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Methods in theory of one-dimensional systems

M.Yu. Lashkevich

Syllabus

1. 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.
2. 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.
3. 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.
4. 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.
5. 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.
6. 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.
7. 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.
8. 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.
9. 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.
10. 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.
11. 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.
12. 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.