Bound states around vacuum in scalar ModMax model

In this work, we consider a two-dimensional scalar field model inspired by the dimensional reduction of a four-dimensional ModMax theory. Upon projecting out the 4D theory down to a 2D theory we obtain a theory which presents a constant electric field and two scalar fields. In order to investigate kinks, we include the presence of a potential and consider the static case with one of the fields in the vacuum, showing that the solutions for the non-uniform field can be mapped into the ones arising from the canonical model. By studying the linear stability of the model, we show that fluctuations around the uniform field are described by a Sturm-Liouville eigenvalue equation whose weight function depends on the non-uniform solution and the parameter of the ModMax model. Remarkably, the presence of the aforementioned weight may bring bound states to light, contrary to what occurs in the canonical model.