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arXiv:1610.02605 · Journal of Superconductivity and Novel Magnetism

Exact Time Evolution of the Asymmetric Hubbard Dimer

Shankar Balasubramanian, James K. Freericks

Published Sep 29, 2016• 4 citations
arXivHTMLPDF
DOI
10.1007/s10948-016-3811-9

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In this dissertation, I introduce both ground-state and thermal density functional theory. Throughout I use the asymmetric two-site Hubbard model, called the Hubbard dimer for short, to better understand and/or develop these theories. This model is used because it can be solved analytically and it contains all the necessary physics while still being conceptually simple enough to tease apart the various aspects of density functional theory. Ground-statedensity functional theory has seen broad use in many disciplines including physics, chemistry, geology, and material science and has led to a number of important physical and technological successes. In the first two chapters I elucidate the behavior of the ground-state theory using the Hubbard dimer. The simplicity of the model allows me to showcase aspects of the theory that are common points of confusion within the electronic structure community, e.g. the fundamental gap problem. The next two chapters focus on thermal density functional theory which has been coming to prominence as the study of warm dense matter has becomea growing interest at the national laboratories and in the astronomical body community. The Hubbard dimer allows me to do the first ever exact thermal density functional theory calculation. In this work I am better able to understand the approximations used in thermal density functional theory and can point to why they succeed and fail. This also allows me to illustrate old conditions and derive new ones. I conclude with an overview of the work and a few different directions in which the asymmetric Hubbard dimer could be used further.
We show that propagation with the best possible adiabatic approximation in time-dependent density-functional theory fails to properly transfer charge in an asymmetric two-site Hubbard model when beginning in the ground state. The approximation is adiabatic but exact otherwise, constructed from the exact ground-state exchange-correlation functional that we compute via constrained search. The model shares the essential features of charge-transfer dynamics in a real-space long-range molecule, so the results imply that the best possible adiabatic approximation, despite being able to capture nonlocal ground-state step features relevant to dissociation and charge-transfer excitations, cannot capture fully time-resolved charge-transfer dynamics out of the ground state.
The exact functionals associated with the (singlet) ground state and the two singlet excited states of the asymmetric Hubbard dimer at half-filling are calculated using both Levy's constrained search and Lieb's convex formulation. While the ground-state functional is, as is commonly known, a convex function with respect to the density, the functional associated with the doubly excited state is found to be concave. Also, because the density-potential mapping associated with the first excited state is noninvertible, its "functional" is a partial, multivalued function composed of one concave and one convex branch that correspond to two separate domains of the external potential. Remarkably, it is found that, although the one-to-one mapping between density and external potential may not apply (as in the case of the first excited state), each state-specific energy and corresponding universal functional are "functions" whose derivatives are each other's inverse, just as in the ground state formalism.