Phase diagram of Ising models constructed from hypergraph-product codes*

We study the phase diagram of random bond Ising models with
multi-spin couplings related to the finite rate hypergraph-product
(HP) quantum LDPC codes obtained from random (3, 4)- and (5,
6)-regular Gallager code ensembles [1]. On the Nishimori line, the
corresponding partition functions are related to maximum-likelihood
(ML) decoding of these codes, with the multicritical point
corresponding to the ML decoding threshold. From the specific heat
calculations, we find that the transition in the random bond Ising
model is numerically close to the self-dual line obtained in the
replica limit by Nishimori [2]. The transition is discontinuous (1st
order) away from the multicritical point, with the discontinuity
gradually disappearing as one approaches the multicritical point.
We also construct a simple analytic lower bound for the transition
temperature. [1] J-P. Tillich and G. Zemor, IEEE Transactions on
Information Theory 60, 1193–1202 (2014). [2] H. Nishimori and
K. Nemoto, J. Phys. Soc. Jpn.71, 1198 (2002).