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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 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.

  • 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.