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\centerline{\Large Methods of the theory of one-dimensional quantum systems}
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\leftline{\bf Abstract}
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The purpose of the course is to acquaint students with methods for obtaining exact results in the theory of one\-/dimensional quantum systems and some of there
applications.

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\item $O(2)$-model and Berezinskii\--Kosterlitz\--Thouless transition.

{\it Vortices in the $O(2)$\-/models and the Coulomb gas. Expression in terms of a non\-/local field and equivalence to the sine\-/Gordon model. Plasma --- gas transition. The scaling dimension of the perturbation operator and the exact value of the transition point.}

\item Bosonization of the Thirring model.

{\it Representation of fermions in terms of bosonic fields (bosonization). Cancellation of divergent parts in the Lagrangian and the exact relation between the coupling constants.}

\item Renormalization group for the Berezinskii\--Kosterlitz\--Thouless transition

{\it Renormalization of coupling constants in the sine\-/Gordon model in the second order of the perturbation theory. Renormalization group and renormalization group flows.}

\item $O(3)$-model: mass generation by instantons.

{\it Topological properties of the $O(3)$\-/model, topologically nontrivial solutions in the Euclidean plane. Qualitative description of the mass generation by instantons.}

\item $O(N)$\-/model: $1/N$\-/expansion.

{\it $1/N$ perturbation theory for the $O(N)$ \-/models. Mass generation. Kinematic scattering conditions and calculation of the $S$\-/matrix via the perturbation theory.}

\item $O(N)$-model: integrability and exact $S$\-/matrix.

{\it Higher integrals of motion and factorization $S$\-/matrices. Young\--Baxter equation. Calculation of the $S$\-/matrices of the $O(N)$\-/models from factorization conditions and comparison with the perturbative result.}

\item Thirring model: Bethe Ansatz solution.

{\it Pseudovacuum and wave functions of the Thirring model in Bethe Ansatz. Bethe equations and their thermodynamic limit. Renormalization of charge. Spectrum and $S$\-/model matrix.}

\item Heisenberg spin chain and its scaling limit.

{\it XYZ\-/model. Jordan\--Wigner transformation and XY\-/model. Scaling limit and the relation to the Thirring/Sine\-/Gordon model.}

\item Young\--Baxter equation and Bethe Ansatz.

{\it XXZ\-/model and six\-/vertex model. Young equation\--Baxter and commuting transfer\-/matrices. Pseudovacuum and eigenstates in the
algebraic Bethe Ansatz. Bethe equations and their solution in the thermodynamic limit. Calculation of the free energy of the six\-/vertex model.}

\item Kondo problem: derivation of the Bethe Ansatz.

{\it Kondo effect. Reducing to a one\-/dimensional problem. Primary and secondary Bethe Ansatz. The system of Bethe equations for the Kondo problem.}

\item Kondo problem: solving Bethe equations.

{\it Ground state in the zero magnetic field. Speculation on the derivation of magnetization for a system in an external magnetic field. A brief discussion of the final temperature case.}

\end {enumerate}


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