Astroid Spinodal Boundary in Phase-Based Ising Machines

Oscillator Ising machines (OIMs) and dynamical Ising machines (DIMs) encode binary spins in phase states stabilized by second-harmonic injection (SHI).

In a coupled network, the competition between SHI and the instantaneous local network field reshapes each oscillator's conditional energy landscape.

We show that this competition drives a transition between monostable and bistable regimes through an astroid spinodal boundary.

Near this boundary, the barrier scales as $\Delta E_i\propto\mu_i^{3/2}$ at generic smooth points and as $\Delta E_i\propto\mu_i^{2}$ at the longitudinal cusp.

OIMs and DIMs obey the same spinodal geometry, with their conditional landscapes related by a reversal of the transverse field.

Finally, the first-harmonic conditional landscape is mathematically equivalent, up to an additive constant, to the Stoner--Wohlfarth energy of a uniaxial magnetic particle.