ConstrainedPath Quantum Monte Carlo Approach for the Nuclear

Constrained-Path Quantum Monte. Carlo Approach for the Nuclear Shell Model Jérémy Bonnard 1, 2, Olivier Juillet 2 1 INFN, section of Padova 2 University of Caen, LPC Caen

The nuclear shell model Independent nucleons Interacting nucleons pf Configuration mixing sd Active major shell p Inert magic core s Spectroscopy, electromagnetic transition and decay probabilities, deformation … Extended • No-Core Shell Model: ab initio calculations (light nulcei) frameworks • Continuum/Gamow Shell Model: unified description of structure and reactions

Motivations Applicability strongly restricted by the exponential scaling of the size of the Hilbert space with the number of nucleons/shells Quantum Monte-Carlo (QMC) methods represent attractive alternatives to the direct diagonalization of the Hamiltonian Shell Model Monte Carlo Koonin, Dean, & Langanke Phys. Rept. 278, 1 (1997) ü Ground-state properties ü Finite-temperature properties Spectroscopy Sign/phase problem (Except in specific cases) Objecti ve A QMC method allowing to reach the spectroscopy of nuclei with a wellcontrolled sign/phase problem !

Theoretical foundations of QMC methods Configuration-mixing approaches with QMC approaches Many-Body Hilbert Space with any Real & positive Exact wave function reformulated in terms of the average of independentparticle states:

Imaginary-time propagation Many-Body Hilbert Space The initial wave function is projected onto the ground state with the same symmetries Walkers that randomly explore the overcomplete basis The orbitals undergo a Brownian motion reproducing in average the exact many-

Importance of the initial state Many-Body Hilbert Space The initial wave function is projected onto the ground state with the same symmetries The statistical fluctuations are reduced by initializing the Brownian motion with a good approximation of the exact

Principle of the Importance-Sampling Technique Standard sampling Gaussian distribution Importance sampling Probability distribution dedicated to the function Efficiency improved by adaptating the distribution

The stochastic scheme with guided dynamic Idea: Importance sampling incorporated within the S. Zhang, H. Krakauer, PRL 90, 1336401 (2003) Brownian motion Quadratic form of one-body operators: Drift guided by the trial state Diffusion

The sign problem: Origin Many-Body Hilbert Space If the centroids and coincinde, the contributions to the sampling of the two populations cancel each other out: ! . All these trajectories only contribute to the statistical errors and, hence, only degrade the signal-to-noise ratio.

The sign problem: concrete manifestation Stoitcheva et al. , nucl-th/0708, 2945 (2007) Shell Model Monte-Carlo Exact USD Effective interaction

The sign problem: Control Many-Body Hilbert Space All the resulting walkers are divided into a population and a population. having exactly opposite contributions Sign problem ! Finally, the sign problem is controlled by requiring Selection via a trial Standard approximationstate used in nuclear ab initio calculations and in condensed matter physics Constrained Path QMC S. Zhang, et al. , PRL 74, 3652 (1995) Fixed-Node DMC, GFMC D. M. Ceperley, B. Alder, PRL 45, 566 (1980)

From sign to phase problem: Phaseless approximation Sign problem Phase problem Many-Body Hilbert Space Constrained-Path approximation Phaseless QMC S. Zhang, H. Krakauer, PRL 90, 1336401 (2003)

Variational trial state: The VAP method What trial wave function to initiate, guide, and constrainthe Brownian motion? The better the trial state, the more reduced the bias due to the constraint Spectroscopy Quantum numbers:

Variational trial state: The VAP method What trial wave function to initiate, guide, and constrainthe Brownian motion? The better the trial state, the more reduced the bias due to the constraint Spectroscopy Quantum numbers: VAP method: Energy minimization after restoration of quantum numbers Yrast states , Projection operator onto spin , Product of determinants Similar to the VAMPIR approach without direct consideration of pairing Variation After Mean-field Projection In Realistic model space, K. W. Schmid et al. , PRC 29, 291 (1984) Extension for non-yrast states Example: , Projector onto the subspace orthogonal to the. lower states previously obtained

Phaseless QMC results Stoitcheva et al. , nucl-th/0708, 2945 (2007) Shell Model Monte-Carlo Exact • VAP • QMC • Exact JB & O. Juillet, PRL 111, 012502 (2013) (yrast states) JB & O. Juillet, in preparation(non-yrast states) USD Effective interaction

Phaseless QMC results • VAP • QMC • Exact JB & O. Juillet, PRL 111, 012502 (2013) (yrast states) JB & O. Juillet, in preparation(non-yrast states) USD/GXPF 1 A Effective interactions

Summary & persepectives Objective Spectroscopy of nuclei through the shell model via a stochastic reformulation of the Schrödinger equation : Methods : A QMC approach initialized, steereed, and constrained by a Hartree-Fock state with symmetry restoration before variation Results: sd- and pf-shell results proving the ability of the method to yield nearly exact spectroscopies for any nuclei with any interaction Perspective • • s: Treatment of 3 -body interactions Possibility to apply the phaseless QMC formalism to continuum/Gamow shell model? • Real time/finite temperature implementation

Thank you for your attention FUSTIPEN Topical Meeting « New Directions for Nuclear Structure and Reaction Theories » March 16 -20, 2015, GANIL, Caen, France