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Numerical Study of Topology
on the Lattice
Hidenori Fukaya （YITP）
Collaboration with T.Onogi
1. Introduction
2. Instantons in 2-dimensional QED
3. Atiyah-Singer index theorem
4.
vacuum & U(1) problem
5. Summary
1.
Introduction

Exact symmetry on the lattice
– Gauge symmetry
 Broken symmetry on the lattice
– Lorentz inv.
– Chiral symmetry
– SUSY
・・・・・
To improve these symmetries is
important !!
1.1 Chiral Symmetries on the Lattice

Ginsparg Wilson relation Phys.Rev.D25,2649
(1982)

gives a redefinition of chiral symmetries on the
lattice without fermion doublings.
→ chiral symmetry at classical level.
..
Luscher’s admissibility condition
Nucl.Phys.B549,295 (1999)
realizes topological charges on the lattice.
→ understanding of quantum anomalies.
1.2 Effects of Admissibility Condition

Topological charge
（QED on T2）
（SU（2）theory on T4）

Improvement of locality of Dirac operator
 Topological Charge in QED on T
2
Continuum

Lattice
LatticeCase
caseCase
Topologically
non-trivial
background
on
T2multi（L×L）
Topological
charge
canfor
be
defined
this
No
patch is necessary
the
latticeby
gauge
fields.
valued
field strength.
The
multi-valued
vector potential includes the
effects of transition functions.
e.g.
On the boundary,
anothersmoothly.
patch and
transition
But
is not defined
Following
function are
areallowed.
needed to define
changes
and
consistently.
Admissibility can
these
transformations.
Classification
of prevent
transition
functions
⇒ Conservation
of topologicalcharge
charge .
⇒ Topological
 Continuum limit of admissibility condition
 action
 without admissibility condition
 under admissibility condition
The continuum
is thecharge
same !!for any ε.
⇒ exact limit
topological
1.3 Our Work
Numerical Simulation under
the admissibility condition
 Instantons on the lattice
 Improvement of chiral symmetry
 θvacuum effects
 U（1） problem
We studied 2-dimensional vectorlike QED.
1.4 Numerical Simulation

..
Luscher’s action
Admissibility is satisfied automatically by this
action. (ε＝１．０) We use the domain-wall fermion
action for the fermion part.

Algorism
Hybrid Monte Carlo method (HMC)
Configurations are updated by small changes of
link variables.
⇒ The initial topological charge is conserved.
2. Instantons in 2-dimensional QED
2.1 Topological charge

.. configuration
The initial
Luscher’splaquette
action (( Q=0
Q
=2) )
Luscher’s
action
Wilson’s
action
This configuration gives the constant
electric background solution with
topological charge Q.
2.2 Multi-instantons on the lattice

Instanton-antiinstanton pair？

2 instantons and 1 antiinstanton ?
..
Luscher’s action can generate
configurations including multi-instantons
without changing the topological charge!!!
3.
Atiyah Singer index theorem
# of zero modes with chirality +
# of zero modes with chirality －
The lattice Dirac operators are large matrices.
We can compute the eigenvalues and the
eigenvectors of the domain-wall Dirac operator
numerically with Householder method
and QL method .
(lattice size = 16×16×6)
3.1 1 instanton
Eigenvalues
Eigen
function
(lower spinor）
spinor)
function（upper
1 (# of Instantons) ＝ 1 (# of zeromode with chirality +)
⇒consistent with Atiyah-Singer index theorem.
3.2 Instanton-antiinstanton pair
Eigenvalues
Eigen
function （lower
（upper spinor）
→annihilation
3.3 2 instantons + 1antiinstanton
→1 instanton
Eigenvalues
Eigen
function （lower
（upper spinor）
3.4 Configurations at strong coupling
Luscher action
（β＝0.1）
Q
=
Q=１
０ ０
Q=０
Q=１
Plaquette action
Q = １（β=2.0）
Q=０
Q=１
Admissibility realizes A-S theorem very well !!
4.
θvacuum and U(1) problem
4.1 θ dependence and reweighting
 total expectation value in θ vacuum
reweighting factor
Generating functional in each sector
Expectation value in each sector
 The reweighting factors
 Advantages of our method
 We can generate
configurations
↓ differenciated
by β（＝１/ｇ ）.in any
２
topological sector very efficiently.
 θvacuum↓ effects
can be evaluated
integrated again.
without simulating with complex actions.
Moreover, one set of configurations can
be used at different θ.
The
factors
β= 1.0
(2-flavor fermions).
 reweighting
Improvement
of at
chiral
symmetry.
※Details are shown in H.F,T.Onogi,Phys.Rev.D68,074503.
4.2 Simulation of 2-flavor QED
(The massive Schwinger model)
S.R.Coleman,Annal Phys.101,239(1976)
 parameters
 size
Y.Hosotani,R.Rodriguez, J.Phys.A31,9925(1998)
:J.E.Hetrick,Y.Hosotani,S.Iso,
16×16 (×6)Phys.Lett.B350,92(1995) etc.

g
:
1.0 , 1.4
 Confinement

:
-5 ～+5
Q
θvacuum

:
0.1 , 0.15 , 0.2 , 0.25 , 0.3
m
U(1) problem

sampling
per 10
trajectory
of HMC
The
massiveconfig.
Schwinger
model
has many
properties
updatingsimilar to QCD and it has been studied
analytically very well.
 Isotriplet meson mass
coupling constant
fermion mass
fermion mass dependence at θ= 0：consistent with
θdependence
（m=0.2）:
consistent
continuum
theory
and chiral
limit iswith
also good.
continuum theory at small θ
 isosinglet meson mass
Comparison with results of
“Y.Hosotani,R.Rodriguez,J.Phys.A31,9923(1998)”:
qualitatively consistent with continuum theory.
5.
Summary
..
 Luscher’s gauge action can generate
configurations in each sector and multiinstantons are also allowed.
 Atiyah-Singer index theorem is well realized on
the lattice under admissibility condition.
 θ vacuum effects can be evaluated by
reweighting and the results are consistent with
the continuum theory.
..
Luscher’s admissibility condition keeps topological
properties of the gauge theory very well !
 Prospects
 Theory
More studies of “subtraction” topology .
→ “subtraction” version of Wess-Zumino condition.
→ Non-perturbative classification of anomalies.
→ Construction of chiral gauge theories on the lattice.
 Simulation
Application of Luscher’s gauge action to 4-d QCD
→ Improvement of chiral symmetries.
→ Understanding of multi-instanton effects.
→ Non-perturbative analysis of θ vacuum effects.
 Chiral condensation
θ dependence of