Exact Pseudofermion Action for Hybrid
Monte Carlo Simulation of One-Flavor
Domain-Wall Fermion
Yu-Chih Chen
(for the TWQCD Collaboration)
Physics Department
National 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
𝐷𝑑𝑤𝑓 𝑚 = 𝝆𝒔 𝐷𝑤 + 𝐼 + 𝝈𝒔 𝐷𝑤 − 𝐼 𝐿 𝑚
where 𝝆𝒔 = 𝒄 + 𝒅𝝎𝒔 , 𝝈𝒔 = 𝒄 − 𝒅𝝎𝒔 and 𝜔 = diag 𝜔1 , 𝜔2 , ⋯ , 𝜔𝑁𝑠 , and
𝐷𝑤 is the standard Wilson-Dirac operator with −𝑚0 0 < 𝑚0 < 2 ,
𝐿 𝑚 = 𝑃+ 𝐿+ 𝑚 + 𝑃− 𝐿− 𝑚 =
𝐿+ 𝑚
𝑠,𝑠′
𝐿+ 𝑚
0
0
𝐿− 𝑚
𝐷𝑖𝑟𝑎𝑐
𝛿𝑠′ ,𝑠−1 , 1 < 𝑠 ≤ 𝑁𝑠
=
−𝑚𝛿𝑠′ ,𝑁𝑠 , 𝑠 = 1, with 𝑚 = 𝑟𝑚𝑞 , 𝑟 = 1/ 2𝑚0 1 − 𝑑𝑚0
𝐿− 𝑚 = 𝐿+ 𝑚
𝑇
𝐿± (𝑚) are the matrices in the fifth dimension and depend on the quark
mass.
Lattice Dirac Operator for Domain-Wall Fermion (DWF)
For 𝑁𝑠 = 4, the form of 𝐿± are
 0

1

L 
 0

 0
0
0
0
0
1
0
0
1
 m

0

0 

0 
 0

0

L 
 0

 m
1
0
0
1
0
0
0
0
0 

0

 1

0 
Using 𝜌𝑠 = 𝑐 + 𝑑𝜔𝑠 and 𝜎𝑠 = 𝑐 − 𝑑𝜔𝑠 , the 𝐷𝑑𝑤𝑓 (𝑚) can be written as
𝐷𝑑𝑤𝑓 (𝑚) = 𝑫𝒘 𝒄𝝎 𝑰 + 𝑳(𝒎) + 𝒅 𝑰 − 𝑳(𝒎)
Matrix in 4D
Matrices in 5-th dimension
𝑸+ 𝒎
𝟎
𝟎
𝑸− 𝒎
+ 𝑰 − 𝑳(𝒎)
𝑫𝒊𝒓𝒂𝒄
Lattice Dirac Operator for Domain-Wall Fermion (DWF)
𝑫𝒅𝒘𝒇 (𝒎) = 𝑫𝒘 𝒄𝝎 𝑰 + 𝑳(𝒎) + 𝒅 𝑰 − 𝑳(𝒎)
+ 𝑰 − 𝑳(𝒎)
1. If 𝝎𝒔 are the optimal weights given in Ref. [1], it gives
Optimal Domain-Wall Fermion
𝐻
𝐻2
≈ 𝑆(𝐻) = 𝐻
𝑙
𝑏𝑙
∶ Zolotarev Optimal Rational Approximation
𝐻 2 + 𝑑𝑙
𝑆(𝐻) =
1−
1+
𝑁𝑠
𝑠=1 𝑇𝑠
𝑁𝑠
𝑠=1 𝑇𝑠
→
𝐻
𝐻2
,
as 𝑁𝑠 → ∞
where 𝑇𝑠 = (1 − 𝐻𝑠 ) (1 + 𝐻𝑠 ), 𝐻𝑠 = 𝜔𝑠 𝐻 and 𝐻 = 𝑐𝐻𝑤 1 + 𝑑𝛾5 𝐻𝑤
[1] T. W. Chiu, Phys. Rev. Lett. 90, 071601 (2003)
−1
𝐻𝑤 = 𝛾5 𝐷𝑤
Lattice Dirac Operator for Domain-Wall Fermion (DWF)
𝑫𝒅𝒘𝒇 (𝒎) = 𝑫𝒘 𝒄𝝎 𝑰 + 𝑳(𝒎) + 𝒅 𝑰 − 𝑳(𝒎)
+ 𝑰 − 𝑳(𝒎)
2. If 𝝎𝒔 = 𝟏, 𝒄 = 𝟎. 𝟓, 𝒅 = 𝟎. 𝟓, it gives
Domain-Wall Fermion with Shamir Kernel
3. If 𝝎𝒔 = 𝟏, 𝒄 = 𝟏. 𝟎, 𝒅 = 𝟎. 𝟓, it gives
Domain-Wall Fermion with Scaled (𝜶 = 𝟐) Shamir Kernel
𝐻
𝐻2
≈ 𝑆(𝐻) = 𝐻
𝑙
𝑆(𝐻) =
𝑏′𝑙
∶ Polar Approximation
2
𝐻 + 𝑑′𝑙
1−
1+
𝑁𝑠
𝑠=1 𝑇𝑠
𝑁𝑠
𝑠=1 𝑇𝑠
→
𝐻
𝐻2
,
as 𝑁𝑠 → ∞
where 𝑇𝑠 = (1 − 𝐻) (1 + 𝐻), and 𝐻 = 𝑐𝐻𝑤 1 + 𝑑𝛾5 𝐻𝑤
−1
Lattice Dirac Operator for Domain-Wall Fermion (DWF)
For a physical observable 𝒪(𝑈)
1
𝒪(𝑈) =
𝑍
1
=
𝑍
𝐷𝑞𝐷𝑞𝐷𝑈𝒪(𝑈)exp −𝑞𝐷𝑓 𝑈 𝑞 − 𝑆𝑔 𝑈
𝐷𝑈𝒪(𝑈)det[𝐷𝑓 (𝑈)]exp −𝑆𝑔 𝑈
If 𝐷𝑓 (𝑈) = 𝐾 ∗ 𝐷(𝑈), where the matrix K is independent of the gauge field
𝐷𝑈𝒪(𝑈)det[𝐷𝑓 (𝑈)]exp −𝑆𝑔 𝑈
𝐷𝑈det[𝐷𝑓 (𝑈)]exp −𝑆𝑔 𝑈
=
𝐷𝑈𝒪(𝑈)det[𝐷(𝑈)]exp −𝑆𝑔 𝑈
𝐷𝑈det[𝐷(𝑈)]exp −𝑆𝑔 𝑈
Lattice Dirac Operator for Domain-Wall Fermion (DWF)
Using the redefined operator 𝐷(𝑈)
1
𝒪(𝑈) =
𝑍
1
=
𝑍
𝐷𝑈𝒪(𝑈)det[𝐷(𝑈)]exp −𝑆𝑔 𝑈
𝐷𝜙𝐷𝜙 † 𝐷𝑈𝒪(𝑈) exp −𝜙 † 𝐻−1 𝑈 𝜙 − 𝑆𝑔 𝑈
where 𝐻 satisfies:
1) det 𝐻 = det 𝐷
2) H is Hermitian
3) H is positive-definite
Lattice Dirac Operator for Domain-Wall Fermion (DWF)
For DWF, since 𝜔 and 𝐿 are independent of the gauge field,
+ 𝑑𝐼 𝐼+−𝐿(𝑚)
𝐿(𝑚) + 𝑑
+ 𝐼 𝐼−−𝐿(𝑚)
𝐿(𝑚)
𝐷𝑑𝑤𝑓 𝑚 = 𝐷𝑤 𝑐𝜔 𝐼 + 𝐿(𝑚) 𝑐𝜔
𝑫𝒅𝒘𝒇 → 𝑫 𝒎 = 𝑫𝒘 + 𝑴 𝒎 = 𝑫𝒘 +
𝑴+ 𝒎
𝟎
𝑀± 𝑚 = 𝜔 −1/2 𝑐𝑁± 𝑚 + 𝜔−1 𝑑
−1 𝜔 −1/2
𝑁± 𝑚 = 1 + 𝐿± (𝑚) 1 − 𝐿± (𝑚)
−1
𝟎
𝑴− 𝒎
−1
𝑫𝒊𝒓𝒂𝒄
Two-Flavor Algorithm (TFA) [2]
For the DWF Dirac operator
𝐷 𝑚 = 𝐷𝑤 + 𝑀(𝑚) = 𝐷𝑤 + 𝑃+ 𝑀+ 𝑚 + 𝑃− 𝑀− (𝑚)
we can apply the Schur decomposition with the even-odd preconditioning
4 − 𝑚0 + 𝑀(𝑚)
𝐷 𝑚 =
𝑜𝑒
𝐷𝑤 (𝑈)
where
𝐼
𝑜𝑒
=
𝐷𝑤 𝑀5 (𝑚)
0
𝐼
𝐷𝑤𝑒𝑜(𝑈)
𝑀5 (𝑚)−1
≡
4 − 𝑚0 + 𝑀(𝑚)
𝐷𝑤𝑜𝑒
𝑀5 (𝑚)−1
0
0
𝑀5 (𝑚)−1 𝐶(𝑚)
𝑒𝑜
𝐷𝑤
𝑀5 (𝑚)−1
𝐼
0
𝑒𝑜
𝑀5 (𝑚)𝐷𝑤
𝐼
𝑜𝑒
𝐶 𝑚 = I − 𝑀5 (𝑚)𝐷𝑤
𝑀5 (𝑚)𝐷𝑤𝑒𝑜
We 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) .
Two-Flavor Algorithm (TFA)
The pseudofermion action for HMC simulation of 2-flavor QCD with DWF is
𝑆𝑝𝑓
1
† †
= 𝜙 𝐶 (1)
𝐶(1)𝜙
†
𝐶 𝑚 𝐶 𝑚
𝐶(𝑚)
det
𝐶(1)
2
Two flavor
The field 𝜙 can be generated by the Gaussian noise field 𝜂
𝑆𝑝𝑓 =
𝜙†𝐶 †
1
1
1 †
𝐶 1 𝜙 = 𝜂†𝜂
𝐶 𝑚 𝐶 𝑚
1
1
𝜂=
𝐶 1 𝜙 ⟺ 𝜙=
𝐶(𝑚)𝜼
𝐶 𝑚
𝐶(1)
generated from Gaussian noise
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(𝑚)]
det[𝐷 (𝑚, 1)]
=
×
det[𝐷(1)]
det[𝐷(1)]
det[𝐷 (𝑚, 1)]
where 𝐷 𝑚 = 𝐷𝑤 + 𝑀(𝑚) = 𝐷𝑤 + 𝑃+ 𝑀+ 𝑚 + 𝑃− 𝑀− (𝑚)
In Dirac space
𝑊 − 𝑚0 + 𝑀+ (𝑚)
𝐷 𝑚 =
− 𝜎⋅𝑡 †
𝜎⋅𝑡
𝑊 − 𝑚0 + 𝑀+ (𝑚)
𝑊 − 𝑚0 + 𝑀+ (𝑚)
𝐷 𝑚, 1 =
− 𝜎⋅𝑡 †
𝜎⋅𝑡
𝑊 − 𝑚0 + 𝑀+ (1)
TWQCD’s One Flavor Algorithm (TWOFA)
Use type I Schur decomposition to 𝐷 𝑚, 1 , and 𝐷(𝑚)
𝐴
𝐶
𝐵
𝐼
=
𝐷
𝐶𝐴−1
0
𝐼
𝐴
0
0
𝐷 − 𝐶𝐴−1 𝐵
𝐼 𝐴−1 𝐵
0
𝐼
we then have
det 𝐷 𝑚, 1
det 𝐷 𝑚
det[𝐻 𝑚 + Δ− (𝑚)]
1
=
= det 𝐼 + Δ− (𝑚)
det[𝐻 𝑚 ]
𝐻 𝑚
where
𝐻 𝑚 = 𝑅5 𝑊 − 𝑚0 + 𝑀− 𝑚 + 𝜎 ⋅ 𝑡
†
1
𝜎⋅𝑡
𝑊 − 𝑚0 + 𝑀+ (𝑚)
Δ− 𝑚 = 𝑅5 𝑀− 1 − 𝑀− (𝑚) , 𝑅5 = 𝛿𝑠+𝑠′,𝑁𝑠
TWQCD’s One Flavor Algorithm (TWOFA)
Use type II Schur decomposition to 𝐷 1 , and 𝐷 (𝑚, 1)
𝐴
𝐶
−1
𝐵
𝐼
𝐵𝐷
=
𝐷
0
𝐼
𝐴 − 𝐵𝐷−1 𝐶
0
0
𝐷
𝐼
𝐷−1 𝐶
0
𝐼
we then have
det 𝐷 1
det 𝐷 𝑚, 1
det[𝐻(1)]
1
=
= det 𝐼 + Δ+ (𝑚)
det[𝐻 1 − Δ+ (𝑚)]
𝐻 1 − Δ+ (𝑚)
where
1
𝐻 1 = 𝑅5 𝑊 − 𝑚0 + 𝑀+ 1 + 𝜎 ⋅ 𝑡
+ 𝜎⋅𝑡
𝑊 − 𝑚0 + 𝑀− (1)
Δ+ 𝑚 = 𝑅5 𝑀+ 1 − 𝑀+ (𝑚) , 𝑅5 = 𝛿𝑠+𝑠′,𝑁𝑠
†
TWQCD’s One Flavor Algorithm (TWOFA)
By using the Sherman-Morrison formula, we have found that the fifth
dimensional matrices 𝑀± can be rewritten as
𝑴± 𝒎 =
𝝎−𝟏/𝟐 𝑨± −𝟏 𝝎−𝟏/𝟐
𝟐𝒄𝒎
+
𝑹𝟓 𝝎−𝟏/𝟐 𝒗± 𝒗± 𝑻 𝝎−𝟏/𝟐
𝟏 + 𝒎 − 𝟐𝒄𝒎𝝀
where 𝜆 is a scalar function of 𝑐, 𝑑 and 𝜔, 𝑣± are the vector functions of 𝑐,
𝑑 and 𝜔, and here we have defined
𝐴± = 𝑐𝑁± 0 + 𝜔−1 𝑑
With these form of 𝑀± (𝑚), Δ± (𝑚) can be rewritten as
Δ± 𝑚 = 𝑅5 𝑀± 1 − 𝑀± (𝑚) = 𝑘𝜔 −1/2 𝑣± 𝑣± 𝑇 𝜔 −1/2
𝑐
1−𝑚
𝑘=
1 − 𝑐𝜆 1 + 𝑚(1 − 2𝑐𝜆)
TWQCD’s One Flavor Algorithm (TWOFA)
Next we use det 𝐼 + 𝐴𝐵 = det[𝐼 + 𝐵𝐴], one then has
det 𝐷 𝑚, 1
det 𝐷 𝑚
1
= det 𝐼 + Δ− (𝑚)
𝐻 𝑚
=det 𝐼 +
𝑘𝜔 −1/2 𝑣− 𝑣− 𝑇 𝜔 −1/2
=det 𝐼 + 𝑘𝑣−
𝑇 𝜔 −1/2
1
𝐻 𝑚
1
𝜔 −1/2 𝑣−
𝐻 𝑚
Positive definite and Hermitian
TWQCD’s One Flavor Algorithm (TWOFA)
Again, with det 𝐼 + 𝐴𝐵 = det[𝐼 + 𝐵𝐴], one also has
det 𝐷 1
det 𝐷 𝑚, 1
1
= det 𝐼 + Δ+ (𝑚)
𝐻 1 − Δ+ (𝑚)
=det 𝐼 +
𝑘𝜔 −1/2 𝑣+ 𝑣+ 𝑇 𝜔 −1/2
=det 𝐼 + 𝑘𝑣+
𝑇 𝜔 −1/2
1
𝐻 1 − Δ+ (𝑚)
1
𝜔 −1/2 𝑣+
𝐻 1 − Δ+ (𝑚)
Positive definite and Hermitian
TWQCD’s One Flavor Algorithm (TWOFA)
𝑆1 = 𝜙1 † 𝐼 + 𝑘𝑣− 𝑇 𝜔 −1/2
𝑆2 = 𝜙2
†
𝐼 + 𝑘𝑣+
𝑇 𝜔 −1/2
×
1
𝜔 −1/2 𝑣− 𝜙1
𝐻 𝑚
1
𝜔 −1/2 𝑣+ 𝜙2
𝐻 1 − Δ+ (𝑚)
det 𝐷 𝑚, 1
det 𝐷 1
det 𝐷 𝑚
det 𝐷 𝑚, 1
det[𝐷(𝑚)]
=
det[𝐷(1)]
One flavor determinant
TWQCD’s One Flavor Algorithm(TWOFA)
Use 𝑆1 , 𝑆2 and some algebra, the pseudofermion action of one-flavor
domain-wall fermion can be written as
1
0
†
𝑇 −1/2
−1/2
𝑆𝑝𝑓 = 0 𝜙1
𝐼 − 𝑘𝑣− 𝜔
𝜔
𝑣−
𝜙1
𝐻(𝑚)
+ 𝜙2
†
0 𝐼 + 𝑘𝑣+
𝑇 𝜔 −1/2
1
𝜔 −1/2 𝑣+
𝐻 1 − Δ+ (𝑚)𝑃+
where 𝐻 𝑚 = 𝛾5 𝑅5 𝐷(𝑚) , 𝑅5 = 𝛿𝑠+𝑠′,𝑁𝑠
Δ± 𝑚 = 𝑘𝜔 −1/2 𝑣± 𝑣± 𝑇 𝜔 −1/2
𝑐
1−𝑚
𝑘=
1 − 𝑐𝜆 1 + 𝑚(1 − 2𝑐𝜆)
𝜙2
0
TWQCD’s One Flavor Algorithm(TWOFA)
The initial pseudofermion fields of each HMC trajectory are generated by
Gaussian noises as follows.
𝑆1 = 𝜙1
†
𝑇
𝐼 + 𝑘𝑣− 𝜔
−1/2
1
𝜔 −1/2 𝑣− 𝜙1 = 𝜂1 † 𝜂1
𝐻 𝑚
1
𝑇
−1/2
⇒ 𝜙1 = 𝐼 + 𝑘𝑣− 𝜔
𝜔 −1/2 𝑣−
𝐻 𝑚
𝑆2 = 𝜙2
†
𝐼 + 𝑘𝑣+
𝑇 𝜔 −1/2
−1/2
𝜂1
1
𝜔 −1/2 𝑣+ 𝜙2
𝐻 1 − Δ+ (𝑚)
1
𝑇
−1/2
⇒ 𝜙2 = 𝐼 + 𝑘𝑣+ 𝜔
𝜔 −1/2 𝑣+
𝐻 1 − Δ+ (𝑚)
Gaussian noise
−1/2
𝜂2
TWQCD’s One Flavor Algorithm (TWOFA)
The invesre square root can be approximated by the Zolotarev optimal
rational approximation
𝜉1
𝜙1
𝑁𝑝
=
𝑙=1
𝑏𝑙
𝐼
1 + 𝑑𝑙
𝜙2
𝜉2
𝑁𝑝
=
𝑙=1
Gaussian noise
𝑏𝑙
𝐼
1 + 𝑑𝑙
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.
1/4𝑛
1/4𝑛
1
†
†
†
𝑆𝑝𝑓 =
𝜙𝑛 𝐶(1)𝐶 1
𝐶(1)𝐶 1
𝜙𝑛
1/2𝑛
𝐶(𝑚)𝐶 † 𝑚
𝑛
1. det
𝐶(1)
𝐶 𝑚
=
𝑛 det
𝐶(1)𝐶 †
1
1/4𝑛
1
𝐶(𝑚)𝐶 † 𝑚
1/2𝑛
𝐶(1)𝐶 †
1
1/4𝑛
2. Positive definite and Hermitian
The fields 𝜙𝑛 are generated by the Gaussian noise fields 𝜂𝑛
𝜙𝑛 =
1
𝐶(1)𝐶 † 1
1/4𝑛
𝐶(𝑚)𝐶 †
𝑚
1/4𝑛
Gaussian noise
𝜂𝑛
[3] M. A. Clark and A. D. Kennedy, Phys. Rev. Lett. 98, 051601 (2007)
Rational Hybrid Monte Carlo(RHMC) Algorithm
𝑆𝑝𝑓 =
𝜙𝑛
†
†
𝐶(1)𝐶 1
𝑛
𝑁𝑝
𝐴1/𝛼
= 𝑏0 +
𝑙=1
𝑏𝑙
𝐴 + 𝑑𝑙
1/4𝑛
1
𝐶(𝑚)𝐶 † 𝑚
1/2𝑛
†
𝐶(1)𝐶 1
1/4𝑛
𝜙𝑛
rational approximation
where 𝑁𝑝 is the number of poles for rational approximation.
The 𝑁𝑝 numbers of inverse matrices 1 (𝐴 + 𝑑𝑙 ) 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) 𝑀𝑆 ≡ 8 ∗ 𝑁𝑥 3 ∗ 𝑁𝑡
2) 𝑀𝑈 = 48 ∗ 𝑀𝑆 , link variables
3) 𝑀𝑃 = 32 ∗ 𝑀𝑆 , momentum
4) 𝑀𝑉 = 24 ∗ 𝑁𝑠 ∗ 𝑀𝑆 , 5D vector
Then the ratio of the memory usage for RHMC and TWOFA is
20 + 3 3 + 2𝑁𝑝 𝑁𝑠
𝑀𝑅𝐻𝑀𝐶
=
𝑀𝑇𝑊𝑂𝐹𝐴
32 + 10.5𝑁𝑠
where 𝑁𝑝 is the number of poles for MMCG in RHMC algorithm
TWOFA vs. RHMC with DWF
20 + 3 3 + 2𝑁𝑝 𝑁𝑠
𝑀𝑅𝐻𝑀𝐶
=
𝑀𝑇𝑊𝑂𝐹𝐴
32 + 10.5𝑁𝑠
TWOFA vs. RHMC with DWF
II. Efficiency:
The lattice setup is
𝛃 = 𝟓. 𝟗𝟓,
𝒎𝒒 = 𝟎. 𝟎𝟏,
𝑳𝟑 = 𝟖𝟑 ,
𝑻 = 𝟏𝟔,
𝑵𝒔 = 𝟏𝟔,
HMC Steps: (Gauge, Heavy, Light) = (1, 1, 10), 𝑵𝒑 = 𝟏𝟐 for RHMC
We compare RHMC and TWOFA for the following cases:
1) DWF with 𝒄 = 𝟏. 𝟎, 𝒅 = 𝟎. 𝟎 and 𝝀𝒎𝒊𝒏 𝝀𝒎𝒂𝒙 = 𝟎. 𝟎𝟓 𝟔. 𝟐 (Optimal DWF)
2) DWF with 𝒄 = 𝟎. 𝟓, 𝒅 = 𝟎. 𝟓 and 𝝎𝒔 = 𝟏 (Shamir)
3) DWF with 𝒄 = 𝟏. 𝟎, 𝒅 = 𝟎. 𝟓 (𝜶 = 𝟐) and 𝝎𝒔 = 𝟏 (Scaled Shamir)
TWOFA vs. RHMC with DWF on the 𝟖𝟑 × 𝟏𝟔 × 𝟏𝟔 Lattice
Maximum Forces
1. Optimal Domain-Wall Fermion : Maximum Forces
𝒎𝒉 = 𝟎. 𝟒
𝒎𝒒 = 𝟎. 𝟎𝟏
𝒎𝒉 = 𝟎. 𝟒
𝒎𝒒 = 𝟎. 𝟎𝟏
TWOFA vs. RHMC with DWF on the 𝟖𝟑 × 𝟏𝟔 × 𝟏𝟔 Lattice
1. Optimal Domain-Wall Fermion: ∆𝑯
∆𝐻
𝒆−∆𝑯 = 𝟎. 𝟗𝟗𝟗𝟐(𝟏𝟔)
erfc
𝚫𝑯
= 𝟎. 𝟗𝟖𝟏(𝟏𝟏)
𝟐
Accept = 𝟎. 𝟗𝟖𝟎(𝟎𝟖)
𝒆−∆𝑯 = 𝟏. 𝟎𝟎𝟎𝟑(𝟏𝟔)
erfc
𝚫𝑯
= 𝟎. 𝟗𝟗𝟒(𝟏𝟖)
𝟐
Accept = 𝟎. 𝟗𝟖𝟕(𝟎𝟕)
TWOFA vs. RHMC with DWF on the 𝟖𝟑 × 𝟏𝟔 × 𝟏𝟔 Lattice
1. Optimal Domain-Wall Fermion :
𝛃 = 𝟓. 𝟗𝟓,
𝒎𝒒 = 𝟎. 𝟎𝟏,
𝑳 = 𝟖,
𝑻 = 𝟏𝟔,
HMC Steps: (Gauge, Heavy, Light) = (1, 1, 10),
𝑵𝒔 = 𝟏𝟔,
𝑵𝒑 = 𝟏𝟐 for RHMC
ODWF (kernel 𝑯𝒘 ) with 𝒄 = 𝟏. 𝟎, 𝒅 = 𝟎. 𝟎 and 𝝀𝒎𝒊𝒏 𝝀𝒎𝒂𝒙 = 𝟎. 𝟎𝟓 𝟔. 𝟐
Algorithm
𝒎𝟎
𝒎𝒉
Plaquette
Force (Gauge)
Force (heavy)
Force (light)
TWOFA
1.3
0.4
0.58051(09)
5.15555(34)
0.18971(13)
0.01401(29)
RHMC
1.3
0.4
0.58100(10)
5.15762(36)
0.35334(11)
0.06946(44)
Algorithm
Accept erfc( ∆𝑯 𝟐)
𝒆−∆𝑯
Memory
𝑻𝒕𝒓𝒂𝒋.
𝑻𝒕𝒓𝒂𝒋. (sec.)
TWOFA
0.980(8)
0.981(11)
0.9992(16)
1.00
1.00
0𝟖𝟖𝟏𝟓(𝟐𝟑𝟗)
RHMC
0.987(7)
0.994(18)
1.0003(16)
6.58
1.21
𝟏𝟎𝟔𝟑𝟗(𝟓𝟏𝟏)
TWOFA vs. RHMC with DWF on the 𝟖𝟑 × 𝟏𝟔 × 𝟏𝟔 Lattice
Maximum Forces
2. Shamir Kernel (𝝎𝒔 = 𝟏) : Maximum Forces
𝒎𝒉 = 𝟎. 𝟏
𝒎𝒉 = 𝟎. 𝟏
𝒎𝒒 = 𝟎. 𝟎𝟏
𝒎𝒒 = 𝟎. 𝟎𝟏
TWOFA vs. RHMC with DWF on the 𝟖𝟑 × 𝟏𝟔 × 𝟏𝟔 Lattice
2. Shamir Kernel (𝝎𝒔 = 𝟏) : ∆𝑯
∆𝐻
𝒆−∆𝑯 = 𝟎. 𝟗𝟗𝟗𝟗 𝟏𝟔
erfc
𝚫𝑯
= 𝟎. 𝟗𝟖𝟕(𝟏𝟑)
𝟐
Accept = 𝟎. 𝟗𝟖𝟕(𝟎𝟕)
𝒆−∆𝑯 = 𝟏. 𝟎𝟎𝟕𝟒 𝟏𝟖
erfc −
𝚫𝑯
= 𝟏. 𝟎𝟒𝟕(𝟎𝟓)
𝟐
Accept = 𝟎. 𝟗𝟗𝟕(𝟎𝟑)
TWOFA vs. RHMC with DWF on the 𝟖𝟑 × 𝟏𝟔 × 𝟏𝟔 Lattice
2. Shamir Kernel (𝝎𝒔 = 𝟏) :
𝛃 = 𝟓. 𝟗𝟓,
𝒎𝒒 = 𝟎. 𝟎𝟏,
𝑳 = 𝟖,
𝑻 = 𝟏𝟔,
𝑵𝒔 = 𝟏𝟔,
HMC Steps: (Gauge, Heavy, Light) = (1, 1, 10), 𝑵𝒑 = 𝟏𝟐 for RHMC
DWF (Shamir kernel) with 𝒄 = 𝟎. 𝟓, 𝒅 = 𝟎. 𝟓 and 𝝎𝒔 = 𝟏
Algorithm
𝒎𝟎
𝒎𝒉
Plaquette
Force (Gauge)
Force (heavy)
Force (light)
TWOFA
1.8
0.1
0.59061(09)
5.17686(34)
0.14663(45)
0.03578(18)
RHMC
1.8
0.1
0.59094(14)
5.17866(35)
0.28522(68)
0.10757(06)
Algorithm
Accept erfc( ∆𝑯 𝟐)
𝒆−∆𝑯
Memory
𝑻𝒕𝒓𝒂𝒋.
𝑻𝒕𝒓𝒂𝒋. (sec.)
TWOFA
0.987(7)
0.987(13)
0.9999(16)
1.00
1.00
𝟔𝟔𝟒𝟒(𝟒𝟑)
RHMC
0.997(3)
0.953(06)
1.0074(18)
6.58
1.00
𝟔𝟔𝟐𝟗(𝟐𝟒)
TWOFA vs. RHMC with DWF on the 𝟖𝟑 × 𝟏𝟔 × 𝟏𝟔 Lattice
Maximum Forces
3. Scaled Shamir Kernel (𝝎𝒔 = 𝟏 and 𝜶 = 𝟐) : Maximum Forces
𝒎𝒉 = 𝟎. 𝟏
𝒎𝒉 = 𝟎. 𝟏
𝒎𝒒 = 𝟎. 𝟎𝟏
𝒎𝒒 = 𝟎. 𝟎𝟏
TWOFA vs. RHMC with DWF on the 𝟖𝟑 × 𝟏𝟔 × 𝟏𝟔 Lattice
3. Scaled Shamir Kernel (𝝎𝒔 = 𝟏 and 𝜶 = 𝟐) : ∆𝑯
∆𝐻
𝒆−∆𝑯 = 𝟏. 𝟎𝟎𝟎𝟎 𝟏𝟓
erfc
𝚫𝑯
= 𝟎. 𝟗𝟗𝟎(𝟏𝟒)
𝟐
Accept = 𝟎. 𝟗𝟖𝟑(𝟎𝟕)
𝒆−∆𝑯 = 𝟏. 𝟎𝟎𝟑𝟖 𝟏𝟖
erfc −
𝚫𝑯
= 𝟏. 𝟎𝟑𝟑(𝟎𝟗)
𝟐
Accept = 𝟎. 𝟗𝟗𝟕(𝟎𝟑)
TWOFA vs. RHMC with DWF on the 𝟖𝟑 × 𝟏𝟔 × 𝟏𝟔 Lattice
3. Scaled Shamir Kernel (𝝎𝒔 = 𝟏 and 𝜶 = 𝟐) :
𝛃 = 𝟓. 𝟗𝟓,
𝒎𝒒 = 𝟎. 𝟎𝟏,
𝑳 = 𝟖,
𝑻 = 𝟏𝟔,
𝑵𝒔 = 𝟏𝟔,
HMC Steps: (Gauge, Heavy, Light) = (1, 1, 10), 𝑵𝒑 = 𝟏𝟐 for RHMC
DWF (scaled Shamir kernel) with 𝒄 = 𝟏. 𝟎, 𝒅 = 𝟎. 𝟓 (𝜶 = 𝟐)and 𝝎𝒔 = 𝟏
Algorithm
𝒎𝟎
𝒎𝒉
Plaquette
Force (Gauge)
Force (heavy)
Force (light)
TWOFA
1.8
0.1
0.59061(09)
5.17854(34)
0.14646(13)
0.03359(13)
RHMC
1.8
0.1
0.59032(09)
5.17670(32)
0.28559(39)
0.10775(06)
Algorithm
Accept erfc( ∆𝑯 𝟐)
𝒆−∆𝑯
Memory
𝑻𝒕𝒓𝒂𝒋.
𝑻𝒕𝒓𝒂𝒋. (sec.)
TWOFA
0.983(7)
0.990(14)
1.0000(15)
1.00
1.00
𝟔𝟑𝟔𝟎(𝟑𝟖)
RHMC
0.997(3)
0.967(07)
1.00038(18)
6.58
1.17
𝟕𝟒𝟒𝟕(𝟑𝟖)
Concluding Remarks
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 323 × 64 × 16, and 243 × 48 × 16 lattices.
Thank You