# Complex Langevin simulation of a random matrix model at nonzero chemical potential

## Abstract

In this study we test the complex Langevin algorithm for numerical simulations of a random matrix model of QCD with a first order phase transition to a phase of finite baryon density. We observe that a naive implementation of the algorithm leads to phase quenched results, which were also derived analytically in this article. We test several fixes for the convergence issues of the algorithm, in particular the method of gauge cooling, the shifted representation, the deformation technique and reweighted complex Langevin, but only the latter method reproduces the correct analytical results in the region where the quark mass is inside the domain of the eigenvalues. In order to shed more light on the issues of the methods we also apply them to a similar random matrix model with a milder sign problem and no phase transition, and in that case gauge cooling solves the convergence problems as was shown before in the literature.

- Authors:

- Univ. of Regensburg, Regensburg (Germany)
- Swansea Univ., Swansea (United Kingdom)
- Stony Brook Univ., Stony Brook, NY (United States)
- Heidelberg Univ., Heidelberg (Germany); The College of William and Mary, Williamsburg, VA (United States); Thomas Jefferson National Accelerator Facility (TJNAF), Newport News, VA (United States)

- Publication Date:

- Research Org.:
- Thomas Jefferson National Accelerator Facility, Newport News, VA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Nuclear Physics (NP) (SC-26)

- OSTI Identifier:
- 1434242

- Report Number(s):
- JLAB-THY-18-2691; DOE/OR/23177-4423; arXiv:1712.07514

Journal ID: ISSN 1029-8479; PII: 7727

- Grant/Contract Number:
- AC05-06OR23177; NSF PHY-1516509

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2018; Journal Issue: 3; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Lattice QCD; Lattice Quantum Field Theory; Matrix Models

### Citation Formats

```
Bloch, Jacques, Glesaaen, Jonas, Verbaarschot, Jacobus J. M., and Zafeiropoulos, Savvas.
```*Complex Langevin simulation of a random matrix model at nonzero chemical potential*. United States: N. p., 2018.
Web. doi:10.1007/JHEP03(2018)015.

```
Bloch, Jacques, Glesaaen, Jonas, Verbaarschot, Jacobus J. M., & Zafeiropoulos, Savvas.
```*Complex Langevin simulation of a random matrix model at nonzero chemical potential*. United States. doi:10.1007/JHEP03(2018)015.

```
Bloch, Jacques, Glesaaen, Jonas, Verbaarschot, Jacobus J. M., and Zafeiropoulos, Savvas. Tue .
"Complex Langevin simulation of a random matrix model at nonzero chemical potential". United States.
doi:10.1007/JHEP03(2018)015. https://www.osti.gov/servlets/purl/1434242.
```

```
@article{osti_1434242,
```

title = {Complex Langevin simulation of a random matrix model at nonzero chemical potential},

author = {Bloch, Jacques and Glesaaen, Jonas and Verbaarschot, Jacobus J. M. and Zafeiropoulos, Savvas},

abstractNote = {In this study we test the complex Langevin algorithm for numerical simulations of a random matrix model of QCD with a first order phase transition to a phase of finite baryon density. We observe that a naive implementation of the algorithm leads to phase quenched results, which were also derived analytically in this article. We test several fixes for the convergence issues of the algorithm, in particular the method of gauge cooling, the shifted representation, the deformation technique and reweighted complex Langevin, but only the latter method reproduces the correct analytical results in the region where the quark mass is inside the domain of the eigenvalues. In order to shed more light on the issues of the methods we also apply them to a similar random matrix model with a milder sign problem and no phase transition, and in that case gauge cooling solves the convergence problems as was shown before in the literature.},

doi = {10.1007/JHEP03(2018)015},

journal = {Journal of High Energy Physics (Online)},

number = 3,

volume = 2018,

place = {United States},

year = {Tue Mar 06 00:00:00 EST 2018},

month = {Tue Mar 06 00:00:00 EST 2018}

}

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