The tight binding model is a minimalistic electronic structure model for predicting properties of materials and molecules. For insulators at zero Fermi-temperature we show that the potential energy surface of this model can be decomposed into exponentially localised site energy contributions, thus providing qualitatively sharp estimates on the interatomic interaction range which justifies a range of multi-scale models. For insulators at finite Fermi-temperature we obtain locality estimates that are uniform in the zero-temperature limit. A particular feature of all our results is that they depend only weakly on the point spectrum. Numerical tests confirm our analytical results. This work extends and strengthens (Chen, Ortner 2016)Link opens in a new window and (Chen, Lu, Ortner 2018)Link opens in a new window for finite temperature models.
We show that the local density of states (LDOS) of a wide class of tight-binding models has a weak body-order expansion. Specifically, we prove that the resulting body-order expansion for analytic observables such as the electron density or the energy has an exponential rate of convergence both at finite Fermi-temperature as well as for insulators at zero Fermi-temperature. We discuss potential consequences of this observation for modelling the potential energy landscape, as well as for solving the electronic structure problem.
Markus Bachmayr, Geneviève Dusson, Christoph Ortner, and Jack Thomas. Polynomial approximation of symmetric functions. arXiv:2109.14771 (2021). [arXivLink opens in a new window | abstract].
We study the polynomial approximation of symmetric multivariate functions and of multi-set functions. Specifically, we consider $f(x_1,…,x_N)$, where $x_i \in \mathbb R^d$, and $f$ is invariant under permutations of its $N$ arguments. We demonstrate how these symmetries can be exploited to improve the cost versus error ratio in a polynomial approximation of the function $f$, and in particular study the dependence of that ratio on $d,N$ and the polynomial degree. These results are then exploited to construct approximations and prove approximation rates for functions defined on multi-sets where $N$ becomes a parameter of the input.
The tight binding model is a minimalistic electronic structure model for predicting properties of materials and molecules. For insulators at zero Fermi-temperature we show that the potential energy surface of this model can be decomposed into exponentially localised site energy contributions, thus providing qualitatively sharp estimates on the interatomic interaction range which justifies a range of multi-scale models. For insulators at finite Fermi-temperature we obtain locality estimates that are uniform in the zero-temperature limit. A particular feature of all our results is that they depend only weakly on the point spectrum. Numerical tests confirm our analytical results. This work extends and strengthens (Chen, Ortner 2016) and (Chen, Lu, Ortner 2018) for finite temperature models.
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Education:
Sept 2018 - Sept 2021: PhD in Mathematics and Statistics, University of Warwick
Supervised by Christoph Ortner
Thesis: Analysis of an ab initio Potential Energy Landscape
Prize: Faculty Thesis Prize 2022 (joint winner)
Sept 2017 - Aug 2018: MSc in Mathematics and Statistics, University of Warwick
March 2019. Solid Mechanics Working Group Meeting, University of Warwick. Seminar Talk: Zero Temperature Limit of the Tight Binding Model for Point Defects
First Year Supervisor: three groups of Maths & Stats students Modules covered: Sets & Numbers, Mathematical Analysis (Terms 1&2) and Linear Algebra.
This year I also helped out marking Mathematical Analysis (first year module for external maths students)
2018/19:
First Year Supervisor: one group of Discrete Mathematics students (as above)
Second Year Supervisor: two groups of Mathematics students Modules covered: Analysis III, Algebra I: Advanced Linear Algebra, Multivariable Calculus (Term 1) & Algebra II: Groups and Rings, Norms Metrics & Topologies (Term 2).
2017/18:
First Year Supervisor: one group of MORSE, Data Science and Maths & Stats students (as above)
This year I also helped out marking Mathematical Analysis (first year module for external maths students)
2016/17:
First Year Supervisior: one group of MORSE and Maths & Stats students (as above)
