Emergent Topological Universality and Marginal Replica Symmetry Breaking in Gauge-Correlated Spin Glasses

Abstract

Recent tensor-network samplings of modified Nishimori spin glasses have revealed robust finite-temperature critical transitions in two dimensions, defying the standard Edwards-Anderson lower critical dimension boundary ($d_{l}\approx2.5$). We present a theoretical framework demonstrating that the discrete $Z_{2}$ gauge constraints utilized to bypass Monte Carlo kinetic traps fundamentally alter the system's universality class. By mapping the algorithmic disorder distribution to the 2D Ising Conformal Field Theory (CFT), we prove the emergent spatial variance generates a fractional momentum operator that drives the dynamic upper critical dimension to zero ($d_{u}\rightarrow0$). This marginal topology dynamically suppresses the replica-coupling vertices, yielding an infinite-order Berezinskii-Kosterlitz-Thouless (BKT) transition and a non-integrable replicon divergence that predicts a massive instability toward 1-step Replica Symmetry Breaking (1-RSB). Leveraging a spectral Corner Transfer Matrix Renormalization Group (CTMRG) architecture up to macroscopic scales ($L=1024$), we quantitatively validate the topological scaling argument $\mathcal{G}((T-T_{c})\ln(L/L_{0}))$. By isolating the continuum field theory from microscopic lattice artifacts, we recover the fundamental lattice metric $L_{0}\approx 0.94$, unequivocally confirming the existence of a distinct, topologically driven spin-glass phase.