Quantum and Classical Potts Criticality in Driven-Dissipative Bosonic Lattices

What the math gives to ML

The paper identifies an exact global \(\mathbb{Z}_3\) symmetry generated by the three-photon terms: under \(\hat a_j\mapsto\omega\hat a_j\), with \(\omega=e^{2\pi i/3}\), the cubic drive is invariant. This gives a concrete algebraic recipe for neural modules whose features transform in charge sectors modulo 3, rather than using unconstrained real channels. The most transferable asset is not the critical-exponent result itself, but the charge-conservation rule for linear maps and multiplicative nonlinearities. A practical test is a \(\mathbb{Z}_3\)-equivariant graph or transformer block on data with ternary phase, color, or cyclic-label symmetry, comparing parameter efficiency and out-of-distribution rotation accuracy against a standard architecture.