  In [Van Beeumen, et. al, HPC Asia 2020,
https://www.doi.org/10.1145/3368474.3368497] a scalable and matrix-free
eigensolver was proposed for studying the many-body localization (MBL)
transition of two-level quantum spin chain models with nearest-neighbor $XX+YY$
interactions plus $Z$ terms. This type of problem is computationally
challenging because the vector space dimension grows exponentially with the
physical system size, and averaging over different configurations of the random
disorder is needed to obtain relevant statistical behavior. 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 $L = 24$. The matrix-free
approach does not suffer from this memory bottleneck, however, its scalability
is limited by a computation and communication imbalance. We present a few
strategies to reduce this imbalance and to significantly enhance the
scalability of the matrix-free eigensolver. To optimize the communication
performance, we leverage the consistent space runtime, CSPACER, and show its
efficiency in accelerating the MBL irregular communication patterns at scale
compared to optimized MPI non-blocking two-sided and one-sided RMA
implementation variants. The efficiency and effectiveness of the proposed
algorithm is demonstrated by computing eigenstates on a massively parallel
many-core high performance computer.
