# Gradient-based stochastic estimation of the density matrix

## Abstract

Fast estimation of the single-particle density matrix is key to many applications in quantum chemistry and condensed matter physics. The best numerical methods leverage the fact that the density matrix elements f(H) _{ij} decay rapidly with distance rij between orbitals. This decay is usually exponential. However, for the special case of metals at zero temperature, algebraic decay of the density matrix appears and poses a significant numerical challenge. Here, we introduce a gradient-based probing method to estimate all local density matrix elements at a computational cost that scales linearly with system size. For zero-temperature metals, the stochastic error scales like S ^{-(d+2)/2d}, where d is the dimension and S is a prefactor to the computational cost. The convergence becomes exponential if the system is at finite temperature or is insulating.

- Authors:

- Univ. of Tennessee, Knoxville, TN (United States). Dept. of Physics and Astronomy
- Univ. of Virginia, Charlottesville, VA (United States). Dept. of Physics
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Univ. of Tennessee, Knoxville, TN (United States). Dept. of Physics and Astronomy; Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Quantum Condensed Matter Division and Shull-Wollan Center

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)

- OSTI Identifier:
- 1481135

- Alternate Identifier(s):
- OSTI ID: 1423724

- Report Number(s):
- LA-UR-17-30801

Journal ID: ISSN 0021-9606

- Grant/Contract Number:
- AC52-06NA25396

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of Chemical Physics

- Additional Journal Information:
- Journal Volume: 148; Journal Issue: 9; Journal ID: ISSN 0021-9606

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; 37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CHEMISTRY

### Citation Formats

```
Wang, Zhentao, Cher, Gia-Wei, Barros, Kipton Marcos, and Batista, Cristian D.
```*Gradient-based stochastic estimation of the density matrix*. United States: N. p., 2018.
Web. doi:10.1063/1.5017741.

```
Wang, Zhentao, Cher, Gia-Wei, Barros, Kipton Marcos, & Batista, Cristian D.
```*Gradient-based stochastic estimation of the density matrix*. United States. doi:10.1063/1.5017741.

```
Wang, Zhentao, Cher, Gia-Wei, Barros, Kipton Marcos, and Batista, Cristian D. Mon .
"Gradient-based stochastic estimation of the density matrix". United States.
doi:10.1063/1.5017741.
```

```
@article{osti_1481135,
```

title = {Gradient-based stochastic estimation of the density matrix},

author = {Wang, Zhentao and Cher, Gia-Wei and Barros, Kipton Marcos and Batista, Cristian D.},

abstractNote = {Fast estimation of the single-particle density matrix is key to many applications in quantum chemistry and condensed matter physics. The best numerical methods leverage the fact that the density matrix elements f(H)ij decay rapidly with distance rij between orbitals. This decay is usually exponential. However, for the special case of metals at zero temperature, algebraic decay of the density matrix appears and poses a significant numerical challenge. Here, we introduce a gradient-based probing method to estimate all local density matrix elements at a computational cost that scales linearly with system size. For zero-temperature metals, the stochastic error scales like S-(d+2)/2d, where d is the dimension and S is a prefactor to the computational cost. The convergence becomes exponential if the system is at finite temperature or is insulating.},

doi = {10.1063/1.5017741},

journal = {Journal of Chemical Physics},

number = 9,

volume = 148,

place = {United States},

year = {Mon Mar 05 00:00:00 EST 2018},

month = {Mon Mar 05 00:00:00 EST 2018}

}