# The Art of Mean-Field Theory

We take our first steps with a deep study of the art of mean-field theory which is a ubiquitous technique in many-body physics, and lays the cornerstones to understand the rôle of the 2-body interactions and their manifold consequences.

The presumed time span is ca. 2 months to cover all the topics.

### Plan

- construction of mean-field (canonical/operator formalism), its physical meaning
- order parametre and its physical meaning
fluctuations and how they destroy the mean-field

mean-field for Ising model

-- order parametre and its behaviour wrt. fluctuations: magnetization

-- thermodynamic quantities: ‘first-’ and ‘second-order’

--- entropy

--- magnetic susceptibility (static, position dependent)

--- specific heat

-- gap equation and its vanishing

-- first study of the phase transitionmean-field for Hubbard/Anderson model

(*choose one between Hubbard or Anderson*)

-- Stoner criterion

-- spontaneous magnetization

-- Pauli parmagnetism:*cf.*Curie paramagnetismmean-field in Lagrangian/action formalism

-- saddle-point approximationvariational mean-field method

anomalous mean-fields

-- Bose-Einstein condensation (BEC)

-- superconductivity à la Bogoliubov–de-Gennes (BdG)above mean-field (brief overview)

-- Gaussian fluctuations around the saddle-point

-- Broken symmetry and its restoration: ergodicity breaking, Goldstone modesinformation-theoretical view of mean-field (optional)

mean-field in Green-function language (brief overview)

brief introduction to

*dynamical*mean-field theory

### Recommended references

for full bibliography, refer to the References node.

- Bruus & Flensberg (2004), chp. 4.
- Goldenfeld (1992), chp. 2.
- Kopietz et al. (2010), chps. 1--2.
- Chaikin & Lubensky (2000), chp. 4.
- Schwabl (2006), chp. 6.
- Altland & Simons (2010), chp. 6.
- Sethna (2006), chp. 6.
- Scheck (2016), chp. 5.
- Sólyom (2007).
- Uzunov (1993).
- Tong (2012).
- Nagaosa (1999), chp. 4.
- Merhav (2010).
- Decoster (2004).
- McKay (2003), chp. 33.

*Notice: This section is subject to updates.*