Minimal Doubling Fermion and Hermiticity

We analyze the lattice fermion kinetic term using PT symmetry, R-hermiticity, and $\gamma_{5}$-hermiticity. R-hermiticity is a condition for Hermite action and it is related to $\gamma_{5}$-hermiticity and PT symmetry. Assuming that a translation-invariant kinetic term with continuum and periodic function does not have PT symmetry, it can have R-hermiticity or $\gamma_{5}$-hermiticity. We prove that a kinetic term with continuum and periodic function that is PT symmetric does not reduce doublers. As a simple example, we analyze the two-dimensional two-flavor Gross-Neveu model with minimal doubling fermions. The minimal doubling fermions break PT symmetry and R-hermiticity, hence complex or non-Hermite coupling constants are caused by quantum correction.

The "doublers" appear around each zero-modes as D(p) = D(p + q) with D(p) = 0, and they contribute observables.
In the case of NA, the half doublers that half doublers have same chirality and the others have opposite.
In cases of MDAs, they have opposite chirality each other. The NA and MDAs have γ 5 Hermiticity: For massless, they also have chiral symmetry: The MDAs violate (hyper-)cubic symmetry and some discrete symmetries. We define charge conjugation(C), parity transformation(P), time reflaction(T) and their combinational transformation laws acting on a fermion kinetic term 2 : We present these symmetric properties of the NA and MDAs in Tab 1 3 .

N-flavors Gross-Neveu model and Renormalization Group Flow in Two Dimension
In this section, we describe the N-flavors Gross-Neveu(GN) model [20] and calculate Wilsionian renormalization group flows(RGFs) using the NA and MDAs numerically. Firstly we will review the N-flavors GN model and then we will calculate the RGFs.

Action and Symmetry
We define the continuum Euclidian Lagrangian of N-flavors GN model as follows: where m is a fermion mass and g is a coupling constant of four fermi interaction. We omit flavor indices if we do not have to write explicitly,ψψ ≡ N i=1ψ i ψ i , where "i" means flavor degrees of freedom.
This Lagrangian has U (1) symmetry: In the case of massless fermions, this Lagrangian has chiral Z 4 symmetry: In the case of massive fermions, chiral Z 4 symmetry reduces to chiral Z 2 symmetry(n = 0, 2). In addition, if all flavors have same masses it has the SU (N ) F symmetry: It is convenient to redefine the GN action using an auxiliary scalar field σ instead of (ψψ): According to this manipulation, we can obtain the action which involves Yukawa interaction instead of four fermi interaction. According to perterbative calculation, the NG model has asymptotic freedom property [21,22].

Wilsonian Renormalization Group Flow
Using the Wilsonian method [23], we calculate numerically the RGFs for the mass and coupling constant starting from the trivial fixed point (m = g 2 = 0).
We explain how to calculate Wilsonian RGFs in Appendix A.
In the case of MDAs, we use each zero-modes as the different flavor fermions, and in the case of NA we use only two zero-modes,p = (0, 0) and (π, π). We now represent the spinor indices explicitly, and we distinguish 0, 1 from 2, 3 as different flavors. We assume that high frequency modes of fields are not effective, therefore we neglect their contributions, ψ(1 < |k|),ψ(1 < |k|) and σ(1 < |k|). 4 We choose initial conditions for the mass as m 00 = m 11 = m 22 = m 33 = 0, ±0.25, ±0.5, and for the coupling constant as g 2 = 0, 0.2, 0.4.
The off-diagonal mass components equal to zero in all cases. We will calculate numerically up to one-loop quantum effects and RGFs which toward IR from the initial conditions 4 . In our calculation we define γ-matrices as follows: The results are referred in fig.1. The RGFs of the NA and MDAs are similar forms and differences among each values are O(10 −3 ).
In cases of MDAs, however, off-diagonal mass components are generated by the RGFs, except the initial value which is trivial fixed point(m = 0, g 2 = 0). The obtained results do not have Hermiticity.
We show this fact in fig.2 and fig.3 whose initial conditions are m = 0, g 2 = 0.2. The fig.2(a) and (b) are the RGFs of D md1 (p) and D md2 (p) respectively. The fig.3(a) and (b) are relations between the off-diagonal mass components and iterations using D md1 and D md2 respectively. Though the off-diagonal mass components amplify as the flows go to IR because of scaling effect, they do not break chiral Z 4 symmetry. This fact means that the MDAs do not have Hermiticity and it seems to have any problems. In the following section, we will discuss this phenomenon and Hermiticity.

γ 5 -Hermiticity, R-Hermiticity and PT symmetry
In the previous section, we observed non-Hermiticity contributions in fermion effective mass components. In Euclidian formulation, a fermion kinetic term in continuum limit does not have Hermiticity but anti-Hermiticity. The NA also has an anti-Hermiticity kinetic term. In the case of NA, however, an effective mass does not have anti-Hermite components explicitly, therefore the cause of them is clearly added even function terms, (cos p − 1)γ. We will consider Hermiticity and PT symmetry, and what added terms generate explicit non-Hermite effective coupling constants. Hermicitity is discussed in ref. [18] at the case of the minimal doubling fermion on the Hyperdiamond lattice. We will focus on only kinetic terms from now on and discuss In the case of two dimension 5 .
We will focus on only kinetic terms from now on. In the case of two dimension, we define a translation invariant kinetic term in momentum space as follows: where f µ (k) are complex numbers in general. We define three conditions, γ 5 -Hermiticity, R-Hermiticity and PT symmetry 6 : According to estimate integrating part of one-loop calculation we used the sectional measurement method. We took the length of division ∆pµ = 0.01 (O(0.01 2 ) error in the integrating part and it seems that accuracy is not very well. 5 We can apply this discussion to the case of any even dimension. 6 In following discussion, we can use C symmetry instead of PT symmetry because of CPT theorem. These conditions are not independent of one another. We can easily lead to the fact that another condition is automatically satisfied if a kinetic term satisfy two of the three conditions. This is a sufficient condition but not a necessary condition. If f µ (k) is pure imaginary, γ 5 -Hermiticity guarantees anti-Hermite condition for a kinetic term. R-Hermiticity is also used as (anti-)Hermite condition, for example in ref. [19], however it is not well-defined because forward-derivative, D fd (k) = µ e ikµ − 1 γ µ , satisfy this condition. We will show that R-Hermiticity is a condition for real effective coupling constants. We assume that a fermion kinetic term has R-Hermiticity and effective coupling constants is the following form: with where g 0 is a real bare coupling constant, g ef f is an effective coupling constant, S αβ (k) is a fermion propagator and I (n) is n-loop quantum effect which is constructed from r-fermion propagertors. We  act Hermite conjugate to the second term of r.h.s. The effective parameter is real if the following condition is satisfied, Ordinary Euclidian actions are constructed from Hermite terms except a fermion kinetic term, so that I (n) is also constructed from only Hermite ingredients. Therefore we can use R-Hermiticiy as a reality condition for parameters, as long as (4.8) is satisfied. If γ 5 -Hermiticity is not satisfied and R-Hermiticity is satisfied, one-loop fermion propagator produces diagonal mass components. In the continuum Euclidian action case, these components are not produced. Therefore both γ 5 -Hermiticity and R-Hermiticity should be satisfied for consistency. Next, we will show that PT symmetry is always broken if we add extra kinetic terms to a NA to reduce to doublers 7 .

Statement
In even dimension, a PT symmetric kinetic term which is assumed periodicity and continuity always has more than or equals to 2 d poles.

Proof
For simplicity, we also assume translation invariant 8 . A general 2π periodic and continuum D(k) is the following form: where A µ (n), B µ (n), C µ (n), D µ (n) are real numbers and E µ are complex constants. From PT symmetry, A µ (n) = B µ (n) = E µ = 0, for all µ, n.  The D(k) always has two poles at k = 0 and π for each dimension. Therefore D(k) has equal or more than 2 d poles.
This statement equals to that we can not reduce to the number of doublers using PT symmetric kinetic terms 9 . In numerical simulation context, γ 5 -Hermiticity is a very important condition to avoid sign problem. Assumed translation invariance, R-Hermiticity is not satisfied if D(k) satisfy γ 5 Hermiticity and does not PT symmetry. Therefore effective parameters have explicit non-Hermiticity. In a process of rewriting from Minkovskian to Euclidian, Hermite fermion kinetic terms transmute anti-Hermite ones. A general lattice fermions which is reduced the number of doublers have possibilities of generating this anti-Hermite effective coupling constants. The R-Hermiticity is a criterion to remove non-Hermiticity. Because MDAs have only γ 5 -Hermiticity, the fermion effective mass which is constructed from odd fermion propagators has non-Hermiticity components explicitly in RGFs.

Conclusion and Discussion
We have investigated the one-loop Wilsonian renormalization group flows(RGFs) of the twoflavors Gross-Neveu(GN) model in two dimension with naive action(NA) and two minimal doubling actions(MDAs). We observed that in cases of MDAs off-diagonal mass components which are non-Hermiticity are generated even massless initial conditions. In order to understand the reason for non-Hermiticity mass components, we considered γ 5 Hermiticity, R-Hermiticity and PT (or C) symmetry conditions. These conditions are not independent, satisfied two of the three conditions is a sufficient condition for that another condition is satisfied automatically. However it is not a necessary condition. We suggested that R-Hermiticity is a condition for removing non-Hermicity. However, both γ 5 -Hermiticity and R-Hermiticity should be satisfied. Becausethe diagonal mass components which is not consistent are generated if only R-Hermiticy is satisfied. Therefore we can not reduce the numbers of doubler with modified kinetic terms which have periodicity and continuity because of a relation between doublers and PT symmetry.
We applied this relation to the MDAs which have only γ 5 -Hermiciticy. Because of the non-R-Hermite kinetic terms, explicit non-Hermite effective mass components appear.
In lattice perturbative, theory we have to care whether we can remove relevant and marginal symmetry breaking terms with counter-terms 10 . In massive two-flavors GN model, non-Hermite masses are small in comparision with diagonal masses and we can remove them with counter-terms. In massless case, non-Hermite masses appear however they do not break chiral Z 4 symmetry. Therefore it is not so trouble at one-loop level. In more higher loop or non-perturbative case, we do not know how far parameters are involved the non-Hermite effects and should be fine-tuned.

Appendix A Wilsonian Renormalization Group
In this appendix, we review a method to calculate the Wilsonian renormalization group flow in the case of GN model in two dimension [23].
We define the partition function of the GN model in momentum space 11 : and L GN is given in (3.5). We can treat of N as a mass parameter of auxiliary field σ. Here we assumed that the high frequency modes had already integrated and they do not contribute effectivelly. Then we split the field configurations as follows: where σ l (p) = σ(p) if 0 < |p| < d 2 k (2π) 2 (S(k)S(k)) αβ D(k) · δ αβ , (A.12) 11 We omit the subscript which means flavors. 12 On account of numerical efficiency, we choose a division which split between σ l and σ h as p = 4 5 .

10
where S(k) and D(k) are propagators of each fields presented below, "tr" is a trace operation of the fermionic indices and η ψ and η σ are rescaling parameters of fermion and auxiliary field respectively. We can define these parameters with dimensional analysis in the following values: (A.14) We can obtain propagators from the GN action: