Phase transitions in coupled Ising chains and SO($N$)-symmetric spin chains

We investigate the nature of quantum phase transitions in a (1+1)-dimensional field theory composed of $N$ copies of the Ising conformal field theory interacting via competing relevant perturbations. The field theory governs the competition between a mass term and an interaction involving the product of $N$ order-parameter fields, which is realized, e.g. in coupled Ising chains, two-leg spin ladders, and SO($N$)-symmetric spin chains. By combining a perturbative renormalization group analysis and large-scale matrix-product state simulations, we systematically determine the nature of the phase transition as a function of $N$. For $N=2$ and $N=3$, we confirm that the transition is continuous, belonging to the Ising and four-state Potts universality classes, respectively. In contrast, for $N \ge 4$, our results provide compelling evidence that the transition becomes first order. We further apply these findings to specific lattice models with SO($N$) symmetry, including spin-$1/2$ and spin-$1$ two-leg ladders, that realize a direct transition between an SO($N$) symmetry-protected topological phase and a trivial phase. Our results refine a recent conjecture regarding the criticality of transitions between SPT phases.