## UC Berkeley / Lawrence Berkeley Laboratory

#### A Scalable Matrix-Free Eigensolver for Studying Many-Body Localization

**Roel Van Beeumen, LBNL**

We present a scalable and matrix-free eigensolver for studying nearest-neighbor Heisenberg spin chain plus random on-site disorder models that undergo a many-body localization (MBL) transition. This type of problem is computationally challenging because its dimension grows exponentially with the physical system size, and the solve must be iterated many times to average over different configurations of the random disorder. For each eigenvalue problem, eigenvalues from different regions of the spectrum and their corresponding eigenvectors need to be computed. Traditionally, the interior eigenstates for a single eigenvalue problem are computed via the shift-and-invert Lanczos algorithm. Due to the extremely high memory footprint of the LU factorizations, this technique is not well suited for large number of spins $L$, *e.g.*, one needs thousands of compute nodes on modern high performance computing infrastructures to go beyond $L = 24$. We propose a new matrix-free approach that does not suffer from this memory bottleneck and even allows for simulating spin chains up to $L = 24$ spins on a single compute node. We discuss the OpenMP and hybrid MPI–OpenMP implementations of matrix-free block matrix-vector operations that are the key components of the new approach. The efficiency and effectiveness of the proposed algorithm is demonstrated by computing eigenstates in a massively parallel fashion, and analyzing their entanglement entropy to gain insight into the MBL transition.