Behavior of the model for negative nonlinearity parameter (b < 0)

Determine the phase structure, critical lines, and order of phase transitions of the two-parameter tensor-network model for negative values of the nonlinearity parameter b<0, including how disfavored domain-wall crossings in the loop-gas/Ising dual picture alter the topology of the phase diagram and the stability of the ferromagnetic, paramagnetic, and anti-ferromagnetic phases.

Background

The paper analyzes a minimal two-parameter model defined via a tensor network that is dual to both a classical Ising-like statistical mechanics system and a fermionic path-integral. The parameters a and b represent, respectively, nearest-neighbor spin interactions (or intra-cell hopping) and a local quartic nonlinearity (or ring-exchange term). The authors develop the phase diagram primarily for b≥0, identifying three phases and their transition lines, including a multicritical point.

In the loop gas interpretation, b controls the weight of domain-wall crossings (with b>0 favoring crossings), while in the Ising interpretation it corresponds to a four-spin plaquette interaction. The paper focuses on b>0 and indicates that the case b<0—where crossings are disfavored and could qualitatively change the phase diagram—has not been analyzed and is left open.