Large scale neural quantum states reveal the interplay between superconductivity and quantum criticality in the Hofstadter-Hubbard model

Understanding how a parent insulating state shapes the superconductivity that emerges upon doping is a long-standing problem dating back to Anderson's resonating-valence-bond proposal. The triangular-lattice Hofstadter-Hubbard model with $\pi/2$ flux per plaquette offers an ideal setting for this question: at half filling it hosts two distinct parent states---an integer quantum Hall insulator at weak coupling and a chiral spin liquid at intermediate coupling---separated by a topological phase transition. Using neural quantum states on tori of up to $432$ sites, we present strong evidence that the transition is continuous, with a vanishing $2e$ charge gap and critical charge fluctuations. Upon doping, we find a topological superconductor with off-diagonal long-range order consistent with $d+id$ pairing on either side of the transition. The two ingredients of superconductivity, pair formation and phase coherence, respond to the parent state in sharply different ways: while the pairing order parameter remains nearly unchanged across the transition, the superfluid stiffness is strongly enhanced near the critical point. The energy scale of the superconductor is therefore set not by which parent state is doped, but by proximity to the transition between them. Our results establish neural quantum states as a powerful tool for understanding the delicate interplay between long-ranged electronic correlations and superconductivity.