In this thesis, I study the local charges
{Qk} 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
Qk for
k>5, where
k denotes the support, had been unknown in previous studies. I find that
Qk 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
{Qk}. 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.
Absence of local conserved quantities, or nonintegrability, is often assumed when discussing various phenomena in quantum many-body systems, such as thermalization and transport. However, no concrete proof of this property is known in electron--phonon coupled systems, a typical setting for condensed matter physics. In this paper, we show that the one-dimensional Holstein model has no nontrivial local conserved quantities other than the Hamiltonian itself and the total fermion number operator. We further show that the absence of nontrivial local conserved quantities also holds for the more general Holstein--Hubbard model. We also establish the absence of nontrivial local conserved quantities in these models on higher-dimensional hypercubic lattices. Our result has accomplished an advance in nonintegrability proofs by expanding their scope to systems in which particles with different statistical properties are mixed.
The one-dimensional Hubbard model is integrable in the sense that it has an infinite family of conserved currents. We explicitly construct a ladder operator which can be used to iteratively generate all of the conserved current operators. This construction is different from that used for Lorentz invariant systems such as the Heisenberg model. The Hubbard model is not Lorentz invariant, due to the separation of spin and charge excitations. The ladder operator is obtained by a very general formalism which is applicable to any model that can be derived from a solution of the Yang-Baxter equation.