标题： 一维Hubbard模型中局部守恒量的研究 显示英文标题

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

摘要： 在本论文中，我研究了一维Hubbard模型中的局部电荷$\{Q_k\}$，这是一个用于强关联电子系统中非微扰效应理论研究的可积系统。我得到了它们的显式闭合形式表达式。对于$k$-局部电荷$Q_k$在$k > 5$中的表达式，其中$k$表示支持范围，之前的研究所未知。我发现$Q_k$是跳跃项特定类型的乘积的线性组合。我引入了一种图示符号来高效地表示这些乘积，这使得可以预测局部电荷的一般公式。这些线性组合对于$k > 5$具有非平凡的系数，其中一些系数被证明与广义Catalan数相同，这些数也出现在Heisenberg链的局部电荷中。我推导了这些系数的递归方程。我还证明了所得到的局部电荷是完备的。 因此，任何局部电荷都可以表示为$\{Q_k\}$的线性组合。 尽管局部电荷的完备性在转移矩阵表述中是一个自然的猜想，但几乎所有的案例都没有给出证明。 最后，我强调这是首次研究展示了缺乏提升算符的电子系统局部电荷的一般显式表达式。 显示英文摘要

摘要： 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.