Hunting Anomalous Excitations in BCC Helium-4

Hunting Anomalous Excitations in BCC Helium-4

Jaron T. Krogel1

Saad Khairallah2

David Ceperley1

1Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL

2Lawrence Livermore National Laboratory, Livermore, CA

Neutron Scattering ExperimentsNeutron Scattering Experiments

.

Markovich, et al. PRL 88, 19 (2002) Pelleg, et al. PRB 73, 180301 (2006)

Discovery of HOB along 110 Discovery of LOB & HOB along 111

Aims of this study1. Calculate excitation spectrum from first principles2. Explore the nature of the excitations, i.e. are they

related to vacancies, defects, localized modes, …

Goals and MotivationGoals and Motivation

Why Correlation Function Quantum Monte Carlo?1. Used to obtain excitation energies for molecular

vibrations and homogeneous electron gas2. Both energies and excited state wavefunctions are

available, providing more microscopic detail

Carleo, et al., PRB 80, 094301 (2009)

Variational Theorem Imaginary Time Projection

Many Body Basis

Generalized Eigenvalue Problem

Rayleigh Quotient

Projected Basis

Projected Eigenvalues

Correlation Function Quantum Monte Carlo

Correlation Function Quantum Monte Carlo

are strict upper bounds to

, for t largeJ.K.L. MacDonald, PR 43, 830 (1933)

Brief Overview of CFQMC Implementation

•Single random walk samples guiding function

•Basis states and local energies saved in imaginary time histories

•Matrix elements appear as 2-point correlation functions

Correlation Function Quantum Monte Carlo

Correlation Function Quantum Monte Carlo

D..M. Ceperley & B. Bernu, J. Chem. Phys. 89, 6316 (1988)

Interactions Many Body BasisPair Potential

Aziz, et al., Metrologia 27, 211 (1990)

Aziz HFD-B2 Potential

Site Excitations

L.H. Nosanow, PR 146, 120 (1966)

Modeling Crystalline HeliumModeling Crystalline Helium

1/r10

10.9 K

1/r6

Trial Ground State

Crystal Symmetries

Translation Symmetries

Point Group Symmetries

Modeling Crystalline HeliumModeling Crystalline Helium

Crystal MomentumCrystal Momentum Operator K

Basis Representation

Crystal Momentum Eigenvalues

Simultaneous Diagonalization

Results: Eigenvalue ConvergenceResults: Eigenvalue Convergence

54 atom cell (3x3x3 unit cells)

Results: Dispersion RelationResults: Dispersion Relation

Composite 54 (3x3x3)128 (4x4x4)250 (5x5x5)

LegendBlack (Exp Ac)Red (Exp Opt)Blue o (CFQMC)

Results: Dispersion RelationResults: Dispersion Relation

Composite 54 (3x3x3)128 (4x4x4)250 (5x5x5)

LegendBlack (Exp Ac)Red (Exp Opt)Blue o (CFQMC)

Conclusions•A site local basis appears to sufficiently describe acoustic modes•Lower optic branch unobserved, perhaps qualitative differences•Possible sighting of higher optic modes

Future Work•Investigate higher optic mode with longer projection in smaller cell•Compute real space density to assess the nature of the excitations

*Supported by DOE Endstation Grant: DOE-DEFG05-08OR23336

Conclusions and Future WorkConclusions and Future Work