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Uncle Hamiltonian was built to show the complex relationship between MPS (Matrix Product State) states and Hamiltonians, which claims that for a block injective MPS state, we can build a local frustration free uncle Hamiltonian with a continuous spectrum and the MPS state is a ground state of it.

I am a little confused about the physical picture of it. The procedure of building the uncle Hamiltonian is:

(1) For a MPS state with parameter $A$, the original parent Hamiltonian $H(A)$ has degenerated ground states but the Hamiltonian is gapped.

(2) A perturbation on the MPS $A\rightarrow A+\sigma P$ leads to a correspondent perturbed parent Hamiltonian $H(A+\sigma P)$. If the limit $lim_{\sigma \rightarrow 0}H(A+\sigma P)$ exists, we get a so-called uncle Hamiltonian.

Questions:

(1)No matter with or without the perturbation, $H(A)$ and $H(A+\sigma P)$ are both parent Hamiltonians of MPS states so that they are both gapped. Then how can the uncle Hamiltonian, as a limit $lim_{\sigma \rightarrow 0}H(A+\sigma P)$, be gapless? How and when is the gap closed here? Is there an intuitive picture of the mechanism (the change of the spectrum)?

(2) What's the meaning of introducing the uncle Hamiltonian? Is it just a descriptive artifact so that there is no real physical effect? Does this mean the uncle Hamiltonian is only a temporary phenomenon and therefore we do not need to worry about it? What can we learn from the construction of the uncle Hamiltonian?

(3) If it's really an artifact or a very fine-tuned situation, can we resolve it for example by adding noise to the Hamiltonian? Is it possible that a minor random perturbation on the uncle Hamiltonian will open the gap again?

(4) For a given MPS state, the Hamiltonians(gapped and gapless) that taking it as its ground state can be regarded as a manifold. What's the structure of this manifold? Or we can construct a fibre bundle taking the Hamiltonians as the fibre and the MPS states as the base space. Then what's the structure of the fibre bundle?

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