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Real Space Migdal–Kadanoff Renormalisation of Glassy Systems: Recent Results and a Critical Assessment

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Abstract

In this manuscript, in honour of L. Kadanoff, we present recent progress obtained in the description of finite dimensional glassy systems thanks to the Migdal–Kadanoff renormalisation group (MK-RG). We provide a critical assessment of the method, in particular discuss its limitation in describing situations in which an infinite number of pure states might be present, and analyse the MK-RG flow in the limit of infinite dimensions. MK-RG predicts that the spin-glass transition in a field and the glass transition are governed by zero-temperature fixed points of the renormalization group flow. This implies a typical energy scale that grows, approaching the transition, as a power of the correlation length, thus leading to enormously large time-scales as expected from experiments and simulations. These fixed points exist only in dimensions larger than \(d_L>3\) but they nevertheless influence the RG flow below it, in particular in three dimensions. MK-RG thus predicts a similar behavior for spin-glasses in a field and models of glasses and relates it to the presence of avoided critical points.

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Notes

  1. Unfortunately disordered nearest neighbour Potts glasses do not display glassy phenomenology in three dimension. For this reason we introduced SP model in order to bypass the problems (weak frustration and lack of glassy behaviour) found for disordered Potts model.

  2. Residual multi-\(\sigma _i\) interactions between closest blocks, if present, can be safely neglected remain since they become negligible with respect to the pairwise interaction between nearest neighbour blocks if the size of the blocks is taken much larger than \(\ell _{I}\).

  3. Let’s just point out a subtlety: the field presents in Eq. (1) are site-fields while the ones present in Eq. (2) are link-fields, i.e. associated to a link. The renormalisation of Eq. (1) can produce additional link-fields, in addition to the site ones, exactly as happens for the M-value models.

  4. See the Supplementary information of Ref. [49].

  5. Note that by symmetry the variance of \(H(\sigma )\) does not depend on \(\sigma \) and likewise for \(J(\sigma ,\tau )\).

  6. Besides having the same mean-field theory, disordered models of glasses and realistic models of supercooled liquids possibly also share the same effective field theory for the overlap field, as first argued in [63].

  7. The relation between spin-glasses in a field and glasses in three dimensions was first advocated by Moore and collaborators on field theoretical basis and then supported by MK-RG computations. Although their analysis didn’t discuss this in terms of avoided fixed point their conclusions are very similar to ours.

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Acknowledgements

We thank C. Cammarota, M. Moore, G. Tarjus, P. Urbani for discussions. We acknowledge support from the ERC Grants NPRGGLASS and from the Simons Foundation (N. 454935, Giulio Biroli).

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Correspondence to Giulio Biroli.

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Dedicated to the memory of Leo Kadanoff.

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Angelini, M.C., Biroli, G. Real Space Migdal–Kadanoff Renormalisation of Glassy Systems: Recent Results and a Critical Assessment. J Stat Phys 167, 476–498 (2017). https://doi.org/10.1007/s10955-017-1748-4

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