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

TL;DR

This paper investigates the continuum limit of the two-dimensional $ =2$ Landau-Ginzburg model using lattice simulations, providing precise measurements of scaling dimensions and exploring the conjectured correspondence with superconformal field theories.

Contribution

It introduces an extrapolation method to reach the continuum limit and employs a supersymmetric-invariant algorithm for accurate scaling dimension measurements.

Findings

Successful continuum extrapolation of the model.

Precise measurement of the scaling dimension.

Support for the conjectured superconformal correspondence.

Abstract

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 lattice field theory; the scaling dimension and the central charge have been directly measured. We study a single superfield with a 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.

Tables

Table 1. Table 1: Scaling dimension  1 − h − h ¯ 1 ℎ ¯ ℎ 1-h-\bar{h} measured in preceding studies.

	Reference	$1-h-\overline{h}$	Expected value
$A_2$	Kawai–Kikukawa Kawai:2010yj	$0.660\left(11\right)$	$2/3=0.666\dots$
	Kamata–Suzuki Kamata:2011fr	$0.616\left(25\right)\left(13\right)$
	Morikawa–Suzuki Morikawa:2018ops	$0.682\left(10\right)\left(7\right)$
$A_3$	Morikawa–Suzuki Morikawa:2018ops	$0.747\left(11\right)\left(12\right)$	$0.75$

Table 2. Table 2: Classification of the configurations obtained for the  A 2 subscript 𝐴 2 A_{2} -type theory. 𝒩 conf subscript 𝒩 conf \mathcal{N}_{\text{conf}} denotes the total number of configurations for each setup. In the upper half of the table, the number of configurations for  L 𝐿 L is shown; in the lower half, that for  L ′ = 2 ​ L superscript 𝐿 ′ 2 𝐿 L^{\prime}=2L is shown. The symbol  ( n , m ) 𝑛 𝑚 (n,m) implies that, for a configuration  N ​ ( p ) 𝑁 𝑝 N(p) , we find ( n + m ) 𝑛 𝑚 (n+m) solutions, A i ​ ( p ) subscript 𝐴 𝑖 𝑝 A_{i}(p) ( i = 1 𝑖 1 i=1 , …, n + m 𝑛 𝑚 n+m ); the  n 𝑛 n solutions take  det sign ⁡ ∂ ( N , N ∗ ) ∂ ( A , A ∗ ) = + 1 sign 𝑁 superscript 𝑁 𝐴 superscript 𝐴 1 \det\operatorname{sign}\frac{\partial(N,N^{*})}{\partial(A,A^{*})}=+1 and the  m 𝑚 m solutions take  − 1 1 -1 .

$L/a$	$a\lambda$	${𝒩}_{\text{conf}}$	$(2,0)$	$(3,1)$	$(4,2)$	$(1,0)$	$(2,1)$	$(3,2)$	$(3,0)$	$(4,1)$
10	0.1780	7680	7680	0	0	0	0	0	0	0
12	0.2135	5120	5119	1	0	0	0	0	0	0
14	0.2538	5120	5119	1	0	0	0	0	0	0
16	0.3000	5120	5112	8	0	0	0	0	0	0
18	0.3420	5120	5093	27	0	0	0	0	0	0
20	0.3888	5120	5070	50	0	0	0	0	0	0
22	0.4500	5120	5023	97	0	0	0	0	0	0
24	0.5100	5120	4961	156	3	0	0	0	0	0
26	0.5705	5120	4909	204	6	0	0	0	1	0
20	0.1780	5120	5117	3	0	0	0	0	0	0
24	0.2135	5120	5104	16	0	0	0	0	0	0
28	0.2538	5120	5075	44	1	0	0	0	0	0
32	0.3000	4320	4236	83	1	0	0	0	0	0
36	0.3420	2592	2514	77	1	0	0	0	0	0
40	0.3888	2592	2472	118	0	0	1	1	0	0
44	0.4500	2592	2458	131	2	0	0	0	0	1
48	0.5100	2592	2433	157	2	0	0	0	0	0
52	0.5705	1512	1392	107	4	1	1	1	6	0

Table 3. Table 3: Quality of the configurations obtained for the  A 2 subscript 𝐴 2 A_{2} -type theory. The Witten index of Eq.  Δ Δ \Delta  ( 2.12 ) and the one-point function of Eq.  δ 𝛿 \delta  ( 3.4 ) are numerically computed for  L 𝐿 L and  L ′ = 2 ​ L superscript 𝐿 ′ 2 𝐿 L^{\prime}=2L ; Δ Δ \Delta should be identical to  2 2 2 , and  δ 𝛿 \delta should identically vanish. For  L ′ / a = 52 superscript 𝐿 ′ 𝑎 52 L^{\prime}/a=52 , the quality of the configurations obtained is poorer due to the computational cost.

$L/a$	$L^{\prime }/a$	$a\lambda$	$\Delta \left(L\right)$	$\Delta \left(L^{\prime }\right)$	$\delta \left(L\right)$	$\delta \left(L^{\prime }\right)$
10	20	0.1780	2	2	$-$0.00099(104)	$-$0.00005(67)
12	24	0.2135	2	2	$-$0.00063(107)	$+$0.00046(56)
14	28	0.2538	2	2	$-$0.00019(94)	$-$0.00030(48)
16	32	0.3000	2	2	$-$0.00024(81)	$-$0.00004(46)
18	36	0.3420	2	2	$-$0.00109(74)	$+$0.00020(52)
20	40	0.3888	2	1.9992(5)	$-$0.00078(67)	$+$0.00053(55)
22	44	0.4500	2	2.0004(4)	$-$0.00005(62)	$+$0.00031(48)
24	48	0.5100	2	2	$+$0.00041(56)	$+$0.00000(41)
26	52	0.5705	2.0002(2)	2.002(2)	$-$0.00058(52)	$+$0.00073(110)

Table 4. Table 4: Scalar susceptibility, u = 3.9175 𝑢 3.9175 u=3.9175 .

$L/a$	$L^{\prime }/a$	$a\lambda$	$\ln \chi \left(L\right)$	$\ln \chi \left(L^{\prime }\right)$	$\Sigma (u,a/L)$
10	20	0.1780	3.9174(59)	4.6338(72)	4.6338(93)
12	24	0.2135	3.9175(73)	4.6642(69)	4.6642(100)
14	28	0.2538	3.9193(70)	4.6844(66)	4.6844(97)
16	32	0.3000	3.9171(69)	4.6913(68)	4.6913(97)
18	36	0.3420	3.9166(68)	4.7223(83)	4.7223(107)
20	40	0.3888	3.9215(65)	4.7251(81)	4.7251(104)
22	44	0.4500	3.9162(62)	4.7400(76)	4.7400(97)
24	48	0.5100	3.9186(60)	4.7610(70)	4.7610(93)
26	52	0.5705	3.9175(56)	4.7823(91)	4.7823(107)

Table 5. Table 5: Scaling dimension measured at finite volumes. The results in the last two rows are obtained by reading the slope of  ln ⁡ χ 𝜒 \ln\chi for  ( L / a , L ′ / a ) = ( 24 , 48 ) 𝐿 𝑎 superscript 𝐿 ′ 𝑎 24 48 (L/a,L^{\prime}/a)=(24,48) or  ( L / a , L ′ / a ) = ( 26 , 52 ) 𝐿 𝑎 superscript 𝐿 ′ 𝑎 26 52 (L/a,L^{\prime}/a)=(26,52) in Table  4 .

	Fit range of $L$	$a\lambda$	$1-h-\overline{h}$
Kamata–Suzuki Kamata:2011fr	From 24 to 36	0.3000	0.603(19)
	From 26 to 36	0.3000	0.609(25)
	From 24 to 48	0.5100	0.6076(66)
	From 26 to 52	0.5705	0.6238(77)

Table 6. Table 6: ln ⁡ χ ​ ( L ′ ) 𝜒 superscript 𝐿 ′ \ln\chi(L^{\prime}) with  u = 3.9175 𝑢 3.9175 u=3.9175 and L / a = 52 𝐿 𝑎 52 L/a=52 when the number of configurations, 𝒩 conf subscript 𝒩 conf \mathcal{N}_{\text{conf}} , varies.

${𝒩}_{\text{conf}}$	$\ln \chi \left(L^{\prime }\right)$
1512	4.7823(91)
756	4.7950(133)
378	4.8087(193)

Equations

$$S$$

$$\displaystyle\qquad\qquad\qquad+\left(\bar{\psi}_{\dot{1}},\psi_{2}\right)_{I}\sum_{J=1}^{N_{\Phi}}\begin{pmatrix}2\delta_{IJ}\partial&\frac{\partial^{2}W(\{A\})^{*}}{\partial A^{*}_{I}\partial A^{*}_{J}}\\ \frac{\partial^{2}W(\{A\})}{\partial A_{I}\partial A_{J}}&2\delta_{IJ}\bar{\partial}\end{pmatrix}\begin{pmatrix}\psi_{1}\\ \bar{\psi}_{\dot{2}}\end{pmatrix}_{J}\Biggr{]},$$

$$1-h-\bar{h}$$

$$W(\Phi )=\frac{\lambda }{n+1}{\Phi }^{n+1},$$

$$\phi (x)$$

$$p_{\mu }=\frac{2\pi }{L_{\mu }}{n}_{\mu }(n_{\mu }=0,\pm 1,\pm 2,\dots ),$$

$$S=S_B+\frac{1}{L_0{L}_1}\sum _p\left({\bar{\psi }}_{\dot{1}},{\psi }_2\right)(-p)\left(\begin{array}{cc}2i{p}_z& W^{\prime \prime }(A{)}^{\ast }\ast \\ W^{\prime \prime }(A)\ast & 2i{p}_{\bar{z}}\end{array}\right)\left(\begin{array}{c}{\psi }_1\\ {\bar{\psi }}_{\dot{2}}\end{array}\right)(p),$$

$$({\phi }_1\ast {\phi }_2)(p)\equiv \frac{1}{L_0{L}_1}\sum _q{\phi }_1(q){\phi }_2(p-q),$$

$$S_B$$

$$|p_{\mu }|\le \Lambda \text{for μ=0 and 1}.$$

$$\Lambda \equiv \frac{\pi }{a},$$

$$\mathcal{Z}$$

$$=\int \prod _{|p_{\mu }|\le \frac{\pi }{a}}\left[dN(p)d{N}^{\ast }(p)\right]e^{-S_B}\sum _i{\mathrm{sign}\det \frac{\partial (N,N^{\ast })}{\partial (A,A^{\ast })}|}_{A=A_i,A^{\ast }=A_i^{\ast }},$$

$$2i{p}_{z}A(p)+W^{\prime }(A{)}^{\ast }(p)-N(p)=0,$$

$$\det \left(\begin{array}{cc}2i{p}_z& W^{\prime \prime }(A{)}^{\ast }\ast \\ W^{\prime \prime }(A)\ast & 2i{p}_{\bar{z}}\end{array}\right)=\det \frac{\partial (N,N^{\ast })}{\partial (A,A^{\ast })}.$$

$$\Delta =⟨\sum _i{\mathrm{sign}\det \frac{\partial (N,N^{\ast })}{\partial (A,A^{\ast })}|}_{A=A_i,A^{\ast }=A_i^{\ast }}⟩,$$

$$W(\Phi )$$

$$\sqrt{\frac{{\sum }_p|2i{p}_{z}A(p)+W^{\prime }(A{)}^{\ast }(p)-N(p){|}^2}{{\sum }_q|N(q){|}^2}}<\{\begin{array}{ll}10^{-14}& \text{for L<52a}\\ 10^{-13}& \text{for L=52a}.\end{array}$$

$$\sqrt{\frac{{\sum }_p|A_1(p)-A_2(p){|}^2}{{\sum }_q|A_1(q){|}^2}}<\{\begin{array}{ll}10^{-11}& \text{for L<52a}\\ 10^{-10}& \text{for L=52a}.\end{array}$$

$$\delta \equiv \frac{⟨S_B⟩}{(L+1{)}^2}-1=0.$$

$$\chi (L_{\mu })=\frac{1}{a^2}{\int }_{L_0{L}_1}{d}^{2}x⟨A(x)A^{\ast }(0)⟩=\frac{1}{a^2{L}_0{L}_1}⟨|A(p=0){|}^2⟩.$$

$$⟨A(x)A^{\ast }(0)⟩=\frac{1}{z^{2h}{\bar{z}}^{2\bar{h}}},$$

$$\chi \propto (L_0{L}_1{)}^{1-h-\bar{h}}.$$

$$\Sigma (s,u,a/L)=\ln \chi (sL){|}_a.$$

$$1-h-\bar{h}$$

$$\Sigma (3.9175,a/L)$$

$$1-h-\bar{h}$$

$${\Sigma (3.9175,a/L)|}_{L/a<52}$$

$$1-h-\bar{h}=0.6612(86).$$

$$1-h-\bar{h}=0.6699(77)(87).$$

$$\tilde{\chi }(L)=\frac{1}{a^2}{\int }_{|x|\ge {\lambda }^{-1}}{d}^{2}x⟨A(x)A^{\ast }(0)⟩.$$

Full text

\subjectindex

B16, B24, B34, B38

1. 1 Introduction

2. 2 SUSY-preserving formulation

3. 3 Numerical setup

4. 4 Scaling dimension

5. 4.1 Susceptibility of the scalar field

6. 4.2 Continuum limit of the susceptibility

7. 4.3 Numerical measurement of the scaling dimension

8. 4.4 Discussion on the fit function

9. 5 Conclusion

1 Introduction

A Lagrangian realization of a conformal field theory (CFT) provides an important tool to clarify the conformal-invariant system. As a famous example, the Feigin–Fuks (integral) representation Feigin:1981st ; Feigin:1982tg gives a free-field Lagrangian on the curved spacetime. Feigin and Fuks employed this to explore the unitary representation of the Virasoro algebra, and proved the Kac determinant formula in an elegant way. Their technique has come in useful Dotsenko:1984nm ; Felder:1988zp for performing many computations explicitly and understanding the system intuitively. Although the existence of such a Lagrangian is not always obvious, one can extract more information from the Lagrangian by using techniques based on quantum field theory.

A strongly interacting CFT Lagrangian is realized by the Landau–Ginzburg (LG) model (or the LG description), which is expected to become conformal invariant in extremely low-energy regions. This realization is characterized as a critical behavior under the renormalization group flow; CFT would be a scale-invariant theory on the non-trivial infrared (IR) fixed point under the flow. Such critical phenomena are of great interest in a wide range of physics. Originally, the idea of the LG description was introduced as a phenomenological model to describe superconductivity Ginzburg:1950sr ; in this context, the Lagrangian is replaced by the free energy. To understand the critical behavior in an LG model it is important to classify the critical exponent, that is, the scaling of observables in the quantum field theory.

Let us consider one such example of LG models, the two-dimensional ($2$D) massless $\mathcal{N}=2$ Wess–Zumino (WZ) model Wess:1974tw with a quasi-homogeneous superpotential. From the dimensional reduction of the $4$D $\mathcal{N}=1$ WZ model, the $2$D $\mathcal{N}=2$ WZ action with $N_{\Phi}$ superfields is given by111Here, we consider the $2$D $\mathcal{N}=(2,2)$ supersymmetry, and not $\mathcal{N}=(2,0)$.

[TABLE]

where $A_{I}$ ($I=1$, …, $N_{\Phi}$) are complex scalars, $\begin{pmatrix}\psi_{1}\\ \bar{\psi}_{\dot{2}}\end{pmatrix}_{I}$ are $2$D Dirac fermions, and $F_{I}$ are auxiliary fields; we work in the Euclidean space, and use the complex coordinate $z=x_{0}+ix_{1}$ ($\bar{z}=x_{0}-ix_{1}$) and the corresponding derivative $\partial=(\partial_{0}-i\partial_{1})/2$ ($\bar{\partial}=(\partial_{0}+i\partial_{1})/2$). The model is believed to become an $\mathcal{N}=2$ superconformal field theory (SCFT) in the IR limit DiVecchia:1985ief ; DiVecchia:1986cdz ; DiVecchia:1986fwg ; Boucher:1986bh ; Gepner:1986ip ; Cappelli:1986hf ; Cappelli:1986ed ; Gepner:1986hr ; Gepner:1987qi ; Cappelli:1987xt ; Kato:1987td ; Gepner:1987vz . Much evidences of this conjectured WZ/SCFT correspondence has been given in Refs. Kastor:1988ef ; Vafa:1988uu ; Martinec:1988zu ; Lerche:1989uy ; Howe:1989qr ; Cecotti:1989jc ; Howe:1989az ; Cecotti:1989gv ; Cecotti:1990kz ; Howe:1990cg ; Witten:1993jg and so on; for example, Refs. Kastor:1988ef ; Howe:1989qr ; Howe:1989az discuss the renormalization group flow for the $N_{\Phi}=1$ WZ model with the monomial superpotential, $W(\Phi)\propto\Phi^{n+1}$ ($n=2$, $3$, …), which corresponds to the $A_{n}$ minimal model of the $\mathcal{N}=2$ SCFT. See Refs. Polchinski:1998rr ; Hori:2003ic ; Tachikawa:2018sae for reviews. However, we have no complete proof of the conjectured LG correspondence to SCFT. This is because the $2$D $\mathcal{N}=2$ WZ model is strongly coupled in low-energy regions, and perturbation theory possesses IR divergences. It is difficult to directly observe the critical behavior in the WZ model.

Recently, the conjectured WZ/SCFT correspondence has been non-perturbatively studied by using numerical techniques based on lattice field theory. In the case of a single superfield with cubic and quartic superpotentials, which corresponds to the $A_{2}$ and $A_{3}$ minimal models, respectively, the authors of Refs. Kawai:2010yj ; Kamata:2011fr ; Morikawa:2018ops numerically measured the scaling dimension $h+\bar{h}$ of the primary fields (see Table 1).

The first remarkable study Kawai:2010yj is based on the lattice formulation by Kikukawa and Nakayama Kikukawa:2002as , which preserves one nilpotent supersymmetry (SUSY) exactly;222In the continuum limit, the full SUSY in the formulation Kikukawa:2002as is automatically restored to all orders of perturbation theory Giedt:2004qs ; Kadoh:2010ca . Reference Kadoh:2016eju is a review of SUSY on the lattice, which refers to lattice formulations of the $2$D $\mathcal{N}=2$ WZ model. the others are on the SUSY-preserving momentum-cutoff regularization by Kadoh and Suzuki Kadoh:2009sp . Both non-perturbative formulations make essential use of the existence of the Nicolai or Nicolai–Parisi–Sourlas mapping Nicolai:1979nr ; Nicolai:1980jc ; Parisi:1982ud ; Cecotti:1983up . In particular, by applying the latter formulation to the WZ model with multiple superfields, the central charge in ADE-type minimal models Vafa:1988uu had also been measured quite straightforwardly Kamata:2011fr ; Morikawa:2018ops ; Morikawa:2018zys . One can observe good agreement of the scaling dimension $h+\bar{h}$ in Table 1 and the central charge Kamata:2011fr ; Morikawa:2018ops ; Morikawa:2018zys with those of the expected minimal models. These studies achieved a triumph for lattice field theory, and enable us to study more general $\mathcal{N}=2$ SCFTs.

Although the corresponding SCFT is defined as the continuum theory with infinite volume, the above results are not extrapolated to the thermodynamic and continuum limits. Moreover, it was noted Morikawa:2018ops that the computation of $h+\bar{h}$ in Ref. Kamata:2011fr is quite sensitive to a UV ambiguity because of the locality breaking in the Kadoh–Suzuki formulation with a finite cutoff. To justify numerical studies based on the formulation, such a UV ambiguity should disappear in the infinite-volume and continuum limits. It is important and helpful to analyze the limits 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. Kamata:2011fr ; Kawai:2010yj is developed into an analysis method with continuum-limit extrapolation. The extrapolation also carries out the thermodynamic limit. Then, we numerically simulate the IR behavior of a scalar correlator, extrapolate it to the continuum limit, and perform a precision measurement of the scaling dimension; we have the scaling dimension

[TABLE]

This more reliable result is rather consistent with the conjectured $A_{2}$-type correspondence. Our computation would support the restoration of the locality in the continuum limit. In this regard, the theoretical background of the formulation is still not clear, so the restoration of the locality should be observed more carefully. One can apply our extrapolation method to other non-perturbative formulations. We hope that the numerical approaches, when further developed, will be useful to investigate a superstring theory through the LG/Calabi–Yau correspondence Martinec:1988zu ; Cecotti:1990wz ; Greene:1988ut ; Witten:1993yc .

2 SUSY-preserving formulation

We consider the $A$-type theory, that is, the $N_{\Phi}=1$ WZ model of Eq. (1.1) with the superpotential

[TABLE]

where $n$ is a positive integer, $\lambda$ 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 $2$D Euclidean box of physical size $L_{0}\times L_{1}$. Then, the Fourier transformation of each field $\varphi(x)$ is defined by

[TABLE]

Here, the momentum $p$ is discretized as

[TABLE]

where the Greek index $\mu$ runs over [math] 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,

[TABLE]

where $p_{z}=(p_{0}-ip_{1})/2$ ($p_{\bar{z}}=(p_{0}+ip_{1})/2$), the symbol $*$ denotes the convolution

[TABLE]

and the boson part of the action, $S_{B}$, is given by

[TABLE]

The field products in $W^{\prime}(A)$ and $W^{\prime\prime}(A)$ are understood as the convolution. The new variable $N(p)$ in Eq. (2.6) specifies the so-called Nicolai mapping Nicolai:1979nr ; Nicolai:1980jc ; Parisi:1982ud ; Cecotti:1983up ; 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. Kadoh:2009sp . In the formulation, a momentum cutoff $\Lambda$ is introduced as

[TABLE]

Then, we also define a “lattice spacing” $a$ by

[TABLE]

and all dimensionful quantities are measured in units of $a$. Although an underlying lattice space is not always assumed Kamata:2011fr , we will use this parameter to take the “continuum limit” $a\to 0$, which implies that we remove the UV cutoff as $\Lambda\to\infty$. The partition function is then given by

[TABLE]

where $A_{i}$ ($i=1$, $2$, …) are solutions of the equation

[TABLE]

and $A_{i}^{*}$ are their complex conjugates. In the second line of Eq. (2.9), we have used the Nicolai mapping in Eq. (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:

[TABLE]

The simulation algorithm is summarized in Refs. Kamata:2011fr ; Morikawa:2018ops ; Morikawa:2018zys .

This regularized system, Eq. (2.9), possesses some remarkable features:

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 $\mathcal{N}=2$ superconformal multiplet Morikawa:2018ops . This fact enables us to numerically compute such Noether currents directly and easily Kamata:2011fr ; Morikawa:2018ops ; Morikawa:2018zys .333See Refs. Caracciolo:1989pt ; Caracciolo:1991cp for a general construction of the energy-momentum tensor in lattice field theory. Recently, a regularization-independent construction of such Noether currents has been developed in terms of the gradient flow Narayanan:2006rf ; Luscher:2009eq ; Luscher:2010iy ; Luscher:2011bx ; see also Ref. Suzuki:2016ytc for a review. 2. 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 $p_{\mu}$. This algorithm is completely free from any undesired autocorrelation and the critical slowing down. 3. 3.

The normalized partition function,

[TABLE]

can be computed numerically, which gives the Witten index, $\operatorname{tr}(-1)^{F}$ Witten:1982df ; Cecotti:1981fu . When the superpotential is a polynomial of degree $n$, e.g. $W(A)\propto A^{n+1}$, we should have $\Delta=n$.

Unfortunately, there are some difficulties for the algorithm; see, e.g., Ref. Morikawa:2018ops . In particular, the momentum cutoff breaks the locality of the theory. When the numbers $L_{\mu}/a$ are taken as odd integers, this formulation is nothing but the dimensional reduction of the lattice formulation of the $4$D WZ model Bartels:1983wm based on the SLAC derivative Drell:1976bq ; Drell:1976mj ; this is plagued by the pathology that the locality is not automatically restored in the continuum limit Dondi:1976tx ; Karsten:1979wh ; Kato:2008sp ; Bergner:2009vg . On the other hand, for the massive $2$D $\mathcal{N}=2$ WZ model, one can argue the restoration of it as $a\to 0$ within perturbation theory Kadoh:2009sp . For the massless case, since perturbation theory possesses IR divergences, it is not clear whether its restoration is automatically accomplished. Nevertheless, the numerical results in the preceding studies and ours below suggest the validity of the approach.

3 Numerical setup

We summarize the numerical setup that we will use in this paper. Our setup is based on the simulation setup in Ref. Morikawa:2018ops . We consider the $2$D $\mathcal{N}=2$ WZ model with the superpotential of Eq. (2.1) of degree $2$,

[TABLE]

which corresponds to the $A_{2}$ minimal model. Here, the coupling constant $\lambda$ 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 $L\times L$, where $L/a$ is taken as even integers in the interval $[10,52]$.

To numerically compute observables, e.g.  Eq. (2.12), we first generate Gaussian random numbers $N(p)$ for each $p_{\mu}$. Then we solve the multi-variable algebraic equation in Eq. (2.10) with respect to $A(p)$; we should ideally find all the solutions $A_{i}(p)$ ($i=1$, $2$, $\dots$) numerically. To do this, we employ the Newton–Raphson method and set the convergence threshold as

[TABLE]

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 not relatively high). 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

[TABLE]

Finally, we tabulate the classification of the configurations obtained in Table 2, where the coupling $a\lambda$ has already been tuned in accordance with an argument given in the next section. In Table 3, we list the numerical results of the Witten index in Eq. (2.12), $\Delta=2$, and the one-point SUSY Ward–Takahashi identity Catterall:2001fr (see also Ref. Morikawa:2018ops )

[TABLE]

Whether $\Delta$ and $\delta$ are numerically reproduced indicates the quality of our configurations.

4 Scaling dimension

4.1 Susceptibility of the scalar field

To numerically determine the scaling dimension, we first explain the finite-size scaling analysis in Refs. Kawai:2010yj ; Kamata:2011fr , which is compatible with the continuum limit as we will develop later. Let us consider the susceptibility of the scalar field $A$, defined by Kawai:2010yj

[TABLE]

In the IR limit, the scalar field is expected to behave as a chiral primary field with the conformal dimensions $(h,\bar{h})$; the two-point function of $A$ behaves as

[TABLE]

for large $|x|=\sqrt{x^{2}}$. Note that $h+\bar{h}$ is called the scaling dimension, and $h-\bar{h}$ is the spin. Now suppose that the field $A$ is spinless, $h=\bar{h}$. Then, we observe the finite-volume scaling of the scalar susceptibility for large $L_{\mu}$, as

[TABLE]

Numerically simulating the scalar correlator for some different volumes but the same value of the coupling, one can read the exponent, $1-h-\bar{h}$, from the slope of $\ln\chi(L_{\mu})$ 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 $L=L_{0}=L_{1}$.

4.2 Continuum limit of the susceptibility

As already announced, we consider the thermodynamic and continuum limits, $a/L\to 0$. No extrapolation has been done in the preceding numerical studies. In Refs. Kawai:2010yj ; Kamata:2011fr ; Morikawa:2018ops ; Morikawa:2018zys , the grid size $L/a$ is expected to be taken as sufficiently large values, while the coupling $\lambda$ in the superpotential in Eq. (2.1) is fixed by $a\lambda=0.3$; 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 obvious. In fact, we will find that the susceptibility, $\chi(L)$, 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 thermodynamic and continuum limits.

We have also recognized the pathology of the locality in the lattice formulation that is based on the SLAC derivative; the computation of $\ln\chi(L)$ with finite $L/a$ is quite sensitive to this problem Kamata:2011fr ; Morikawa:2018ops (see also Sect. 4.4). A proposal given in Ref. Morikawa:2018ops is to directly study the correlation function in the momentum space, $\langle A(p)A^{*}(-p)\rangle$. 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 $L/a\to\infty$ limit appears not quite smooth Morikawa:2018ops .444The central charge, which can be measured by computing the energy-momentum tensor correlator $\langle T_{zz}(p)T_{zz}(-p)\rangle$, appears to possess a higher convergence speed than the scaling dimension, though the approach to $L/a\to\infty$ is also not quite smooth Morikawa:2018ops ; Morikawa:2018zys . We would need a more systematic method for the infinite-volume and continuum limits, while the locality should be restored in the limits.

Our strategy for the continuum limit is very similar to that in Ref. Luscher:1991wu . We regard $\ln\chi(L)$ as the same kind of running coupling $\bar{g}^{2}(L)$ 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 $a_{i}\lambda$ is tuned so that $\ln\chi(L)$ (or $\bar{g}^{2}(L)$) is kept fixed; we denote $u=\ln\chi(L)$. A system with a different grid size $L^{\prime}/a^{\prime}\neq L/a_{i}$ and the same parameter $a^{\prime}\lambda^{\prime}=a_{i}\lambda$ possesses the physical box size $L^{\prime}\times L^{\prime}$ with $a^{\prime}=a_{i}$. Then, we compute $\ln\chi(L^{\prime})$ ($\bar{g}^{2}(L^{\prime})$) for $L^{\prime}/a_{i}$ and $a_{i}\lambda$; we observe the $a$-dependence of $\ln\chi(L^{\prime})|_{a}$ ($\bar{g}^{2}(L^{\prime})|_{a}$), and attempt to extrapolate this in the continuum limit, $\lim_{a\to 0}\ln\chi(L^{\prime})|_{a}$.

To be more specific, we introduce the scaling function $\Sigma$ as

[TABLE]

The statistical error of $\Sigma$ would be given by the square root of the sum of squared errors of $\ln\chi(L)$ and $\ln\chi(sL)$, owing to the long-distance behavior in Eq. (4.3). As a consequence of the continuum limit with a to-be-determined fit function, we can obtain the scaling dimension

[TABLE]

The cutoff dependence will be determined from numerical results. Note that the unique mass scale $\lambda$ in this model should be sufficiently larger than $1/L$ to study the conformal behavior Kawai:2010yj , hence $\lambda L\to\infty$ as the continuum limit. This indicates that the extrapolation carries out the thermodynamic limit at the same time. We can apply our extrapolation method to the continuum limit to other non-perturbative formulations, for example the lattice formulation in Ref. Kawai:2010yj .

4.3 Numerical measurement of the scaling dimension

In this subsection, we perform precision measurement of the scaling dimension for the $A_{2}$-type theory with the cubic superpotential $\Phi^{3}$ by using the above continuum-limit analysis. In Sect. 3 we had already summarized our parameter set and the classification of the obtained configurations.

We tabulate the numerical results of the scalar susceptibility with the various box sizes of $L$ and $L^{\prime}=2L$ in Table 4. The third column is devoted to the tuned values of the coupling, $a\lambda$, so that $\ln\chi(L)$ in the fourth column is kept almost fixed. The results of $\Sigma(u,a/L)$ are shown in the last column, where we have omitted the first argument $s=2$ of $\Sigma(s,u,a/L)$, while we set $u=\ln\chi(L)$ as $3.9175$. The error of $\Sigma(u,a/L)$ is given by the square root of the sum of the squared errors of $\ln\chi(L)$ and $\ln\chi(L^{\prime})$.

In Ref. Kamata:2011fr the scaling dimension was obtained from the slope of the susceptibility in the formulation by using data for $24\leq L/a\leq 36$ or $26\leq L/a\leq 36$ with a fixed coupling; we have a similar slope of $\ln\chi$ for $(L/a,L^{\prime}/a)=(24,48)$, though we have used different values of $a\lambda$ (see Table 5). We will find a significant difference between such numerical results at a finite cutoff and our result below at $a/L=0$.

Now we have enough data to clarify the $(a/L)$-dependence of $\Sigma(u,a/L)$. Figure 1 shows $\Sigma(u,a/L)$ as a function of $a/L$ given in Table 4. From the plot, we simply apply a linear function of $a/L$ in order to take the continuum limit; then we have

[TABLE]

with $\chi^{2}/\text{d.o.f.}=1.417$. From Eq. (4.5), the scaling dimension is given by

[TABLE]

This result is consistent with the expected exact value $1-h-\bar{h}=2/3=0.6666\dots$ within the statistical error.

Because the quality of configurations with $L/a=52$ is poorer due to the computational cost (see Sect. 3), the computation of $\ln\chi$ could be less accurate. In fact, the above result in Fig. 1 implies that there is a discrepancy between the central values of $\ln\chi(L)$ and the fit function at $L/a=52$. To make sure that this discrepancy comes from statistical fluctuations, we show the behavior of $\ln\chi(L)$ for $L/a=52$ when the number of configurations varies in Table 6; the deviation of the central values decreases.

To estimate the systematic error, we may omit the configurations for $L/a=52$; that is,

[TABLE]

with $\chi^{2}/\text{d.o.f.}=0.807$; we obtain

[TABLE]

The main result of the scaling dimension in this paper is given by

[TABLE]

Here, the 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).

4.4 Discussion on the fit function

We found that a linear fit of $\Sigma(s,u,a/L)$ 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 for the scaling dimension from same data. If the two results are 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 $\langle A(x)A(0)\rangle$ in $\ln\chi(L)$ in Eq. (4.1) Kawai:2010yj . The modified scalar susceptibility $\tilde{\chi}$ is defined by

[TABLE]

The coupling $\lambda$ is the unique mass scale in the WZ model with the superpotential in Eq. (2.1), and the correlations at short lengths $\sim\lambda^{-1}$ would not affect the scaling in Eq. (4.3) of $\chi(L)$ in low-energy regions. Note that the shape of the excised space is slightly different from those in Refs. Kawai:2010yj ; Kamata:2011fr , 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 sensitivity to the excised space size; this is the problem that the susceptibility in Ref. Kamata:2011fr is quite sensitive to the UV ambiguity. In terms of the Fourier modes of $A$, we have

[TABLE]

where $|p|=\sqrt{p^{2}}$ and $J_{1}$ is the Bessel function of the first kind.

The parameter tuning above indicates that the dimensionless coupling $a\lambda$ becomes large as $L/a\to\infty$, while $\ln\chi(L)$ is kept fixed. That is, in the small-$a$ limit, the volume of the excised space becomes smaller and smaller; we must have completely the same result of the scaling dimension as in the method of Eq. (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 $\ln\tilde{\chi}(L)$ and to determine the fit function, one can justify the numerical determination of the scaling dimension from $\Sigma$. In the same way as $\ln\chi(L)$, we define the new scaling function $\tilde{\Sigma}$ by

[TABLE]

Here, $u$ is given by the fixed number $\ln\chi(L)$, which is identical to the value of $\ln\tilde{\chi}(L)$ in the continuum limit, that is, $\lambda^{-1}\to 0$. Similarly, one can measure the scaling dimension by Eq. (4.5) with $\tilde{\Sigma}$ and another to-be-determined fit function.

From the $\tilde{\Sigma}(u,a/L)$-$(a/L)$ plot in Fig. 2 we obtain the fitted quadratic curves

[TABLE]

with $\chi^{2}/\text{d.o.f.}=1.600$, or

[TABLE]

with $\chi^{2}/\text{d.o.f.}=1.423$. These fitting results give the scaling dimension as

[TABLE]

respectively. These two results are consistent with our previous result in Eq. (4.10). We have obtained the precise and reliable result in Eq. (4.10) through the finite-size scaling with the continuum-limit extrapolation.

5 Conclusion

In this paper we numerically studied the IR behavior of the $2$D $\mathcal{N}=2$ WZ model with the cubic superpotential, which is believed to provide the Landau–Ginzburg description of the $A_{2}$ minimal model of the $2$D $\mathcal{N}=2$ SCFT. To take the continuum and infinite-volume limits, we developed a systematic extrapolation method for the scalar susceptibility $\chi(L)$; this method is applicable to various non-perturbative formulations of the model. Then, from the numerical simulation of $\chi(L)$ on the basis of the SUSY-invariant formulation with a momentum cutoff, we performed the precision measurement of the scaling dimension through the finite-size scaling analysis. The result of the scaling dimension in Eq. (4.10) is rather consistent with the conjectured WZ/SCFT correspondence.

As shown in Table 5 and Fig. 1, we observed a significant difference between our net result and the ones at any finite $L/a$. The scalar susceptibility takes a slow approach to the $a/L=0$ limit, at least in the present formulation. By using our extrapolation analysis, we can get down to the target SUSY continuum theory with the infinite volume; from a numerical simulation based on the formulation by Kadoh and Suzuki, we obtained the limiting value for the simplest $A_{2}$ theory. This result not only has a smaller margin of error in the numerical value, but also would be much more reliable than those of preceding studies, which were computed at finite $L/a$; it shows a coherence picture being quite consistent with the theoretical conjecture.

Our result seems to support the restoration of the locality in the continuum limit. The UV ambiguity in $\chi(L)$ with finite $L/a$, that is, the sensitivity to the excised space size $\sim\lambda^{-1}$ around the contact point, has disappeared because $\lambda^{-1}\to 0$ in the limit. We indeed found that the results in Eq. (4.16) based on the excision prescription are consistent with Eq. (4.10) without the excision. Also, in addition to the earlier numerical simulations based on the present formulation, it would be exemplified by good agreement between Eq. (4.10) and the expected value that the momentum-cutoff regularization in the $2$D theory works quite well. However, the theoretical background of our computational approach is still not clear, so we should observe the locality restoration more carefully; this is a future problem.

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.

Acknowledgments

We would like to thank Sinya Aoki, Daisuke Kadoh, Yoshio Kikukawa, Taichiro Kugo, Katsumasa Nakayama, and Hiroshi Suzuki for helpful discussions and comments. The numerical computations were partially carried out by the supercomputer system ITO of the Research Institute for Information Technology (RIIT) at Kyushu University. This work was supported by JSPS KAKENHI Grant Number JP18J20935.