More about the Grassmann tensor renormalization group
PPrepared for submission to JHEP
UTHEP-751
More about the Grassmann tensor renormalizationgroup
Shinichiro Akiyama, a Daisuke Kadoh b,c a Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8571,Japan b Physics Division, National Center for Theoretical Sciences, National Tsing-Hua University,Hsinchu, 30013, Taiwan c Research and Educational Center for Natural Sciences, Keio University,Yokohama 223-8521, Japan
E-mail: [email protected] , [email protected] Abstract:
We discuss a tensor network formulation for relativistic lattice fermions. The Grass-mann tensor is concretely defined with the auxiliary Grassmann fields which play a role ofbond degrees of freedom. This formulation is immediately applicable to any lattice theorywith only nearest neighbor interactions. We introduce a general formula to derive thetensor network representation for lattice fermions, which is expressed by the singular valuedecomposition for a given Dirac matrix. We also test this formulation numerically for thefree Wilson fermion. a r X i v : . [ h e p - l a t ] M a y ontents Tensor renormalization group (TRG) is a promising computational approach to study thelattice field theory. The dynamics of the theory can be investigated by the TRG mostlyin the thermodynamic limit without suffering from the negative sign problem since it doesnot employ any stochastic process. The TRG was originally proposed by Levin and Naveas a real space renormalization group for the two-dimensional Ising model [1]. Extensionsto fermionic systems were firstly discussed by Gu et al. [2, 3], and the Grassmann TRGhas been applied to many models such as the Schwinger model with and without θ term[4–6], the Gross-Neveu model with finite density [7], and N = 1 Wess-Zumino model [8]. These earlier works have verified that the TRG is also useful to evaluate the path integralover the Grassmann variables.In this paper, we introduce a different way from these studies to derive a tensor networkrepresentation and to implement the Grassmann TRG. The tensor network is derivedwith auxiliary Grassmann fields. As an advantage over the conventional tensor networkformulation, this method allows us to find a general formula, characterized by the singularvalue decomposition for the Dirac matrix, to obtain the tensor network representation forany lattice fermion theory with only nearest neighbor interactions.This paper is organized as follows. In Sec. 2, we define a Grassmann tensor and itscontraction rule. We explain how to construct the tensor network with auxiliary Grassmannfields in Sec. 3. To see the validity of the current formulation, numerical results for thetwo-dimensional free Wilson fermion are provided in Sec. 4. Sec. 5 is devoted to summaryand outlook. See Refs. [9–12] for other related studies. – 1 –
Formalism of the Grassmann tensor
The variables η i ( i = 1 , · · · , N ) are single component Grassmann numbers which satisfy theanti-commutation relation { η i , η j } = 0. We begin with defining a Grassmann tensor and itscontraction rule with single component index η i . Then those with multi component indexΨ = ( η , η , · · · , η N ) are defined by extending the single component case straightforwardly.The Grassmann tensor T of rank N is defined as T η η ··· η N = (cid:88) i =0 1 (cid:88) i =0 · · · (cid:88) i N =0 T i i ··· i N η i η i · · · η i N N , (2.1)where η i are single component Grassmann numbers and T i i ··· i N is referred to as a coeffi-cient tensor, whose rank is also N , with complex entries. Fig. 1 (a) represents a Grassmanntensor, where the external lines correspond to the indices η i .We consider a Grassmann contraction among Grassmann tensors. Let A η ...η N and B ζ ...ζ M be two Grassmann tensors of rank N and M , respectively. The Grassmann con-traction has an orientation which comes from the anti-commutation relation of Grassmannvariables. We define a Grassmann contraction from η to ζ as (cid:90) d ¯ ξ d ξ e − ¯ ξξ A ξη ...η N B ¯ ξζ ...ζ M . (2.2)Eq. (2.2) itself is a Grassmann tensor, and the coefficient tensor of Eq.(2.2) is given by acontraction of two coefficient tensors of A and B with some sign factors. One can considera contraction from η i to ζ j as a straightforward extension of Eq.(2.2) with keeping theweight factor e − ¯ ξξ . In Fig. 1 (b), the Grassmann contraction is shown as a shared link withthe arrow. Note that (cid:82) d ¯ ξ d ξ e − ¯ ξξ A ¯ ξη ...η N B ξζ ...ζ M should be represented as Fig. 1 (b) withthe 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 componentvariables Ψ a ( a = 1 , · · · , K ) as Ψ a = ( η ( a − m +1 , η ( a − m +2 , · · · , η am ). The Grassmanntensor Eq. (2.1) is also expressed as T Ψ Ψ ··· Ψ K ≡ T η η ··· η N (2.3)The rank of a Grassmann tensor should be carefully read from the dimension of indicessince both sides of Eq. (2.3) have the same rank. Fig. 2 (a) represents an example ofGrassmann tensor with multi component indices, where the external links shown as solidlines correspond to the multi component indices Ψ a . Other cases are straightforwardlygeneralized from Eq.(2.3) which is the case of N = 4 m and m component indices.To define the Grassmann contraction with multi dimensional indices, we consider thecase of N = mK, M = mL in Eq.(2.2) for simplicity. The N and M rank tensors A η η ··· η N and B ζ ζ ··· ζ M are expressed as A Ψ Ψ ··· Ψ K and B Φ Φ ··· Φ L where Ψ a and Φ a are m component We assume that either of A and B is a commutative tensor whose coefficient tensor T i i ··· i N = 0 for( i + i + · · · i N ) mod 2 = 1. In this case, A η ...η N B ζ ...ζ M = B ζ ...ζ M A η ...η N and Eq. (2.2) can also beexpressed as (cid:82) d¯ ξ d ξ e − ¯ ξξ B ¯ ξζ ...ζ M A ξη ...η N . – 2 – (a) (b) 𝒜 ℬ 𝜉 ҧ𝜉 𝜂 𝜂 𝜂 𝑁 𝜁 𝜁 𝜁 𝑀 𝜂 𝜂 𝜂 𝜂 𝑁 𝒯 (𝑎) (𝑏) Figure 1 . Graphical representations of (a) Grassmann tensor Eq.(2.1) and (b) Grassmann con-traction Eq. (2.2). The external lines specify uncontracted indices. The arrow in the internal linerepresents a contracted direction from ξ to ¯ ξ . (a) (b) 𝒜 ℬ Ξ തΞ Ψ Ψ Ψ 𝐾 Φ Φ Φ 𝐿 Ψ Ψ Ψ Ψ 𝐾 𝒯 (𝑎) (𝑏) Figure 2 . Graphical representations of (a) Grassmann tensor Eq.(2.3) and (b) Grassmann con-traction Eq. (2.4). The external lines specify uncontracted multi component indices. The arrow inthe internal line represents a contracted direction from Ξ to ¯Ξ. indices defined as in Eq. (2.3). Then the Grassmann contraction is given for the multicomponent case: (cid:90) d¯ΞdΞ e − ¯ΞΞ A ΞΨ ... Ψ K B ¯ΞΦ ... Φ L (2.4)where Ξ = ( ξ , ξ , · · · , ξ m ) , ¯Ξ = ( ¯ ξ m , · · · , ¯ ξ , ¯ ξ ) andd¯ΞdΞ e − ¯ΞΞ ≡ m (cid:89) n =1 d ¯ ξ n d ξ n e − ¯ ξ n ξ n . (2.5)The case of m = 1 reproduces Eq. (2.2). We should note that ¯Ξ contains ¯ ξ n in a reverseorder so that the coefficient tensor of Eq.(2.4) is simply given by a tensor contraction of– 3 –oefficient tensors of A and B without extra sign factors. Fig. 2 (b) shows the Grassmanncontraction with multi component indices.It is easy to define a Grassmann tensor network with these notations. Let T n beGrassmann tensors. Then the tensor network is defined by a product of T T · · · where allindices are contracted as Eq. (2.4). We prove that the partition function of lattice fermion theory with nearest neighbor inter-actions is expressed as a Grassmann tensor network. We assume that the theory has thetranslational invariance on the lattice. The d -dimensional hypercubic lattice is defined bya set of integer lattice sites Λ = { ( x , x , · · · , x d ) | x i ∈ Z for i = 1 , , · · · , d } where thelattice spacing a is set to a = 1. Although the proof is given for the infinite volume lattice,one can easily extend it to the case of a finite volume lattice.Consider lattice fermion fields ψ an and ¯ ψ an for n ∈ Λ where a runs from 1 to N , whichis the degree of freedom of the internal space such as the spinor or the flavor space. Thenthe lattice fermion action is formally given by S = (cid:88) n ∈ Λ ¯ ψ n ( Dψ ) n (3.1)where D is the Dirac operator acting on the femrion field as ( Dψ ) an = (cid:80) m ∈ Λ (cid:80) Nb =1 D abnm ψ bm .We may consider that D takes a form of D abnm = W ab δ nm + d (cid:88) µ =1 ( X µ ) ab δ n +ˆ µ,m + d (cid:88) µ =1 ( Y µ ) ab δ n − ˆ µ,m (3.2)without loss of generality. 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 partition function is defined as Z = (cid:90) (cid:2) d ψ d ¯ ψ (cid:3) e − S (3.3)where [d ψ d ¯ ψ ] = (cid:81) n ∈ Λ (cid:81) Na =1 d ψ an d ¯ ψ an with single component Grassmann measures d ψ an andd ¯ ψ an .Let us firstly condier the term ¯ ψ n X µ ψ n +ˆ µ in the action, dropping the spacetime index n, µ in the following for simplicity. The SVD of X ab is given by X ab = (cid:80) Nc =1 U ac σ c ( V † ) cb where σ c ≥ U, V are unitary matrices. Then we have¯ ψXψ = N (cid:88) c =1 σ c ¯ χ c χ c (3.4)where ¯ χ = ¯ ψU and χ = V † ψ . See Refs. [4–8] and the discussion in Ref. [13] for the similardeformation. Using an identity,e − σ c ¯ χ c χ c = (cid:90) d¯ η c d η c exp [ − ¯ η c η c + ¯ χ c ¯ η c + σ c η c χ c ] , (3.5)– 4 –e can easily show thate − ¯ ψ n X µ ψ n +ˆ µ = K µ (cid:89) c =1 (cid:90) d¯ η cn,µ d η cn,µ exp (cid:104) − ¯ η cn,µ η cn,µ − ( ¯ ψ n U X µ ) c η cn,µ + ( σ X µ ) c ¯ η cn,µ ( V † X µ ψ n +ˆ µ ) c (cid:105) , (3.6)where σ X µ and U X µ , V X µ are singular values and singular vectors of X µ . Here n, µ depen-dences are explicitly shown. Similary,e − ¯ ψ n + µ Y µ ψ n = L µ (cid:89) c =1 (cid:90) d¯ ζ cn,µ d ζ cn,µ exp (cid:104) − ¯ ζ cn,µ ζ cn,µ + ( ¯ ψ n + µ U Y µ ) c ¯ ζ cn,µ + ( σ Y µ ) c ζ cn,µ ( V † Y µ ψ n ) c (cid:105) , (3.7)where σ Y µ and U Y µ , V Y µ are singular values and singular vectors of Y µ .Thus, we have Z = (cid:90) (cid:2) d ¯ΨdΨ (cid:3) e − (cid:80) n ∈ Λ (cid:80) dµ =1 ¯Ψ µ ( n )Ψ µ ( n ) (cid:89) n ∈ Λ T Ψ ( n ) ··· Ψ d ( n ) ¯Ψ d ( n − ˆ d ) ··· ¯Ψ ( n − ˆ1) (3.8)where T Ψ ··· Ψ d ¯Ψ d ··· ¯Ψ = (cid:90) (cid:32) N (cid:89) a =1 d ψ a d ¯ ψ a (cid:33) exp (cid:2) − ¯ ψW ψ (cid:3) × exp d (cid:88) µ =1 K µ (cid:88) c =1 (cid:110) − ( ¯ ψU X µ ) c η cµ + ( σ X µ ) c ¯ η cµ ( V † X µ ψ ) c (cid:111) × exp d (cid:88) µ =1 L µ (cid:88) c =1 (cid:110) ( ¯ ψ n U Y µ ) c ¯ ζ cµ + ( σ Y µ ) c ζ cµ ( V † Y µ ψ n ) c (cid:111) , (3.9)where Ψ µ = ( η µ , · · · , η K µ µ , ζ µ , · · · , ζ L µ µ ) and ¯Ψ µ = (¯ ζ L µ µ , · · · , ¯ ζ µ , ¯ η K µ µ , · · · , ¯ η µ ). Note that T is uniformly defined for the spacetime. It is easy to show Eq. (3.8) by inserting Eq. (3.9)into it with identities Eq. (3.6) and Eq. (3.7). Fig. 3 shows Eq. (3.9) in three dimensions.Eq. (3.8) is a Grassmann tensor network since a pair of Ψ( n ) and ¯Ψ( n ) appears once inEq. (3.8) under (cid:81) n ∈ Λ and they are contracted with the weight e − ¯Ψ( n )Ψ( n ) . We denoteEq. (3.8) as Z = gTr (cid:34) (cid:89) n ∈ Λ T Ψ ( n ) ··· Ψ d ( n ) ¯Ψ d ( n − ˆ d ) ··· ¯Ψ ( n − ˆ1) (cid:35) (3.10)where gTr is defined as contractions of all possible pairs with the weight e − ¯ΨΨ . Thesituation is quite similar with the tensor network representation for spin models, whichis denoted by tTr over tensor contractions on lattice. This tensor network formulation isimmediately applicable to any lattice model of relativistic fermions. We have assumed the periodic boundary condition. – 5 – 𝒯 ഥΨ Ψ ഥΨ Ψ ഥΨ Figure 3 . Graphical representations of Eq. (3.9) in three dimensions.
The current Grassmann tensor network can be evaluated by coarse-graining algorithmswith a truncation of degrees of freedom, such as the original Levin-Nave TRG [1] andsome variations of the TRG [14–17]. In this section, we consider the higher-order TRG(HOTRG) [14] applicable to any dimensional lattices for a Grassmann tensor network. (A) (B)(C)
Grassmann
HOSVD GrassmannContractionIteration
Figure 4 . Schematic picture of the Grassmann HOTRG. (A) Grassmann tensor network in twodimensions. (B) Grassmann isometries are inserted in the whole network. (C) Tensor network isrenormalized so that the lattice size is reduced by a factor of 2.
Let us consider a two dimensional case as an example. The Grassmann tensor network– 6 –s made of a 4 K -rank Grassmann tensor, which is identified as the Grassmann tensor T XY ¯ Y ¯ X of rank four with respect to K component indices X, ¯ X, Y, ¯ Y . Hereafter we countthe rank of a Grassmann tensor in terms of K component index. We assume that X and Y live on the links ( n, n + ˆ µ ) for µ = 1 ,
2, respectively and ¯ X and ¯ Y live on the links ( n, n − ˆ µ )for µ = 1 ,
2, respectively. In the initial Grassmann tensor network, K = N which is thenumber of components in the original fermion field.Fig. 4 schematically illustrates the algorithm of the current Grassmann HOTRG , whichemploys the higher-order singular value decomposition (HOSVD) for the coefficient tensorof M X X Y ¯ Y ¯ X ¯ X = (cid:90) d¯ΞdΞ e − ¯ΞΞ T X Ξ ¯ Y ¯ X T X Y ¯Ξ ¯ X . (4.1) M is identified as a Grassmann tensor of rank 6, and the coefficient tensor M which isread 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, M X X Y ¯ Y ¯ X ¯ X = (cid:32) (cid:89) k =1 (cid:90) d¯Ξ k dΞ k e − ¯Ξ k Ξ k (cid:33) U AX X Ξ U BY Ξ U C ¯ Y Ξ U D ¯ X ¯ X Ξ S ¯Ξ ¯Ξ ¯Ξ ¯Ξ . (4.2)This decomposition is referred to as the Grassmann HOSVD , which is equivalent to theHOSVD for the coefficient tensor M . The Grassmann HOSVD gives us a Grassmannisometry , (cid:90) d ¯ΦdΦ e − ¯ΦΦ U ¯ X ¯ X Φ U ¯Φ X X , (4.3)which is inserted into the Grassmann tensor network to truncate the bond degrees offreedom (Fig. 4(B)). U is chosen from U A and U D in Eq. (4.2), following the algorithmof the HOTRG [14]. Formally, Φ and ¯Φ in Eq. (4.3) are ( k + 1)-component Grassmannnumbers, where k is an integer such that 2 k < D < k +1 with D the bond dimension.However, a decimal numeral system defined in Ref. [18] (or see Appendix A) allows us justto pick up D × D elements in the coefficient tensor in U . This is practically useful toimplement the current Grassmann HOTRG.Then, the coarse-graining renormalization is accomplished by the Grassmann contrac-tion, T (cid:48) XY ¯ Y ¯ X = (cid:32) (cid:89) i =1 (cid:90) d ¯Φ i dΦ i e − ¯Φ i Φ i (cid:90) d ¯Φ (cid:48) i dΦ (cid:48) i e − ¯Φ (cid:48) i Φ (cid:48) i (cid:33) U ¯Φ ¯Φ X M Φ Φ Y ¯ Y ¯Φ (cid:48) ¯Φ (cid:48) U ¯ X Φ (cid:48) Φ (cid:48) . (4.4)Iterating the above procedure, we can evaluate Z by Z = (cid:90) d ¯ X d X e − ¯ XX (cid:90) d ¯ Y d Y e − ¯ Y Y T XY ¯ Y ¯ X (4.5)with the periodic boundary condition. If one imposes the anti-periodic boundary condition in 2-direction, the multi Grassmann number in T on the link ( n, n + ˆ µ ), say Y , should be replaced by − Y before carry out the integration in Eq. (4.5) – 7 –e examine the above Grassmann HOTRG by benchmarking with the one-flavor col-orless free Wilson fermion on a square lattice. We assume the anti-periodic boundarycondition in 2-direction. This free lattice fermion theory has been employed to test theefficiency of conventional Grassmann TRGs [7, 9]. The initial tensor is given by Eq.(3.9)derived as a formula in the previous section. The explicit form is shown in Appendix B.Fig. 5 shows the free energy per site against the mass M on 2 × D = 16.With the choice of D ≥
16, the calculation by the current Grassmann HOTRG agrees withthe exact results up to the machine precision. This situation is completely same with theconventional Grassmann HOTRG [9].Fig. 6 plots the relative error of the free energy on 1024 × δ = (cid:12)(cid:12)(cid:12)(cid:12) ln Z ( L = 1024 , D ) − ln Z exact ( L = 1024)ln Z exact ( L = 1024) (cid:12)(cid:12)(cid:12)(cid:12) . (4.6)It is confirmed that the current Grassmann HOTRG has achieved the same accuracy withthe conventional one both for massless and massive fermions. Hence, the current tensornetwork formulation enables us to study the lattice fermion theory with the TRG approach. M l n Z ( L = ) ExactThis work
Figure 5 . Free energy density for the free Wilson fermions against the mass M on 2 × D = 16. It agrees with the exact values up to the machine precision. A tensor network formulation for lattice fermions has been constructed, based on the intro-duction of the auxiliary fermion fields. This formulation is immediately applicable to manytypes of the lattice fermions with nearest neighbor interaction. We have also introduced a– 8 –
10 20 30 40 50 D -8 -6 -4 -2 d This work ( M =0)This work ( M =1)Ref. [9] ( M =0)Ref. [9] ( M =1) Figure 6 . Relative error of the free energy on 1024 × D . general formula to derive the tensor network representation for lattice fermions. We havealso implemented the Grassmann HOTRG, whose accuracy is exactly same with Ref. [9].It is worth noting that the current formulation depends on the introduction of auxiliaryGrassmann fields both in spatial and temporal directions for the path integral Z . On theother hand, the path integral is derived from Z = Tr( e − β ˆ H ) inserting a complete setof coherent fermion states. This implies that a tensor network representation for latticefermions could be derived directly from Z = Tr( e − β ˆ H ). Fig. 7 shows a possible relationshipbetween the path integral, operator formalism and tensor network. This viewpoint wouldbe useful in establishing tensor network formulations not only for lattice fermions, but alsofor scalar or gauge fields on lattice. A Truncation technique
We introduce the singular value decomposition of a Grassmann tensor, which is equivalentto the decomposition for the corresponding coefficient tensor. Let T ΨΦ be a Grassmanntensor whose rank is 2 N . We represent the coefficient tensor of T ΨΦ as 2 N × N matrix T IJ with I = ( i , · · · , i N ) and J = ( i N +1 , · · · , i N ). Since the Grassmann parity of T ψφ iseven, T IJ takes a non-zero value if and only if N (cid:88) k =1 i k mod 2 = 0 (A.1)– 9 – = න 𝐝𝝍𝐝 ഥ𝝍 𝐞 −𝑺 𝝍,ഥ𝝍 Path integral
𝒁 = 𝐓𝐫 𝐞 −𝜷𝑯
Partition function
𝒁 = 𝐠𝐓𝐫 ෑ 𝒏∈𝚲 𝓣 𝚿 𝒏 ⋯𝚿 𝒅 𝒏 ഥ𝚿 𝒅 (𝒏−𝒅)ഥ𝚿 (𝒏−𝟏) Tensor network
Auxiliary Grassmann fields in spatial and temporal directionsInserting a complete set in temporal direction ?
Figure 7 . A possible relationship between the path integral, partition function and tensor network. is satisfied. This condition allows us to obtain a block diagonal matrix representation for T IJ . According to Ref. [18], we now define the following decimal numeral system, I = (cid:40)(cid:80) Nk =1 k − i k ( i + · · · + i N mod 2 = 0)1 − i + (cid:80) Nk =2 k − i k ( i + · · · + i N mod 2 = 1) . (A.2)Thanks to this decimal numeral system, the parity of (cid:80) Nk =1 i k corresponds with that of I .Applying this system for I and J in T IJ , one obtains the block diagonal matrix T = (cid:34) T E T O (cid:35) . (A.3)The singular value decomposition for T is obtained from that for T E and T O , T E IJ = (cid:88) K :even U E IK σ E K V E JK , (A.4) T O IJ = (cid:88) K :odd U O IK σ O K V O JK . (A.5)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. (4.3), the Grassmann isometry defines new bond Grassmann numbers Φ and ¯Φ. It– 10 –s worth emphasizing that the parity of these Grassmann numbers correspond to the parityof K in Eqs. (A.4) and (A.5). This property is significantly useful in developing the currentGrassmann HOTRG. B Tensor network formulation for two-dimensional Wilson fermions
The Dirac matrix with the Wilson parameter r = 1 is given by D n,m = ( M + 2) (cid:34) δ n,m δ n,m (cid:35) − (cid:34) δ n,m +ˆ1 + δ n,m − ˆ1 δ n,m +ˆ1 − δ n,m − ˆ1 δ n,m +ˆ1 − δ n,m − ˆ1 δ n,m +ˆ1 + δ n,m − ˆ1 (cid:35) − (cid:34) δ n,m +ˆ2 δ n,m − ˆ2 (cid:35) . (B.1)Applying the singular value decomposition, we have D n,m = ( M + 2) (cid:34) δ n,m δ n,m (cid:35) + U (cid:34) δ n,m +ˆ1 δ n,m − ˆ1 (cid:35) V † + U (cid:34) δ n,m +ˆ2 δ n,m − ˆ2 (cid:35) V † , (B.2)where U = 1 √ (cid:34) − − − (cid:35) , U = (cid:34) − − (cid:35) , (B.3)and V † µ = − U µ for µ = 1 ,
2. Following Eq. (3.9), the Grassmann tensor T Ψ Ψ ¯Ψ ¯Ψ withthe ordering Ψ µ = ( η µ , η µ ) and ¯Ψ µ = (¯ η µ , ¯ η µ ) for µ = 1 , T Ψ Ψ ¯Ψ ¯Ψ = det D n,n + (cid:104) D n,n U ( V † ) + D n,n U ( V † ) (cid:105) η ¯ η + (cid:104) D n,n U ( V † ) + D n,n U ( V † ) (cid:105) η ¯ η + D n,n U ( V † ) η ¯ η + D n,n U ( V † ) η ¯ η − (cid:104) D n,n U ( V † ) + D n,n U ( V † ) (cid:105) η η + (cid:104) D n,n U ( V † ) + D n,n U ( V † ) (cid:105) ¯ η ¯ η − D n,n U ( V † ) η η + D n,n U ( V † ) ¯ η ¯ η + D n,n U ( V † ) η ¯ η + D n,n U ( V † ) η ¯ η + D n,n U ( V † ) η η − D n,n U ( V † ) ¯ η ¯ η + D n,n U ( V † ) η ¯ η + D n,n U ( V † ) η ¯ η − (cid:2) det U η ¯ η + U U η ¯ η + U U ¯ η ¯ η + U U η η + U U η ¯ η + det U η ¯ η (cid:3) × (cid:104) det V † η ¯ η + ( V † ) ( V † ) η ¯ η − ( V † ) ( V † ) η η − ( V † ) ( V † ) ¯ η ¯ η + ( V † ) ( V † ) η ¯ η + det V † η ¯ η (cid:105) . (B.4)– 11 – cknowledgments We are grateful to Yoshinobu Kuramashi, Ryo Sakai and Shinji Takeda for insightful dis-cussions. S. A. also thanks Yusuke Yoshimura for many helpful comments about the im-plementation of the Grassmann HOTRG in Ref. [9]. This work is supported by the JSPSKAKENHI Grant JP19K03853.
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