Lecture 1:
Fredholm integral operators of a special form will be introduced. Their main properties will be described. Sometimes these integral oprators are called 'integrable' integral operators. The algebra of integrable integral operators will be described. These integral operators are related to Riemann-Hilbert problems and also to classical completely integrable PDE's. Different applications of 'integrable' integral operators will be presented. In these lectures the main application will be the description of quantum correlation functions of integrable models. This is the only method which permits evaluation of correlation functions depending on all of the important physical parameters: temperature, chemical potential and magnetic field. Explicit formulae for large time and long distance asymptotics can be obtained.
Lecture 2:
The quantum inverse scattering method and the algebraic Bethe ansatz will be briefly recalled. The six vertex model will be studied. We shall concentrate on the evaluation of the partition function on the finite square lattice with domain wall boundary conditions. This partition function can be represented as the determinant of a matrix. The dimension of the matrix coincides with the dimension of the lattice. The actual derivation will be presented in the simplified rational situation [isotropic case of the six vertex model]. Different appications of this determinant representation will be mentioned. In these lectures the main application will be the derivation of the determinant represention of quantum correlation functions. First we shall obtain the determinant represention for the scalar products.
Lecture 3:
One can use the algebraic Bethe Ansatz in order to define the scalar
products. These scalar products are building blocks of correlation
functions. The main purpose of the lecture will be to derive the
determinant representation for scalar products. Several ideas will be
involved.
i) Laplace summation formula will be used . It expresses a
determinant of a sum of two matrices in terms of their minors.
ii) quantum dual fields will be introduced. These are linear
combinations of canonical Bose fields. They help to represent double
products as single products. This in turn reduces the dynamics of Bethe
Ansatz solvable models to the 'free fermionic' one.
iii) The partition function of the six vertex model with domain wall
boundary conditions is a highest coefficient in the represention for the
scalar product. All these ideas help to sum up the represention for the
scalar product and to obtain the determinant representation. The same methods
repeated again help to go from the determinant reperesention of the
scalar products to the determinant reperesention for correlation
functions
Lecture 4: Time and Temperature Dependent Correlation Functions: The Nature of Asymptotics
We shall start with the impenetrable Bose gas.
Asymptotics of correlation functions depend on all the parameters:
temperature and density (chemical potential).
Depending on the sign of the chemical potential, the model can be in two
different phases: Luttinger liquid phase (conformal phase) and gas phase.
The nature of the asymptotics is essentially different in these two phases.
In the Luttinger liquid phase, asymptotics approach the
standard expressions of conformal field theory at low temperatures.
In the gas phase asymptotics looks like quantum mechanical Green's function
with corrections.
In the penetrable Bose gas the qualitative nature of the asymptotics is the
same.
Other models will also be considered. For example the Hubbard model
(which can be viewed as a discreet version of the two component Fermionic
nonlinear Schroedinger). At zero magnetic field it also has two phases .
1) Korepin, V. E.; Bogoliubov, N. M., Izergin, A. G. Quantum inverse scattering method and correlation functions. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 1993.