## Numerical methods for many-particle systems - part I: basics

*Mi, 16:00- 17:30; Do. 16:00- 17:30*

Seminarraum des Instituts für Theoretische Physik

**schedule:**
- Thursday, July 17th, 16:00 - 17:30

**Topic:**
Basic numerical methods and concepts for the investigation
of quantum-mechanical many-particle systems. One of the
topics is the calculation of the Green functions of correlated
fermionic and bosonic systems.
Part II of the lecture will deal with more
advanced methods such as quantum Monte-Carlo, numerical
renormalization group, etc.

The lecture is aiming at
students with some background in quantum-field theory;
experience with modern programming languages (c, fortran, ...)
is useful, but not absolutely necessary.

**Literature:**
- W. Krauth

Statistical Mechanics: Algorithms
and Computations

Oxford University Press (2006)

www.smac.lps.ens.fr

Highly recommended as an introduction for how to develop
algorithms for physical problems. The topics
are, however, quite different to the ones covered in this
lecture.

### I. Single-particle and many-particle spectra

#### I.1 a single level

#### I.2 many levels (no interactions)

#### I.3 tight-binding models

two-site model, one-dimensional chain (periodic and open boundary
conditions), diagonalization of the Hamiltonian versus diagonalization
of the Hamiltonian matrix, diagonalization via Fourier-Transformation, density
of states

### II. Green functions

#### II.1 some definitions

imaginary-time Green function G(tau), spectral function A(omega),
how to calculate A(omega) from a given G(tau)
#### II.2 Green functions for the single-impurity Anderson model

#### II.3 broadening of discrete spectral functions

#### II.4 continued fraction expansion

### III Exact diagonalization of small clusters

#### III.1 general remarks

#### III.2 single-impurity Anderson model

#### III.3 Hubbard model

#### III.4 spin-boson model

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