I have been doing some simple experiments with the transverse field Ising model and its Jordan Wigner transformed version, the Kitaev chain, in the Julia version of ITensor. I am trying to understand how DMRG deals with ground state degeneracies/ topological degeneracies. I have a few questions about this:

When the transverse field is zero or small, DMRG ends up picking one of the two ferromagnetically ordered states at random. Why does it not pick linear combinations? My guess would be: Linear combinations of the ordered states have long range entanglement. This cannot be captured by DMRG because it arrives at the ground state by optimizing local bonds. Is that correct?

Also, the MPS representation of these linear combinations would have a tiny bond dimension of 2. So what is the signature of long range entanglement in the MPS representation if it is not a large bond dimension?

Is there any way though to force the DMRG to pick up one of the cat states?

One possibility is to Jordan Wigner transform the Ising model to a Kitaev chain. When I compute ground states for the Kitaev chain with open boundary conditions in the topological phase, DMRG always picks up a uniform superposition of the even and odd parity states. This would correspond to the ordered states in the spin language. Again, why does DMRG do this? Does the long range entanglement argument translate into the fermion language as well?

A side comment/ question about this: Even though DMRG picks up the "ordered" state, when I compute the expectation value of the Jordan Wigner spin sz, I get zero instead of +1 or -1 which is expected of the ordered state.

Upon adding periodic or anti-periodic boundary terms, DMRG ends up picking the even or odd parity state (and this would correspond to picking up a cat state in the spin language). My question about this is the following: For the DMRG to successfully pick a definite parity state, I need a large enough bond dimension for the initial state psi0 (the number of sweeps does not matter so much). Why is this the case when the final bond dimension of the cat state is just 2? And how should the bond dimension of the initial state scale with system size to enable the DMRG to converge to a definite parity ground state?

Thank you!