Spin-glass theory for pedestrians

By Tommaso Castellani, Andrea Cavagna. 2005

384 citations recorded by the source corpus.

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.

Subfields: Condensed Matter Physics, Economics and Econometrics, Materials Chemistry

View Paper: https://www.semanticscholar.org/paper/4e07376e980cf0bc56ad495ae30594b107ed7ff0

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Citations in the corpus

Comparing dynamics: deep neural networks versus glassy systems (2018, methodology): “…of degrees of freedom comparable to the number M of weights that are used in deep learning (in our examples M = 10 4 − 10 7 ) would take an exponentially long time to go beyond the highest and widest minima and reach the bottom of the landscape (Castellani & Cavagna, 2005; Berthier & Biroli, 2011).”
Reinforcement Learning in Different Phases of Quantum Control (2017, methodology): “To study the correlations between the infidelity minima as a function of the total protocol duration T, we define the correlator qðTÞ closely related to the Edwards-Anderson order parameter for the existence of spin-glass order [69,70] as”
Mean-field theory of hard sphere glasses and jamming (2008, methodology): “In principle the two methods should be equivalent (see (Castellani and Cavagna, 2005) for a very pedagogical discussion in the case of mean field models), still it seems that the replica method gives more accurate quantitative results and for this reason we will focus on this method in the…”
Extreme value statistics of correlated random variables: A pedagogical review (2019, neutral): “[264] T.”
Spectral bias and task-model alignment explain generalization in kernel regression and infinitely wide neural networks (2020, methodology): “[51] Tommaso Castellani and Andrea Cavagna.”
Dipole blockade in a cold Rydberg atomic sample [Invited] (2010, background): “Analogy can be made with excitation diffusion of excitons [41] or spin glasses systems [42].”
Theoretical perspective on the glass transition and amorphous materials (2010, methodology): “Technically, the thermodynamics of the p-spin can be solved, for p > 2, using a one-step replica symmetry breaking ansatz, see (Castellani and Cavagna, 2005) for a review.”
Universality of the SAT-UNSAT (jamming) threshold in non-convex continuous constraint satisfaction problems (2017, methodology): “We do not discuss in details the properties of this transition, for which we refer to [ 4, 30, 32 ] .”