Grassmann tensor renormalization group for one-flavor lattice Gross-Neveu model with finite chemical potential

We apply the Grassmann tensor renormalization group (GTRG) to the one-flavor lattice Gross-Neveu model in the presence of chemical potential. We compute the fermion number density and its susceptibility and confirm the validity of GTRG for the finite density system. We introduce a method analogous to the reweighting method for Monte Carlo method and test it for some parameters.


Introduction
The tensor renormalization group (TRG) is one of the numerical renormalization methods. It was originally introduced by Levin and Nave in the triangular lattice Ising model [1], and then has been applied to bosonic models on square lattice: X-Y model [2], O(3) model [3] and φ 4 theory [4]. Recently, Xie et al. developed the second TRG [5] to improve this method by using global optimization instead of local one. An extension to higher dimensional system, named higher order TRG, was also introduced by Xie et al. and it was examined in the 3D Ising model [6]. Furthermore, a generalization to fermion system called Grassmann tensor renormalization group (GTRG) was proposed by Gu et al. [7,8]. Then it has been applied to the two-dimensional QED [9], which is a gauge-fermion system, and a study including θ-term [10] was also given by Shimizu and Kuramashi.
An advantage of the TRG is that this method can be applied to any systems suffering from the sign problem with the Monte Carlo method. The sign problem occurs in for example, finite fermion density systems, θ-term included systems, lattice chiral gauge theories and so on. The purpose of this paper is to apply the GTRG to a simple finite fermion density system on the lattice, namely the Gross-Neveu model [11] containing four-fermion interaction in the presence of chemical potential with the Wilson fermion lattice formulation. This model is known to share important properties, asymptotically free and spontaneous symmetry breaking, with QCD and considered to be its toy model. Usually, large N -expansion is used to analyze the model. In this paper, nevertheless, we restrict to one flavor just for a simplicity although generalization to many flavors is straightforward. This work provides a benchmark for future study of complicated and higher-dimensional model, say QCD with finite density eventually.
This paper is organized as follows. In Sec. 2, after defining the lattice Gross-Neveu model with chemical potential, we review the derivation of the tensor network representation for the model and explain the GTRG procedure as well as how to implement the anti-periodic boundary condition in this representation. Numerical results are presented in Sec. 3. In Sec. 4, we propose a method analog to the reweighting method in the Monte Carlo method. Sec. 5 is devoted to summary and outlook.
In the following, we consider 2-dimensional lattice box N 1 × N 2 .

Grassmann TRG
In this subsection, we explain the GTRG for the tensor network representation (28). In the same way as the usual TRG, we decompose the bosonic part on a site n by the SVD and truncate at D cut , where U 1,3 are unitary matrix, σ 13 is singular value and n * is the coarse-grained lattice site where D 1 and D 3 are defined by and new exponent x n * −1,f with one-component is introduced with constraints, Then the tensor T xntnx n−1 t n−2 is decomposed and approximated as where x n * = (x n * ,b , x n * ,f ) and Fig. 1 The decomposition of tensor. The horizontal (vertical) axis corresponds to 1direction (2-direction).
For another decomposition rotated 90 degree (See Figure 1), by introducing new variables ξ n * and ξ n * −2 * , it is similarly given by where S 2 and S 4 are defined by The bosonic part S 2 and S 4 are determined by the SVD as follows A coarse-grained tensor is obtained by Note that constraint δ 0,x n * ,f +t n * ,f +x n * −1 * ,f +t n * −2 * ,f mod 2 is imposed for the coarse-grained tensor. Figure 2 shows the contraction for the original indices in this renormalization step. Repeat this renormalization step until the number of lattice point reaches 2 × 2, namely four reduced tensors. From these tensors, the parition function is computed by full index contractions.
Computational costs of a standard SVD routine are proportional to the third power of the matrix size, thus the cost of the decomposition of a tensor is of order D 6 cut . On the other hand, the cost of the contraction is of order D 6 cut . Therefore the total cost of GTRG is proportional to D 6 cut .
Therefore the once renormalized partition function is obtained by where the site indices are replaced by n * → n for a readability and another boundary matrices are given by Similarly, a twice coarse-grained tensor contracted on n 1 = n 2 is obtained by and the twice renormalized partition function is obtained by Therefore, in this formulation, the boundary condition returns to the original one every 2 renormalization steps.

Numerical Results
First, we compare the numerical results of ln Z with the exact value ln Z exact in the free massless case. Figure 3 shows the relative deviation as a function of D cut . The convergence behavior is roughly observed although it is not so smooth. The convergence rate at µ = 1 is slower than that of µ = 2. For µ = 2, lattice volume dependence is not seen while for µ = 1 larger volume is strongly affected by truncation error.
To see the convergence issue in more detail, we investigate the spectrum of bosonic tensor in Figure 4. Clear hierarchy is observed for µ = 2 while nearly degenerated structure is seen for µ = 1 especially after several iterations. Figure 5 shows the relative deviation as a function of µ with fixed D cut = 64. The deviation rapidly increases around µ ≈ 0.3 and 1 where transition-like behavior is actually observed as shown later.  Fig. 3 The relative deviation δ as a function of D cut for free massless case.  Next, we compute the fermion number density defined as  Fig. 5 The relative deviation δ as a function of µ with fixed D cut = 64 for free massless case. Around µ = 0.3 and 1, the deviation becomes large. For N 1 = N 2 = 4, the TRG result becomes exact, thus the relative deviation is exactly zero up to machine precision thus this shows a validity of our calculation. Figure 6 plots the fermion number density as a function of µ for some non-trivial sets of parameters. Since the model is in two dimensional system with one-flavor, the saturation density for fermion number is one. We observe that the fermion density saturates to this value for larger chemical potential. Finally, we perform the finite size scaling analysis for the quark number susceptibility defined as The susceptibility as a function of µ is shown in Figure 7 for various spatial volumes with two sets of parameter (m, g) = (0, 0) and (0, 0.7). For both cases, we observe that there is a peak around µ = 1 and the peak height shows no volume dependence, therefore we conclude that this transition is cross-over. For lower µ 0.6, the TRG results develop some peaks for both couplings. In order to check whether these peaks are fake or not, we compare with the exact results at g = 0 shown as curves for each volume N 1 = 32, 64, 96 where for larger volume the peaks disappear in the lower µ region. From the comparison, we find that the TRG results at g = 0, shown as dots, tend to deviate from these curves for larger volume. Thus we conclude that these peaks at g = 0 of TRG results especially with larger volume are fake. For g = 0.7, since we cannot directly compare with the exact results, we are content with being comparing two results obtained by different resolution of the chemical potential in the numerical derivative. And then the difference is barely seen thus we expect that the peak around µ = 0.4 for g = 0.7 is not a fake but of course further study is required to make solid our expectation. For free massless case, around the peak positions (µ ≈ 1), the  Fig. 6 Fermion number density n as a function of µ with fixed N 1 = N 2 = 32 and D cut = 64. For larger chemical potential, the number density for all cases of (m, g) we investigated saturates to unity as expected. relative deviation in Figure 5 becomes large. It has been known that the approximation for TRG gets worse near a critical point, while we observe that such a behavior occurs even for cross-over case. Needless to say, on another parameters (N 1 , N 2 , m, g), real phase transition can occur and the strength of transition may change, thus the source of the loss of accuracy we observed here could be a remnant of the real phase transition.

Formulation
In the TRG calculation, one usually computes the partition function at several parameter points (mesh). Then numerical derivative of partition function with respect to the parameter is made by using a few points and one needs a fine mesh to reduce a discretization error. To reduce the computational time, we propose a method to obtain an approximated coarsegrained tensor at one parameter by using another set of singular values at different parameter. Using an analogy from Monte Carlo method, we refer to this method as the reweighting method.

Fig. 8
The relative deviation between reweighting method and exact value in eq.(69) as a function of µ for free massless case with fixed D cut = 64. From top to bottom, the original value of µ is given by µ = 0, 1 and 2 respectively.

Summury and Outlook
We have applied the GTRG to the one-flavor lattice Gross-Neveu model with chemical potential in the Wilson fermion formulation. At some non-trivial parameter set at finite density, we found a transition-like behavior and the finite size scaling study shows that this transition is a cross-over but not a real phase transition. Furthermore, we observed that around the "transition" point the approximation of TRG gets worse, although this is not a critical point.
We introduced the reweighting method for TRG and demonstrated for some parameters. As a result, the errors increase as the distance from original parameters and the lattice size. Furthermore we observed that the reweighting from around "transition" point quickly deteriorates compared with reweighting from off-transition region. This is the first application of the GTRG to finite density sistem. We hope that the formulation given in this work is extended another finite density systems.