Study On Integrability Of Some Lattice System And String Models And Related Research

There have profound relations between the string theory and condensed matter physics. They both offer each other in methods, mechanics and principles. They are mutually promotional and influencing. There have close relations among the classical integrability of the string, the lattice models and the classical field theory. Constructing integrable lattice models and obtaining the eigenstates, energy spectrum and Bethe ansatz equations of the systems may provide a tool for studying classical solutions of string, supersymmetric gauge theory and integrable systems. Studying the integrability and solution transformation of the string Sigma model may benefit us greatly to understand the AdS/CFT correspondence. The study on these aspects have already attracted much attention.This thesis comes in two parts. In part I(chapters II and III), we mainly deal with the exact solutions of multi-component Bariev model for correlated hopping under general boundary conditions and the multi-component Bariev model with a hardcore repulsion under open boundary conditions in one dimension. Bariev model is one of the most significant models in condensed matter physics, which is relevant to the high-Tc superconductivity. It has been extensively studied in the periodic case. Although there are a few works about the solution of open Bariev models(two- and three-component Barive model and muti-component Bariev model with fixed boundary conditons). The solution of the one-dimensional muti-component Bariev model under general open boundary conditions has not been given out yet. We construct the general form of the Hamiltonian of the model and study the integrability of the model through the coordinate Bethe ansatz method. We also obtain the energy spectrum, the specific integrable boundary conditions and the corresponding Bethe ansatz equations. The multi-component Bariev model for correlated hoping and a hard-core repulsion under open boundary condition is also shown to be integrable in one dimension. The energy spectrum, the Bethe ansatz equations and the integrable boundary conditions are derived. Due to the hard core repulsion, the distribution of particles was compressed effectively. Thus we hope the new model has different properties from the standard Bariev model under open boundary conditions. Using the results of this paper and the thermodynamical Bethe ansatz method, one can study the thermodynamical properties of the two models such as specific heat and susceptibility.In part II(chapters IV,V and VI), we devote to study the integrability and solution transformation of some string Sigma models. We find one parameter flat currents of the Sigma model on supercoset targets with Z_2m grading given by Young satisfaction equations of motion and the Virasoro constraint. This means that one can generate a series of classical solutions from the original one. For these new solutions one can also construct flat currents and conserved charges, which form the same set with the original one. Although these solutions have the same infinite set of conserved quantities, they are not equivalent to the original solution. Each conserved quantity has different physical meanings for different solutions. Since this type of Sigma model can be used in the pure-spinor description of the superstring, the study of the solution transformation of the model will be helpful to the study of the string. We construct actions and flat currents of Green-Schwarz Sigma models on supercoset targets whose kinetic terms only contain the target-space bosons in contrast with hybrid type Sigma models. This type of Sigma models can be used to describe the Green-Schwarz superstring. First, we construct the Green-Schwarz type Sigma models with flat currents with Z₄, Z₆ and Z₈ grading. We also give the explicit form of the action and the flat currents. Then we consider a simple case of the kinetic term of the model with Z_4m grading and show that there exist a one-parameter family of flat currents of the model by requiring a suitable choice of the Wess-Zumino term. Such flat currents naturally lead to a hierarchy of classical conserved nonlocal charges. The existence of these charges is an indication of the integrability for the model. We also find that the one-parameter flat currents of the model satisfy equations of motion and the Virasoro constraint. This implies that one can generate a series of classical solutions from an existing one. Especially in the Z₄ case, our model coincide with the well known model given by Metsaev and Tseytlin on a supercoset PSU(2,2|4)/[SO(4,1)×SO(5)] and similar models. In the sixth chapter, we obtain the string action in theγ-deformed AdS₃×S³ background under the light cone gauge by the TST transformation. We also construct the Lax connection for the fixed uniform light-cone gauge theory and prove that the Lax connection is flat. Thus the model is integrable.