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We analyze the possible quantum phase transition patterns occurring withinthe $O(N) \times {\mathbb{Z}_2}$ scalar multi-field model at vanishingtemperatures in $(1+1)$-dimensions. The physical masses associated with the twocoupled scalar sectors are evaluated using the loop approximation up to secondorder. We observe that in the strong coupling regime, the breaking $O(N) \times{\mathbb{Z}_2} \to O(N)$, which is allowed by theMermin-Wagner-Hohenberg-Coleman theorem, can take place through a second-orderphase transition. In order to satisfy this no-go theorem, the $O(N)$ sectormust have a finite mass gap for all coupling values, such that conformality isnever attained, in opposition to what happens in the simpler ${\mathbb{Z}_2}$version. Our evaluations also show that the sign of the interaction between thetwo different fields alters the transition pattern in a significant way. Theseresults may be relevant to describe the quantum phase transitions taking placein cold linear systems with competing order parameters. At the same time thesuper-renormalizable model proposed here can turn out to be useful as aprototype to test resummation techniques as well as non-perturbative methods.

Quantum phase transitions in a bidimensional $O(N) \times {\mathbb{Z}_2}$ scalar field model