In integrable one-dimensional quantum systems an infinite set of local conserved quantities exists which can prevent a current from decaying completely. For cases like the spin current in the
XXZ model at zero magnetic field or the charge current in the attractive Hubbard model at half filling, however, the current operator does not have overlap with any of the local conserved quantities. We show that in these situations transport at finite temperatures is dominated by a diffusive contribution with the Drude weight being either small or even zero. For the
XXZ model we discuss in detail the relation between our results, the phenomenological theory of spin diffusion, and measurements of the spin-lattice relaxation rate in spin chain compounds. Furthermore, we study the Haldane-Shastry model where a conserved spin current exists.
We investigate the exact integrability of the one-dimensional (1D) Bariev model in the framework of the quantum inverse scattering method (QISM). Using the Jordan - Wigner transformation, the 1D Bariev model can be regarded as a coupled spin model. We construct the higher conserved currents which commute with the Hamiltonian. The explicit form of the conserved currents helps us to infer the L-operator of the 1D Bariev model. From the L-operator, we construct a transfer matrix which is a generating function of the conserved currents. We also find the corresponding R-matrix which satisfies the Yang - Baxter relation. Thus the exact integrability of the 1D Bariev model is established. The R-matrix does not have the `difference property' for the spectral parameter, as in the case of the 1D Hubbard model. We also provide the Lax representation and the fermionic formulation of the Yang - Baxter relation.
One of the phenomena associated with quantum integrable systems is the possibility of persistent currents, i.e. currents which do not decay away entirely, but have some portion that continues to flow undiminished and indefinitely. These residual currents are shown to be the conserved part of the current operator, and calculable from the conservation laws of the system. In a particular system, previous attempts to calculate a known residual current from the conservation laws have failed. A numerical investigation is undertaken, and this disparity with the formal results is resolved by the inclusion of a previously overlooked conservation law. An important corollary to these results is that requiring the mutual commutativity of the conservation laws of a quantum integrable system, previously assumed by analogy with the classical case, is an unnecessary and potentially disastrous restriction. Methods of generating the local conservation laws of a quantum integrable system are investigated, and the current method of using a Boost operator is shown to be subtly flawed. The method is discovered to implicitly require additional knowledge in the form of Hamiltonian identities in order to avoid otherwise unphysical terms. A new method is proposed based on the idea that the logarithm of the Transfer matrix of a system generates these local conservation laws. The method is applicable to a wide class of systems whose Lax operator obeys a certain condition, and the majority of the work required to generate the local conservation laws is entirely general and thus only needs to be done once. This new method is then applied to two quite different spin-chain Hamiltonians, the XXZ and Hubbard models, and shown to successfully generate all of the known local conservation laws of these models and some new ones.