# Domain wall fermion QCD with the exact one flavor algorithm

## Abstract

Lattice QCD calculations including the effects of one or more nondegenerate sea quark flavors are conventionally performed using the rational hybrid Monte Carlo (RHMC) algorithm, which computes the square root of the determinant of $${\mathcal{D}}^{\dagger{}}\mathcal{D}$$, where $$\mathcal{D}$$ is the Dirac operator. The special case of two degenerate quark flavors with the same mass is described directly by the determinant of $${\mathcal{D}}^{\dagger{}}\mathcal{D}$$—in particular, no square root is necessary—enabling a variety of algorithmic developments, which have driven down the cost of simulating the light (up and down) quarks in the isospin-symmetric limit of equal masses. As a result, the relative cost of single quark flavors—such as the strange or charm—computed with RHMC has become more expensive. This problem is even more severe in the context of our measurements of the $$\mathrm{{\Delta}}I=1/2$$ $$K{\rightarrow}{\pi}{\pi}$$ matrix elements on lattice ensembles with $G$-parity boundary conditions, since $G$-parity is associated with a doubling of the number of quark flavors described by $$\mathcal{D}$$ , and thus RHMC is needed for the isospin-symmetric light quarks as well. In this paper we report on our implementation of the exact one flavor algorithm (EOFA) introduced by the TWQCD Collaboration for simulations including single flavors of domain wall quarks. We have developed a new preconditioner for the EOFA Dirac equation, which both reduces the cost of solving the Dirac equation and allows us to reuse the bulk of our existing high-performance code. Coupling these improvements with careful tuning of our integrator, the time per accepted trajectory in the production of our $2+1$ flavor $G$-parity ensembles with physical pion and kaon masses has been decreased by a factor of 4.2.

- Authors:

- Brookhaven National Lab. (BNL), Upton, NY (United States). Dept. of Physics
- Columbia Univ., New York, NY (United States). Dept. of Physics

- Publication Date:

- Research Org.:
- Brookhaven National Lab. (BNL), Upton, NY (United States); Columbia Univ., New York, NY (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25); Intel Corporation (United States)

- OSTI Identifier:
- 1425994

- Alternate Identifier(s):
- OSTI ID: 1438197

- Report Number(s):
- BNL-205694-2018-JAAM

Journal ID: ISSN 2470-0010

- Grant/Contract Number:
- SC0012704; SC0011941

- Resource Type:
- Journal Article: Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Volume: 97; Journal Issue: 5; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; lattice QCD; mesons; quarks; CP violation; hybrid Monte Carlo algorithm; path-integral Monte Carlo

### Citation Formats

```
Jung, C., Kelly, C., Mawhinney, R. D., and Murphy, D. J..
```*Domain wall fermion QCD with the exact one flavor algorithm*. United States: N. p., 2018.
Web. doi:10.1103/PhysRevD.97.054503.

```
Jung, C., Kelly, C., Mawhinney, R. D., & Murphy, D. J..
```*Domain wall fermion QCD with the exact one flavor algorithm*. United States. doi:10.1103/PhysRevD.97.054503.

```
Jung, C., Kelly, C., Mawhinney, R. D., and Murphy, D. J.. Tue .
"Domain wall fermion QCD with the exact one flavor algorithm". United States.
doi:10.1103/PhysRevD.97.054503.
```

```
@article{osti_1425994,
```

title = {Domain wall fermion QCD with the exact one flavor algorithm},

author = {Jung, C. and Kelly, C. and Mawhinney, R. D. and Murphy, D. J.},

abstractNote = {Lattice QCD calculations including the effects of one or more nondegenerate sea quark flavors are conventionally performed using the rational hybrid Monte Carlo (RHMC) algorithm, which computes the square root of the determinant of ${\mathcal{D}}^{\dagger{}}\mathcal{D}$, where $\mathcal{D}$ is the Dirac operator. The special case of two degenerate quark flavors with the same mass is described directly by the determinant of ${\mathcal{D}}^{\dagger{}}\mathcal{D}$—in particular, no square root is necessary—enabling a variety of algorithmic developments, which have driven down the cost of simulating the light (up and down) quarks in the isospin-symmetric limit of equal masses. As a result, the relative cost of single quark flavors—such as the strange or charm—computed with RHMC has become more expensive. This problem is even more severe in the context of our measurements of the $\mathrm{{\Delta}}I=1/2$ $K{\rightarrow}{\pi}{\pi}$ matrix elements on lattice ensembles with $G$-parity boundary conditions, since $G$-parity is associated with a doubling of the number of quark flavors described by $\mathcal{D}$ , and thus RHMC is needed for the isospin-symmetric light quarks as well. In this paper we report on our implementation of the exact one flavor algorithm (EOFA) introduced by the TWQCD Collaboration for simulations including single flavors of domain wall quarks. We have developed a new preconditioner for the EOFA Dirac equation, which both reduces the cost of solving the Dirac equation and allows us to reuse the bulk of our existing high-performance code. Coupling these improvements with careful tuning of our integrator, the time per accepted trajectory in the production of our $2+1$ flavor $G$-parity ensembles with physical pion and kaon masses has been decreased by a factor of 4.2.},

doi = {10.1103/PhysRevD.97.054503},

journal = {Physical Review D},

number = 5,

volume = 97,

place = {United States},

year = {Tue Mar 13 00:00:00 EDT 2018},

month = {Tue Mar 13 00:00:00 EDT 2018}

}