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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) - PowerPoint PPT Presentation
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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