Equivalence of Chiral Fermion Formulations
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Transcript of Equivalence of Chiral Fermion Formulations
Saturday 22 April 2023Saturday 22 April 2023Workshop on Computational Hadron PhysicsWorkshop on Computational Hadron Physics
Hadron Physics I3HP Topical WorkshopHadron Physics I3HP Topical Workshop
Equivalence of Chiral Equivalence of Chiral Fermion FormulationsFermion Formulations
A D KennedyA D KennedySchool of Physics, The University of EdinburghSchool of Physics, The University of Edinburgh
Robert Edwards, Robert Edwards, Bálint JoóBálint Joó, , Kostas OrginosKostas Orginos ((JLabJLab) ) Urs Wenger Urs Wenger (ETHZ)(ETHZ)
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Contents
On-shell chiral symmetryNeuberger’s OperatorInto Five Dimensions
Kernel
Schur ComplementConstraintApproximation
tanhЗолотарев
RepresentationContinued Fraction Partial FractionCayley Transform
Chiral Symmetry BreakingNumerical StudiesConclusions
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Chiral Fermions
Conventions
We work in Euclidean spaceWe work in Euclidean space
γγ matrices are Hermitianmatrices are Hermitian
We writeWe write
We assume all Dirac We assume all Dirac
operators are operators are γγ55 HermitianHermitian
†5 5D D
D D
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It is possible to have chiral symmetry on the lattice without doublers if we only insist that the symmetry holds on shell
On-shell chiral symmetry: I
Such a transformation should be of the form
(Lüscher)
is an independent field from
has the same Spin(4) transformation properties as
does not have the same chiral transformation
properties
as in Euclidean space (even in the continuum)
y†y
y
551 1;i aD i aDe e
y
†y
†y†y
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On-shell chiral symmetry: II
For it to be a symmetry the Dirac
operator must be invariant 5 51 1i aD i aDD e De D
5 51 1 0aD D D aD For an infinitesimal transformation this implies that
5 5 52D D aD D
Which is the Ginsparg-Wilson relation
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5 5 5†sgnˆ W
w
W W
D MD M
D M D M
Both of these conditions are satisfied if(f?) we define
(Neuberger)
Neuberger’s Operator: I
We can find a solution of the Ginsparg-Wilson relation as follows
† †15 5 5 5 5 52 1 ;ˆ ˆ ˆaD aD aD
Let the lattice Dirac operator to be of the form
This satisfies the GW relation iff 25 1ˆ
It must also have the correct continuum limit
Where we have defined where W WD Z O a 2WZM aZ
2 25 5 52 1 1ˆ WD
D Z aZ O a O aM
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Into Five Dimensions
H Neuberger hep-lat/9806025A Boriçi hep-lat/9909057, hep-lat/9912040, hep-lat/0402035A Boriçi, A D Kennedy, B Pendleton, U Wenger hep-lat/0110070R Edwards & U Heller hep-lat/0005002趙 挺 偉 (T-W Chiu) hep-lat/0209153, hep-lat/0211032, hep-lat/0303008R C Brower, H Neff, K Orginos hep-lat/0409118 Hernandez, Jansen, Lüscher hep-lat/9808010
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Is DN local?It is not ultralocal (Hernandez, Jansen, Lüscher)
It is local iff DW has a gap
DW has a gap if the gauge fields are smooth enough
q.v., Ben Svetitsky’s talk at this workshop (mobility edge, etc.)
It seems reasonable that good approximations to DN
will be local if DN is local and vice versa
Otherwise DWF with n5 → ∞ may not be local
Neuberger’s Operator: II
1, 1 1 sgn
52D H H
N 10 μ
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Four dimensional space of algorithms
Neuberger’s Operator: III
Representation (CF, PF, CT=DWF)
Constraint (5D, 4D)
,
( )sgn( ) ( )
( )n
n mm
P HH H
Q HApproximation
5 WH D MKernel
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Kernel
Shamir kernel
55
5
;2
WT T T
W
a D MH D aD
a D M
Möbius kernel
5 5
55 5
;2
WM M M
W
b c D MH D aD
b c D M
5W WH D M
Wilson (Boriçi) kernel
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1 0 0 111 11 0 0 1
A A B
CA D CA B
1 0 0 1
1 0 0 1
1 0
1 11 0
A B
CA D CA B
Schur Complement
It may be block diagonalised by an LDU factorisation (Gaussian elimination)
1det det
A BAD ACA B
C DIn particular
A B
C D
Consider the block matrixEquivalently a matrix over a skew field = division ring
The bottom right block is the Schur complement
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1
2
2
1
n
n
D
00
00
1
2
52
1
n
n
D
Constraint: I
1
2
2
1
n
n
DU
So, what can we do with the Neuberger operator represented as a Schur complement?Consider the five-dimensional system of linear
equations00
00
1L 1LL
The bottom four-dimensional component is, Nn n DD
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Constraint: II
Alternatively, introduce a five-dimensional pseudofermion field 1 12 n
Then the pseudofermion functional integral is
† 15†
5 ,1
det det det detn
Dj j
j
d d e D LDU D D
So we also introduce n-1 Pauli-Villars fields
†,
11 1
†,
1 1
detj j jj
n nD
j j j jj j
d d e D
and we are left with just det Dn,n = det DN
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Approximation: tanh
Pandey, Kenney, & Laub; Higham; NeubergerFor even n (analogous formulæ for odd n)
11 1
1,11
1tanh tanh
1
nxx
n n nxx
x n x
2
2 2
2
2 2
1
tan
1
12tan
1
n
n
kx
nk
kx
nk
xn
2
2 2 21 12 2cos sin1
2 1n
x k kk n n
xn
ωj
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Approximation: Золотарев
2
2/ 2
21
212
sn ;1
sn / ; sn 2 / ;1sn ; sn ;
1sn 2 / ;
n
m
z k
z M iKm n k
z k M z k
iK m n k
sn(z/M,λ)
sn(z,k)
ωj
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Approximation: Errors
The fermion sgn problem
Approximation over 10-2 < |x| < 1Rational functions of degree (7,8)
ε(x) – sgn(x)
log10 x
0.01
0.005
-0.01
-0.005
-2 1.5 -1 -0.5 0.50
Золотарев tanh(8 tanh-1x)
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Representation: Continued Fraction I
0
1
2
3
1 0 0
1 1 0
0 1 1
0 0 1
A
A
A
A
Consider a five-dimensional matrix of the form
10
01 10 11
1 121 2
1 32
1 0 0 0 1 0 00 0 01 0 0 0 1 00 0 0
0 1 0 0 0 0 0 0 10 0 0
0 0 1 0 0 0 1
SSS SS
S S SS
S
Compute its LDU decomposition
0 01
1; n n
n
S A S AS
where
3 3 3 32
2 21
10
1 1 11 1
1
S A A AS A A
S AA
then the Schur complement of the matrix is the
continued fraction
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Representation: Continued Fraction II
We may use this representation to linearise our rational approximations to the sgn function
1
1 21
1
1 0
1
22
, ( )
n
n
n
n
n
n
cc
H HH
HH
cc
c cc
c
0
01
0 0
0 02
01
21
21 1
22 1 2
21 2 1
0
1
10
c c cn n n
c c cn n n
c c c
c c c c
Hn
Hn
H
H
Hc
as the Schur complement of the five-dimensional matrix
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Representation: Partial Fraction I
2
22
1
1
1
2
2
0 0
0 0 0
0 0
0 0 0
0 0
1 1
1 1
1
1
11
x
p
p x
x
p
q
p
x
q
R
Consider a five-dimensional matrix of
the form (Neuberger & Narayanan)
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Compute its LDU decomposition
So its Schur complement is
12 2
1
22 2
2
p x
x q
p xR
x q
2
2 2 2
2 2
2
1
1 1 1
2
1 2
2
1
1 0 0 0 0
1 0 0 0
0 0 1 0 0
0 0 1 0
1
p
x
p q x q
x x q x q
p
xp q x q
x x q x q
12 2
1 1
2
1
1
22 2
2 2
2
2
2
2 2
2
2 2
1
(
( )
)
0 0 0 0
0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0
x
p
p x q
xq
p x
x q
x
p
p x q
R
xqp x
x q
1
2
1
1 1
2 2
1 1
2
2 2
2 2
2 2
1 0 0
0 1 0 0
0 0 1
0 0 0 1
0 0 0 0 1
p
p
x
x
p
xq x q
x q x qp
xq x q
x q x q
1 11
11 01
1
1 12
21 02
1 0
0 0
0 0
0 0
0 0
0
x
pp x
q
x
R
x
pp
q
Representation: Partial Fraction II
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Representation: Partial Fraction III
This allows us to represent the partial
fraction expansion of our rational function as
the Schur complement of a five-dimensional
linear system
1, 2 2
1
( )n
jn n
j j
pH H
H q
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1
2 1
3 2 1
4 3 2 1
1 0 0 0 1 0 0
0 1 0 0 0 1 0
0 0 1 0 0 0 1
0 0 0 0 0 0 1
A
A A
A A A
C A A A A
1
2 1
3 2 1
4 3 2 1
1 0 0
0 1 0
0 0 1
0 0 0
A
A A
A A A
C A A A A
2
3
4
1 0 0 0
1 0 0
0 1 0
0 0 1
A
A
A
Representation: Cayley Transform I
1
2
3
4
1 0 0
1 0 0
0 1 0
0 0
A
A
A
A C
Consider a five-dimensional matrix of the
formCompute its LDU decomposition
CT 1 2 1n nS C A A A A So its Schur complement is
Neither L nor U depend on C
1CT 2
1 112 51
11 1
1P P T
TS
T
If where , and , then
1 nT T T 1s sA T
1C P P
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Representation: Cayley Transform II
The Neuberger operator is
CTN
CT
,1
P P SD H
S
1 1
( ) ;1 1
T x xx T x
T x x
T(x) is the Euclidean Cayley transform of, ( ) sgn( )n m x x
1
x x T xT x
0 0 0 1T jj
j
xT x
x
For an odd function we have
In Minkowski space a Cayley transform maps between Hermitian (Hamiltonian) and unitary (transfer) matrices
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(1) (1) (1)
(2) (2) (2)
(3) (3) (3)
(4) (4) (4)
00
00
D D P D PD P D D P
D P D D PD P D P D
The Neuberger operator with a general Möbius kernel is related to the Schur complement of
D5 (μ) (1) (1) (1) (1)
(2) (2) (2) (2)
(3) (3) (3) (3)
(4) (4) (4) (4) (4)
0 0 0 00 0 0 0
0 0 0 00 0 0
D D D DD D D D
P PD D D D
D D D D D
Representation: Cayley Transform III
with and( ) ( )
5
( ) ( )5
( ) 1
( ) 1
s ss W
s ss W
D b D M
D c D M
152 1P
( ) ( ) ( ) ( )5 55 5 5 5 5 5;s s s s
s
b cb c b c b c
P-μP-
P+ μP+
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(1) (1) (1) (1)
(2) (2) (2) (2)
5 (3) (3) (3) (3)
(4) (4) (4) (4)
0 0 0 0
0 0 0 0( )
0 0 0 0
0 0 0 0
D D D D
D D D DD P P
D D D D
D D D D
P
1 0 0 0 0 0 0 1
0 1 0 0 1 0 1 0
0 0 1 0 0 1 0 0
0 0 0 1 0 0 1 0
P P
P
Cyclically shift the columns of the right-handed part where
5 5D D P
Representation: Cayley Transform IV
P+P- μP+μP-
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( ) ( ) ( )s s sQ D P D P
Representation: Cayley Transform V
1 0 0 0
0 1 0 0
0 0 1 0
10 0 0 P P
C
(1)
(2)
(3)
(4)
0 0 0
0 0 0
0 0 0
0 0 0
Q
Q
Q
Q
Q1( ) ( )s s s M
ss M
HT Q Q
H
With some simple rescaling
11
12
13
14
1 0 0
1 0 01
0 1 05
10 0
( )
T
T
T
T P P
D
Q PCx
The domain wall operator reduces to the form introduced before
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Representation: Cayley Transform VI
It therefore appears to have exact off-shell chiral symmetryBut this violates the Nielsen-Ninomiya theorem
q.v., Pelissetto for non-local versionRenormalisation induces unwanted ghost doublers, so we cannot use DDW for dynamical (“internal”) propagatorsWe must use DN in the quantum action instead
We can us DDW for valence (“external”) propagators, and thus use off-shell (continuum) chiral symmetry to manipulate matrix elements
5 5( ) (1)D D We solve the equation
Note that satisfies1 1DW ND D a
1 1 1 15 5 5 5 5, , 2 , 2 0DW N N N ND D a D D D a
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Chiral Symmetry Breaking
Ginsparg-Wilson defect5 5 5 52 LD D aD D Using the approximate Neuberger operator 1
52 1aD H
L measures chiral symmetry breaking 212 1La H
The quantity is essentially the usual
domain wall residual mass (Brower et al.)
†
res †
tr
trLG G
mG G
mres is just one moment of L
G is the quark propagator
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Numerical Studies
Used 15 configurations from the RBRC dynamical DWF dataset
316 32 12 2
0.8 1.8 0.02
V n L ns fM
Matched π mass for Wilson and Möbius kernelsAll operators are even-odd preconditionedDid not project eigenvectors of HW
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mres per Configuration
mres is not sensitive to this small eigenvalue
But mres is sensitive to this one
ε
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Cost versus mres
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Conclusions
Relatively goodZolotarev Continued FractionRescaled Shamir DWF via Möbius (tanh)
Relatively poor (so far…)Standard Shamir DWFZolotarev DWF ( 趙 挺 偉 )
Can its condition number be improved?
Still to doProjection of small eigenvalues
HMC5 dimensional versus 4 dimensional dynamicsHasenbusch acceleration
5 dimensional multishift?Possible advantage of 4 dimensional nested Krylov solvers
Tunnelling between different topological sectorsAlgorithmic or physical problem (at μ=0)Reflection/refraction Assassination of Peter of Assassination of Peter of Lusignan Lusignan
(1369)(1369)(for use of wrong chiral formalism?)(for use of wrong chiral formalism?)