Department for General and Applied Physics, 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.