Numerical Study of Topology on the Lattice

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Numerical Study of Topologyon the Lattice
Hidenori Fukaya (YITP)
Collaboration with T.Onogi

• Introduction
• Instantons in 2-dimensional QED
• Atiyah-Singer index theorem
• vacuum U(1) problem
• Summary

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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 !!
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1.1 Chiral Symmetries on the Lattice

• Ginsparg Wilson relation
• gives a redefinition of chiral symmetries on the
  lattice without fermion doublings.
• ? chiral symmetry at classical level.
• Luschers admissibility condition
• realizes topological charges on the lattice.
• ? understanding of quantum anomalies.

Phys.Rev.D25,2649 (1982)
..
Nucl.Phys.B549,295 (1999)
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1.2 Effects of Admissibility Condition

• Topological charge

(QED on T2)
(SU(2)theory on T4)

• Improvement of locality of Dirac operator

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• Topological Charge in QED on T2

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• Continuum limit of admissibility condition
• without admissibility condition
• under admissibility condition

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1.3 Our Work
Numerical Simulation under the admissibility
condition

• Instantons on the lattice
• Improvement of chiral symmetry
• ?vacuum effects
• U(1) problem

We studied 2-dimensional vectorlike QED.
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1.4 Numerical Simulation
..

• Luschers action
• Admissibility is satisfied automatically by this
  action. (e1.0) 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.

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2. Instantons in 2-dimensional QED
2.1 Topological charge
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2.2 Multi-instantons on the lattice

• Instanton-antiinstanton pair?

• 2 instantons and 1 antiinstanton ?

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3. Atiyah Singer index theorem
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 16166)
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3.1 1 instanton
1 ( of Instantons) 1 ( of zeromode with
chirality ) ?consistent with
Atiyah-Singer index theorem.
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3.2 Instanton-antiinstanton pair
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3.3 2 instantons 1antiinstanton
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3.4 Configurations at strong coupling
Admissibility realizes A-S theorem very well !!
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4. ?vacuum and U(1) problem
4.1 ? dependence and reweighting

• total expectation value in ? vacuum

reweighting factor
Generating functional in each sector
Expectation value in each sector
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• Advantages of our method
• We can generate configurations in any
• topological sector very efficiently.
• ?vacuum effects can be evaluated
• without simulating with complex actions.
• Moreover, one set of configurations can
• be used at different ?.
• Improvement of chiral symmetry.

?Details are shown in H.F,T.Onogi,Phys.Rev.D68,074
503.
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4.2 Simulation of 2-flavor QED (The massive
Schwinger model)

• parameters
• size 1616 (6)
• g 1.0 , 1.4
• Q -5 5
• m 0.1 , 0.15 , 0.2 , 0.25 , 0.3
• sampling config. per 10 trajectory of HMC
• updating

S.R.Coleman,Annal Phys.101,239(1976) Y.Hosotani,R.
Rodriguez, J.Phys.A31,9925(1998) J.E.Hetrick,Y.Hos
otani,S.Iso, Phys.Lett.B350,92(1995) etc.

• Confinement
• ?vacuum
• U(1) problem
• The massive Schwinger model has many
• properties similar to QCD and it has been studied
• analytically very well.

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• isosinglet meson mass

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5. Summary

• Luschers gauge action can generate
• configurations in each sector and multi-
• instantons are also allowed.
• Atiyah-Singer index theorem is well realized on
• the lattice under admissibility condition.
• ? vacuum effects can be evaluated by
• reweighting and the results are consistent
  with
• the continuum theory.

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• 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 Luschers gauge action to 4-d
  QCD
• ? Improvement of chiral symmetries.
• ? Understanding of multi-instanton effects.
• ? Non-perturbative analysis of ? vacuum
  effects.

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• Chiral condensation