Critical couplings of two dimensional Ising model on various lattices

What the math gives to ML

The paper provides a constructive Kac–Ward fermionic representation in which a lattice model becomes a finite-dimensional momentum-dependent matrix \(\mathcal A(\mathbf{k})\). Criticality is detected by the exact zero-mode condition \(\det \mathcal A(0)=0\), while the low-momentum expansion gives a mass gap and dispersion relation. This suggests a directed-edge neural architecture whose propagation operator includes non-backtracking connectivity and orientation-dependent turning phases, with training controlled by the smallest singular value of \(\mathcal A(0)\). The transferable mechanism is a computable spectral critical surface separating contractive, near-critical, and amplifying propagation regimes.