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.
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.
We examine the nature, number, and interrelation of conservation laws in the one-dimensional Hubbard model. In previous work by Shastry [Phys. Rev. Lett. 56, 1529 (1986); 56, 2334 (1986); 56, 2453 (1986); J. Stat. Phys. 50, 57 (1988)], who studied the model on a large but finite number of lattice sites
(Na), only
Na+1 conservation laws, corresponding to
Na+1 operators that commute with themselves and the Hamiltonian, were explicitly identified, rather than the $\ensuremath{\sim}{2N}_{a}$ conservation laws expected from the solvability and integrability of the model. Using a pseudoparticle approach related to the thermodynamic Bethe ansatz, we discover an additional
Na+1 independent conservation laws corresponding to nonlocal, mututally commuting operators, which we call transfer-matrix currents. Further, for the model defined in the whole Hilbert space, we find there are two other independent commuting operators (the squares of the \ensuremath{\eta}-spin and spin operators) so that the total number of local plus nonlocal commuting conservation laws for the one-dimensional Hubbard model is
2Na+4. Finally, we introduce an alternative set of
2Na+4 conservation laws which assume particularly simple forms in terms of the pseudoparticle and Yang-particle operators. This set of mutually commuting operators lends itself more readily to calculations of physically relevant correlation functions at finite energy or frequency than the previous set.