Methods in theory of one-dimensional systems
The problems are given at the end of each lecture.
- O(2) model and the Kosterlitz-Thouless transition
- Vortices in the O(2) model and Coulomb gas.
- Formulation in terms of a nonlocal field and its equivalence to the sine-Gordon model.
- Plasma-gas 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 Kosterlitz-Thouless transition
- Renormalization of coupling constants of the sine-Gordon 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.
- Yang-Baxter 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 Jordan-Wigner transformation and the XY model.
- The scaling limit and the relation with the Thirring/sine-Gordon model.
- Yang-Baxter equation and Bethe Ansatz
- The XXZ model and the six-vertex model.
- The Yang-Baxter 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 six-vertex model.
- Kondo problem: derivation of the Bethe Ansatz
- The Kondo effect.
- Reduction to a one-dimensional 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.