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.
- 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’
--- magnetic susceptibility (static, position dependent)
--- specific heat
-- gap equation and its vanishing
-- first study of the phase transition
mean-field for Hubbard/Anderson model
(choose one between Hubbard or Anderson)
-- Stoner criterion
-- spontaneous magnetization
-- Pauli parmagnetism: cf. Curie paramagnetism
mean-field in Lagrangian/action formalism
-- saddle-point approximation
variational mean-field method
-- 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 modes
information-theoretical view of mean-field (optional)
mean-field in Green-function language (brief overview)
brief introduction to dynamical mean-field theory
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.