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Numerical Study of Topology on the Lattice

Numerical Study of Topology on the Lattice. Hidenori Fukaya （YITP）. Collaboration with T.Onogi. Introduction Instantons in 2-dimensional QED Atiyah-Singer index theorem 　　vacuum & U(1) problem Summary. 1. Introduction. Exact symmetry on the lattice Gauge symmetry

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Numerical Study of Topology on the Lattice

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  1. Numerical Study of Topologyon theLattice 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. • Luscher’s 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. Continuum Case • Topologically non-trivial background on T2 （L×L） • On the boundary, another patch and transition • function are needed to define and • consistently. • Classification of transition functions • ⇒　Topological charge • Lattice Case • Topological charge can be defined by this multi- • valued field strength. • But is not defined smoothly. Following • changes are allowed. • Admissibility can prevent these transformations. • ⇒　Conservation of topological charge .　 • Lattice case • No patch is necessary for the lattice gauge fields. • The multi-valued vector potential includes the • effects of transition functions. • e.g. • Topological Charge in QED on T2

  6. action • The continuum limit is the same for any ε. ⇒　exact topological charge !! • 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 • θvacuum effects • U（1） problem We studied 2-dimensional vectorlike QED.

  8. 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.

  9. 　　.. • The initial configuration • This configuration gives the constant electric background solution with topological charge Q. .. 　　Luscher’s action ( Q =2 ) Luscher’s action ( Q=0 ) Wilson’s plaquette action 2. Instantons in 2-dimensional QED 2.1 Topological charge

  10. .. Luscher’s action can generate configurations including multi-instantons without changing the topological charge!!! 2.2 Multi-instantons on the lattice • Instanton-antiinstanton pair？ • 2 instantons and 1 antiinstanton ?

  11. # of zero modes with chirality + # of zero modes with chirality － 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 = 16×16×6)

  12. Eigenvalues Eigen function（upper spinor） Eigen function (lower spinor) 3.1 1 instanton 1 (# of Instantons)　＝　1 (# of zeromode with chirality +) ⇒consistent with Atiyah-Singer index theorem.

  13. →annihilation Eigenvalues Eigen function （upper spinor） Eigen function （lower spinor） 3.2 Instanton-antiinstanton pair

  14. →1 instanton Eigenvalues Eigen function （upper spinor） Eigen function （lower spinor） 3.3 　2 instantons + 1antiinstanton

  15. Luscher action （β＝0.1） Plaquette action （β=2.0） Q = ０ Q = １ Q = ０ Q = １ Q = ０ Q = １ Q = ０ Q = １ 3.4 Configurations at strong coupling Admissibility realizes A-S theorem very well !!

  16. 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

  17. The reweighting factors ↓　differenciated by β（＝１/ｇ２）. ↓　integrated again. The reweighting factors at β= 1.0 (2-flavor fermions). • Advantages of our method • We can generate configurations in any • topological sector very efficiently. • θvacuum effects can be evaluated • without simulating with complex actions. • Moreover, oneset of configurations can • be used at different θ. • Improvement of chiral symmetry. ※Details are shown in H.F,T.Onogi,Phys.Rev.D68,074503.

  18. 4.2 Simulation of 2-flavor QED (The massive Schwinger model) • parameters • size : 16×16 (×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.Hosotani,S.Iso, Phys.Lett.B350,92(1995) etc. • Confinement • θvacuum • U(1) problem • The massive Schwinger model has many • properties similar to QCD and it has been studied • analytically very well.

  19. Isotriplet meson mass coupling constant fermion mass fermion mass dependence at θ= 0：consistent with continuum theory and chiral limit is also good. θdependence （m=0.2）: consistent with continuum theory at small θ

  20. Comparison with results of “Y.Hosotani,R.Rodriguez,J.Phys.A31,9923(1998)”: qualitatively consistent with continuum theory. • isosinglet meson mass

  21. 5. Summary .. .. 　Luscher’s admissibility condition keeps topological properties of the gauge theory very well ! • Luscher’s 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. • θ 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 Luscher’s gauge action to 4-d QCD • →　Improvement of chiral symmetries. • 　　→　Understanding of multi-instanton effects. • 　　→　Non-perturbative analysis of θ vacuum effects.

  23. 　θ dependence of • Chiral　condensation

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