Investigation of complex $\phi^{4}$ theory at finite density in two dimensions using TRG

We study the two-dimensional complex $\phi^{4}$ theory at finite chemical potential using the tensor renormalization group. This model exhibits the Silver Blaze phenomenon in which bulk observables are independent of the chemical potential below the critical point. Since it is expected to be a direct outcome of an imaginary part of the action, an approach free from the sign problem is needed. We study this model systematically changing the chemical potential in order to check the applicability of the tensor renormalization group to the model in which scalar fields are discretized by the Gaussian quadrature. The Silver Blaze phenomenon is successfully confirmed on the extremely large volume $V=1024^2$ and the results are also ensured by another tensor network representation with a character expansion.


I. INTRODUCTION
The tensor network (TN) is a promising approach to study lattice models with a sign problem. Coarse-graining algorithms of tensor networks such as the tensor renormalization group (TRG) [1] do not have any stochastic process unlike the Monte Carlo method which is based on the stochastic interpretation of the Boltzmann factor in path integrals. So a development of this approach could lead to deep understanding of quantum field theories that suffer from the sign problem such as QCD at finite chemical potential, finite θ angle, chiral gauge theories and SUSY theories. Although the TRG algorithm has been already introduced into the research of lattice quantum field theories [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18], further studies are desirable to confirm if the TRG properly works for theories with a severe sign problem.
The complex φ 4 theory at finite chemical potential is the simplest model that suffers from a severe sign problem. This model exhibits the so-called Silver Blaze phenomenon in which bulk observables do not depend on the chemical potential below the critical point. Since it is directly related to the imaginary part of the action, various methods that could overcome the sign problem, such as the complex Langevin approach [19], the thimble method [20][21][22], and the worldline representation [23,24], have been used to study the model. In case of TRG, it is not straightforward to apply the algorithm to the scalar field theory because the tensor indices are given by the field variable which takes any real or complex number and numerical computation is not directly applied to such an infinite dimensional tensor.
In refs. [15,17] we have proposed a methodology of defining a finite dimensional tensor in the scalar field theory. We employed the Gaussian quadrature rule to discretize the scalar field so that a critical coupling constant of the Z 2 symmetry breaking in the twodimensional real φ 4 theory is evaluated with the TRG procedure. The result was consistent with those obtained with other conventional methods. Namely, our discretization method effectively works for the real scalar field theory. This implies that the TRG approach with the discretized field variables can be also effective for a complex scalar field theory.
In this paper, we study the two-dimensional complex φ 4 theory at finite chemical potential using the TRG method with the Gauss quadrature discretization for the scalar field. The expectation values for the scalar field and the number density are evaluated to investigate the Silver Blaze phenomenon. Furthermore, in order to confirm that the TRG method properly works, we compare the results to those obtained from another TN representation with the character expansion.
The rest of this paper is organized as follows: In Sec. II we define the target model and construct the TN representation for the partition function. Numerical results are presented in Sec. III, where the Silver Blaze phenomenon is confirmed. We also make a comparison of the results obtained from the naive TN representation of the partition function and another TN representation. Section IV is devoted to summary and future perspectives.
The Euclidean continuum action of the two-dimensional complex φ 4 theory at finite chemical potential is defined by with a complex scalar field φ(x), the bare mass m, the quartic coupling constant λ > 0, and the chemical potential µ. This theory describes a relativistic Bose gas with finite chemical potential. The action is complex for µ = 0 because the third term of eq. (1) is a pure imaginary number.
In the lattice theory, the scalar field denoted as φ n lives on a site n of a lattice Γ = {(n 1 , n 2 ) | n ν = 1, 2, . . . , N i } with the lattice volume V = N 1 × N 2 . The lattice spacing a is set to 1. We assume that the scalar field satisfies the periodic boundary condition, φ n+Nνν = φ n for ν = 1, 2, whereν is the unit vector of the ν-direction. The lattice action is given by Note that the chemical potential is introduced as a pure imaginary constant vector potential in the temporal direction [25]. Since the lattice action also satisfies (S(µ)) * = S(−µ), it is difficult to apply a naive Monte Carlo method to this model.
The partition function is defined as a standard manner: where the complex field φ n is represented in terms of two real fields as φ n = 1 √ 2 (A n + iB n ) and the integral measure is given by Dφ ≡ n∈Γ dA n dB n . In the following we show that Z is represented as a tensor network according to refs. [15,17]. The expectation value of any local field can also be represented as a tensor network in a similar way.
The Boltzmann weight e −S is expressed as a product of local factors: where for w, z ∈ C. It is possible to decompose the Boltzmann weight in this way as long as the lattice action contains only the nearest-neighbor interaction.
The continuous scalar field is discretized by the Gauss-Hermite quadrature rule to introduce a finite dimensional tensor as in refs. [15,17]. For one-variable integration of a proper function g(x), the quadrature provides a discretization as follows: where y α and w α are the α-th root of the K-th Hermite polynomial H K (x) and the corresponding weight defined as w α = 2 K−1 K! √ π/(K 2 H K−1 (y α ) 2 ), respectively. Here K dictates the order of approximation and for large K the accuracy of approximation is expected to be better 7 .
For the two-variable case (φ = 1 where Applying eq. (7) to each complex field, Z is approximated by Z(K) as where {α,β} ≡ n∈Γ K αn=1 K βn=1 . As a result of the discretization, f ν can be regarded as a K 2 × K 2 complex valued matrix: with the row index α, β = 1, 2, . . . , K and the column index α ′ , β ′ = 1, 2, . . . , K. Note that φ(α, β) is given by discretized points y α , y β in eq. (8). Then the singular value decomposition is applied to the matrix: where σ k is k-th singular value sorted in the descending order, and U [ν] and V [ν] are K 2 ×K 2 unitary matrices with the row index α, β and the column index k. Plugging eq. (11) into eq. (9), we find that Z(K) can be expressed as a tensor network, where and {x,t} ≡ n∈Γ K 2 xn,tn=1 . We obtain a finial expression by truncating the summation in eq. (12) up to D (≤ K 2 ) to reduce the computational complexity: where ′ {x,t} ≡ n∈Γ D xn,tn=1 . This truncation keeps a better precision when σ k in eq. (11) has a sharp hierarchy structure. We should note that the initial tensor T depends on K.
D becomes the bond dimension of tensors which is fixed throughout computations, and the convergence of results for K and D are checked numerically.

III. NUMERICAL RESULTS
Numerical results of two-dimensional complex φ 4 theory at finite chemical potential are presented in this section. The TRG [1] is employed to coarse-grain the tensor network eq. (14) on a periodic lattice with the volume V = N 2 (N = 2 m , m ∈ Z) and the lattice spacing a = 1. The coarse-graining procedure of partition function is briefly described in our previous paper [17] in which a procedure for the expectation value of a local field is also given. In the TRG algorithm, the SVD is truncated up to a fixed integer D, which is the bond dimension of tensors.  Then the expectation value of an operator O may be expressed as where e −S = e −Re(S) e iθ . Using the TRG, Z pq and O pq for a local operator O can also be evaluated from a tensor dropping the last two terms in eq. (5).
The sign problem appears as a difficulty in evaluating the ratio of eq. (16). For large µ, since the phase factor e iθ has a large fluctuation, both the average phase factor, and Oe iθ pq approach zero. Then, in the Monte Carlo method, it becomes difficult to evaluate O due to a 0/0 problem. In other words, the severeness of the sign problem is measured by the numerical value of eq. (17). Figure 3 shows the average phase factor evaluated by the TRG for various µ and V . We use m 2 = 0.01 and λ = 1 which are the same parameters as [24]. As clearly seen, the average phase factor decreases as µ increases for fixed space-time volume V while it also decreases as V increases for fixed µ. We thus confirm that, in the zero temperature and large spacial volume limits, severe sign problems happen even for small values of µ.

B. Silver Blaze phenomenon
In the thermodynamic limit, bulk observables are independent of µ below a critical µ c as well as finite density QCD. This is called the Silver Blaze phenomenon which is a direct outcome of an imaginary part of the action. Although the computational cost of the Monte Carlo method has a large volume dependence, the TRG is suitable for observing the Silver Blaze phenomenon clearly since its cost scales with the logarithm of the lattice volume and the thermodynamic limit can be easily taken. Figure 4 shows the µ-dependence of particle number density, The differentiation with respect to µ in the above equation is estimated by numerical differentiation. The Silver Blaze phenomenon is clearly observed for large volumes. The density does not depend on µ for small µ region, and it begins to increase at µ ≈ 0.94. In particular, the cusp structure around µ ≈ 0.94 tends to be sharper for larger volumes.
In fig. 5, we compare the result of the number density to that of the phase quenched  Figure 7 shows |φ| 2 as a function of µ for the same parameters as those of fig. 4, which is evaluated by the TRG with an impurity tensor [17]. As in the case of the density, the result is independent of µ for µ 0.94 and a sharp rise is seen around µ ≈ 0.94.
In order to study the stability of the Silver Blaze phenomenon against changing the physical parameters (m and λ), we also compute the particle number density for (m 2 , λ) = (0.01, 0.1) and (0.1, 0.1) as shown in fig. 8. Note that, for smaller m or λ, the exponential damping in the Boltzmann weight is weaker. Even for such cases, the Silver Blaze phenomenon is clearly observed.

C. Comparison with another tensor network representation
We have represented the partition function as a TN using the Gauss-Hermite quadrature for both the real and imaginary parts of each scalar field but one may use another representation, for instance, with a polar coordinate and the character expansion given in Appendix A. It is known that the partition function in the case does not have an imaginary part. This formulation is also useful for the TN method. The Gaussian quadrature is needed only for the radial variable because the angular variable is transcribed into a tensor index with the character expansion. Thus, the cost of making the initial tensor is basically cheaper than making eq. (13). See Appendix A for the details.
In figure 9, two representations are compared by showing |φ 2 | against µ. As a result they agree with each other well and it is hard to see the difference between them at this resolution. Thus we can conclude that choices of TN representation are basically irrelevant to our conclusion.

IV. SUMMARY
In this paper we have derived a TN representation for the complex scalar field theory discretizing the continuous scalar fields with the Gauss-Hermite quadrature rule. Using the TRG procedure for the TN representation of partition function, the average phase factor, the particle number density and |φ| 2 were evaluated.
As a result, the Silver Blaze phenomenon is clearly observed for the extremely large volume V = 1024 2 which is essentially in the zero temperature and the large spacial volume limits. We also examine another TN representation using the character expansion. Then, our numerical results of two representations do not have a visible difference, and the conclusion does not change for the other TN representation. Thus we confirm that the TN method is effective for a quantum field theory with the severe sign problem. using the character expansion of e x cos z : where I p is the p-th modified Bessel function of the first kind. In the polar coordinate, the lattice action eq. (2) is written as Using the above formulas, a dual formulation of the partition function is obtained as Integrating the angular variables turns out to be constraints for p and q variables with Kronecker's delta. Note that all entries of (A3) are real and non-negative.
To define a finite dimensional tensor, we truncate the summation for p n , q n and discretize the radial variable r n with Gauss-Hermite quadrature. In this case, since r n ∈ [0, ∞), we use the 2K-point Gauss-Hermite quadrature with only the positive K nodes 9 . Then the discrete version of (A3) is given by 2πy αn w αn e y 2 α × h pn (y αn , y α n+1 )h qn (y αn , y α n+2 ) δ (pn+qn−p n−1 −q n−2 ),0 e µqn , h p (y α , y β ) with fixed p is now regarded as a K × K matrix to which the SVD can be applied: Plugging eq. (A6) into eq. (A4) leads to a TN representation of Z (N CE , K): l e µb δ a+b,c+d K α=1 y α w α e y 2 α U [a] αi U Note that (x n , p n ), (t n , q n ), (x n−1 , p n−1 ), (t n−2 , q n−2 ) may be interpreted as four index pairs defined on four different links stemmed from the site n. ThusT may be interpreted as a rank-4 tensor whose bond dimension is K × (2N CE + 1) since x n , t n = 1, 2 . . . , K and x of eq. (A8) into the computation. Let us arrange σ [p] x in the descending order for all x and p and suppose that the nth largest singular value is σ x ′ . Then a one-to-one mapping f between n and (x ′ , p ′ ) can be given, that is, n = f (x ′ , p ′ ). 10 Using this mapping, the combined index X n and T n are given by X n = f (x n , p n ), T n = f (t n , q n ). (A9) Once f is given, X n is uniquely given for x n , p n and vice versa. Then the tensor is represented as T (CE) with (x, p) = f −1 (X), (t, q) = f −1 (T ) and the same identifications for X ′ , T ′ . Then truncated version of the discretized partition partition function is given by