It is known that the ground state of some quantum spin models is non-degenerate. For example, the ground states of the quantum Ising model and the ferromagnetic Heisenberg model on the subspace of a fixed value of the z-component of the total spin are non-degenerate. For certainty, the Hamiltonian operators of these models are written as $$ \hat{H}_{qI} = -h\sum_j \hat{S}_j^x - \frac12\sum_{i,j}J_{i,j}\ \hat{S}^z_i\hat{S}^z_j $$ and $$ \hat{H}_{Heis} = -\frac12\sum_{i,j} J_{i,j}\ \hat{\vec{S}}_i \hat{\vec{S}}_j,\quad J_{i,j} \geq 0. $$ I have a strong feeling that the non-degeneracy property of the ground states in these cases is a consequence of some theorem. Alas, I have not seen any references to such a theorem in the books and articles I have read. Can someone help and give a link where such a theorem is mentioned?
For the case of the Schrodinger equation for a particle in a potential well, such a theorem is well known. See, for example, the discussion here
Update. Response to NorbertSchuch's comments. My question is about finite-size systems. Otherwise, I would not say that the ground state of the quantum Ising model is non-degenerate. The Heisenberg ferromagnetic model is discussed in this book. The model with the interaction of the nearest neighbors is discussed there on pages 265-266. I think the same arguments are valid for the general ferromagnetic Heisenberg model. For the case when $\hat{S}^\alpha_j$ are spin-s operators, the statement is made that the ground state of the ferromagnetic Heisenberg model is $2sN+1$-fold degenerate, where $N$ is the number of spins. This number coincides with the number of possible different values of $z$-component of total spin, $S^z_{tot}$. Therefore I think that the ground state of the finite-size ferromagnetic Heisenberg model is non-degenerate at a fixed $S^z_{tot}$.
The question of what properties of the Heisenberg and Ising model lead to non-degeneracy of the ground state can be considered as part of my question. Although, after looking at the link provided by Tobias Funke, I think that this feature is the non-positivity of elements and irreducibility of the non-diagonal part of the Hamiltonian operator in some basis. Then a previously unknown to me version of the Perron Frobenius theorem is applied. I didn't know before about generalization of this theorem to the case of matrices whose elements can be zero.
