1. Syo Kamata Rikkyo University In collaboration with Hidekazu Tanaka

2. Motivation ～ Fermion on Lattice ～ Continuum fermion Naïve fermion Nielsen-Ninomiya theorem [ Nilsen et al. ] Non-doubling fermion cannot simultaneously satisfy …  Translation invariance  Locality Lattice fermion [ Wilson ] [ Kogut et al. ] [ Kersten ] [ Creutz ] … ● Wilson fermion: breaks chiral symmetry ● KS fermion: doublers ⇒ flavors ● MD fermion: preserves exact chiral symmetry but breaks some discrete and cubic symmetries. Naïve discretize  Chiral symmetry  hermiticity 16 Doublers in four dim.

3. Motivation ～ hermiticity ～ hermiticity is needed for reality of the Dirac determinant. index theorem with GW relation. … We dare to break hermiticity for these motivation: ➢ To understand lattice fermion structure. ➢ We have the possibility for single pole fermion in anomaly free case. ➢ It is helpful for understanding sign problem. For these motivation, we define and investigate Non- hermiticity fermion based on MDF in two dimensions

4. Talk Plan Definition: Non- hermiticity fermion Eigenvalue and pole distribution Gross-Neveu model Parity phase diagram Chiral phase diagram ( imaginary chemical potential model ) Summary

5. Samples As samples, we define five Dirac operators in two dimensions. Translation invariance × hermiticity Locality × discrete symmetry Exact chiral symmetry × cubic symmetry We firstly examine Eigenvalue distribution Number of poles

6. Eigenvalue and pole distribution 4 poles for 2 poles for Space enclosed eigenvalues In continuum limit, eigenvalues are distributed along the imaginary axis.

7. Eigenvalue and pole distribution 6 poles for any Eigenvalues are distributed like “cloud”. In continuum limit, eigenvalues are distributed to the entire plane. ⇒ Can not use because det D = 0.

8. Eigenvalue and pole distribution 7 poles for 3 poles for Eigenvalues are distributed like “cloud”. In continuum limit, eigenvalues are distributed to the entire plane. ⇒ Can not use because det D = 0.

9. Eigenvalue and pole distribution 7 poles for any Eigenvalues are distributed like “cloud”. In continuum limit, eigenvalues are distributed to the entire plane. ⇒ Can not use because det D = 0.

10. Eigenvalue and pole distribution 3 poles for any Eigenvalues are distributed like “cloud”. In continuum limit, eigenvalues are distributed to the entire plane. ⇒ Can not use because det D = 0.

11. Eigenvalue Distribution From these analysis, we see that D1 has the possibility for estimation of observable. And D1 has two poles at.

12. Why many doublers are generated ? We have a question … Why many doublers are generated ? Because its dispersion relation is complex. The third term has opposite signature to the first and second term. Therefore there is a possibility of generating more than four doublers. 6 poles for any

13. Test: Gross-Neveu model in Two dimensions We adapt non- hermiticity fermion to a concrete model [Gross et. al. ’79 ] Toy model of QCD In large N limit, we can solve with saddle point approximation. Parity broken phase (Aoki phase) [Aoki ’84] We capture parity broken phase diagrams. And examine whether coupling constants are real values or not. We adapt our fermion,

14. Outline for getting phase diagram We define the continuum action Introducing auxiliary fields, We discretize space-time and define lattice action

15. Outline for getting phase diagram Integrated out fermion, we obtain effective action Supposing translation invariance for auxiliary fields (index 0). large N limit, we can use saddle point approximation, We obtain the following equations,

16. Gross-Neveu model & Aoki phase From the equation, we can write down gap eq. The blue line is similar to the red line. And coupling constants are real values despite of non- hermiticity. ⇒ We can capture parity symmetry breaking phase diagram using non- hermiticity fermion. Near the critical line,

17. Gross-Neveu model & Aoki phase Flavored mass case : We introduce flavored mass and add to mass term. Then we modify the gap equations. We split spectra between doublers.

18. Imaginary Chemical potential We also adapt non- hermiticity fermion to imaginary chemical potential model. We can not use because its determinant is complex. We fix m=0, then we capture chiral symmetry broken phase diagram.

19. Imaginary Chemical potential Gap eq: We fix parameters at Near the critical line,

20. Summary We constructed non- hermiticity fermion in two dimensions  As same symmetries as minimal doubling fermion  There are two poles at As a simple test, we capture phase diagrams for two dimensional Gross-Neveu model using non- hermiticity fermion.  We cauture parity and chiral symmetry breaking phase diagrams.  The coupling constants are real.

21. Gross-Neveu model & Aoki phase Pion mass on critical line Massless pion on critical line

22. Reflection Positivity Link Reflection Reflection sym.

23. Reflection Positivity

24. Eigenvalue Distribution

25. Pole Distribution

26. Minimal doubling fermion Chiral sym. ○ (hyper) Cubic sym. × Discrete sym.