Variational Monte Carlo method for shell-model calculations in odd-mass nuclei and restoration of symmetry

Abstract

We investigate two kinds of extensions for the variational Monte Carlo (VMC) method with the Pfaffian in the nuclear shell-model calculations. One is the extension to odd-mass nuclei, for which we find a new Pfaffian expression of the VMC matrix elements. We can, thereby, give a unified VMC framework both for even and odd mass nuclei. The other is the extension of the variation after angular-momentum projection. We successfully implement the full angular-momentum projected trial state into the VMC method, which can provide us with the precise yrast energies. We also find a unique characteristic, namely that this angular-momentum projection in the VMC can be even "approximately" performed. This characteristic is useful not only for efficient computation but also for precise estimation of the yrast energies through the energy-variance extrapolation.

Symmetry projection to coupled-cluster singles and doubles wave function through the Monte Carlo method

“…( 13) by the Markov chain Monte Carlo (MCMC) method. Note that this technique has been presented in cases with pair-condensates [14][15][16][17]. Therefore, by applying the Monte Carlo sampling, the symmetry projected energy can be estimated as…”

“…(13). To perform the MCMC, we take the basis states specified by s as a random walker, and we move this s-basis to another nearby s ′ -basis under the control of the Metropolis-Hasting (MH) algorithm, keeping the detailed balance as in [16,17].…”

$SO(3)$ quadratures in angular-momentum projection

“…A mean-field method often provides us with a spontaneously symmetry-broken wave function, whose symmetry can be restored by the projection method. The projection method plays an important role in various mean-field and beyond-mean-field approaches such as the projected shell model [2], the VAMPIR method [3], the projected configuration interaction method [4], the Fermionic molecular dynamics [5], the Monte Carlo shell model (MCSM) [6], the quasi-particle vacua shell model [7], the hybrid multi-determinant method [8], the projected variational Monte Carlo method [9], and the generator coordinate method (GCM) with angular-momentum projection (e.g. [10][11][12]).…”

Alpha correlations in terms of the hyper-Pfaffian

“…Recently the variational Monte Carlo (VMC) method with the pair-condensate trial state has been applied to fermion many-body systems [7,8], including the nuclear shell model [9,10]. The VMC trial wave function with the pair-condensation state is given as…”

“…where P f is the Pfaffian. (For its detail, see [9,10] for example.) Note that the Pfaffian has been recently studied in the overlap between the HFB wave functions, for examples, [11][12][13].…”