We extend T. Prosen's construction of quasilocal conserved quantities for the XXZ model [Phys. Rev. Lett. 106, 217206 (2011)] to the case of periodic boundary conditions. These quasilocal operators stem from a two-parameter transfer matrix which employs a highest-weight representation of the quantum group algebra inherent in the Yang-Baxter algebra. In contrast with the open chain, where the conservation law is weakly violated by boundary terms, the quasilocal operators in the periodic chain exactly commute with the Hamiltonian and other local conserved quantities.
We extend T. Prosen's construction of quasilocal conserved quantities for the XXZ model [Phys. Rev. Lett. 106, 217206 (2011)] to the case of periodic boundary conditions. These quasilocal operators stem from a two-parameter transfer matrix which employs a highest-weight representation of the quantum group algebra inherent in the algebraic structure of the Bethe ansatz. In contrast with the open chain, where the conservation law is weakly violated by boundary terms, the quasilocal operators in the periodic chain exactly commute with the Hamiltonian and other local conserved quantities.
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.
Integrable models are characterized by the existence of an extensive number of local conserved quantities.Their existence is closely related to various exact methods, such as the Bethe ansatz and quantum inverse scattering.They also play an important role in nonequilibrium dynamics in the integrable models.Their steady states and transport phenomena exhibit exotic properties, which are also related to this point.Local conserved quantities in a quantum integrable model in one dimension are generated from the transfer matrix of the corresponding classical two-dimensional model.However, it is still non-trivial to obtain their explicit expression.The meaning of "local" here is that each conserved quantity consists of the sum of observables acting on a finite range of sites called its support.We note that the local conserved quantities in this context are not O (1)-body observables in general.Their explicit expressions enable us to analyze theories of nonequilibrium dynamics using the local conserved quantities in detail, for example, the generalized Gibbs ensemble, the Mazur bound, and the generalized hydrodynamics.In this thesis, we have studied the local conserved quantities in the XYZ spin-1/2 chain, which is an integrable model corresponding to the classical eight-vertex model.We have obtained their explicit expression for arbitrary coupling constants of the XYZ chain.The model at the isotropic point called the Heisenberg or XXX chain is an exceptional case in which the expression had been known.We have checked that our results reproduce the known expression for the XXX chain as a special case.To obtain the explicit expression, we have employed the notation of doubling-product operators, which was introduced by Shiraishi to study the non-integrability of the XYZ spin-1/2 chain in a magnetic field.Using this notation, we have found recursion relations of the coefficients among different terms in the conserved quantities in the XYZ chain.Then, we have solved the recursion relations and obtained the explicit expression.We have shown that the local conserved quantity Q k for a fixed support k is essentially unique i ii ABSTRACT if 2 ≤ k ≤ L/2 with L being the number of sites.We have obtained the explicit expression for all {Q k } for 1 ≤ k ≤ L/2 and one Q k for each k with L/2 < k ≤ L.We have also studied the currents of these conserved quantities based on the respective continuity equations.We have obtained their explicit expression and numerically investigated their time evolution.In particular, we have studied the effects of an integrabilitybreaking perturbation on the time evolution of the currents and found that this term drastically changes their long-time behaviors as follows.As for the initial state, we have chosen several energy eigenstates of the XXX chain with a nonzero energy current, which is also known to be conserved.In particular, we have studied the time evolution with the Hamiltonian of the XXZ chain with and without a next-nearest-neighbor interaction and seen the effects of this integrability-breaking perturbation.We have found that, for these initial states, the long-time average of the total currents is nonzero for the integrable cases while zero for the non-integrable cases.We have also studied this behavior analytically and shown that the degeneracy of the energy eigenstates is essential.Our results for the XYZ spin-1/2 chain have revealed the structure of the conserved quantities and currents in detail.They provide materials for the development of algebraic studies in integrable models and will be useful for detailed analyses of nonequilibrium dynamics in these models.
A class of quantum chains possessing a family of local conserved charges with a Catalan tree pattern is studied. Recently, we have identified such a structure in the integrable SU(N)-invariant chains. In the present work we find sufficient conditions for the existence of a family of charges with this structure in terms of the underlying algebra. Two additional systems with a Catalan tree structure of conserved charges are found. One is the spin 1/2 XXZ model with Δ=−1. The other is a new octonionic isotropic chain, generalizing the Heisenberg model. This system provides an interesting example of an infinite family of noncommuting local conserved quantities.