arXiv:1907.12124 - Githubissues

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Title: From eigenstate to Hamiltonian: new prospects for ergodicity and localization
Reason: inverse problem: from eigenstates to parent Hamiltonian
Some results:
- Quantum localization (interacting or not) leads to the non-uniqueness of parent Hamiltonians (due to the fact that e3 for covariance matrix is approaching zero algebraically with system size)
- On the contrary, ergodic many-body eigenstates fully and uniquely encode their parent Hamiltonian. (No nontrivial eigenvalues for C approaching zero, the minimal one approaching 1/4 instead)
- The new fields configurations for parent Hamiltonian is segmental correlated with the original fileds. With domain walls lying in the cut for the minimal value of EE.
- The energy for new parent Hamiltonian is centered at epsilon=0.5 no matter what the initial energy is.

Relevant works on this inverse problem

Xiao-Liang Qi and Daniel Ranard, “Determining a local hamiltonian from a single eigenstate,” arXiv:1712.01850.
Eli Chertkov and Bryan K. Clark, "Computational Inverse Method for Constructing Spaces of Quantum Models from Wave Functions", Phys. Rev. X 8, 031029 (2018).
James R. Garrison and Tarun Grover, “Does a single eigenstate encode the full hamiltonian?” Phys. Rev. X 8, 021026 (2018).
Maxime Dupont and Nicolas Laflorencie, “Many-body localization as a large family of localized ground states,” Phys. Rev. B 99, 020202 (2019). Similar to the main paper, using ground states to construct new parent Hamiltonians and the states is turning to the highly excited states. Perhaps the other direction is more useful, i.e. directly mapping excited states to ground states of some Hamiltonian, such that Lanczos can be used to extract the eigenstates which are in general computational harder for excited states. Observables to compare the ground state with MBL excited states: <ln \lambda_i> where lambda_i is sorted SVD spectrum for half chain reduced density matrix, and histogram of S(i|i+1), where the cut is on i, such histogram has two peaks. See Fig 4c in this paper. (Possible difference between and ln<>?)

Simply minimize the energy variance on given parameter manifold. It may indicate different things when choosing different subspace of Hamiltonians. (Cannot determine whether it is ground state or excited states for the new parent Hamiltonian?)