More about the Grassmann tensor renormalization group

We derive a general formula of the tensor network representation for $d$-dimensional lattice fermions with ultra-local interactions, including Wilson fermions, staggered fermions, and domain-wall fermions. The Grassmann tensor is concretely defined with auxiliary Grassmann variables that play a role in bond degrees of freedom. Compared to previous works, our formula does not refer to the details of lattice fermions and is derived by using the singular value decomposition for the given Dirac matrix without any ad-hoc treatment for each fermion. We numerically test our formula for free Wilson and staggered fermions and find that it properly works for them. We also find that Wilson fermions show better performance than staggered fermions in the tensor renormalization group approach, unlike the Monte Carlo method.


Introduction
Tensor renormalization group (TRG) is a promising computational approach to study the lattice field theory. The dynamics of the theory can be investigated by the TRG mostly in the thermodynamic limit without suffering from the sign problem since it does not employ any stochastic process. The TRG was originally proposed by Levin and Nave as a real space renormalization group for the two-dimensional Ising model [1]. Extensions to fermionic systems were firstly discussed by Gu et al. [2,3], and the Grassmann TRG has been applied to many models such as the Schwinger model with and without θ term [4][5][6], the Gross-Neveu model with finite density [7], and N = 1 Wess-Zumino model [8]. 1 These earlier works have verified that the TRG is also useful to evaluate the path integral over the Grassmann variables.
In this paper, we introduce a different way from these studies to derive a tensor network representation and to implement the Grassmann TRG. The initial tensor network and renormalized ones are treated in a unified manner. We derive a general tensor network formula, characterized by the singular value decomposition (SVD) of a given Dirac matrix, for any local lattice fermion such as Wilson fermions, staggered fermions, or domain-wall fermions. Our method has several advantages over previous methods. As seen later, it is useful in comparing tensor networks for various lattice fermions from common aspects (for instance, computational times and errors). In previous works, the subscript structure of initial tensors is different from those of renormalized tensors. 2 Such an inhomogeneity of tensors requires separate handling of tensors for the initial and the renormalized steps. That kind of extra handling is absent in our method. With our method, several proposed TRG methods applicable for the Ising model are naturally transcribed to ones with fermions. This paper is organized as follows. In Sec. 2, we define a Grassmann tensor and its contraction rule. We explain how to construct the tensor network with auxiliary Grassmann fields in Sec. 3. To test our method, numerical results for the two-dimensional free Wilson and staggered fermions are provided in Sec. 4. Sec. 5 is devoted to summary and outlook. Truncation technique is explained in Appendix A. Concrete forms of Grassmann tensor for 2D Wilson and staggered fermions are shown in Appendix B.

Formalism of the Grassmann tensors
The variables η i (i = 1, · · · , N ) are single-component Grassmann numbers which satisfy the anti-commutation relation {η i , η j } = 0. We begin with defining a Grassmann tensor and its contraction rule with single-component index η i . 3 Then those with multi-component index Ψ = (η 1 , η 2 , · · · , η N ) are defined by extending the single component case straightforwardly.
The Grassmann tensor T of rank N is defined as where η i are single-component Grassmann numbers and T i 1 i 2 ···i N is referred to as a coefficient tensor, whose rank is also N , with complex entries. Fig. 1 (a) represents a Grassmann tensor, where the external lines correspond to the indices η i . We consider a Grassmann contraction among Grassmann tensors. Let A η 1 ...η N and B ζ 1 ...ζ M be two Grassmann tensors of rank N and M , respectively. 4 The Grassmann contraction has an orientation which comes from the anti-commutation relation of Grassmann variables. We define a Grassmann contraction from η 1 to ζ 1 as Eq. (2.2) itself is a Grassmann tensor, and the coefficient tensor of Eq.(2.2) is given by a contraction of two coefficient tensors of A and B with some sign factors. One can consider a contraction from η i to ζ j as a straightforward extension of Eq.(2.2) with keeping the weight factor e −ξξ . In Fig. 1 (b), the Grassmann contraction is shown as a shared link with the arrow. Note that dξdξ e −ξξ Aξ η 2 ...η N B ξζ 2 ...ζ M should be represented as Fig. 1 (b) with the opposite arrow.
Let us now move on to the multi-component case. For simplicity of explanation, we take N = mK for Eq. (2.1). Then N Grassmann numbers η n are divided into m component 3 The Grassmann contraction is also discussed in Refs. [2,3] and the appendix B of Ref. [13]. Eqs. (2.1) and (2.2) defined below generalize the results of section 3.3 of Ref. [11]. 4 We assume that either of A and B is a commutative tensor whose coefficient tensor A(B)i 1 i 2 ···i N (M ) = 0 for (i1 + i2 + · · · i N (M ) ) mod 2 = 1.  variables Ψ a (a = 1, · · · , K) as Ψ a = (η (a−1)m+1 , η (a−1)m+2 , · · · , η am ). The Grassmann tensor Eq. (2.1) is also expressed as The rank of a Grassmann tensor should be carefully read from the dimension of indices since both sides of Eq. To define the Grassmann contraction with multi-dimensional indices, we consider the case of N = mK, M = mL in Eq.(2.2) for simplicity. The N and M rank tensors A η 1 η 2 ···η N and B ζ 1 ζ 2 ···ζ M are expressed as A Ψ 1 Ψ 2 ···Ψ K and B Φ 1 Φ 2 ···Φ L where Ψ a and Φ a are m-component indices defined as in Eq. (2.3). Then the Grassmann contraction is given for the multi-component case: where Ξ = (ξ 1 , ξ 2 , · · · , ξ m ),Ξ = (ξ m , · · · ,ξ 2 ,ξ 1 ) and dΞdΞ ≡ m n=1 dξ n dξ n e −ξnξn . (2.5) The case of m = 1 reproduces Eq. (2.2). We should note thatΞ containsξ n in a reverse order so that the coefficient tensor of Eq.(2.4) is simply given by a tensor contraction of coefficient tensors of A and B without extra sign factors. Fig. 2 (b) shows the Grassmann contraction with multi-component indices. It is easy to define a Grassmann tensor network with these notations. Let T n be Grassmann tensors. Then the tensor network is defined by a product of T 1 T 2 · · · where all indices are contracted as Eq. (2.4).

General formula of tensor network for arbitrary lattice fermions
We prove that the path integral of lattice fermion theory with nearest-neighbor interactions is expressed as a Grassmann tensor network. We assume that the theory has translational invariance on the lattice. The d-dimensional hypercubic lattice is defined by a set of integer lattice sites Λ = {(n 1 , n 2 , · · · , n d ) | n i ∈ Z for i = 1, 2, · · · , d} where the lattice spacing a is set to a = 1. Although we begin with a lattice action defined by quadratic forms of Grassmann variables, it is straightforward to include four-fermion interactions. Nextnearest-neighbor and higher interactions are also easily included because these terms are expressed as nearest-neighbor ones using auxiliary Grassmann variables.
Consider lattice fermion fields ψ a (n) andψ a (n) for n ∈ Λ where a runs from 1 to N , which is the degree of freedom of the internal space such as the spinor or the flavor space. Then the lattice fermion action is formally given by where D is the Dirac operator acting on the fermion field as We may consider that D takes a form of without loss of generality. Here, δ(n, m) is the Kronecker delta. X µ , Y µ , W are matrices with respect to the internal space. The W term is an on-site interaction and the X µ and Y µ terms are nearest-neighbor interactions. The path integral is defined as Let us firstly consider X µ term in the action, dropping the spacetime index n, µ in the following for simplicity. The SVD of X ab is given by whereχ =ψU and χ = V † ψ. See Refs. [4][5][6][7][8] and the discussion in Ref. [14] for the similar deformation. Using an identity, we can easily show that where σ Xµ and U Xµ , V Xµ are singular values and singular vectors of X µ . Here n, µ dependences are explicitly shown. Similarly, Note that T is uniformly defined for the spacetime. It is easy to show Eq. (3.9) by inserting Eq. (3.10) into it with identities Eq. (3.7) and Eq. (3.8). Fig. 3 shows Eq. (3.10) in three dimensions. Eq. (3.9) is a Grassmann tensor network since a pair of Ψ(n) andΨ(n) appears once in Eq. (3.9) under n∈Λ and they are contracted with appropriate weights. We denote Eq. (3.9) as where gTr means all possible Grassmann contractions defined in Eqs. (2.4) and (2.5). The situation is quite similar with the tensor network representation for spin models, which is denoted by tTr over tensor contractions on lattice. This tensor network formulation is immediately applicable to any model with lattice fermions. It is also straightforward to extend the formulation to the models with nextnearest-neighbor and higher interactions because Eq. (3.6) allows us to express a nextnearest-neighbor term as nearest-neighbor ones. Other on-site terms such as four-fermion interactions can also be included in Eq. (3.10) with no difficulty.

Numerical applications
The current Grassmann tensor network can be evaluated by coarse-graining algorithms with a truncation of degrees of freedom, such as the original Levin-Nave TRG [1] and some variations of the TRG [15][16][17][18]. In this section, we consider the higher-order TRG (HOTRG) [15], which is applicable to any dimensional lattices for a Grassmann tensor network. Let us consider a two-dimensional case as an example. The Grassmann tensor network is made of a 4K-rank Grassmann tensor, which is identified as the Grassmann tensor T XYȲX with four K-component indices X,X, Y,Ȳ . Here, K is the number of hopping terms. Hereafter we count the rank of a Grassmann tensor in terms of K-component index. We assume that X and Y live on the links (n, n +μ) for µ = 1, 2, respectively and X andȲ live on the links (n, n −μ) for µ = 1, 2, respectively. Fig. 4 schematically illustrates the algorithm of the Grassmann HOTRG, which employs the higher-order singular value decomposition (HOSVD) for the coefficient tensor of M is identified as a Grassmann tensor of rank 6, and the coefficient tensor M which is read from Eq. (4.1) is given by a contraction of coefficient tensor T with some sign factors. We can decompose M in a formal way, This decomposition is referred to as the Grassmann HOSVD, which is equivalent to the HOSVD for the coefficient tensor M . The Grassmann HOSVD gives us a Grassmann isometry, which is inserted into the Grassmann tensor network to truncate the bond degrees of freedom ( Fig. 4(B)). U is chosen from U A and U D in Eq. (4.2), following the algorithm of the HOTRG [15]. Note that the dimension of Φ (Φ) is originally the square of that of X m because UX 2X1 Φ is a square matrix with the column X 1 , X 2 and the row Φ. For a given bond dimension D, we truncate U so that the effective dimension of coefficient tensor associated with Φ (Φ) runs up to D, setting extra elements of U to zero. 5 However, a decimal numeral system defined in Ref. [19] (or see Appendix A) allows us just to pick up D 2 × D elements in the coefficient tensor in U. This is useful to implement the current Grassmann HOTRG in practice. Renormalized Grassmann tensor is finally defined by Repeating the above procedure, Z can be approximated by Here, we assume that the lattice theory is defined on a finite lattice of V = 2 n with the periodic boundary condition 6 and T (n) is the renormalized tensor at nth renormalization step.
It is worth emphasizing that in the above procedure, the original Grassmann tensor T XYȲX is converted into the coarse-grained one T (1) XYȲX . Since both of them are defined via Eq. (2.1), the current formulation recursively introduces the Grassmann tensor under the TRG procedure. Thanks to this property, we can avoid any ad-hoc treatment in defining the initial tensor network and coarse-grained one as in Ref. [9].
We examine the above Grassmann HOTRG by benchmarking with the one-flavor colorless free Wilson fermion and free staggered fermions on a two-dimensional square lattice. Unless otherwise noted, we assume the anti-periodic boundary condition in 2-direction. Initial tensors for these fermions are obtained via Eq. (3.10). See Appendices B.1 and B.2 for their concrete forms. It is confirmed that the current Grassmann HOTRG has achieved the same accuracy as the conventional one [9] both for massless and massive fermions. Fig. 7 shows the computational time as a function of the bond dimension for the Wilson or staggered fermions with various conditions. The solid curve represents the theoretical scaling of the computational time of the HOTRG, which is O(D 7 ) in 2 dimensions, and the current Grassmann HOTRG computation well reproduces it.
Finally, we evaluate the relative error defined via Eq. (4.6) as a function of the computational time for both Wilson and staggered fermions, varying values of mass, and boundary conditions. As shown in Fig. 8, with the vanishing mass, the Grassmann HOTRG achieves the higher accuracy for the Wilson fermions compared to staggered fermions with the fixed computational time. This may be attributed to the chiral symmetry in staggered fermions, which makes the hierarchy of the singular values in the Grassmann tensor milder. When  these fermions are massive, the Grassmann HOTRG reaches slightly higher accuracy for Wilson fermions within the fixed computational time. We should note that the situation is quite different from the Monte Carlo simulation, where the computational time of staggered fermions is faster than that of the other lattice fermions. Fig. 8 suggests that Wilson fermions show better performance than staggered fermions in the TRG method with the fixed execution time, unlike the Monte Carlo method.

Summary and outlook
A tensor network formulation for fermion theories was discussed, based on the introduction of the auxiliary fermion fields. We derived a general formula of the tensor network representation for lattice fermions. This formula is immediately applicable for many types of local lattice fermions such as Wilson fermions, staggered fermions and domain-wall fermions. Our method is useful in practice because it allows us to recursively introduce the coarsegrained Grassmann tensor throughout the TRG calculation and provides a fair comparison among various lattice fermions from a common aspect. Implementing the Grassmann HOTRG, whose accuracy are the same as in Ref. [9], our numerical results suggest that Wilson fermions show better performance than staggered fermions in the TRG method with the fixed execution time. The situation is quite different from the Monte Carlo method, and these results would be interesting as a starting point of further studies, not only for free field theories but also interacting ones.
It is worth noting that the current formulation depends on the introduction of auxiliary Grassmann fields both in spatial and temporal directions for the path integral Z. On the other hand, the path integral is derived from Z = Tr(e −βĤ ) inserting a complete set of coherent fermion states. This implies that a tensor network representation for lattice fermions could be derived directly from Z = Tr(e −βĤ ). Fig. 9 shows a possible relationship between the path integral, operator formalism, and tensor network. This viewpoint may be useful in extending our method to interacting theories with gauge fields.  Figure 9. A possible relationship between the path integral, operator formalism, and tensor network.

A Truncation technique
We introduce the SVD of a Grassmann tensor, which is equivalent to the SVD for the corresponding coefficient tensor. Let T ΨΦ be a Grassmann tensor whose rank is 2N . We represent the coefficient tensor of T ΨΦ as 2 N × 2 N matrix T IJ with I = (i 1 , · · · , i N ) and J = (i N +1 , · · · , i 2N ). Since the Grassmann parity of T ψφ is even, T IJ takes a non-zero value if and only if is satisfied. This condition allows us to obtain a block diagonal matrix representation for T IJ . According to Ref. [19], we now define the following decimal numeral system, Thanks to this decimal numeral system, the parity of 2N k=1 i k corresponds with that of I. Applying this system for I and J in T IJ , one obtains the block diagonal matrix The SVD for T is obtained from that for T E and T O , Picking up the largest D numbers of singular values and corresponding singular vectors, T is approximated with a lower-rank matrix. We apply the above technique for M M † , where M is the coefficient tensor in Eq. (4.1). In Eq. where and V † µ = −U µ for µ = x, t. Following Eq. (3.10), the Grassmann tensor T ΨxΨtΨtΨx with the ordering Ψ µ = (η µ , ζ µ ) andΨ µ = (ζ µ ,η µ ) for µ = x, t is obtained immediately. One finds where D ab ≡ D ab (n, n), A = U x , B = U t .

B.2 Staggered fermions
The action of the two-dimensional staggered fermions is given by S = n=(nx,nt)∈Λχ where χ(n) andχ(n) are single-component Grassmann fields and p µ (n) is the staggered sign function defined by p x (n) = 1 and p t (n) = (−1) nx . Since there is no Dirac structure in the action of the staggered fermion, we can immediately derive the tensor network representation for the path integral without applying SVD. Employing Eqs. (3.7) and (3.8), we can obtain with Ψ µ = (η µ , ζ µ ) andΨ µ = (ζ µ ,η µ ) for µ = x, t. Note that the resulting Grassmann tensor depends on the parity of n x .