Extracting Quantum Many-Body Scarred Eigenstates with Matrix Product States

This repository provides the source code to implement the DMRG-S algorithm proposed in ¹, which accurately extracts quantum many-body scarred eigenstates (a small fraction of non-thermal excited eigenstates of non-integrable Hamiltomians). The DMRG-S algorithm can access system sizes far beyond the scope of exact diagonalization and assist analytical studies in discovering exact MPS representations of new scars for generic Hamiltonians.

The DMRG-S algorithm is implemented based on the ITensor library ² in Julia programming language. The environment setup requires the installation of the ITensor.jl package (with version $\ge$ v0.3.24). The changes to the ITensors library consist of three Julia scripts:

• projmpo.jl
• abstractprojmpo.jl
• dmrgs.jl

projmpo.jl and abstractprojmpo.jl include some changes to the original files in the folder "src/mps/" of the ITensors pakage, which are listed below:

• projmpo.jl includes “product_label” in the struct ProjMPO to add another mode for the product function in abstractprojmpo.jl;
• abstractprojmpo.jl includes some new methods to contract local tensors from MPO $(H-\xi)^2$ and MPS $|\psi_t\rangle$ in order to obtain $\mathcal{A_t}^{[i,i+1]}$ and $\tilde{\psi_t}^{[i,i+1]}$ ¹. In addition, the original function product is added with another mode to implement the operation for the matrix $\mathcal{A}_{t,\text{eff}}^{[i,i+1]}$ mutiplying a vector;

dmrgs.jl contains the main function of DMRG-S and SIMPS method with two-site optimization.

Instructions for setup and usage:

1. Replace the original files projmpo.jl and abstractprojmpo.jl in the folder src/mps/ of the ITensors pakage (which is typically located in .julia/packages/ITensors) with our revised version.
2. Add the file dmrgs.jl into the folder src/mps/. Then add a line include("mps/dmrgs.jl") in the file src/ITensors.jl of the ITensors pakage.
3. Add the methods dmrgs and simps in the file src/exports.jl of the ITensors pakage.
4. Using julia PXP_dmrgs.jl in the terminal to run the code.

where PXP_dmrgs.jl include several adjustable parameters:

• initial_energy for the initial setting of $\xi_0$
• psi0 for the initial setting of $|\psi_0\rangle$
• N is the system size
• maxD sets the maximum bond dimension
• minD sets the minimum bond dimension to start with
• var_thre sets threshold for the variance to update the $\xi_t$

The output files are stored in the fold data in the form of .h5 to store the MPSs during the optimization.

References:

- [1] S.-Y. Zhang, D. Yuan, T. Iadecola, S. Xu, and D.-L. Deng, Extracting quantum many-body scarred eigenstates with matrix product states (2022), arXiv:2211.05140.

- [2] M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor Software Library for Tensor Network Calculations, SciPost Phys. Codebases , 4 (2022).