Factorization in large-scale many-body calculations

Abstract

One approach for solving interacting many-fermion systems is the configurationinteraction method, also sometimes called the interacting shell model, where one finds eigenvalues of the Hamiltonian in a many-body basis of Slater determinants (antisymmetrized products of single-particle wavefunctions). The resulting Hamiltonian matrix is typically very sparse, but for large systems the nonzero matrix elements can nonetheless require terabytes or more of storage. An alternate algorithm, applicable to a broad class of systems with symmetry, in our case rotational invariance, is to exactly factorize both the basis and the interaction using additive/multiplicative quantum numbers; such an algorithm recreates the many-body matrix elements on the fly and can reduce the storage requirements by an order of magnitude or more. We discuss factorization in general and introduce a novel, generalized factorization method, essentially a 'double-factorization' which speeds up basis generation and set-up of required arrays. Although we emphasize techniques, we also place factorization in the context of a specific (unpublished) configuration-interaction code, BIGSTICK, which runs both on serial and parallel machines, and discuss the savings in memory due to factorization.

Large-scale shell-model calculations on the spectroscopy ofN<126Pb isotopes

“…The computational limit for contemporary shell-model calculations is around 10 10 (in the M -scheme) for systems with roughly the same numbers of protons and neutrons. This is possible by applying the so-called factorization technique [59,60]. Systems with only identical particles are more difficult to treat numerically.…”

Realistic calculations for c coefficients of the isobaric mass multiplet equation in 1p0f shell nuclei

“…We performed a series of shell-model calculations using the program BIGSTICK [14,15] to compute the ccoefficients of the IMME for odd-odd N = Z nuclei and their T = 1 analogs in the 1p0f shell with 42 ≤ A ≤ 54. Calculations were performed with the full 1p0f -shell model space, except for A = 54 where up to five particles excited from the 0f 7/2 orbit were permitted with M -scheme dimensions ∼ 500 M [16].…”

Toward scalable many-body calculations for nuclear open quantum systems using the Gamow Shell Model

“…[13]. In the traditional SM, the |φ states are harmonic oscillator states [10,11], which are well suited for well-bound nuclear states, but not for loosely bound and resonant nuclei. Hence, we use Berggren basis states instead [14], which are generated by a finite depth potential, and contain bound, resonant and scattering states (see Fig.(1)).…”

“…This can be effected because theĴ 2 operator is closed in M -scheme, as it connects Slater determinants whose one-body states differ through their m quantum number only, i.e. belonging to the same configuration (also called partition) [10,11]. A configuration enumerates its occupied shells without consideration of the m quantum numbers of the occupied one-body states.…”

“…In GSM, configurations and Slater determinants involve few valence particles, many shells and many one-body states, whereas one has many valence particles, few shells and few one-body states in SM. Thus, we use an implementation based on bit storage which is different than traditional SM [11]. As a state can be occupied by at most one nucleon due to antisymmetry, and previously mentioned SM features, it is convenient in SM to associate a Slater determinant to a binary number.…”