Difference in observable value for spins with fully symmetric Hamiltonian

DMRG and Numerical Methods
1

Hello,

This is more of a conceptual question, and I think I might know the answer, but I’m still interested in knowing what you guys think.

I have discovered that very often, even with fermionic Hamiltonian that’s symmetric with up and down spins (all terms are the same for both spins symmetric terms, or we have terms like U\hat{n}_{\uparrow\downarrow}), and with the same number of up and down particles using QN conserving basis, especially in time evolution (TDVP), I will still get observables that start to differ.

My questions is, how exactly does this happen? I can see a heuristic argument, that maybe when the entanglement in the system start to become significant, the algorithm behind truncation of SVDs do not explicit ‘know’ that they have to respect up and down spin, and maybe the sectors corresponding to up and down spins can start to take different dimensions (or something similar).

But I guess my question is if the states (up and down) start out as identical, what mechanism can cause them to start to diverge (i.e. if I’m diagonalizing the same effective Hamiltonian with the same input vectors, then they should deterministically give me the same results). I can definitely see errors accumulating as system evolves, but I guess in my mind the errors should also be the same for the up and down spins.

2

I’m not totally sure what observables you are referring to, but the total S^z of your system should not change at all if you are conserving total S^z in your calculations. If that is changing, please check that you have that QN set to be conserved, or else please report a bug if that’s still not happening.

If you are referring to a more local property, like whether the expected value of S^z on some site is expected to be zero in the exact limit, and you are seeing deviations from this, then it is probably a combination of finite-size / finite-boundary effects and finite bond dimension effects.