# Accuracy of the microcanonical Lanczos method to compute real-frequency dynamical spectral functions of quantum models at finite temperatures

## Abstract

We examine the accuracy of the microcanonical Lanczos method (MCLM) developed by Long et al. [Phys. Rev. B 68, 235106 (2003)] to compute dynamical spectral functions of interacting quantum models at finite temperatures. The MCLM is based on the microcanonical ensemble, which becomes exact in the thermodynamic limit. To apply the microcanonical ensemble at a fixed temperature, one has to find energy eigenstates with the energy eigenvalue corresponding to the internal energy in the canonical ensemble. Here in this paper, we propose to use thermal pure quantum state methods by Sugiura and Shimizu [Phys. Rev. Lett. 111, 010401 (2013)] to obtain the internal energy. After obtaining the energy eigenstates using the Lanczos diagonalization method, dynamical quantities are computed via a continued fraction expansion, a standard procedure for Lanczos-based numerical methods. Using one-dimensional antiferromagnetic Heisenberg chains with S = 1/2, we demonstrate that the proposed procedure is reasonably accurate, even for relatively small systems.

- Authors:

- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Materials Science & Technology Division
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Center for Nanophase Materials Science (CNMS); Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Computational Science and Engineering Division
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Materials Science & Technology Division; Univ. of Tennessee, Knoxville, TN (United States). Dept. of Physics and Astronomy
- Tokyo Univ. of Science, Chiba (Japan). Dept. of Applied Physics

- Publication Date:

- Research Org.:
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21); USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)

- OSTI Identifier:
- 1435237

- Alternate Identifier(s):
- OSTI ID: 1434183

- Grant/Contract Number:
- AC05-00OR22725

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Physical Review E

- Additional Journal Information:
- Journal Volume: 97; Journal Issue: 4; Journal ID: ISSN 2470-0045

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; 97 MATHEMATICS AND COMPUTING

### Citation Formats

```
Okamoto, Satoshi, Alvarez, Gonzalo, Dagotto, Elbio, and Tohyama, Takami.
```*Accuracy of the microcanonical Lanczos method to compute real-frequency dynamical spectral functions of quantum models at finite temperatures*. United States: N. p., 2018.
Web. doi:10.1103/PhysRevE.97.043308.

```
Okamoto, Satoshi, Alvarez, Gonzalo, Dagotto, Elbio, & Tohyama, Takami.
```*Accuracy of the microcanonical Lanczos method to compute real-frequency dynamical spectral functions of quantum models at finite temperatures*. United States. doi:10.1103/PhysRevE.97.043308.

```
Okamoto, Satoshi, Alvarez, Gonzalo, Dagotto, Elbio, and Tohyama, Takami. Fri .
"Accuracy of the microcanonical Lanczos method to compute real-frequency dynamical spectral functions of quantum models at finite temperatures". United States.
doi:10.1103/PhysRevE.97.043308.
```

```
@article{osti_1435237,
```

title = {Accuracy of the microcanonical Lanczos method to compute real-frequency dynamical spectral functions of quantum models at finite temperatures},

author = {Okamoto, Satoshi and Alvarez, Gonzalo and Dagotto, Elbio and Tohyama, Takami},

abstractNote = {We examine the accuracy of the microcanonical Lanczos method (MCLM) developed by Long et al. [Phys. Rev. B 68, 235106 (2003)] to compute dynamical spectral functions of interacting quantum models at finite temperatures. The MCLM is based on the microcanonical ensemble, which becomes exact in the thermodynamic limit. To apply the microcanonical ensemble at a fixed temperature, one has to find energy eigenstates with the energy eigenvalue corresponding to the internal energy in the canonical ensemble. Here in this paper, we propose to use thermal pure quantum state methods by Sugiura and Shimizu [Phys. Rev. Lett. 111, 010401 (2013)] to obtain the internal energy. After obtaining the energy eigenstates using the Lanczos diagonalization method, dynamical quantities are computed via a continued fraction expansion, a standard procedure for Lanczos-based numerical methods. Using one-dimensional antiferromagnetic Heisenberg chains with S = 1/2, we demonstrate that the proposed procedure is reasonably accurate, even for relatively small systems.},

doi = {10.1103/PhysRevE.97.043308},

journal = {Physical Review E},

number = 4,

volume = 97,

place = {United States},

year = {Fri Apr 20 00:00:00 EDT 2018},

month = {Fri Apr 20 00:00:00 EDT 2018}

}