This is actually a common problem that makes very much sense is comes from the fact that DMRG optimizes locally (just two sites at once) and the special initial state - product states are very special, having no entanglement at all...
People often say that DMRG is "stuck in a local minima"
You can understand what's going on analytically by considering the effective Hamiltonian, which is diagonalized:
Say you initalize the product state |0101010101> with half filling and want to optimize two sites in the center.
The effective Hamiltonian is the Hamiltonian projected on the current state except on the two sites to be optimized.
If \( H = \sum_{i} (a^\dagger_i a_{i+2} + h.c) + diagonal terms \), you can see the projection makes the off-diagonal terms disappear, because
each offdiagonal term of the Hamiltonian contains a site in the projected region (not being optimized), and there \(<0|a|0> = 0 = <1|a|1>\).
This means the effective Hamiltonian does not contain any of the offdiagonal terms and is already diagonalized, so your state does not get updated - it stays a product state.
As soon as you start from a state which has nonzero expectation value of \(a,a^\dagger\), this problem is avoided.
If you have a non-zero nearest-neighbor hopping, the hopping on the two optimized sites leads to a ground state there with a nonzero expectation value of \(a,a^\dagger\), so you also avoid the problem.
The general solution to this problem is to use the ominous "Mixer" in the DMRG, in the simplest case just by setting
mixer=True
in the DMRG parameters. The mixer "perturbs" the state you have by applying (parts of) the offdiagonal terms in the Hamlitonian to the state, helps to get out of the local minima and is switched off after a few sweeps. It was introduced by Steve White in
arXiv:1805.00055.
Even when you have the nearest neighbor interaction, the mixer might help to get a better final ground state Energy, especially in models with long-range interactions.