Problem with the Rank of a matrix after a FindRoot

I am solving a boundary problem defined by a set of four-component vectorial functions. Without going into many details, I have two functions on the left and two on the right of an interface. The boundary problem is defined by requiring that the sum of the two functions on the left is equal to the sum of the two on the right when the functions are evaluated at the position of the interface. The various components of the functions are given by confluent hypergeometric functions of the first and second kind. A similar problem is studied in this work but with two-components functions "L. Cohnitz, A. De Martino, W. Häusler, and R. Egger, Phys. Rev. B 94, 165443 (2016) []".

The problem admits a unique solution when the determinant of the matrix associated with the 4 functions is zero. There is a free parameter $ j$ and for a given value of it, I can determine with FindRoot the associated value of the energy $ \mathcal{E}$ . For this pair of number, the determinant is smaller than $ 10^{-10}$ , however, the rank of the associated matrix is not reduced to 3 but is still 4. To reduce the Rank of the matrix, I need to work around the solution $ \mathcal{E}$ , until the determinant is not smaller than $ 10^{-16}$ !

Is there any solution around this problem. I am happy to give additional details if necessary, I am not posting the code because is pretty long at the moment.