# 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:

- 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; PRVDAQ; 054503

- Grant/Contract Number:
- SC0011941; SC0012704

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Name: Physical Review D 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 = {2018},

month = {3}

}

DOI: 10.1103/PhysRevD.97.054503

Works referenced in this record:

##
$2+1$ flavor domain wall QCD on a $(2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{fm}{)}^{3}$ lattice: Light meson spectroscopy with ${L}_{s}=16$

journal, July 2007

- Allton, C.; Antonio, D. J.; Blum, T.
- Physical Review D, Vol. 76, Issue 1

##
Chronological inversion method for the Dirac matrix in hybrid Monte Carlo

journal, January 1997

- Brower, R. C.; Ivanenko, T.; Levi, A. R.
- Nuclear Physics B, Vol. 484, Issue 1-2

##
Möbius Fermions: Improved Domain Wall Chiral Fermions

journal, March 2005

- Brower, Richard C.; Neff, Hartmut; Orginos, Kostas
- Nuclear Physics B - Proceedings Supplements, Vol. 140

##
Continuum limit physics from $2+1$ flavor domain wall QCD

journal, April 2011

- Aoki, Y.; Arthur, R.; Blum, T.
- Physical Review D, Vol. 83, Issue 7

##
Speeding up lattice QCD simulations with clover-improved Wilson fermions

journal, May 2003

- Hasenbusch, M.; Jansen, K.
- Nuclear Physics B, Vol. 659, Issue 1-2

##
Chiral fermions from lattice boundaries

journal, September 1993

- Shamir, Yigal
- Nuclear Physics B, Vol. 406, Issue 1-2

##
Domain wall QCD with near-physical pions

journal, May 2013

- Arthur, R.; Blum, T.; Boyle, P. A.
- Physical Review D, Vol. 87, Issue 9

##
New class of variance-reduction techniques using lattice symmetries

journal, November 2013

- Blum, Thomas; Izubuchi, Taku; Shintani, Eigo
- Physical Review D, Vol. 88, Issue 9

##
Renormalization group improved action for SU(3) lattice gauge theory and the string tension

journal, August 1984

- Iwasaki, Y.; Yoshié, T.
- Physics Letters B, Vol. 143, Issue 4-6

##
The kaon semileptonic form factor in N f = 2 + 1 domain wall lattice QCD with physical light quark masses

journal, June 2015

- Boyle, P. A.; Christ, N. H.; Flynn, J. M.
- Journal of High Energy Physics, Vol. 2015, Issue 6

##
Low energy constants of $SU(2)$ partially quenched chiral perturbation theory from ${N}_{f}=2+1$ domain wall QCD

journal, March 2016

- Boyle, P. A.; Christ, N. H.; Garron, N.
- Physical Review D, Vol. 93, Issue 5

##
A method for simulating chiral fermions on the lattice

journal, August 1992

- Kaplan, David B.
- Physics Letters B, Vol. 288, Issue 3-4

##
Topology of the SU (2) vacuum: a lattice study using improved cooling

journal, August 1997

- de Forcrand, Philippe; García Pérez, Margarita; Stamatescu, Ion-Olimpiu
- Nuclear Physics B, Vol. 499, Issue 1-2

##
The BAGEL assembler generation library

journal, December 2009

- Boyle, Peter A.
- Computer Physics Communications, Vol. 180, Issue 12

##
The Möbius domain wall fermion algorithm

journal, November 2017

- Brower, Richard C.; Neff, Harmut; Orginos, Kostas
- Computer Physics Communications, Vol. 220

##
Exact pseudofermion action for Monte Carlo simulation of domain-wall fermion

journal, November 2014

- Chen, Yu-Chih; Chiu, Ting-Wai
- Physics Letters B, Vol. 738

##
Möbius Fermions

journal, March 2006

- Brower, R. C.; Neff, H.; Orginos, K.
- Nuclear Physics B - Proceedings Supplements, Vol. 153, Issue 1

##
Domain wall QCD with physical quark masses

journal, April 2016

- Blum, T.; Boyle, P. A.; Christ, N. H.
- Physical Review D, Vol. 93, Issue 7

##
$\mathrm{K}\to \pi \pi $ $\mathrm{\Delta}\mathrm{I}=3/2$ decay amplitude in the continuum limit

journal, April 2015

- Blum, T.; Boyle, P. A.; Christ, N. H.
- Physical Review D, Vol. 91, Issue 7

##
Standard Model Prediction for Direct $CP$ Violation in $K\to \pi \pi $ Decay

journal, November 2015

- Bai, Z.; Blum, T.; Boyle, P. A.
- Physical Review Letters, Vol. 115, Issue 21

##
Lattice QCD with N f = 2 + 1 + 1 domain-wall quarks

journal, April 2017

- Chen, Yu-Chih; Chiu, Ting-Wai
- Physics Letters B, Vol. 767

##
Glueball masses and string tension in lattice QCD

journal, June 1987

- Albanese, M.; Costantini, F.; Fiorentini, G.
- Physics Letters B, Vol. 192, Issue 1-2