Dynamical exchange-correlation field in the Hubbard-Holstein model
(2026) FYSK04 20261Department of Physics
Mathematical Physics
- Abstract
- This study aims to analyse the nature of the dynamical exchange correlation field in the Hubbard-Holstein model. The goal being to find the dependence of such a field on the model parameters much like in analogy with Density Functional Theory. In order to do so, we begin by studying the analytical solutions of simple systems, such as the Hubbard dimer and small Holstein systems. This will then be used as a reference point for a numerical analysis of the exchange correlation field in the Hubbard-Holstein model. From this, one expects to be able to deduce a general form of such a field and find a correlation between the Hubbard, the Holstein, and the Hubbard-Holstein model.
- Popular Abstract
- Unlike what we were taught in school, small particles such as electrons do not only move through materials like little billiard balls, but also undulate like waves, leading to a much more complicated description of their behaviour. As a result, systems with many electrons quickly become too complex to describe by tracking each electron individually. This has led to a nearly century-long search for accurate approximations of such systems, a pursuit that continues to this day.
In fact, to make progress, physicists often turn to simplified models that focus on the most important effects. One such model considers how electrons bump into each other, slow one another down and react to the vibrations of the atoms around them. In materials... (More) - Unlike what we were taught in school, small particles such as electrons do not only move through materials like little billiard balls, but also undulate like waves, leading to a much more complicated description of their behaviour. As a result, systems with many electrons quickly become too complex to describe by tracking each electron individually. This has led to a nearly century-long search for accurate approximations of such systems, a pursuit that continues to this day.
In fact, to make progress, physicists often turn to simplified models that focus on the most important effects. One such model considers how electrons bump into each other, slow one another down and react to the vibrations of the atoms around them. In materials where these effects are strong, a commonly used model is the Hubbard-Holstein model.
However, for larger systems, the use of these simplified models is usually not sufficient. Because of this, one also has to employ additional approximations. One of the most widely used is density functional theory, a method that replaces the nearly impossible task of tracking every electron with a more manageable description of their movements, based on their density.
The challenge that arises in density functional theory is that the detailed influence of one electron on another is only captured approximately. However, in reality, each electron constantly adjusts its motion in response to all the others, leading to a complex, dynamic interplay that is lost in the simplified picture. The exchange correlation potential is one of the tools that allows these missing effects to be reintroduced, and it forms the foundation of this project.
Well-established results for the exchange correlation potential in the Hubbard model (the electrons interacting with each other) as well as the Holstein model (the wiggling of atoms) are already known. The goal of this project is to both extensively characterise the behaviour of the exchange correlation potential in the Hubbard-Holstein model and to find an approximation that combines the known results from the simpler models, offering a tractable approach to this more complex system. This is achieved through extensive numerical computation, by comparing the behaviour of the exchange correlation potential across the different models and parameter regimes, and benchmarking the results against known analytical solutions using the spectral function, a quantity that encodes the energy distribution of electron states.
These results could help us predict the electronic properties of complex materials in a more efficient way than is currently possible, bringing us closer to understanding and designing the materials that shape modern technology. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/student-papers/record/9235766
- author
- Jacob, Sebastian LU
- supervisor
- organization
- course
- FYSK04 20261
- year
- 2026
- type
- M2 - Bachelor Degree
- subject
- keywords
- Dynamical exchange-correlation field Dynamical exchange-correlation potential Hubbard-Holstein model
- language
- English
- id
- 9235766
- date added to LUP
- 2026-06-17 13:40:57
- date last changed
- 2026-06-17 13:40:57
@misc{9235766,
abstract = {{This study aims to analyse the nature of the dynamical exchange correlation field in the Hubbard-Holstein model. The goal being to find the dependence of such a field on the model parameters much like in analogy with Density Functional Theory. In order to do so, we begin by studying the analytical solutions of simple systems, such as the Hubbard dimer and small Holstein systems. This will then be used as a reference point for a numerical analysis of the exchange correlation field in the Hubbard-Holstein model. From this, one expects to be able to deduce a general form of such a field and find a correlation between the Hubbard, the Holstein, and the Hubbard-Holstein model.}},
author = {{Jacob, Sebastian}},
language = {{eng}},
note = {{Student Paper}},
title = {{Dynamical exchange-correlation field in the Hubbard-Holstein model}},
year = {{2026}},
}