphene (TBG) has drawn significant interest due to recent experiments which show that TBG can exhibit strongly correlated behavior such as the superconducting and correlated insulator phases. Much of the theoretical work on TBG has been based on analysis of the Bistritzer-MacDonald model which includes a phenomenological parameter to account for lattice relaxation. In this work, we use a newly developed continuum model which systematically accounts for the effects of structural relaxation. In particular, we model structural relaxation by coupling linear elasticity to a stacking energy that penalizes disregistry. We compare the impact of the two relaxation models on the corresponding many-body model by defining an interacting model projected to the flat bands. We perform tests at charge neutrality at both the Hartree-Fock and Coupled Cluster Singles and Doubles (CCSD) level of theory and find the systematic relaxation model gives quantitative differences from the simplified relaxation model.

math-ph↗
arXiv2025

Learning the local density of states of a bilayer moiré material in one dimension

Recent work of three of the authors showed that the operator which maps the local density of states of a one-dimensional untwisted bilayer material to the local density of states of the same bilayer material at non-zero twist, known as the twist operator, can be learned by a neural network. In this work, we first provide a mathematical formulation of that work, making the relevant models and operator learning problem precise. We then prove that the operator learning problem is well-posed for a family of one-dimensional models. To do this, we first prove existence and regularity of the twist operator by solving an inverse problem. We then invoke the universal approximation theorem for operators to prove existence of a neural network capable of approximating the twist operator.

math-ph↗
arXiv2025

Higher-order continuum models for twisted bilayer graphene

The first-order continuum PDE model proposed by Bistritzer and MacDonald in \cite{bistritzer2011moire} accurately describes the single-particle electronic properties of twisted bilayer graphene (TBG) at small twist angles. In this paper, we obtain higher-order corrections to the Bistritzer-MacDonald model via a systematic multiple-scales expansion. We prove that the solution of the resulting higher-order PDE model accurately approximates the corresponding tight-binding wave function under a natural choice of parameters and given initial conditions that are spectrally localized to the monolayer Dirac points. Numerical simulations of tight-binding and continuum dynamics demonstrate the validity of the higher-order continuum model. Symmetries of the higher-order models are also discussed. This work extends the analysis from \cite{watson2023bistritzer}, which rigorously established the validity of the (first-order) BM model.

math-ph↗
arXiv2024

Radiative transport in a periodic structure with band crossings

We use the Wigner transformation and asymptotic analysis to systematically derive the semi-classical model for the Schrödinger equation in arbitrary spatial dimensions, with any periodic structure. Our particular emphasis lies in addressing the \textit{diabatic} effect, i.e., the impact of Bloch band crossings. We consider both deterministic and random scenarios. In the former case, we derive a coupled Liouville system, revealing lower-order interactions among different Bloch bands. In the latter case, a coupled system of radiative transport equations emerges, with the scattering cross-section induced by the random inhomogeneities. As a specific application, we deduce the effective dynamics of a wave packet in graphene with randomness.

math-ph↗
arXiv2024

Mathematical foundations of phonons in incommensurate materials

In some models, periodic configurations can be shown to be stable under, both, global $\ell^2$ or local perturbations. This is not the case for aperiodic media. The specific class of aperiodic media we are interested, in arise from taking two 2D periodic crystals and stacking them parallel at a relative twist. In periodic media, phonons are generalized eigenvectors for a stability operator acting on $\ell^2$, coming from a mechanical energy. The goal of our analysis is to provide phonons in the given class of aperiodic media with meaning. As rigorously established for the 1D Frenkel-Kontorova model and previously applied by one of the authors, we assume that we can parametrize minimizing lattice deformations w.r.t. local perturbations via continuous stacking-periodic functions, for which we previously derived a continuous energy density functional. Such (continuous) energy densities are analytically and computationally much better accessible compared to discrete energy functionals. In order to pass to an $\ell^2$-based energy functional, we also study the offset energy w.r.t. given lattice deformations, under $\ell^1$-perturbations. Our findings show that, in the case of an undeformed bilayer heterostructure, while the energy density can be shown to be stable under the assumption of stability of individual layers, the offset energy fails to be stable in the case of twisted bilayer graphene. We then establish conditions for stability and instability of the offset energy w.r.t. the relaxed lattice. Finally, we show that, in the case of incommensurate bilayer homostructures, i.e., two equal layers, if we choose minimizing deformations according to the global energy density above, the offset energy is stable in the limit of zero twist angle. Consequently, in this case, one can then define phonons as generalized eigenvectors w.r.t. the stability operator associated with the offset energy.

math-ph↗
arXiv2024

From incommensurate bilayer heterostructures to Allen-Cahn: An exact thermodynamic limit

We give a complete and rigorous derivation of the mechanical energy for twisted 2D bilayer heterostructures without any approximation beyond the existence of an empirical many-body site energy. Our results apply to both the continuous and discontinuous continuum limit. Approximating the intralayer Cauchy-Born energy by linear elasticity theory and assuming an interlayer coupling via pair potentials, our model reduces to a modified Allen-Cahn functional. We rigorously control the error, and, in the case of sufficiently smooth lattice displacements, provide a rate of convergence for twist angles satisfying a Diophantine condition.

math-ph↗
arXiv2024

Modeling of electronic dynamics in twisted bilayer graphene

We consider the problem of numerically computing the quantum dynamics of an electron in twisted bilayer graphene. The challenge is that atomic-scale models of the dynamics are aperiodic for generic twist angles because of the incommensurability of the layers. The Bistritzer-MacDonald PDE model, which is periodic with respect to the bilayer's moiré pattern, has recently been shown to rigorously describe these dynamics in a parameter regime. In this work, we first prove that the dynamics of the tight-binding model of incommensurate twisted bilayer graphene can be approximated by computations on finite domains. The main ingredient of this proof is a speed of propagation estimate proved using Combes-Thomas estimates. We then provide extensive numerical computations which clarify the range of validity of the Bistritzer-MacDonald model.

math-ph↗
arXiv2023

Mathematical aspects of the Kubo formula for electrical conductivity with dissipation

In this expository article, we present a systematic formal derivation of the Kubo formula for the linear-response current due to a time-harmonic electric field applied to non-interacting, spinless charged particles in a finite volume in the quantum setting. We model dissipation in a transparent way by assuming a sequence of scattering events occurring at random-time intervals modeled by a Poisson distribution. By taking the large-volume limit, we derive special cases of the formula for free electrons, continuum and tight-binding periodic systems, and the nearest-neighbor tight-binding model of graphene. We present the analogous formalism with dissipation to derive the Drude conductivity of classical free particles.

math-ph↗
arXiv2022

Bistritzer-MacDonald dynamics in twisted bilayer graphene

The Bistritzer-MacDonald (BM) model, introduced in \cite{Bistritzer2011}, attempts to capture the electronic properties of twisted bilayer graphene (TBG), even at incommensurate twist angles, by an effective periodic model over the bilayer moiré pattern. Starting from a tight-binding model, we identify a regime where the BM model emerges as the effective dynamics for electrons modeled as wave-packets spectrally concentrated at the monolayer Dirac points, up to error that can be rigorously estimated. Using measured values of relevant physical constants, we argue that this regime is realized in TBG at the first "magic" angle.

cond-mat.mes-hall↗
arXiv2022

On the Su-Schrieffer-Heeger model of electron transport: low-temperature optical conductivity by the Mellin transform

We describe the low-temperature optical conductivity as a function of frequency for a quantum-mechanical system of electrons that hop along a polymer chain. To this end, we invoke the Su-Schrieffer-Heeger \emph{tight-binding} Hamiltonian for non-interacting spinless electrons on a one-dimensional (1D) lattice. Our goal is to show via asymptotics how the interband conductivity of this system behaves as the smallest energy bandgap tends to close. Our analytical approach includes: (i) the Kubo-type formulation for the optical conductivity with a nonzero damping due to microscopic collisions; (ii) reduction of this formulation to a 1D momentum integral over the Brillouin zone; and (iii) evaluation of this integral in terms of elementary functions via the three-dimensional Mellin transform with respect to key physical parameters and subsequent inversion in a region of the respective complex space. Our approach reveals an intimate connection of the behavior of the conductivity to particular singularities of its Mellin transform. The analytical results are found in good agreement with direct numerical computations.

math-ph↗
arXiv2021

Computing spectral properties of topological insulators without artificial truncation or supercell approximation

Topological insulators (TIs) are renowned for their remarkable electronic properties: quantised bulk Hall and edge conductivities, and robust edge wave-packet propagation, even in the presence of material defects and disorder. Computations of these physical properties generally rely on artificial periodicity (the supercell approximation), or unphysical boundary conditions (artificial truncation). In this work, we build on recently developed methods for computing spectral properties of infinite-dimensional operators. We apply these techniques to develop efficient and accurate computational tools for computing the physical properties of TIs. These tools completely avoid such artificial restrictions and allow one to probe the spectral properties of the infinite-dimensional operator directly, even in the presence of material defects and disorder. Our methods permit computation of spectra, approximate eigenstates, spectral measures, spectral projections, transport properties, and conductances. Numerical examples are given for the Haldane model, and the techniques can be extended similarly to other TIs in two and three dimensions.

math.NA↗
arXiv2021

Existence and computation of generalized Wannier functions for non-periodic systems in two dimensions and higher

Exponentially-localized Wannier functions (ELWFs) are an orthonormal basis of the Fermi projection of a material consisting of functions which decay exponentially fast away from their maxima. When the material is insulating and crystalline, conditions which guarantee existence of ELWFs in dimensions one, two, and three are well-known, and methods for constructing the ELWFs numerically are well-developed. We consider the case where the material is insulating but not necessarily crystalline, where much less is known. In one spatial dimension, Kivelson and Nenciu-Nenciu have proved ELWFs can be constructed as the eigenfunctions of a self-adjoint operator acting on the Fermi projection. In this work, we identify an assumption under which we can generalize the Kivelson-Nenciu-Nenciu result to two dimensions and higher. Under this assumption, we prove that ELWFs can be constructed as the eigenfunctions of a sequence of self-adjoint operators acting on the Fermi projection. We conjecture that the assumption we make is equivalent to vanishing of topological obstructions to the existence of ELWFs in the special case where the material is crystalline. We numerically verify that our construction yields ELWFs in various cases where our assumption holds and provide numerical evidence for our conjecture.

math-ph↗
arXiv2021

Existence of the first magic angle for the chiral model of bilayer graphene

We consider the chiral model of twisted bilayer graphene introduced by Tarnopolsky-Kruchkov-Vishwanath (TKV). TKV have proved that for inverse twist angles $α$ such that the effective Fermi velocity at the moiré $K$ point vanishes, the chiral model has a perfectly flat band at zero energy over the whole Brillouin zone. By a formal expansion, TKV found that the Fermi velocity vanishes at $α\approx .586$. In this work, we give a proof that the Fermi velocity vanishes for at least one $α$ between $.57$ and $.61$ by rigorously justifying TKV's formal expansion of the Fermi velocity over a sufficiently large interval of $α$ values. The idea of the proof is to project the TKV Hamiltonian onto a finite dimensional subspace, and then expand the Fermi velocity in terms of explicitly computable linear combinations of modes in the subspace, while controlling the error. The proof relies on two propositions whose proofs are computer-assisted, i.e., numerical computation together with worst-case estimates on the accumulation of round-off error which show that round-off error cannot possibly change the conclusion of the computation. The propositions give a bound below on the spectral gap of the projected Hamiltonian, an Hermitian $80 \times 80$ matrix whose spectrum is symmetric about $0,$ and verify that two real 18th order polynomials, which approximate the numerator of the Fermi velocity, take values with definite sign when evaluated at specific values of $α$. Together with TKV's work our result proves existence of at least one perfectly flat band of the chiral model.

math-ph↗