Continuum limit in numerical simulations of the $\mathcal{N}=2$ Landau--Ginzburg model

The $\mathcal{N}=2$ Landau--Ginzburg description provides a strongly-interacting Lagrangian realization of an $\mathcal{N}=2$ superconformal field theory. It is conjectured that one such example is given by the two-dimensional $\mathcal{N}=2$ Wess--Zumino model. Recently, the conjectured correspondence has been studied by using numerical techniques based on the lattice field theory; the scaling dimension and the central charge have been directly measured. We study a single superfield with the cubic superpotential, and give an extrapolation method to the continuum limit. Then, on the basis of a supersymmetric-invariant numerical algorithm, we perform a precision measurement of the scaling dimension through a finite-size scaling analysis.

Although one can observe good agreement of h +h in Table 1 with that of the expected minimal models (and also the central charge [34,35,45]), these results are not extrapolated ones to the continuum limit. Moreover, it was noted [35] that the computation in Ref. [34] is quite sensitive to a UV ambiguity because of the locality breaking in the Kadoh-Suzuki formulation. The restoration of the locality should be observed non-perturbatively in the continuum limit. It is important and helpful to take the continuum limit and precisely determine the scaling dimension.
In this paper, we study a single superfield with the cubic superpotential on the basis of the SUSY-invariant formulation, which is believed to correspond to the A 2 minimal model. The finite-size scaling analysis in Refs. [33,34] is developed into an analysis method with continuum-limit extrapolation. Then, we extrapolate a scalar correlator to the continuum limit, and perform a precision measurement of the scaling dimension; we have the scaling dimension This more reliable result is consistent with the conjectured A 2 -type correspondence. Our computation supports the validity of the formulation, and hence implies the restoration of the locality in the continuum limit. We hope that the numerical approaches, when further developed, will be useful to investigate a superstring theory through the LG/Calabi-Yau correspondence [21,[46][47][48].

SUSY-preserving formulation
We consider the A-type theory, that is, the N Φ = 1 WZ model (1.1) with the superpotential where n is a positive integer, λ is a dimensionful coupling, and we have omitted the index I from the field variable; the theory is conjectured to correspond to the A n minimal model.
Let us suppose that the system is defined in a 2D Euclidean box of physical size L 0 × L 1 . Then, the Fourier transformation of each field ϕ(x) is defined by Here, the momentum p is discretized as where the Greek index µ runs over 0 and 1, and repeated indices are not summed over.
Integrating over the auxiliary field F , we obtain the action in terms of the Fourier modes of the physical component fields, where p z = (p 0 − ip 1 )/2 (pz = (p 0 + ip 1 )/2), the symbol * denotes the convolution 5) and the boson part of the action, S B , is given by The field products in W ′ (A) and W ′′ (A) are understood as the convolution. The new variable N(p) (2.6) specifies the so-called Nicolai mapping [41][42][43][44]; the change of variables from A to N simplifies the path-integral weight drastically as we will see soon.
In what follows, we employ a momentum-cutoff regularization given in Ref. [40]. In the formulation, a momentum cutoff Λ is introduced as |p µ | ≤ Λ for µ = 0 and 1. (2.7) Then, we also define a "lattice spacing" a by Λ ≡ π a , (2.8) and all dimensionful quantities are measured in units of a. Although an underlying lattice space is not always assumed [34], we will use this parameter to take the "continuum limit" a → 0, which implies that we remove the UV cutoff as Λ → ∞. The partition function is then given by where A i (i = 1, 2, . . . ) are solutions of the equation and A * i are their complex conjugate. In the second line in Eq. (2.9), we have used the Nicolai mapping (2.6) and integrated over the fermion fields; note that the fermion determinant coincides with the Jacobian associated with the Nicolai mapping, up to the sign: The simulation algorithm is summarized in Refs. [34,35,45].
This regularized system (2.9) possesses some remarkable features, as follows.
(1) This regularization exactly preserves SUSY, the translational invariance, and the U(1) symmetry. Thus, we can quite straightforwardly construct the appropriate expression for the supercurrent, the energy-momentum tensor, and the U(1) current such that they form the N = 2 superconformal multiplet [35]. This fact enables us to numerically compute such Noether currents directly and easily [34,35,45]. 3 (2) The path-integral weight exp(−S B ) is a Gaussian function of N(p). Thus we can obtain configurations of N(p) by generating Gaussian random numbers for each p µ . This algorithm is completely free from any undesired autocorrelation and the critical slowing down.
(3) The normalized partition function, can be computed numerically, which gives the Witten index, tr(−1) F [56,57]. When the superpotential is a polynomial of degree n, e.g., Unfortunately, there are some difficulties of the algorithm; e.g., see Ref. [35]. In particular, the momentum cutoff breaks the locality of the theory. When the numbers L µ /a are taken as odd integers, this formulation is nothing but the dimensional reduction of the lattice formulation of the 4D WZ model [58] based on the SLAC derivative [59,60]; this is plagued by the pathology that the locality is not automatically restored in the continuum limit [61][62][63][64]. On the other hand, for the massive 2D N = 2 WZ model, one can argue the restoration of it as a → 0 within perturbation theory [40]. For the massless case, since perturbation theory possesses IR divergences, it is not clear whether its restoration is automatically accomplished. We should confirm the restoration of the locality non-perturbatively.

Numerical setup
We summarize the numerical setup that we will use in this paper. Our setup is based on the simulation setup in Ref. [35]. We consider the 2D N = 2 WZ model with the superpotential (2.1) of degree 2, which corresponds to the A 2 minimal model. Here the coupling constant λ is a dimensionful parameter and characterizes the mass scale in this theory. For simplicity, the system is supposed to be defined in the physical box L × L, where L/a is taken as even integers ∈ [10,52].
To numerically compute observables (e.g., Eq. (2.12)), we first generate Gaussian random numbers N(p) for each p µ . Then we solve the multi-variable algebraic equation (2.10) with respect to A(p); we should ideally find all the solutions A i (p) (i = 1, 2, . . . ) numerically. To 7 do this, we employ the Newton-Raphson method, and set the convergence threshold as In the case of L = 52a, which is the most numerically demanding one in this paper, the threshold is less accurate (and also the number of obtained configurations is relatively not many). For a configuration N(p), we randomly generate initial trial configurations of A(p) by Gaussian random numbers with unit variance, so that we obtain 200 solutions for A allowing repetition of identical solutions with L < 52a and 120 solutions with L = 52a. Two solutions A 1 and A 2 are regarded as identical if Finally, we tabulate the classification of obtained configurations in Table 2, where the coupling aλ have been tuned already in accordance with an argument given in the next section. In Table 3, we list the numerical results of the Witten index (2.12), ∆ = 2, and the one-point SUSY Ward-Takahashi identity [66] (see also Ref. [35]) Whether ∆ and δ are numerically reproduced indicates the quality of our configurations.

Susceptibility of the scalar field
To numerically determine the scaling dimension, we first explain the finite-size scaling analysis in Refs. [33,34], which is compatible with the continuum limit as we will develop later. Let us consider the susceptibility of the scalar field A, defined by [33] χ(L µ ) = 1 (4.1) In the IR limit, the scalar field is expected to behave as a chiral primary field with the conformal dimensions (h,h); the two-point function of A behaves as for large |x| = √ x 2 . Note that h +h is called the scaling dimension, and h −h is the spin. Now suppose that the field A is spinless, h =h. Then, we observe the finite-volume scaling of the scalar susceptibility for large L µ , as Numerically simulating the scalar correlator for some different volumes but a same value of the coupling, one can read the exponent, 1 − h −h, from the slope of ln χ(L µ ) as a linear function of ln(L 0 L 1 ). In what follows, for simplicity, we take into account the case of the physical box size L = L 0 = L 1 .

Continuum limit of the susceptibility
As already announced, we consider the continuum limit a → 0, that is, Λ → ∞. Any extrapolation has been not done in the preceding numerical studies. In Refs. [33][34][35]45], the physical box size L is expected to be taken as sufficiently large values, while the coupling λ in the superpotential (2.1) is fixed by aλ = 0.3; good agreement of the scaling dimension with those of the A 2 and A 3 minimal models was observed (Table 1). Unlike in the case of QCD, however, the present model does not possess any dynamical scale, so the "sufficiently small" scale of a is not quite obvious. In fact, we will find that the susceptibility, χ(L), takes a slow approach to a/L = 0. To obtain precise and reliable results, we should extend the above finite-size scaling analysis in order to treat the continuum limit.
We have also recognized the pathology of the locality in the lattice formulation that is based on the SLAC derivative; the computation of ln χ(L) with finite L/a is quite sensitive to this problem [34,35] (see also Sect. 4.4). A proposal given in Ref. [35] is to directly study the correlation function in the momentum space, A(p)A * (−p) . Although the measured scaling dimension with the fixed coupling tends to approach expected values as the grid size L/a increases, the approach to the L/a → ∞ limit appears not quit smooth [35]. 4 We would need a more systematic method of the continuum limit, while the locality should be restored in the limit.
Our strategy of the continuum limit is much similar to that in Ref. [65]. We regard ln χ(L) as the same kind of the running couplingḡ 2 (L) defined on a lattice. To take the continuum limit, various sizes of the lattice spacing, {a i } (i = 1, 2, . . . ), are required; we first prepare various momentum-grid sizes {L/a i }, while the lattice parameter a i λ is tuned so that ln χ(L) (orḡ 2 (L)) is kept fixed; we denote u = ln χ(L). The system with a different grid size L ′ /a ′ = L/a i and the same parameter a ′ λ ′ = a i λ possesses the physical box size L ′ × L ′ with a ′ = a i . Then, we compute ln χ(L ′ ) (ḡ 2 (L ′ )) for L ′ /a i and a i λ; we observe the a-dependence of ln χ(L ′ )| a (ḡ 2 (L ′ )| a ), and attempt to extrapolate this in the continuum limit, lim a→0 ln χ(L ′ )| a .
To be more specific, we introduce the scaling function Σ as Σ(s, u, a/L) = ln χ(sL)| a . (4.4) The statistical error of Σ would be given by a square root of the sum of squared errors of ln χ(L) and ln χ(sL), owing to the long-distance behavior (4.3). As a consequence of the continuum limit with a to-be-determined fit function, we can obtain the scaling dimension (4.5) The cutoff dependence will be determined from numerical results. Note that the unique mass scale λ in this model should be sufficiently larger than 1/L to study the conformal behavior [33], hence λL → ∞ as the continuum limit. We can apply our extrapolation method to the continuum limit to other non-perturbative formulations, for example, the lattice formulation in Ref. [33].

Numerical measurement of the scaling dimension
In this subsection, let us perform the precision measurement of the scaling dimension for the A 2 -type theory with the cubic superpotential Φ 3 , by using the above continuum-limit  Table 5 Scaling dimension measured at finite volumes. The results in the last two rows are obtained by reading the slope of ln χ for (L/a, L ′ /a) = (24,48) or (L/a, L ′ /a) = (26,52) in Table 4. We tabulate the numerical results of the scalar susceptibility with the various box sizes of L and L ′ = 2L in Table 4. The third column is devoted to the tuned values of the coupling, aλ, so that ln χ(L) in the forth column is kept almost fixed. The results of Σ(u, a/L) are shown in the last column, where we have omitted the first argument s = 2 of Σ(s, u, a/L), while we set u = ln χ(L) as 3.9175. The error of Σ(u, a/L) is given by a square root of the sum of the squared errors of ln χ(L) and ln χ(L ′ ).
In Ref. [34], the scaling dimension was obtained from the slope of the susceptibility in the formulation, by using data for 24 ≤ L/a ≤ 36 or 26 ≤ L/a ≤ 36 with a fixed coupling; we have the similar slope of ln χ for (L/a, L ′ /a) = (24, 48) though we have used different values of aλ (see Table 5). Now we have enough data to clarify the (a/L)-dependence of Σ(u, a/L). Fig. 1 shows Σ(u, a/L) as the function of a/L given in Table 4. From the plot, we simply applies a linear function of a/L in order to take the continuum limit, then we have Because the quality of configurations with L/a = 52 is poorer due to the computational cost (see Sect. 3), the computation of ln χ could be less accurate. In fact, the above result in Fig. 1 implies that there is a discrepancy between the central values of ln χ(L) and the fit function at L/a = 52. To make sure that this discrepancy comes from statistical fluctuations, we show the behavior of ln χ(L) for L/a = 52 when the number of configurations varies in Table 6; the deviation of the central values decreases. The main result of the scaling dimension in this paper is given by Here, a number in the second parentheses indicates the systematic error defined by the deviation between the central values of Eq. (4.7) and Eq. (4.9).

Discussion on the fit function
We found that a linear fit of Σ(s, u, a/L) with respect to a/L would be good within the numerical error. To convince ourselves of this fact, let us introduce a slightly modified extrapolation method, by which we obtain another result of the scaling dimension from same data. If two results are much similar, our extrapolation method (or fit function) to the continuum limit works well.
The new method is based on the excision of a small region around the contact point of the integrand A(x)A(0) in ln χ(L) (4.1) [33]. The modified scalar susceptibilityχ is defined byχ The coupling λ is the unique mass scale in the WZ model with the superpotential (2.1), and the correlations at short lengths ∼ λ −1 would not affect the scaling (4.3) of χ(L) in low-energy regions. Note that the shape of the excised space is slightly different from those in Refs. [33,34], but the susceptibility should not be sensitive to such UV details in the continuum limit; if the grid size L/a is not sufficiently large (i.e., L/a is finite), we suffer from the sensitivity to the excised space size. In terms of the Fourier modes of A, we havẽ where |p| = p 2 and J 1 is the Bessel function of the first kind.
The parameter tuning above indicates that the dimensionless coupling aλ becomes large as L/a → ∞, while ln χ(L) is kept fixed. That is, in the small a limit, the volume of the excised space becomes smaller and smaller; we must have the completely same result of the scaling dimension as that in the method (4.5), at least analytically. In numerical simulations, however, it is not known a priori what function we should apply to take the continuum limit. Thus attempting to extrapolate results of lnχ(L) and to determine the fit function, one can justify the numerical determination of the scaling dimension from Σ. In the same way as ln χ(L), we define the new scaling functionΣ bỹ Σ(s, u, a/L) = lnχ(sL). (4.13) Here u is given by the fixed number, ln χ(L), which is identical to the value of lnχ(L) in the continuum limit, that is, λ −1 → 0. Similarly, one can measure the scaling dimension by Eq. (4.5) withΣ and another to-be-determined fit function. respectively. These two results are consistent with our previous result (4.10). We have obtained the precise and reliable result (4.10) through the finite-size scaling with the continuum-limit extrapolation.

Conclusion
In this paper, we numerically studied the 2D N = 2 WZ model with the cubic superpotential, which is believed to provide the Landau-Ginzburg description of the A 2 minimal model of the 2D N = 2 SCFT. On the basis of the SUSY-invariant formulation with a momentum cutoff, we considered the continuum-limit extrapolation of the scalar susceptibility, and then performed the precision measurement of the scaling dimension through the finite-size scaling analysis. The result of the scaling dimension are consistent with the conjectured WZ/SCFT correspondence. The theoretical background of our computational approach is not clear so far, but our result would support the restoration of the locality in the continuum limit. A related issue is the continuum-limit analysis of the central charge.
Such an analysis will be useful to study general SCFTs. It is important to confirm further the theoretical validity of the formulation, in order to investigate superstring theory via the LG/Calabi-Yau correspondence in future. 17