Created over 8 years ago, updated over 8 years ago
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 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
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 modes
information-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.
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 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
- 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 modes
- information-theoretical view of mean-field (optional)
- mean-field in Green-function language (brief overview)
- brief introduction to _dynamical_ mean-field theory
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### Recommended references
for full bibliography, refer to the References [node](/nodes/24135).
* 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.
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*Notice: This section is subject to updates.*