Excitons are naturally occurring quasi-particles associated with the conversion of light to energy (e.g. photosynthesis). Our work applies algebraic topology to study and in particular, to count the number of such excitations in certain systems. We formulate the exciton counting problem in terms of intersection theory in stratified spaces. We generalized the problem to intersections of a smooth curve with a stratified (non-manifold) subspace of the unitary group $U(n)$. Each intersection is weighted with a multiplicity, and the number of excitons can be computed via a topological winding number with an index-like theorem.
The mathematical intersection theory is developed in (Catanzaro et al., 2017). We focus on the intersection of a curve with the subspace of matrices with at least one eigenvalue equal to one inside the manifold of all $n \times n$ unitary matrices. The intersection theory was applied to conjugated molecules in (Li et al., 2014) and certain organic semiconductor systems in (Catanzaro et al., 2015).
This paper introduces an intersection theory problem for maps into a smooth manifold equipped with a stratification. We investigate the problem in the special case when the target is the unitary group U(n)U(n)\textlessmath display="inline" overflow="scroll" altimg="eq-00001.gif"\textgreater\textlessmi\textgreaterU\textless/mi\textgreater\textlessmo class="MathClass-open" stretchy="false"\textgreater(\textless/mo\textgreater\textlessmi\textgreatern\textless/mi\textgreater\textlessmo class="MathClass-close" stretchy="false"\textgreater)\textless/mo\textgreater\textless/math\textgreater and the domain is a circle. The first main result is an index theorem that equates a global intersection index with a finite sum of locally defined intersection indices. The local indices are integers arising from the geometry of the stratification. The result is used to study a well-known problem in chemical physics, namely, the problem of enumerating the electronic excitations (excitons) of a molecule equipped with scattering data.
2015
Counting the number of excited states in organic semiconductor systems using topology
Exciton scattering theory attributes excited electronic states to standing waves in quasi-one-dimensional molecular materials by assuming a quasi-particle picture of optical excitations. The quasi-particle properties at branching centers are described by the corresponding scattering matrices. Here, we identify the topological invariant of a scattering center, referred to as its winding number, and apply topological intersection theory to count the number of quantum states in a quasi-one-dimensional system.
2014
Excited-State Structure Modifications Due to Molecular Substituents and Exciton Scattering in Conjugated Molecules
Attachment of chemical substituents (such as polar moieties) constitutes an efficient and convenient way to modify physical and chemical properties of conjugated polymers and oligomers. Associated modifications in the molecular electronic states can be comprehensively described by examining scattering of excitons in the polymer?s backbone at the scattering center representing the chemical substituent. Here, we implement effective tight-binding models as a tool to examine the analytical properties of the exciton scattering matrices in semi-infinite polymer chains with substitutions. We demonstrate that chemical interactions between the substitution and attached polymer are adequately described by the analytical properties of the scattering matrices. In particular, resonant and bound electronic excitations are expressed via the positions of zeros and poles of the scattering amplitude, analytically continued to complex values of exciton quasi-momenta. We exemplify the formulated concepts by analyzing excited states in conjugated phenylacetylenes substituted by perylene.
