Ising spin glass on random graphs at zero temperature: Not all spins are glassy in the glassy phase

We investigate the replica symmetry broken (RSB) phase of spin glass (SG) models in a random field defined on Bethe lattices at zero temperature. From the properties of the RSB solution we deduce a closed equation for the extreme values of the cavity fields. This equation turns out not to depend on the parameters defining the RSB, and it predicts that the spontaneous RSB does not take place homogeneously on the whole system. Indeed, there exist spins having the same effective local field in all local ground states, exactly as in the replica symmetric (RS) phase, while the spontaneous RSB manifests only on the remaining spins, whose fraction vanishes at criticality. The characterization in terms of spins having fixed or fluctuating local fields can be extended also to the random field Ising model (RFIM), in which case the fluctuating spins are the only responsible for the spontaneous magnetization in the ferromagnetic phase. Close to criticality we are able to connect the statistics of the local fields acting on the spins in the RSB phase with the correlation functions measured in the paramagnetic phase. Identifying the two types of spins on given instances of SG and RFIM, we show that they participate very differently to avalanches produced by flipping a single spin. From the scaling of the number of spins inducing RSB effects close to the critical point and using the M -layer expansion we estimate the upper critical dimension D U ≥ 8 for SG.

[1]  P. Urbani.  Field theory for zero temperature soft anharmonic spin glasses in a field , 2022, Journal of Physics A: Mathematical and Theoretical.

[2]  G. Parisi,et al.  Random-link matching problems on random regular graphs , 2019, Journal of Statistical Mechanics: Theory and Experiment.

[3]  Maria Chiara Angelini,et al.  Loop expansion around the Bethe solution for the random magnetic field Ising ferromagnets at zero temperature , 2019, Proceedings of the National Academy of Sciences.

[4]  M. Mézard,et al.  Journal of Statistical Mechanics: Theory and Experiment , 2011 .

[5]  Maria Chiara Angelini,et al.  Loop expansion around the Bethe approximation through the M-layer construction , 2017, 1707.08499.

[6]  Maria Chiara Angelini,et al.  Real Space Migdal–Kadanoff Renormalisation of Glassy Systems: Recent Results and a Critical Assessment , 2017, Journal of Statistical Physics.

[7]  G. Parisi.  The Marginally Stable Bethe Lattice Spin Glass Revisited , 2016, 1609.05327.

[8]  Sourav Chatterjee,et al.  Absence of Replica Symmetry Breaking in the Random Field Ising Model , 2014, 1404.7178.

[9]  Dmitry Panchenko,et al.  Structure of Finite-RSB Asymptotic Gibbs Measures in the Diluted Spin Glass Models , 2014, 1406.4702.

[10]  G. Parisi,et al.  Replica symmetry breaking in and around six dimensions , 2011, 1111.3313.

[11]  M. Muller,et al.  Avalanches in mean-field models and the Barkhausen noise in spin-glasses , 2010, 1007.2069.

[12]  Guilhem Semerjian,et al.  On the Freezing of Variables in Random Constraint Satisfaction Problems , 2007, ArXiv.

[13]  M. Talagrand,et al.  Bounds for diluted mean-fields spin glass models , 2004, math/0405357.

[14]  Andrea Montanari,et al.  Instability of one-step replica-symmetry-broken phase in satisfiability problems , 2003, ArXiv.

[15]  F. Guerra.  Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model , 2002, cond-mat/0205123.

[16]  G. Parisi,et al.  Spin glasses on Bethe lattices for large coordination number , 2002, cond-mat/0207144.

[17]  V. Akila,et al.  Information , 2001, The Lancet.

[18]  G. Parisi,et al.  Non trivial overlap distributions at zero temperature , 2000, cond-mat/0006188.

[19]  J. Herskowitz,et al.  Proceedings of the National Academy of Sciences, USA , 1996, Current Biology.

[20]  S. Kak.  Information, physics, and computation , 1996 .

[21]  Pik-Yin Lai,et al.  The finite connectivity spin glass: investigation of replica symmetry breaking of the ground state , 1990 .

[22]  Y. Goldschmidt,et al.  Replica symmetry breaking in finite connectivity systems: a large connectivity expansion at finite and zero temperature , 1989 .

[23]  P. Mottishaw,et al.  Replica Symmetry Breaking and the Spin-Glass on a Bethe Lattice , 1987 .

[24]  M. Mézard,et al.  Spin Glass Theory and Beyond , 1987 .

[25]  S. Redner,et al.  Introduction To Percolation Theory , 2018 .

[26]  G. Parisi,et al.  Study of a simple hypothesis for the mean-field theory of spin-glasses , 1981 .

[27]  A. Bray,et al.  Renormalisation-group approach to the spin glass transition in finite magnetic fields , 1980 .

[28]  G. Parisi,et al.  A Simple hypothesis for the spin glass phase of the pnfinite-ranged SK model , 1980 .

[29]  G. Parisi.  A sequence of approximated solutions to the S-K model for spin glasses , 1980 .

[30]  G. Parisi.  The order parameter for spin glasses: a function on the interval 0-1 , 1980 .

[31]  D. Thouless,et al.  Stability of the Sherrington-Kirkpatrick solution of a spin glass model , 1978 .