Spin-glass theory for pedestrians

In these notes the main theoretical concepts and techniques in the field of mean-field spin glasses are reviewed in a compact and pedagogical way, for the benefit of the graduate and undergraduate student. One particular spin-glass model is analysed (the p-spin spherical model) by using three different approaches: thermodynamics, covering pure states, overlaps, overlap distribution, replica symmetry breaking, and the static transition; dynamics, covering the generating functional method, generalized Langevin equation, equations for the correlation and the response, the mode coupling approximation, and the dynamical transition; and finally complexity, covering the mean-field (Thouless–Anderson–Palmer) free energy, metastable states, entropy crisis, threshold energy, and saddles. Particular attention has been paid to the mutual consistency of the results obtained from the different methods.