exact pseudofermion action for hybrid monte carlo simulation of one-flavor domain-wall fermion
DESCRIPTION
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 PresentationTRANSCRIPT
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
Contentβ’ Lattice Dirac Operator for Domain-Wall Fermion (DWF)
β’ Two-Flavor Algorithm (TFA)
β’ TWQCDβs One-Flavor Algorithm (TWOFA)
β’ Rational Hybrid Monte Carlo (RHMC) Algorithm
β’ TWOFA vs. RHMC with Domain-Wall Fermion
β’ 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)
π·ππ€π (π)ΒΏ π«π {ππ [ π°+π³(π)]+π [ π° βπ³(π)] }+ [ π°βπ³(π)]
Using , the can be written as
For , the form of are
010000100001m000
L
000m100001000010
L
Matrix in 4D (πΈ+ΒΏ (π ) ΒΏπΒΏ
πΈβ (π )ΒΏ )π«ππππ
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
π«π ππ (π)ΒΏπ«π {ππ [ π°+π³ (π) ]+π [ π° βπ³ (π) ] }+ [ π°βπ³(π)]
π»βπ»2
βπ (π» )=π»βπ
πππ»2+ππ
:Zolotarev Optimal Rational Approximation
where , and
π»π€=πΎ 5π·π€
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
π«π ππ (π)ΒΏπ«π {ππ [ π°+π³ (π) ]+π [ π° βπ³ (π) ] }+ [ π°βπ³(π)]
π»βπ»2
βπ (π» )=π»βπ
πβ² ππ»2+π β² π
:Polar Approximation
where , and
Lattice Dirac Operator for Domain-Wall Fermion (DWF)
For a physical observable
ΒΏ1πβ«π·ππͺ(π)det [π· π (π) ]exp (βππ (π ) )
If , where the matrix K is independent of the gauge field
β«π·ππͺ(π)det [π· π (π) ]exp (βππ (π ) )β«π·π det [π· π (π)]exp (βππ (π ) )
=β«π·π πͺ(π )det [π· (π)]exp (βππ (π ) )
β«π·π det [π· (π) ]exp (βππ (π ) )
Lattice Dirac Operator for Domain-Wall Fermion (DWF)
Using the redefined operator
ΒΏ1πβ«π·ππ·πβ π·ππͺ(π)exp (βπβ π»β1 (π ) πβππ (π ) )
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,
π Β± (π )=πβ1/2 [ππ Β± (π )+πβ1 π]β 1πβ1 /2
π Β± (π )= [1+πΏΒ±(π)] [1β πΏΒ±(π)]β 1
π·ππ€π (π ) ΒΏπ·π€{ππ [ πΌ+πΏ(π) ]+π [πΌ βπΏ(π)] }+ [πΌ βπΏ(π)]{ππ [ πΌ+πΏ(π) ]+π [πΌ βπΏ(π)] }β 1
Two-Flavor Algorithm (TFA) [2]
π· (π )=π·π€+π (π)=π·π€+π+ΒΏπ+ΒΏ(π )+πβ πβ (π )ΒΏ ΒΏ
For the DWF Dirac operator
we can apply the Schur decomposition with the even-odd preconditioning
where πΆ (π )=I βπ 5(π)π·π€π 5(π)π·π€
ππ ππWe then have
det [π· (π ) ]=det [π5 (π )]β 2Γdet [πΆ (π)]
[2] T. W. Chiu, et al. [TWQCD Collaboration],PoS LAT 2009, 034 (2009); Phys. Lett. B 717, 420 (2012) .
π· (π )=(4βπ0+π (π) π·π€ (π)π·π€ (π ) 4βπ0+π (π))β‘(π 5(π)β1 π·π€
π·π€ π5(π)β 1)ππππ
ΒΏ ( πΌ 0π·π€π5(π) πΌ )(π5(π)β 1 0
0 π5 (π)β 1πΆ(π))( πΌ π5(π)π·π€0 πΌ )ππ
ππ
ππππ
Two-Flavor Algorithm (TFA)
The pseudofermion action for HMC simulation of 2-flavor QCD with DWF is
πππ=πβ πΆβ (1) 1πΆ (π )πΆβ (π )
πΆ (1)π
The field can be generated by the Gaussian noise field
π=1
πΆ (π )πΆ (1 ) πβΊπ=
1πΆ(1)
πΆ (π)πΌ
πππ=πβ πΆβ (1 ) 1πΆβ (π )
1πΆ (π )
πΆ (1 )π=πβ π
generated from Gaussian noise
det [πΆ (π)πΆ (1) ]
2
Two flavor
TWQCDβs 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 [π·(π)] det [π·(1)]
=det [π· (π)]
det [ οΏ½ΜοΏ½(π ,1)]Γdet [ οΏ½ΜοΏ½(π ,1)]det [π· (1)]
π· (π )=(π βπ0+π +ΒΏ(π)ΒΏπ β π‘ΒΏ
π βπ0+π +ΒΏ(π))In Dirac space
οΏ½ΜοΏ½ (π ,1 )=(π βπ0+π+ΒΏ(π )ΒΏπ β π‘ΒΏ
π βπ0+π +ΒΏ(1))
where
TWQCDβs One Flavor Algorithm (TWOFA)
Use type I Schur decomposition to
(π΄ π΅πΆ π·)=( πΌ 0
πΆ π΄β1 πΌ)( π΄ 00 π·βπΆ π΄β1π΅)( πΌ π΄β 1π΅
0 πΌ )we then have
det [ οΏ½ΜοΏ½ (π ,1 ) ]det [π· (π ) ]
=det [~π» (π )+Ξβ(π)]
det [~π» (π )]=det [ πΌ+Ξβ(π) 1
~π» (π ) ]where~π» (π )=π 5 ΒΏ,
TWQCDβs One Flavor Algorithm (TWOFA)
Use type II Schur decomposition to
(π΄ π΅πΆ π·)=(πΌ π΅ π·β1
0 πΌ )(π΄βπ΅π·β1πΆ 00 π·)( πΌ 0
π·β 1πΆ πΌ )we then have
det [π· (1 ) ]det [ οΏ½ΜοΏ½ (π ,1 ) ]
=det [π» (1)]det ΒΏΒΏ
where
π» (1 )=π 5 ΒΏ,
where is a scalar function of , and , are the vector functions of , and , and here we have defined
π΄Β±=ππ Β± (0 )+πβ1π
π΄ Β± (π )=πβπ/ππ¨Β±βππβπ/π+
ππππ+πβππππ πΉπ
βπβπ /ππΒ± πΒ±π»πβπ /π
TWQCDβs 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
ΞΒ± (π )=π 5 [π Β± (1 )βπ Β± (π)]=ππβ1 /2π£Β±π£Β±ππβ1 /2
π= π1βπ π
1βπ1+π(1β2π π)
TWQCDβs One Flavor Algorithm (TWOFA)
Next we use , one then has
det [ οΏ½ΜοΏ½ (π ,1 ) ]det [π· (π ) ]
=det [ πΌ+Ξβ(π) 1~π» (π ) ]
=det [ πΌ+ππ£βππβ 1/2 1~π» (π )
πβ1 /2π£β]
=det [ πΌ+ππβ1 /2π£βπ£βππβ 1/2 1
~π» (π ) ]
Positive definite and Hermitian
TWQCDβs One Flavor Algorithm (TWOFA)
Again, with , one also has
=det ΒΏ
=det ΒΏ
Positive definite and Hermitian
det [π· (1 ) ]det [ οΏ½ΜοΏ½ (π ,1 ) ]
=det ΒΏ
TWQCDβs One Flavor Algorithm (TWOFA)
π1=π1β [ πΌ+ππ£βππβ1 /2 1
~π» (π )πβ 1/2π£β ]π1det [ οΏ½ΜοΏ½ (π ,1 ) ]
det [π· (1 ) ]
π2=π2β ΒΏdet [π· (π ) ]det [ οΏ½ΜοΏ½ (π ,1 ) ]
det [ οΏ½ΜοΏ½ (π ,1 ) ]det [π· (1 ) ]
det [π· (π ) ]det [ οΏ½ΜοΏ½ (π ,1 ) ]
Γ ΒΏdet [π·(π)] det [π·(1)]
One flavor determinant
TWQCDβs One Flavor Algorithm(TWOFA)
Use , and some algebra, the pseudofermion action of one-flavor domain-wall fermion can be written as
πππ=(0 π1β ) [ πΌβππ£βππβ 1/2 1
π» (π)πβ 1/2π£β ]( 0π1)
+(π2β 0 )ΒΏwhere ,
ΞΒ± (π )=ππβ 1/2π£Β±π£Β±ππβ 1/2
π= π1βπ π
1βπ1+π(1β2π π)
TWQCDβs One Flavor Algorithm(TWOFA)The initial pseudofermion fields of each HMC trajectory are generated by Gaussian noises as follows.
π1=π1β [ πΌ+ππ£βππβ1 /2 1
~π» (π )πβ 1/2π£β ]π1=π1β π 1
βπ1=[πΌ+ππ£βππβ 1/2 1~π» (π )
πβ1 /2π£β]β1 /2
π1
Gaussian noiseπ2=π2β ΒΏβπ2=ΒΏΒΏ
TWQCDβs One Flavor Algorithm (TWOFA)
The invesre square root can be approximated by the Zolotarev optimal rational approximation
(π1π1)=βπ=1
π π
[ ππ1+ππ
πΌ+ππ
(1+ππ )2 ππ£β
ππβ1 /2 1
π» (π )β 11+ππ
Ξβ(π)πβπβ 1/2π£β ]( 0π 1)
(π2π2)=βπ=1
ππ
ΒΏΒΏGaussian 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)
πππ=βπππ
β (πΆ (1)πΆβ (1 ) )1 /4 π 1(πΆ (π)πΆβ (π ) )1 /2π
(πΆ (1)πΆβ (1 ) )1/4πππ
1.
2. Positive definite and Hermitian
The fields are generated by the Gaussian noise fields
ππ=1
(πΆ (1)πΆβ (1 ) )1/4π(πΆ (π)πΆβ (π ) )1/4πππ
Gaussian noise
Rational Hybrid Monte Carlo(RHMC) Algorithm
π΄1/πΌ=π0+βπ=1
ππ πππ΄+ππ
rational approximation
πππ=βπππ
β (πΆ (1)πΆβ (1 ) )1 /4 π 1(πΆ (π)πΆβ (π ) )1 /2π
(πΆ (1)πΆβ (1 ) )1/4πππ
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
ππ π»ππΆ
πππππΉπ΄=20+3 (3+2ππ )π π
32+10.5π π
where is the number of poles for MMCG in RHMC algorithm
ππ π»ππΆ
πππππΉπ΄=20+3 (3+2ππ )π π
32+10.5π π
TWOFA vs. RHMC with DWF
II. Efficiency:
The lattice setup is
HMC Steps: (Gauge, Heavy, Light) = (1, 1, 10), for RHMC
π=π .ππ ,ππ=π .ππ ,π³π=ππ ,π»=ππ ,π΅ π=ππ ,
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
ππ=π .πππ=π .π
ππ=π .ππππ=π .ππ
TWOFA vs. RHMC with DWF on the Lattice
1. Optimal Domain-Wall Fermion:
βπ»
)
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
π=π .ππ ,ππ=π .ππ ,π³=π ,π»=ππ ,π΅π=ππ ,
TWOFA vs. RHMC with DWF on the Lattice
2. Shamir Kernel () : Maximum Forces
Max
imum
For
ces
ππ=π .π
ππ=π .π
ππ=π .ππππ=π .ππ
TWOFA vs. RHMC with DWF on the Lattice
2. Shamir Kernel () :
βπ»
TWOFA vs. RHMC with DWF on the Lattice
2. Shamir Kernel () :
DWF (Shamir kernel) with and
for RHMCπ=π .ππ ,ππ=π .ππ ,π³=π ,π»=ππ ,π΅π=ππ ,
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
ππ=π .π
ππ=π .π
ππ=π .ππππ=π .ππ
TWOFA vs. RHMC with DWF on the Lattice
3. Scaled Shamir Kernel ( and ) :
βπ»
TWOFA vs. RHMC with DWF on the Lattice
3. Scaled Shamir Kernel ( and ) :
DWF (scaled Shamir kernel) with and
for RHMCπ=π .ππ ,ππ=π .ππ ,π³=π ,π»=ππ ,π΅π=ππ ,
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