Extended Data Fig. 7: Ramping speed and gauge violation.

a, The phase transition is driven by ramping the mass m and the effective coupling \(\tilde{t}\). We start from a large negative value of mass \(m/\tilde{t}\), retain stronger coupling around the critical point, and end up with a large positive mass. b, Gauge violation against total ramping time calculated with the t-DMRG method in a system with 16 (red), 24 (blue) and 32 (orange) optical-lattice sites. Using the same shape of the ramping curve in a, we change the ramping speed by constraining the total ramping time. The squares points are the maxima of ϵ(t) throughout the dynamics, while the circles represent the gauge violations of the final states. Owing to the coherence of our many-body system, ϵ(t) reaches its maximum around the critical point, and decreases after crossing the critical point (see Fig. 4b).