Compressive Sensing in Heisenberg model Material Science & Engineering

Xinkai Fu1, Fei Zhou2, Vidvuds Ozolins3 Kuang Yaming Honor school, Nanjing University 2 Lawrence Livermore National Laboratory 3 Department of material science and engineering, University of California Los Angeles 1

Generate sensing matrix and select independent identical distributed (iid) structures

Motivation In the Heisenberg model, a model used in statistical physics to model ferromagnetism, the Hamilton is defined as where i and j are neighbors. As the model considers only the coupling between spin neighbors and neglects long range and manybody interactions, it is not accurate. The first principle calculation package VASP for non-collinear magnetic structures requires huge computation time and memory. The interaction coefficients for corresponding clusters contains useful information about the properties of ferromagnetic system (e.g. Curie temperature) and are necessary for Monte Carlo simulation.

The left figure presents the correlation matrix whose off-diagonal entries should be as small as possible. The right figure shows the distribution of the entries.

We generate spin products for each atomic+spin structure and arrange the rows into the sensing matrix. Then we select structures which are as iid as possible. The quality of the selected sensing matrix is measured by computing the N×N cross-correlation matrix. The (i, j) element of the matrix is defined as the normalized product of ith column and jth column.

VASP calculation & Compressive sensing Compressive sensing:

Background Premise: Intuitively, most physics Hamiltonians are approximately sparse in some basis Cluster expanded Heisenberg model

1 2 J CS = arg min µ J 1 + Π J − E 1 J 2

VASP calculation settings: Noncollinear magnetic calculation Fixed lattice volume, relax ions only Convergence: 10-6 eV Results

The figure presents the important efficient cluster interactions (ECI’s). Nearest neighbor pair interactions dominates as expected and the fifth pair iis second most important. There are eight nonzero four-body interaction terms out of the total 171 terms.

The figure shows the reconstruction of energy using the ‘compressed information’ from ECI’s. The predicted energies show well agreement with the original DFT energies.

Compressive sensing for underdetermined linear equation with sparse solution

1 unconstrained minimization min µ u 1 + Au − f 2

2 1

Summary Current results:

From Hart, Blum, Walorski, and Zunger, Nature Materials (2005).

Nearest pair interaction dominates in the model and the fifth nearest pair is also quite important. Fitting with the interactions term give accurate prediction of the energy.

Methods & Results Generating bcc-derivative structures using UNCLE (Universal Cluster Expansion) codes

Cluster expanded Heisenberg model is a great model for ferromagnetic system. It reveals more information about long-range and many-body clusters than the classical Heisenberg model and is a lot faster than the VASP calculation. Selecting structures from derivative super-cells with symmetrically distinct spin arrangements is better than choosing structures with randomly arranged spins. Future work: Relax both ions and volume in VASP calculation to increase the accuracy

N=2

N=3

N=8

a

631 binary structures, 55 distinct structures with up to 8 atoms per cell

Extend the model to different crystal structures

Identifying symmetrically distinct spin arrangements for each structure We developed an algorithm which used a base-6 N-digits labeling to denote each specific spin arrangement. For the first step, we eliminate all translational identical labels by finding the corresponding ‘translational matrix’ so the size of different labels is decreased to approximately 1/n of its original. In the second step, we used space group symmetry operations to eliminate rotational identical spin configurations instead of using geometrically comparing. This algorithm is a O(N) fast one. 5

3

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0

a a

1

2

b a

4

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a

a a

51

The spin on each atomic site is set to +/- x, +/y, +/-z and the arrangements can be labeled (51).

Monte Carlo Simulation

References & Acknowledgements 1.Nelson, Lance J., et al. arXiv preprint arXiv:1307.2938 (2013). 2.Candès, et al. Signal Processing Magazine, IEEE 25.2 (2008): 21-30. 3.Hart, et.al. Physical Review B 77.22 (2008): 224115.

b

b a

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51

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14

The left two labels (51) and (14) are identical under C4 point group symmetry operations.

4.Nelson, Lance J., et al. Physical Review B 87.3 (2013): 035125. This work is supported by UCLA’s CSST program. We acknowledge great help from Yi Xia, Weston Nielson regarding the spin arrangements identification method and its code. We also want to thank SUM for the opportunity to present our work. Contact: Email: