In this talk, we first discuss the central-branch Wilson fermion, which is defined by imposing a specific relation between the mass and the Wilson parameter [1]. This setup gives two massless Dirac fermions in the continuum limit, and it turns out that no fine-tuning of m is required because the extra U(1) symmetry at the central branch prohibits the additive mass renormalization [2,3]. We show that Dirac determinant is positive semi-definite and this formulation is free from the sign problem, so the Monte Carlo simulation of the path integral is possible. By identifying the symmetry at low energy, we find that this lattice model has the mixed ’t Hooft anomaly between the extra U(1) symmetry, lattice translation, and lattice rotation, which means that the trivially gapped phase is forbidden at the central branch [4]. We discuss its relation to the anomaly of half-integer anti-ferromagnetic spin chains, so our lattice gauge theory is suitable for numerical simulation of Haldane conjecture. We also argue that it gives new and strict understanding on parity-broken phase (Aoki phase) of 2d Wilson fermion [4]. Furthermore, we show that our study can be extended to 4d lattice QCD with Wilson fermion, leading to a novel insight into the question which of Aoki-phase or Sharpe-Singleton scenarios is valid.

[References]
[1] M. Creutz, T. Kimura, T. Misumi, Phys. Rev. D83 (2011) 094506, [arXiv:1101.4239].
[2] T. Kimura, S. Komatsu, T. Misumi, T. Noumi, S. Torii, S. Aoki, JHEP 01 (2012) 048, [arXiv:1111.0402].
[3] T. Misumi, PoS LATTICE2012 (2012) 005, [arXiv:1211.6999].
[4] T. Misumi, Y. Tanizaki, [arXiv:1910:09604].