Hi,
I have obtained the Hamiltonian in the MPO form, and my question is whether there is a convenient operation in ITensor to diagonalize H. Specifically, I am referring to the decomposition:
\hat{H} = V^\dagger T V = V^\dagger S^\dagger D S V
where V represents the matrix of Lanczos vectors spanning the Krylov space, and T is the tridiagonal form of the Hamiltonian. After the tridiagonalization process, T is then fully diagonalized to obtain the eigenvalue matrix D and the eigenvector matrix S. This provides a faithful representation of the Hamiltonian when applied to the initial vector.
Do you want the eigenvectors as a set of MPS, and do you want a set of dominant eigenvectors? If you only need dominant eigenvectors in the form of MPS, DMRG and its extensions to excited states is a method for getting dominant eigenvectors of an MPO.
If you want all of the eigenvectors, that will scale exponentially in the size of the MPO.
Thanks Matt.Sorry, I didn’t describe my question clearly. In fact, I am trying to optimize the correction vector based on the paper https://journals.aps.org/pre/abstract/10.1103/PhysRevE.94.053308. I am mainly referring to the content on the second page of the paper. The paper transforms |x(z)\rangle=\frac{1}{E_{0}+\omega-\hat{H}+i \eta}|A\rangle into the form |x(z)\rangle=V^{\dagger} S^{\dagger} \frac{1}{E_{0}+\omega-D+i \eta} S V|A\rangle, where D is the diagonal form of the Hamiltonian operator \hat{H} in the current basis. The expression \hat{H}=V^{\dagger} T V=V^{\dagger} S^{\dagger} D S V is a faithful representation of the Hamiltonian when applied to the starting vector. V represents the matrix of the Lanczos vectors spanning the Krylov space, and T is the representation of the Hamiltonian in tridiagonal form. The tridiagonalization is then followed by a small full diagonalization of T, yielding the matrix of eigenvectors S.I am not yet familiar with the use of ITensor and the MPO, MPS representations, and I am quite confused:
- I think the operation on \hat{H} here is similar to solving for a set of dominant eigenvectors, but how do these MPS act on |A\rangle? It doesn’t seem quite right.
- I want to use the above procedure to compute the Green’s function, but unfortunately, the boundary effects of open boundaries are too strong, and the results do not match the ones I extrapolated to the thermodynamic limit. The results for periodic boundaries are fine. However, when performing DMRG calculations for the ground state with periodic boundaries for more than a hundred sites, convergence becomes difficult. The virtual bond dimension grows quickly, and the subsequent calculations easily run out of memory. As far as I know, this corresponds to adding long-range entanglement. So, I switched to using TDVP to find the ground state and then applied the method described above to decompose \hat{H}, but I’m unclear on how to proceed with the operations.