Spin-glass theory for pedestrians

Abstract

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 spinglass model is analyzed (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 (TAP) free energy, metastable states, entropy crisis, threshold energy, and saddles. Particular attention has been paid on the mutual consistency of the results obtained from the different methods.

Quantitative approximation schemes for glasses

“…This approach was first proposed in [4], but a complete and exact solution of the glass transition of identical hard spheres in d = ∞ was obtained much more recently [5][6][7][8][9][10][11][12]. The phase diagram turns out to be similar to the one of a class of spin glass models (Ising p-spin and Potts glasses [13][14][15][16][17][18][19]), thus confirming the main assumption behind the so-called Random First Order Transition (RFOT) theory of the glass transition [20][21][22].…”

“…(18). We heavily rely on previous work where the physical justification of the recipes we use has been described [6,19,59,67], and here we limit ourselves to give some details on how to apply these recipes. We consider a generic potential v(r), hence the control parameters are temperature T , packing fraction (or density) ϕ, and the other parameters that appear in the potential (a sticky attraction, a long range repulsion, etc.).…”

“…In the rest of this work, we adopt a more technical language and we assume that the reader is familiar with liquid theory [27], with the general formulation of RFOT [19,22,66], with the replica method [67,68] and its application to structural glasses [6,59,60] and with the perturbative methods discussed in [6,69] on which the present work is strongly based.…”

The jamming scenario—an introduction and outlook

“…Such a roughening of the energy landscape is predicted by mean-field models of liquids and spins [131][132][133][134] at the so-called mode-coupling temperature T MCT . However these theoretical approaches lead to a (non-observed) power-law divergence of the viscosity at T MCT , and their interpretation in real space is unclear and actively studied [135,136].…”

Regge theory and statistical mechanics

“…However, the remarkable discovery of Parisi 5 [15] has been that, at low temperatures the procedure does not work (this is the so-called "replica symmetry breaking") because of the presence of many metastable states (only recently [8,9,19] it has been possible to prove rigorously that the Parisi solution is the right one to describe the mean field glassy phase). A very important quantity in spin glass theory (to provide an updated list of references is an hopeless task; for two detailed review, see [14,2]) is the TAP free energy (introduced in [20]) f T AP…”