
Methods in theory of onedimensional systems
Syllabus
 O(2) model and the KosterlitzThouless transition
 Vortices in the O(2) model and Coulomb gas.
 Formulation in terms of a nonlocal field and its equivalence to the sineGordon model.
 Plasmagas transition.
 Scaling dimension of the perturbation operator and the exact value of the transition point.
 Bosonization of the Thirring model
 Representation of fermions in terms of boson fields (bosonization).
 Cancellation of divergent parts in the Lagrangian and an exact relation between the coupling constants.
 Renormalization group for the KosterlitzThouless transition
 Renormalization of coupling constants of the sineGordon model in the second order of the perturbation theory.
 Renormalization group and renormalization group flows.
 O(3) model: generation of mass by instantons
 Topological properties of the O(3) model, topologically nontrivial solution in the Euclidean plane.
 Qualitative description of the generation of mass by instantons.
 O(N) model: 1/N expansion
 Perturbation theory in 1/N for the O(N) model.
 Mass generation.
 Kinematic scattering restrictions and evalutation of the S matrix by means of the perturbation theory.
 O(N) model: integrability and the exact S matrix
 Higher integrals of motion and S matrix factorization.
 YangBaxter equation.
 Evaluation of the S matrix by using the factorization condition and the perturbative result.
 Thirring model: a solution by means of the Bethe Ansatz method
 Pseudovaccum and the wave function of the Thirring model in terms of the Bethe Ansatz.
 Bethe equations and their thermodynamic limit.
 The spectrum and the S matrix of the model.
 Heisenberg spin chain and its Euclidean limit
 XYZ model.
 The JordanWigner transformation and the XY model.
 The scaling limit and the relation with the Thirring/sineGordon model.
 YangBaxter equation and Bethe Ansatz
 The XXZ model and the sixvertex model.
 The YangBaxter equation and commuting transfer matrices.
 The coordinate Bethe Ansatz.
 Algebraic Bethe Ansatz. Solution of Bethe equations
 The pseudovacuum and the eigenstates in the framework of the algebraic Bethe Ansatz.
 The Bethe equations and their solution in the thermodynamic limit.
 Evaluation of the free energy of the sixvertex model.
 Kondo problem: derivation of the Bethe Ansatz
 The Kondo effect.
 Reduction to a onedimensional problem.
 Primary and secondary Bethe Ansatz.
 The system of the Bethe equations for the Kondo problem.
 Kondo problem: solving the Bethe equations
 The ground state in the zero magnetic field.
 Guidelines to derivation of the explicit expression for the magnetization in the magnetic field.
 A short discussion on the finite temperature case.
The problems are given at the end of each lecture.
