Exact Pseudofermion Action for Hybrid Monte Carlo Simulation of One-Flavor

Domain-Wall Fermion

Yu-Chih Chen(for the TWQCD Collaboration)

Physics DepartmentNational Taiwan University

Collaborators: Ting-Wai Chiu

Content• Lattice Dirac Operator for Domain-Wall Fermion (DWF)

• Two-Flavor Algorithm (TFA)

• TWQCD’s One-Flavor Algorithm (TWOFA)

• Rational Hybrid Monte Carlo (RHMC) Algorithm

• TWOFA vs. RHMC with Domain-Wall Fermion

• Concluding Remarks

Lattice Dirac Operator for Domain-Wall Fermion (DWF)

For domain-wall fermion, in general, the lattice Dirac operator reads

where , and is the standard Wilson-Dirac operator with ,

are the matrices in the fifth dimension and depend on the quark mass.

Lattice Dirac Operator for Domain-Wall Fermion (DWF)

𝐷𝑑𝑤𝑓 (𝑚)¿ 𝑫𝒘 {𝒄𝝎 [ 𝑰+𝑳(𝒎)]+𝒅 [ 𝑰 −𝑳(𝒎)] }+ [ 𝑰−𝑳(𝒎)]

Using , the can be written as

For , the form of are

010000100001m000

L

000m100001000010

L

Matrix in 4D (𝑸+¿ (𝒎 ) ¿𝟎¿

𝑸− (𝒎 )¿ )𝑫𝒊𝒓𝒂𝒄

Matrices in 5-th dimension

Lattice Dirac Operator for Domain-Wall Fermion (DWF)

[1] T. W. Chiu, Phys. Rev. Lett. 90, 071601 (2003)

1. If are the optimal weights given in Ref. [1], it gives

Optimal Domain-Wall Fermion

𝑫𝒅𝒘𝒇 (𝒎)¿𝑫𝒘 {𝒄𝝎 [ 𝑰+𝑳 (𝒎) ]+𝒅 [ 𝑰 −𝑳 (𝒎) ] }+ [ 𝑰−𝑳(𝒎)]

𝐻√𝐻2

≈𝑆 (𝐻 )=𝐻∑𝑙

𝑏𝑙𝐻2+𝑑𝑙

:Zolotarev Optimal  Rational   Approximation

where , and

𝐻𝑤=𝛾 5𝐷𝑤

Lattice Dirac Operator for Domain-Wall Fermion (DWF)

2. If , , it gives

Domain-Wall Fermion with Shamir Kernel

3. If , , it gives

Domain-Wall Fermion with Scaled () Shamir Kernel

𝑫𝒅𝒘𝒇 (𝒎)¿𝑫𝒘 {𝒄𝝎 [ 𝑰+𝑳 (𝒎) ]+𝒅 [ 𝑰 −𝑳 (𝒎) ] }+ [ 𝑰−𝑳(𝒎)]

𝐻√𝐻2

≈𝑆 (𝐻 )=𝐻∑𝑙

𝑏′ 𝑙𝐻2+𝑑 ′ 𝑙

:Polar   Approximation

where , and

Lattice Dirac Operator for Domain-Wall Fermion (DWF)

For a physical observable

¿1𝑍∫𝐷𝑈𝒪(𝑈)det [𝐷 𝑓 (𝑈) ]exp (−𝑆𝑔 (𝑈 ) )

If , where the matrix K is independent of the gauge field

∫𝐷𝑈𝒪(𝑈)det [𝐷 𝑓 (𝑈) ]exp (−𝑆𝑔 (𝑈 ) )∫𝐷𝑈 det [𝐷 𝑓 (𝑈)]exp (−𝑆𝑔 (𝑈 ) )

=∫𝐷𝑈 𝒪(𝑈 )det [𝐷 (𝑈)]exp (−𝑆𝑔 (𝑈 ) )

∫𝐷𝑈 det [𝐷 (𝑈) ]exp (−𝑆𝑔 (𝑈 ) )

Lattice Dirac Operator for Domain-Wall Fermion (DWF)

Using the redefined operator

¿1𝑍∫𝐷𝜙𝐷𝜙† 𝐷𝑈𝒪(𝑈)exp (−𝜙† 𝐻−1 (𝑈 ) 𝜙−𝑆𝑔 (𝑈 ) )

where satisfies:

1) H is Hermitian

2) H is positive-definite

Lattice Dirac Operator for Domain-Wall Fermion (DWF)

For DWF, since and are independent of the gauge field,

𝑀 ± (𝑚 )=𝜔−1/2 [𝑐𝑁 ± (𝑚 )+𝜔−1 𝑑]− 1𝜔−1 /2

𝑁 ± (𝑚 )= [1+𝐿±(𝑚)] [1− 𝐿±(𝑚)]− 1

𝐷𝑑𝑤𝑓 (𝑚 ) ¿𝐷𝑤{𝑐𝜔 [ 𝐼+𝐿(𝑚) ]+𝑑 [𝐼 −𝐿(𝑚)] }+ [𝐼 −𝐿(𝑚)]{𝑐𝜔 [ 𝐼+𝐿(𝑚) ]+𝑑 [𝐼 −𝐿(𝑚)] }− 1

Two-Flavor Algorithm (TFA) [2]

𝐷 (𝑚 )=𝐷𝑤+𝑀 (𝑚)=𝐷𝑤+𝑃+¿𝑀+¿(𝑚 )+𝑃− 𝑀− (𝑚 )¿ ¿

For the DWF Dirac operator

we can apply the Schur decomposition with the even-odd preconditioning

where 𝐶 (𝑚 )=I −𝑀 5(𝑚)𝐷𝑤𝑀 5(𝑚)𝐷𝑤

𝑜𝑒 𝑒𝑜We then have

det [𝐷 (𝑚 ) ]=det [𝑀5 (𝑚 )]− 2×det [𝐶 (𝑚)]

[2] T. W. Chiu, et al. [TWQCD Collaboration],PoS LAT 2009, 034 (2009); Phys. Lett. B 717, 420 (2012) .

𝐷 (𝑚 )=(4−𝑚0+𝑀 (𝑚) 𝐷𝑤 (𝑈)𝐷𝑤 (𝑈 ) 4−𝑚0+𝑀 (𝑚))≡(𝑀 5(𝑚)−1 𝐷𝑤

𝐷𝑤 𝑀5(𝑚)− 1)𝑜𝑒𝑒𝑜

¿ ( 𝐼 0𝐷𝑤𝑀5(𝑚) 𝐼 )(𝑀5(𝑚)− 1 0

0 𝑀5 (𝑚)− 1𝐶(𝑚))( 𝐼 𝑀5(𝑚)𝐷𝑤0 𝐼 )𝑜𝑒

𝑒𝑜

𝑒𝑜𝑜𝑒

Two-Flavor Algorithm (TFA)

The pseudofermion action for HMC simulation of 2-flavor QCD with DWF is

𝑆𝑝𝑓=𝜙†𝐶†(1) 1𝐶 (𝑚 )𝐶† (𝑚 )

𝐶 (1)𝜙

The field can be generated by the Gaussian noise field

𝜂=1

𝐶 (𝑚 )𝐶 (1 ) 𝜙⟺𝜙=

1𝐶(1)

𝐶 (𝑚)𝜼

𝑆𝑝𝑓=𝜙†𝐶† (1 ) 1𝐶† (𝑚 )

1𝐶 (𝑚 )

𝐶 (1 )𝜙=𝜂†𝜂

generated from Gaussian noise

det [𝐶 (𝑚)𝐶 (1) ]

2

Two flavor

TWQCD’s One Flavor Algorithm (TWOFA)

For one-flavor of domain-wall fermion in QCD, we have devised an exact pseudofermion action for the HMC simulation, without taking square root.

det [𝐷(𝑚)]  det [𝐷(1)]

=det [𝐷 (𝑚)]

det [ �̂�(𝑚 ,1)]×det [ �̂�(𝑚 ,1)]det [𝐷 (1)]

𝐷 (𝑚 )=(𝑊 −𝑚0+𝑀 +¿(𝑚)¿𝜎 ⋅𝑡¿

𝑊 −𝑚0+𝑀 +¿(𝑚))In Dirac space

�̂� (𝑚 ,1 )=(𝑊 −𝑚0+𝑀+¿(𝑚 )¿𝜎 ⋅𝑡¿

𝑊 −𝑚0+𝑀 +¿(1))

where

TWQCD’s One Flavor Algorithm (TWOFA)

Use type I Schur decomposition to

(𝐴 𝐵𝐶 𝐷)=( 𝐼 0

𝐶 𝐴−1 𝐼)( 𝐴 00 𝐷−𝐶 𝐴−1𝐵)( 𝐼 𝐴− 1𝐵

0 𝐼 )we then have

det [ �̂� (𝑚 ,1 ) ]det [𝐷 (𝑚 ) ]

=det [~𝐻 (𝑚 )+Δ−(𝑚)]

det [~𝐻 (𝑚 )]=det [ 𝐼+Δ−(𝑚) 1

~𝐻 (𝑚 ) ]where~𝐻 (𝑚 )=𝑅5 ¿,

TWQCD’s One Flavor Algorithm (TWOFA)

Use type II Schur decomposition to

(𝐴 𝐵𝐶 𝐷)=(𝐼 𝐵 𝐷−1

0 𝐼 )(𝐴−𝐵𝐷−1𝐶 00 𝐷)( 𝐼 0

𝐷− 1𝐶 𝐼 )we then have

det [𝐷 (1 ) ]det [ �̂� (𝑚 ,1 ) ]

=det [𝐻 (1)]det ¿¿

where

𝐻 (1 )=𝑅5 ¿,

where is a scalar function of , and , are the vector functions of , and , and here we have defined

𝐴±=𝑐𝑁 ± (0 )+𝜔−1𝑑

𝑴 ± (𝒎 )=𝝎−𝟏/𝟐𝑨±−𝟏𝝎−𝟏/𝟐+

𝟐𝒄𝒎𝟏+𝒎−𝟐𝒄𝒎𝝀 𝑹𝟓

❑𝝎−𝟏 /𝟐𝒗± 𝒗±𝑻𝝎−𝟏 /𝟐

TWQCD’s One Flavor Algorithm (TWOFA)

By using the Sherman-Morrison formula, we have found that the fifth dimensional matrices can be rewritten as

With these form of , can be rewritten as

Δ± (𝑚 )=𝑅5 [𝑀 ± (1 )−𝑀 ± (𝑚)]=𝑘𝜔−1 /2𝑣±𝑣±𝑇𝜔−1 /2

𝑘= 𝑐1−𝑐 𝜆

1−𝑚1+𝑚(1−2𝑐 𝜆)

TWQCD’s One Flavor Algorithm (TWOFA)

Next we use , one then has

det [ �̂� (𝑚 ,1 ) ]det [𝐷 (𝑚 ) ]

=det [ 𝐼+Δ−(𝑚) 1~𝐻 (𝑚 ) ]

=det [ 𝐼+𝑘𝑣−𝑇𝜔− 1/2 1~𝐻 (𝑚 )

𝜔−1 /2𝑣−]

=det [ 𝐼+𝑘𝜔−1 /2𝑣−𝑣−𝑇𝜔− 1/2 1

~𝐻 (𝑚 ) ]

Positive definite and Hermitian

TWQCD’s One Flavor Algorithm (TWOFA)

Again, with , one also has

=det ¿

=det ¿

Positive definite and Hermitian

det [𝐷 (1 ) ]det [ �̂� (𝑚 ,1 ) ]

=det ¿

TWQCD’s One Flavor Algorithm (TWOFA)

𝑆1=𝜙1† [ 𝐼+𝑘𝑣−𝑇𝜔−1 /2 1

~𝐻 (𝑚 )𝜔− 1/2𝑣− ]𝜙1det [ �̂� (𝑚 ,1 ) ]

det [𝐷 (1 ) ]

𝑆2=𝜙2† ¿det [𝐷 (𝑚 ) ]det [ �̂� (𝑚 ,1 ) ]

det [ �̂� (𝑚 ,1 ) ]det [𝐷 (1 ) ]

det [𝐷 (𝑚 ) ]det [ �̂� (𝑚 ,1 ) ]

× ¿det [𝐷(𝑚)]  det [𝐷(1)]

One flavor determinant

TWQCD’s One Flavor Algorithm(TWOFA)

Use , and some algebra, the pseudofermion action of one-flavor domain-wall fermion can be written as

𝑆𝑝𝑓=(0 𝜙1† ) [ 𝐼−𝑘𝑣−𝑇𝜔− 1/2 1

𝐻 (𝑚)𝜔− 1/2𝑣− ]( 0𝜙1)

+(𝜙2† 0 )¿where ,

Δ± (𝑚 )=𝑘𝜔− 1/2𝑣±𝑣±𝑇𝜔− 1/2

𝑘= 𝑐1−𝑐 𝜆

1−𝑚1+𝑚(1−2𝑐 𝜆)

TWQCD’s One Flavor Algorithm(TWOFA)The initial pseudofermion fields of each HMC trajectory are generated by Gaussian noises as follows.

𝑆1=𝜙1† [ 𝐼+𝑘𝑣−𝑇𝜔−1 /2 1

~𝐻 (𝑚 )𝜔− 1/2𝑣− ]𝜙1=𝜂1†𝜂 1

⇒𝜙1=[𝐼+𝑘𝑣−𝑇𝜔− 1/2 1~𝐻 (𝑚 )

𝜔−1 /2𝑣−]−1 /2

𝜂1

Gaussian noise𝑆2=𝜙2† ¿⇒𝜙2=¿¿

TWQCD’s One Flavor Algorithm (TWOFA)

The invesre square root can be approximated by the Zolotarev optimal rational approximation

(𝜉1𝜙1)=∑𝑙=1

𝑁 𝑝

[ 𝑏𝑙1+𝑑𝑙

𝐼+𝑏𝑙

(1+𝑑𝑙 )2 𝑘𝑣−

𝑇𝜔−1 /2 1

𝐻 (𝑚 )− 11+𝑑𝑙

Δ−(𝑚)𝑃−𝜔− 1/2𝑣− ]( 0𝜂 1)

(𝜙2𝜉2)=∑𝑙=1

𝑁𝑝

¿¿Gaussian noise

Rational Hybrid Monte Carlo(RHMC) Algorithm

A widely used algorithm to do the one-flavor HMC simulation is the rational hybrid Monte Carlo (RHMC)[3], which can be used for any lattice fermion.

[3] M. A. Clark and A. D. Kennedy, Phys. Rev. Lett. 98, 051601 (2007)

𝑆𝑝𝑓=∑𝑛𝜙𝑛

† (𝐶 (1)𝐶† (1 ) )1 /4 𝑛 1(𝐶 (𝑚)𝐶† (𝑚 ) )1 /2𝑛

(𝐶 (1)𝐶† (1 ) )1/4𝑛𝜙𝑛  

1.

2. Positive definite and Hermitian

The fields are generated by the Gaussian noise fields

𝜙𝑛=1

(𝐶 (1)𝐶† (1 ) )1/4𝑛(𝐶 (𝑚)𝐶† (𝑚 ) )1/4𝑛𝜂𝑛

Gaussian noise

Rational Hybrid Monte Carlo(RHMC) Algorithm

𝐴1/𝛼=𝑏0+∑𝑙=1

𝑁𝑝 𝑏𝑙𝐴+𝑑𝑙

rational approximation

𝑆𝑝𝑓=∑𝑛𝜙𝑛

† (𝐶 (1)𝐶† (1 ) )1 /4 𝑛 1(𝐶 (𝑚)𝐶† (𝑚 ) )1 /2𝑛

(𝐶 (1)𝐶† (1 ) )1/4𝑛𝜙𝑛  

where is the number of poles for rational approximation.

The numbers of inverse matrices can be obtained from the multiple mass shift conjugate gradient.

TWOFA vs. RHMC with DWF

I. Memory Usage :

We list memory requirement (in unit of bytes) for links, momentum and 5D vectors as follows,

1) , link variables

2) , momentum

3) , 5D vector

Then the ratio of the memory usage for RHMC and TWOFA is

𝑀𝑅𝐻𝑀𝐶

𝑀𝑇𝑊𝑂𝐹𝐴=20+3 (3+2𝑁𝑝 )𝑁 𝑠

32+10.5𝑁 𝑠

where is the number of poles for MMCG in RHMC algorithm

𝑀𝑅𝐻𝑀𝐶

𝑀𝑇𝑊𝑂𝐹𝐴=20+3 (3+2𝑁𝑝 )𝑁 𝑠

32+10.5𝑁 𝑠

TWOFA vs. RHMC with DWF

II. Efficiency:

The lattice setup is

HMC Steps: (Gauge, Heavy, Light) = (1, 1, 10), for RHMC

𝛃=𝟓 .𝟗𝟓 ,𝒎𝒒=𝟎 .𝟎𝟏 ,𝑳𝟑=𝟖𝟑 ,𝑻=𝟏𝟔 ,𝑵 𝒔=𝟏𝟔 ,

We compare RHMC and TWOFA for the following cases:

1) DWF with and (Optimal DWF)

2) DWF with and (Shamir)

3) DWF with () and (Scaled Shamir)

TWOFA vs. RHMC with DWF

1. Optimal Domain-Wall Fermion : Maximum Forces

Max

imum

For

ces

𝒎𝒉=𝟎 .𝟒𝒎𝒉=𝟎 .𝟒

𝒎𝒒=𝟎 .𝟎𝟏𝒎𝒒=𝟎 .𝟎𝟏

TWOFA vs. RHMC with DWF on the Lattice

1. Optimal Domain-Wall Fermion:

∆𝐻

)

TWOFA vs. RHMC with DWF on the Lattice

1. Optimal Domain-Wall Fermion :

ODWF (kernel ) with and

Algorithm Plaquette (Gauge) (heavy) (light)

TWOFA 0.58051(09) 5.15555(34) 0.18971(13) 0.01401(29)

RHMC 0.58100(10) 5.15762(36) 0.35334(11) 0.06946(44)

HMC Steps: (Gauge, Heavy, Light) = (1, 1, 10), for RHMC

Algorithm Accept Memory

TWOFA 0.980(8) 0.981(11) 0.9992(16) 1.00 1.00 0

RHMC 0.987(7) 0.994(18) 1.0003(16) 6.58 1.21

𝛃=𝟓 .𝟗𝟓 ,𝒎𝒒=𝟎 .𝟎𝟏 ,𝑳=𝟖 ,𝑻=𝟏𝟔 ,𝑵𝒔=𝟏𝟔 ,

TWOFA vs. RHMC with DWF on the Lattice

2. Shamir Kernel () : Maximum Forces

Max

imum

For

ces

𝒎𝒉=𝟎 .𝟏

𝒎𝒉=𝟎 .𝟏

𝒎𝒒=𝟎 .𝟎𝟏𝒎𝒒=𝟎 .𝟎𝟏

TWOFA vs. RHMC with DWF on the Lattice

2. Shamir Kernel () :

∆𝐻

TWOFA vs. RHMC with DWF on the Lattice

2. Shamir Kernel () :

DWF (Shamir kernel) with and

for RHMC𝛃=𝟓 .𝟗𝟓 ,𝒎𝒒=𝟎 .𝟎𝟏 ,𝑳=𝟖 ,𝑻=𝟏𝟔 ,𝑵𝒔=𝟏𝟔 ,

TWOFA vs. RHMC with DWF on the Lattice

Algorithm Accept Memory

TWOFA 0.987(7) 0.987(13) 0.9999(16) 1.00 1.00

RHMC 0.997(3) 0.953(06) 1.0074(18) 6.58 1.00

Algorithm Plaquette (Gauge) (heavy) (light)

TWOFA 0.59061(09) 5.17686(34) 0.14663(45) 0.03578(18)

RHMC 0.59094(14) 5.17866(35) 0.28522(68) 0.10757(06)

3. Scaled Shamir Kernel ( and ) : Maximum Forces

Max

imum

For

ces

𝒎𝒉=𝟎 .𝟏

𝒎𝒉=𝟎 .𝟏

𝒎𝒒=𝟎 .𝟎𝟏𝒎𝒒=𝟎 .𝟎𝟏

TWOFA vs. RHMC with DWF on the Lattice

3. Scaled Shamir Kernel ( and ) :

∆𝐻

TWOFA vs. RHMC with DWF on the Lattice

3. Scaled Shamir Kernel ( and ) :

DWF (scaled Shamir kernel) with and

for RHMC𝛃=𝟓 .𝟗𝟓 ,𝒎𝒒=𝟎 .𝟎𝟏 ,𝑳=𝟖 ,𝑻=𝟏𝟔 ,𝑵𝒔=𝟏𝟔 ,

TWOFA vs. RHMC with DWF on the Lattice

Algorithm Accept Memory

TWOFA 0.983(7) 0.990(14) 1.0000(15) 1.00 1.00

RHMC 0.997(3) 0.967(07) 1.00038(18

) 6.58 1.17

Algorithm Plaquette (Gauge) (heavy) (light)

TWOFA 0.59061(09) 5.17854(34) 0.14646(13) 0.03359(13)

RHMC 0.59032(09) 5.17670(32) 0.28559(39) 0.10775(06)

1. We have derived a novel pseudofermion action for HMC simulation of one-flavor DWF, which is exact, without taking square root.

2. It can be used for any kinds of DWF with any kernels, and for any approximations (polar or Zolotarev) of the sign function.

3. The memory consumption of TWOFA is much smaller than that of RHMC. This feature is crucial for using GPUs to simulate QCD.

4. The efficiency of TWOFA of is compatible with that of RHMC. For the cases we have studied, TWOFA outperforms RHMC.

5. TWQCD is now using TWOFA to simulate (2+1)-flavors QCD, and (2+1+1)-flavors QCD, on lattices.

Concluding Remarks

Thank You