In the Sherrington-Kirkpatrick (SK) model, one considers an Ising Hamiltonian

$$H = -\sum_{i<j}J_{ij}s_is_j$$

where $J_{ij}$ are drawn independently from a Gaussian distribution with mean $J_0$ and standard deviation $J$.

In their original papers [1,2], SK obtain the phase diagram:

enter image description here

I followed all the steps of the derivation. However I don't understand how the phase diagram for negative $J_0$ is derived, because there is a change of variables involved where the square root of $J_0$ is taken which would give an imaginary result (see line before Eq.(9) in 1). I think some (perhaps trivial) modifications are needed to obtain the phase diagram for negative $J_0$.

So my question is: Assuming replica symmetry (as SK), how does one derive the phase diagram for $J_0<0$?


  1. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.35.1792
  2. https://link.aps.org/doi/10.1103/PhysRevB.17.4384

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