Landau Phystech School of Physics and Research, Moscow Institute of Physics and Technology

Chair for "Problems in theoretical physics"

based at Landau Institute for theoretical physics


Methods in theory of one-dimensional systems

M.Yu. Lashkevich


  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.
The problems are given at the end of each lecture.