Entanglement and R´enyi entropies of (1+1)-dimensional O(3) nonlinear sigma model with tensor renormalization group

: We investigate the entanglement and R´enyi entropies for the (1+1)-dimensional O(3) nonlinear sigma model using the tensor renormalization group method. The central charge is determined from the asymptotic scaling properties of both entropies. We also examine the consistency between the entanglement entropy and the n th-order R´enyi entropy with n → 1.


Introduction
In the past decade the tensor renormalization group (TRG) method 1 , which was originally proposed in the condensed matter physics [1], has been getting applied to the particle physics.Although the target models in the initial stage were restricted to the 2d ones, recent studies cover various four-dimensional (4d) models with the scalar, gauge and fermion fields [7,[10][11][12][13][14].So far much attention has been paid to the sign-problem-free nature of the TRG method [3,[15][16][17][18][19][20][21][22][23][24].On the other hand, there are few studies focusing on an ability of the direct evaluation of the partition function or the path-integral itself, which potentially allows us to measure the entanglement entropy (S A ) and nth-order Rényi entropy (S (n) A ). Up to know only 2d Ising and XY models were investigated [18,25,26].Note that it is difficult for the Monte Carlo method to measure the entanglement and Rényi entropies so that recent lattice QCD studies focused on the so-called entropic C-function, which is an UV-finite observable in proportion to ∂ L S A (L) with L an interval of length, avoiding the direct measurement of the entanglement and Rényi entropies [27][28][29].
In this paper we measure the entanglement and Rényi entropies of the (1+1)d O(3) nonlinear sigma model (O(3) NLSM) using the density matrix without resort to the transfer matrix formalism employed in Ref. [26].This model is massive and shares the property of asymptotic freedom with the (3+1)d non-Abelian gauge theories.We extarct the central charge from the entanglement and nth-order Rényi entropies using the scaling formula for the non-critical (1+1)d models [30].The value of the central charge is comapred with the previous result obtained from the entanglement entropy with the matrix product state (MPS) method [31].We also make a consistency check between the entanglement and Rényi entropies by extrapolating the nth-order Rényi entropy to n = 1.This paper is organized as follows.In Sec. 2, we define the (1+1)d O(3) NLSM on the lattice and give the tensor network representation.We present the numerical results for the entanglement and Rényi entropies in Sec. 3. We determine the central charge and discuss the consistency between the entanglement and Rényi entropies.Section 4 is devoted to summary and outlook.

Formulation and numerical algorithm
Although the definition of the (1+1)d O(3) NLSM and its tensor network representation are already given in the appendix of Ref. [32], we briefly give the relevant expressions for this work to make this paper self-contained.

(1+1)-dimensional O(3) nonlinear sigma model
We consider the partition function of the O(3) NLSM on an isotropic hypercubic lattice The lattice spacing a is set to a = 1 unless necessary.A real three-component unit vector s(n) resides on the sites n and satisfies the periodic boundary conditions s(n + νL) = s(n) (ν = 1, 2).The lattice action S is defined as ( The partition function Z is given by where D[s] is the O(3) Haar measure, whose expression is given later.

Tensor network representation of the model
The vector s(n) in the model can be expressed as The partition function and its measure are written as ) We discretize the integration (2.4) with the Gauss-Legendre quadrature [11,23] after changing the integration variables: ) We obtain with Ω n = (θ(α an ), ϕ(β bn )) ≡ (a n , b n ), where α an and β bn are a-and b-th roots of the K-th Legendre polynomial P K (s) on the site n, respectively.{Ωn} denotes K an=1 K bn=1 .M is a 4-legs tensor defined by (2.9) The weight factor w of the Gauss-Legendre quadrature is defined as . (2.10) After performing the singular value decomposition (SVD) on M : where U and V denotes unitary matrices and σ is a diagonal matrix with the singular values of M in the descending order.We can obtain the tensor network representation of the O(3) NLSM on the site n ∈ Λ 1+1 an,bn where D cut is the bond dimension of tensor T , which controls the numerical precision in the TRG method.The tensor network representation of partition function is given by We employ the higher order tensor renormalization group (HOTRG) algorithm [2] to evaluate Z.

Calculation of entanglement and Rényi entropies
Figure 1 illustrates the calculation procedure of the entanglement entropy.We divide the system to two subsystems A and B, both of which have the same lattice size with L × N t .The density matrix of subsystem A is defined by where Tr B denotes the trace restricted to the subsystem B. We use HOTRG to approximate the density matrix of subsystem A, in which The entanglement entropy is obtained by (2.14)

Numerical results
The density matrix ρ A is evaluated using HOTRG with the bond dimension D cut ∈ [10,130].Note that the correlation length ξ in this model was precisely measured over the range of 1.4 ≤ β ≤ 1.9 with the interval of ∆β = 0.1 in Ref. [33].We list the values of ξ in Table 1 for later convenience.In order to keep the condition a ≪ ξ ≪ L, our results are restricted to 1.4 ≤ β ≤ 1.72 in the following.
Figure 3 shows the N t dependence of the entanglement entropy S A (L) at β = 1.5 with L = 128, where the correlation length is expected to be ξ ∼ 11 [33].The degeneracy of the results for S A (L) with N t = 256, 512 and 1024 indicates the convergence of S A (L) in terms of N t so that N t = 1024 is large enough to be regarded as the zero temperature limit.In Fig. 4 we plot S A (L) with N t = 1024 at β = 1.4,1.5, 1.6 and 1.7.The entanglement entropy shows plateau behavior once the interval L goes beyond the correlation length.This is an expected behavior under the condition of ξ ≪ L [30].As ξ increases for larger β, the plateau of S A (L) starts at larger L and its value is increased according to the theoretical expectation of S A (L) ∼ c 3 ln ξ [30].In Fig. 5 we plot S A (L = 128) at β = 1.4,1.5, 1.6 and 1.7 as a function of 1/D cut .The data of S A (L = 128) shows increasing trend, while slightly fluctuating, for vanishing 1/D cut .This kind of fluctuation is commonly observed in the TRG method.See, e.g., Fig. 11 in Ref. [25] for the Ising model with the HOTRG algorithm.The solid lines express the linear extrapolation of S A (L = 128) at 1/D cut ≤ 0.02 to obtain the value at D cut → ∞, which are listed in Table 1.
The mass gap m in the (1+1)d O(3) NLSM is expressed as [35] where the two-loop expression for the beta function at β → ∞ is used in the last equation.
Since the correlation length is inversely proportional to the mass gap the entanglement entropy is rewritten as in terms of the coupling constant β.In Fig. 6 we plot the β dependence of S A at L = 128 with N t = 1024.We determine the central charge c by fitting the data in the range of 1.4 ≤ β ≤ 1.7 with the function of Eq. (3.2), where the condition of ξ ≪ L is well satisfied.We obtain the value of c = 1.97 (9), which is consistent with c = 2.04 (14) obtained by the MPS method in Ref. [31].We should also note that a recent study of the central charge for the 2d classical Heisenberg model, which is equivalent to the (1+1)d O(3) NLSM on the lattice, yields c ∼ 2 with the tensor-network renormalization method [36].For an instructive purpose Fig. 7 shows an alternative plot of S A (L = 128) with N t = 1024 as a function of ξ measured in Ref. [33].This is motivated by a concern that the (1+1)d O(3) NLSM does not have a good asymptotic scaling property below β ∼ 2.0 [33,37].The use of the fit function S A = c 3 ln ξ + const.gives the central charge c = 2.15(3), which is consistent with c = 1.97(9) obtained above.
Values of correlation length ξ are taken from Ref. [33].16) S (11) A 0.469(2) 0.647(6) 0.870(11) 1.144 (16) Now let us turn to the Rényi entropy.In Fig. 8 we plot the N t dependence of the 2nd-order Rényi entropy S  A (L) is observed in the large N t region so that N t = 1024 is essentially regarded as the zero temperature limit of S 3) The value of c = 2.27( 16) is slightly larger than that determined from S A .We repeat the same calculation for other nth-order Rényi entropy.The n dependence of the central charge c is plotted in Fig. 11, where the error bar of the central charge originates from the 1/D cut extrapolation and the scaling fit with Eq. (3.3) for the Rényi entropy.We observe that the central value of c seems to converge to c = 2 as n increases.
Here we consider the error of the nth-order Rényi entropy stemming from the errors of the eigenvalues in the density matrix.Suppose S(n) A is the true nth-order Rényi entropy and λj denotes the true jth eigenvalue in the density matrix ρ A normalized as Tr A ρ A = 1:

S(n)
where we assume the descending order for the eigenvalue λ1 > λ2 > λ3 , • • • .Introducing the error of λj , which is expressed as δ j , the measured Rényi entropy may be written as Focusing on the error of the Rényi entropy we find δS The error of the Rényi entropy is bounded by the relative error of the maximum eigenvalue of the density matrix in the large n limit.This may explain the convergence behavior of the central charge toward c = 2 observed in Fig. 11.In the Monte Carlo approach it is difficult to calculate the entanglement entropy.

Figure 2
Figure 2 depicts the calculation procedure of the nth-order Rényi entropy defined by
Figure 9 compares S

( 2 )
A (L) at β = 1.4,1.5, 1.6 and 1.7 with N t = 1024 fixed.Our observation is consistent with the theoretical expectation that S (2) A (L) should stay constant in the range of L ≫ ξ according to S 1/n) ln ξ[30].In Fig.10we show D cut dependence of S (2) A (L = 128) at β = 1.4,1.5, 1.6 and 1.7.The extrapolated value of S (2) A (L = 128) at D cut → ∞ is obtained by the linear fit of the data in terms of 1/D cut with 1/D cut ≤ 0.02.The β dependence of S (2) A at L = 128 with N t = 1024 is plotted in Fig. 6 together with S A .We extract the central charge c from the data in 1.4 ≤ β ≤ 1.7 employing the following fit function with n = 2:

Figure 11 :
Figure 11: n dependence of central charge c obtained from nth-ordr Rényi (open) and entanglement (closed) entropies.Solid line denotes c = 2 to guide your eyes.

Table 1 :
Results for S A (L = 128) and S