Study of local conserved quantities in the one-dimensional Hubbard model

Abstract

In this thesis, I study the local charges $\{Q_k\}$ in the one-dimensional Hubbard model, which is an integrable system used for theoretical studies of non-perturbative effects in strongly correlated electron systems. I obtained their explicit expressions in a closed form. An expression of the $k$-local charge $Q_k$ for $k>5$, where $k$ denotes the support, had been unknown in previous studies. I find that $Q_k$ is a linear combination of a particular kind of products of the hopping terms. I introduce a diagrammatic notation to represent these products efficiently, which enables the prediction of the general formula of the local charges. These linear combinations have non-trivial coefficients for $k>5$, and some of the coefficients were proved to be identical to the generalized Catalan numbers, which also appear in the local charges of the Heisenberg chain. I derive the recursion equation for these coefficients. I also prove that the obtained local charges are exhaustive. Thus, any local charge is written as a linear combination of $\{Q_k\}$. Although the completeness of the local charges is a natural conjecture in the transfer-matrix formulation, its proof has not been presented for almost all cases. Lastly, I emphasize that this is the first study demonstrating general explicit expressions of the local charges for an electron system that lacks the boost operator.

arXiv commentDoctoral Dissertation, the University of Tokyo, 216 pages, 12 figures