Remarks on discrete Dirac operators and their continuum limits

We discuss possible definitions of discrete Dirac operators, and discuss their continuum limits. It is well-known in the lattice field theory that the straightforward discretization of the Dirac operator introduces unwanted spectral subspaces, and it is known as the fermion doubling. In oder to overcome this difficulty, two methods were proposed. The first one is to introduce a new term, called the Wilson term, and the second one is the KS-fermion model or the staggered fermion model. We discuss mathematical formulations of these, and study their continuum limits.


Introduction
In a recent paper by Cornean, Garde and Jensen [3], they studied continuum limit of discretized Dirac operators in the sense of norm resolvent convergence, and they found that they do not converges to the (usual) Dirac operators.They found that if one add another term, then these operators converges to the Dirac operators.This corresponds to the Wilson term in the lattice field theory.We discuss this method briefly, and then discuss another method, the KS-fermion (or the staggered fermion) model, which is mathematically ingenious and interesting in itself.Thus this note is partly a survey of these methods, but they are rigorously reformulated in relatively general settings, and we prove some new results on their continuum limits.
The continuum limit of quantum Hamiltonian on the square lattice in the sense of (generalized) norm resolvent convergence was studied by Nakamura and Tadano [8], and several research has been appeared based on the idea of the norm resolvent convergence (see also [9] for the concept of generalized resolvent convergence).Cornean, Garde and Jensen [2] studied the convergence for more general Fourier multipliers, and Exner, Nakamura and Tadano [4] considered continuum limit for quantum graph Hamiltonians.As mentioned above, Cornean, Garde and Jensen [3] considered the continuum limit of discretized Dirac operators, and found difficulty to show the convergence to the continuous Dirac operators.See also Schmidt and Umeda [11] and Isozaki and Jensen [5] for closely related results.
It turned out that such difficulty was widely known in the lattice field (gauge) theory (see, e.g., [1], [10]), and it is generally called the fermion doubling.There are two standard methods to avoid the problem.The first one is adding an additional term to the Hamiltonian (or the Lagrangian), and it is called the Wilson term.The other method is called the KS-fermion model after Kogut and Susskind ([12] and [6]).We try to reformulate these methods, especially the KS-fermion method so that it is appropriate to study the continuum limit in the norm resolvent sense, and prove the convergence of the continuum limit.
The paper is constructed as follows.In Section 2, we prepare several basic tools.At first we explain the notations concerning the square lattices, function spaces, Fourier transforms, and several kinds of difference operators.Then in Subsection 2.2, we introduce a embedding operator from function space on the lattice to the Lebesgues space on the Euclidean space, which is necessary to study the continuum limit following [8].In Subsection 2.3, we recall the definition of the Dirac operators on the Euclidean spaces.In Section 3, we consider the discretization of the Dirac operator using the symmetric difference operators, and explain why it is not appropriate to consider the continuum limit.In Section 4, we introduce the Wilson term, and show the convergence of the continuum limit in the norm resolvent sense for Hamiltonians with the Wilson term under suitable conditions.Section 5 is devoted to the discussion of the KS-fermion model.We introduce the onecomponent KS-Hamiltonian on d-dimensional lattice (with fermion doubling problem), and then transform it to a 2 d components operator without the fermion doubling problem in Subsection 5.1.We briefly examine the spectral properties of this operator in Subsection 5.2, and prove the convergence to a continuum limit in Subsection 5.3.Here the number of components, 2 d , can be higher than those of the standard Dirac operator on R d .We discuss the model for the dimensions 1, 2 and 3 in Section 6, and show that for d = 1 the model is appropriate (and in fact studied in [3] already), and for d = 2 and 3, the continuum limit is decomposed to a direct sum of two standard Dirac operators.

Notations
We denote the square lattice in R d with the lattice spacing h > 0 by hZ d = hn n ∈ Z d .Let {e j } d j=1 be the standard basis of R d , i.e., e j = (δ j,k ) d k=1 ∈ Z d , j = 1, . . ., d, where δ j,k denotes the Kronecker delta symbol.The basis (or the generators) of hZ d is given by {he 1 , . . ., he d }.We recall the dual space (or the dual module) of hZ d is given by h We denote the standard L 2 space on the d dimensional Euclidean space by L 2 (R d ).We use the square summable function space on hZ d , namely ℓ 2 (hZ d ), and we use the norm defined by We denote the Fourier transform on R d by and the inverse Fourier transform by F * .On the lattice hZ d , the Fourier transform for u ∈ ℓ 2 (hZ d ).F h is unitary, and the inverse is given by The partial differential operator on R d , or the momentum operator is denoted by On the lattice hZ d , we set the symmetric difference operators as an approximation of D j on hZ d , where u ∈ ℓ 2 (hZ d ).We also write the forward and backward difference operators by We need an embedding operator J h : ℓ 2 (hZ d ) → L 2 (R d ) when we consider the continuum limit.We employ the following operators ( [8], see also [2]).We need a function where φ = Fϕ.We then set, for z ∈ hZ d , and we define , and the adjoint operator is given by where v ∈ L 2 (R d ).(We remark that our notations are slightly different from [8].In particular, J h = P * h in [8]).In the following, we always suppose Assumption A. ϕ satisfies the condition (2.1), and supp[ φ] ⊂ (−1, 1) d .
We note there exists various such ϕ's, and we simply choose one and fix it here.See [8] for the detail.

Free Dirac operators
We recall the definition of the Dirac operators on R d .See, e.g., Thaller [13] for the survey on Dirac operators.For simplicity, we mainly discuss the free operators without perturbations here.Let α 1 , . . ., α d and β be a set of N × N Hermitian matrices such that We then define the (free) Dirac operator by where D j = −i∂/∂x j , j = 1, . . ., d, and m ≥ 0.

Straightforward discretization of Dirac operators and the fermion doubling
We now discretize the Dirac operators on hZ d .Using D S h;j , we may define the discretized Dirac operator by which is a bounded symmetric operator.The symbol of H S 0;h , ĤS 0;h (ξ), is defined by ⊕N .The eigenvalues of ĤS 0;h (ξ) are given by We note the eigenvalues of H 0 are given by E ± (ξ) = ± |2πξ| 2 + m Hence when h → 0, the resolvent of H S 0;h converges to the direct sum of 2 d copies of the resolvent of H 0 with suitable identification.In particular, H S 0;h cannot converges to the resolvent of H 0 in the norm resolvent sense (see [3] Theorem 4.7, Theorem 5.7).In physics terminology, this implies H S 0;h describes 2 d different fermion particles, and thus this phenomenon is called the fermion doubling ( [1,10]).For this reason, H S 0;h is not considered a reasonable discretization of the Dirac operator.

The Wilson term
One standard procedure to avoid the fermion doubling is adding a term of the form to the Hamiltonian, where △ h is the standard difference Laplacian defined by and ρ > 0 is a small coupling constant.S W is called the Wilson term (see [1], [10] Section 4.3).We set H0;h = H S 0;h + S W .
The Wilson term destroys the fermion doubling for the following simple reason.The symbol of H0;h is given by Ĥ0 and its eigenvalues are given by Ẽ0 . These eigenvalues | Ẽ0;± | still have 2 d local minimal points in the case m > 0, but | Ẽ0;± | ≥ m + ρh −2 at these local minima, except for ξ = 0, and they diverges to +∞ as h → 0. In the case m = 0, these eigenvalues are at least of order O(ρh −2 ) away from any neighborhood of ξ = 0, and hence the absolute values of eigenvalues diverges to +∞ as h → 0. On the other hand, if ρ → 0, the Wilson term is negligible in a neighborhood of ξ = 0 as h → 0. Specifically, we have Theorem 4.1.Suppose ρ → 0 and ρh −2 → ∞ as h → 0. Then for z ∈ C\R, → 0, as h → 0.
Since J h is an isometry, this also implies → 0, as h → 0.
Proof.The proof is essentially the same as the argument in [8] Section 2, and [3] Sections 4.1 and 5.2.We only sketch the argument.We follow the notations of [8], and we write We also note Ĥ0 on the support of φ(hξ).Combining these, we have on the support of φ(hξ).This implies as well as Lemma 2.3 of [8].Combining this with (4.1), we arrive at the conclusion. 5 The KS-fermion model

The construction of the KS-Hamiltonian
Here we describe an interpretation of an idea by Susskind [12] (see also  and [10] Section 4.4), which is called the KS-fermion (or the staggered fermion) model.We write and we also set s 0 (n) = 0. We define operators X h;j and Y h on ℓ 2 (hZ d ) by where u ∈ ℓ 2 (hZ d ) and j = 1, . . ., d.By direct computations, we can easily show X j + mY may be considered as a discrete Dirac operator on ℓ 2 (hZ d ).In particular, we have for u ∈ ℓ 2 (hZ d ).Whereas HKS;h is a scaler operator, i.e., an operator acting on the one-component function space, it still has the fermion doubling problem.In order to solve this problem, we transform the operator HKS;h to an operator H KS;h on [ℓ 2 ((2h)Z d )] ⊕2 d .By doubling the lattice spacing, we reduce the period of the dual space by half, i.e., ((2h)Z d ) ′ = (2h) −1 T d , and remove the problematic periodic critical points.In order to double the lattice spacing, we increase the number of components to 2 d , in the following way: We define the set of indices Λ by and we write a = (a 1 , . . ., a d ) ∈ Λ, where a j ∈ {0, 1}, j = 1, . . ., d.We consider 2 d × 2 d -matrices of the form L = (L a,b ) a,b∈Λ .We denote We define a unitary operator The adjoint operator is given by where w = (w a ) a∈Λ ∈ ℓ 2 ((2h)Z d )] ⊕Λ .Now we define the KS-Hamiltonian by

KS-Hamiltonian in the Fourier space and its eigenvalues
At first we note For simplicity, we denote → 0, as h → 0.
Since J 2h is an isometry, this also implies → 0, as h → 0.
Proof.We first note and on the support of φ(hξ).Combining these, we have on the support of φ(hξ), and then it is straightforward to show the claim as in the proof of Theorem 4.1, or [8].See also [3], Section 3.1.
We note A 1 , . . ., A d and B satisfy the following properties as well as α 1 , . . ., α d and β, i.e., Thus we may consider H KS;0 as a Dirac operator, but the number of components are not necessarily the same as the standard Dirac operators.Namely, if d = 1, then 2 1 = 2 and the number of components is the same as the standard one, but if d = 2, then 2 2 = 4 > 2, and if d = 3 then 2 3 = 8 > 4, and the number of components are twice as that of the standard Dirac operators.We expect H KS;0 is decomposed to a direct sum of the standard Dirac operators, and we confirm it for d ≤ 3 in Section 6.

Examples
Here we consider KS-Hamiltonians and their continuum limit for d = 1, 2 and 3.

1 dimensional case
For d = 1, the model is transparent and easy to understand.It is also essentially the same model discussed in [2] Section 3.1 as the 1D forwardbackward difference model.

2 dimensional case
If d = 2, the one component operator is given by HKS;h u(x, y) = D S h;1 u(x, y) + (−1) x/h D S h;2 u(x, y) + (−1) (x+y)/h mu(x, y) for (x, y) ∈ hZ 2 , where u ∈ ℓ 2 (hZ 2 ).We set for (x, y) ∈ 2hZ 2 and u ∈ ℓ 2 (hZ 2 ), and then we set U h u = (u j ) 4 j=1 .Applying the formula in Section 5.1 we have , and in the continuum limit, we obtain This does not look like the standard 2D Dirac operator, but if we set Thus we arrive at a direct sum of two standard 2D Dirac operators, one of which is simply the complex conjugate.

and α 2 j = β 2 = 1 N
, where N ∈ 2N and 1 N denotes the N × N identity matrix.Typical choices for d = 1, 2, 3 are as follows: We denote a set of Pauli matrices by