1. 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 1 INFN, section of Padova 2 University of Caen, LPC Caen

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

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

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

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

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

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

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

9. 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. 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. !

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

11. 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 Constrained Path QMC S. Zhang, et al., PRL 74,3652 (1995) Fixed-Node DMC,GFMC D.M. Ceperley, B. Alder, PRL 45,566 (1980) Standard approximation used in nuclear ab initio calculations and in condensed matter physics Selection via a trial state

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

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

14. Variational trial state: The VAP method What trial wave function to initiate, guide, and constrain the 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 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), Projection operator onto spin, Product of determinants Extension for non-yrast states Example:, Projector onto the subspace orthogonal to the. lower states previously obtained

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

16. Phaseless QMC results VAP QMC Exact VAP QMC Exact JB & O. Juillet, PRL 111, 012502 (2013) (yrast states) JB & O. Juillet, in preparation (non-yrast states) USD/GXPF1A Effective interactions

17. 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 Perspectives: Treatment of 3-body interactions Treatment of 3-body interactions Possibility to apply the phaseless QMC formalism to continuum/Gamow shell model? Possibility to apply the phaseless QMC formalism to continuum/Gamow shell model? Real time/finite temperature implementation Real time/finite temperature implementation

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