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Numerical Study of Topology on the Lattice

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Instantons in 2-dimensional QED. Atiyah-Singer index theorem ... Algorism. Hybrid Monte Carlo method (HMC) Configurations are updated by small changes of ... – PowerPoint PPT presentation

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Title: Numerical Study of Topology on the Lattice


1
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

2
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 !!
3
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)
4
1.2 Effects of Admissibility Condition
  • Topological charge

(QED on T2)
(SU(2)theory on T4)
  • Improvement of locality of Dirac operator

5
  • Topological Charge in QED on T2

6
  • Continuum limit of admissibility condition
  • without admissibility condition
  • under admissibility condition

7
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.
8
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.

9
2. Instantons in 2-dimensional QED
2.1 Topological charge
10
2.2 Multi-instantons on the lattice
  • Instanton-antiinstanton pair?
  • 2 instantons and 1 antiinstanton ?

11
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)
12
3.1 1 instanton
1 ( of Instantons) 1 ( of zeromode with
chirality ) ?consistent with
Atiyah-Singer index theorem.
13
3.2 Instanton-antiinstanton pair
14
3.3 2 instantons 1antiinstanton
15
3.4 Configurations at strong coupling
Admissibility realizes A-S theorem very well !!
16
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
17
  • 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.
18
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

21
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.

22
  • 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
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