Extracting conserved operators from a projected entangled pair state

Abstract

Given a tensor network state, how can we determine conserved operators (including Hamiltonians) for which the state is an eigenstate? We answer this question by presenting a method to extract geometrically $k$-local conserved operators that have the given infinite projected entangled pair state (iPEPS) in 2D as an (approximate) eigenstate. The key ingredient is the evaluation of the static structure factors of multi-site operators through differentiating the generating function. Despite the approximation errors, we show that our method is still able to extract from exact or variational iPEPS to good precision both frustration-free and non-frustration-free parent Hamiltonians that are beyond the standard construction and obtain better locality. In particular, we find a 4-site-plaquette local Hamiltonian that approximately has the short-range RVB state as the ground state. Moreover, we find a Hamiltonian that has the deformed toric code state at any string tension as excited eigenstates at the same energy, which might be potential candidates for quantum many-body scars.

Figures (9)

• Figure 1: Given a uniform PEPS $|\Psi\rangle$ as input, our method outputs a geometrically $k$-local operator $\hat{H}$ which has the PEPS as an eigenstate.
• Figure 2: Spectrum of the structure factor matrix $\mathcal{S}$ from the 2D AKLT state. (a) Spectrum of the 1-site $\mathcal{S}$. (b) Spectrum of the 2-site $\mathcal{S}$. We compare the spectra from exactly contracting a finite periodic PEPS with a system size $4\times 4$ (blue dots) and approximately contracting the iPEPS using CTMRG with environment bond dimension $\chi=80$ and finite difference step size $\delta=10^{-4}$ (red circle), which gives the same solutions because they are exact and frustration free.
• Figure 3: Spectrum of the structure factor matrix $\mathcal{S}$ from the variational iPEPS for the ground state of the 2D XX model in a staggered $Z$ field. (a) Spectra of the 1-site $\mathcal{S}$ evaluated with $(D,\chi)=(3,60)$ and the finite difference step $\delta=10^{-4}$, where $D$ is the iPEPS bond dimension and $\chi$ is the CTMRG environment bond dimension. 'U(1)' denotes the corresponding eigenvector is the generator $\sum_i Z_i$. When $J=0$, two eigenvalues are $0$ up to precision $10^{-7}$, and the corresponding eigenvectors are $\sum_iZ_i$ (blue) and $\sum_{i}(-1)^i Z_i$ (red), which are consistent with the fact that the ground state is a simple product state. (b) Low lying spectra of the 2-site $\mathcal{S}$ at $J=1/3$ near the critical point, where $\delta=5\times 10^{-3}$.
• Figure 4: Schematics of the tensor network method for evaluating 2-site static structure factors. The black lines are the virtual indices and the purple lines are the physical indices. (a) Expressing the 2-site operator in the generating function in terms of a 2-site MPO. The non-zeros entries of the tensors are defined using the matrices. (b) Obtaining iPEPO tensor from the MPO tensors. (c) Defining the triple-layer tensors with and without open physical indices. The upper and lower iPEPS tensors represented by the green ovals are from $\bra{\Psi}$ and $\ket{\Psi}$, respectively. (d) The matrix $M$ for evaluating one row of $S$, where the tensors represented by squares and rectangles are the corner and edge environment tensors obtained from CTMRG, respectively.
• Figure 5: Schematics of the tensor network method for evaluating 4-site static structure factors. (a) Expressing the 4-site operator in the generating function in terms of a loop MPO. (b) The definition of the non-zero entries of the MPO tensors at the left-top corner in terms of the matrices. The MPO tensors at other corners can be obtained similarity. (c) Obtaining the two iPEPO tensors from the MPO tensors. (d) The matrix $M$ for evaluating one row of $S$. Notice that the tensor represented by the red dot can not be ignored when evaluating the CTMRG environment tensors.