## Course 2: Many-body methods for nuclear physics

Motivation and background.

The course aims at teaching basic theoretical approaches that are used in first-principle calculations of quantum mechanical observables. It requires fluency in the first-level quantum mechanics and gives students expertise in using advanced tools in many-body physics of complex systems. The techniques which are discussed are relevant to many applications in physics and chemistry, in particular for materials science and nanotechnological studies. It is assumed that the students have taken a basic course in many-body theory that includes topics like second quantization, Wick's theorem, many-body perturbation theory, diagrammatic representations of expectation values and Hartree-Fock theory. If this is not the case, the network provides course material which covers these topics. It is then expected that the students will study parts of this material before attending the lectures. The course is however structured so that many of these topics are repeated at the beginning of the course. This material is encompassed in topics 1-4 below.

1. Second quantization

1.1. Hamiltonian of many-fermion system

1.2. Permutation symmetry

1.3. Basis of many-fermion states

1.4. Second-quantization representation for states

1.5. Second-quantization representation for operators

1.6. Second-quantization and angular momentum representation of operators and states

2. Wick's theorem

2.1. Proof of Wick's theorem

2.2. Generalized Wicks theorem

2.3. Normal-ordered operators

2.4. Particle-hole formalism and new reference vacuum

3. Diagrammatic representation

3.1. Derivation of diagram rules, one-particle and two-particle operators

3.2. Applications to expectation values

4. Hartree-Fock theory

4.1. Derivation of Hartree-Fock equations in second quantization

4.2. Thouless' theorem and stability of Hartree-Fock solution

4.3. Brillouin's and Koopman's theorems

4.4. Restricted and unrestricted Hartree-Fock

5. Configuration interaction approaches

5.1. Relation to Hartree-Fock theory and similarity transformations

5.2. Algorithms for diagonalizing eigenvalue problems (Householder transformations and Francis algorithm and Lanczos iterative procedures)

5.3. The nuclear shell model

5.4. Truncated shell-model approaches and no-core shell-model approach.

6. Many-body perturbation theory

6.1. Rayleigh-Schroedinger and Brillouin-Wigner perturbation theory

6.2. Diagrammatic representation of perturbative contributions, computational aspects.

6.3. Time-dependent perturbation theory and linked-diagram theorem.

6.4. Perturbative resummation of diagrams, ladder diagrams and particle-hole diagrams (TDA and RPA).

7. Coupled-cluster theory

7.1. Derivation of basic equations

7.2. Inclusion of correlations beyond two-particle-two-hole excitations, role of triples

7.3. Comparison with many-body perturbation theory and shell-model

7.4. Equation of motion methods for excited states and particle-attached and removed states.

7.5. Discussion of algorithms for solving coupled-cluster equations.

8. Monte Carlo methods

8.1. Numerical integration with Monte Carlo methods

8.2. Metropolis algorithm and detailed balance

8.3. Construction of trial wave functions, Slater determinant and Jastrow factors

8.4. Variational Monte Carlo approaches with importance sampling.

8.5. Applications to simple systems

8.6. Diffusion Monte Carlo methods

8.7. Fixed node approach for fermionic systems

Syllabus of 30+30+60 hours (lectures+exercises+work on projects).