  We study a generalization of the two-dimensional transverse-field Ising
model, combining both ferromagnetic and antiferromagnetic two-body
interactions, that hosts exact global and local Z2 gauge symmetries. Using
exact diagonalization and stochastic series expansion quantum Monte Carlo
methods, we confirm the existence of the topological phase in line with
previous theoretical predictions. Our simulation results show that the
transition between the confined topological phase and the deconfined
paramagnetic phase is of first-order, in contrast to the conventional Z2
lattice gauge model in which the transition maps onto that of the standard
Ising model and is continuous. We further generalize the model by replacing the
transverse field on the gauge spins with a ferromagnetic XX interaction while
keeping the local gauge symmetry intact. We find that the Z2 topological phase
remains stable, while the paramagnetic phase is replaced by a ferromagnetic
phase. The topological-ferromagnetic quantum phase transition is also of
first-order. For both models, we discuss the low-energy spinon and vison
excitations of the topological phase and their avoided level crossings
associated with the first-order quantum phase transitions.
