Bootstrapping the minimal N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 1 superconformal field theory in three dimensions

We develop the numerical bootstrap technique to study the 2 + 1 dimensional N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 1 superconformal field theories (SCFTs). When applied to the minimal N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 1 SCFT, it allows us to determine its critical exponents to high precision. This model was argued in [1] to describe a quantum critical point (QCP) at the boundary of a 3 + 1D topological superconductor. More interestingly, this QCP can be reached by tuning a single parameter, where supersymmetry (SUSY) is realized as an emergent symmetry. We show that the emergent SUSY condition also plays an essential role in bootstrapping this SCFT. But performing a “two-sided” Padé re-summation of the large N expansion series, we calculate the critical exponents for Gross-Neveu-Yukawa models at N =4 and N =8.


Introduction and summary
Conformal field theory (CFT) has applications in a wide range of fields in physics, which includes condensed matter physics, statistical mechanics, and string theory. The conformal bootstrap program was initiated in [2,3]. When applied to the special case of two space-time dimensions, thanks to the Virasoro symmetry, the minimal models can be solved exactly [4]. Since the seminal work of [5], the modern numerical bootstrap has become one of the most powerful tools to study higher dimensional (D>2) conformal field theories. In certain cases, the numerical bootstrap results are the world's most precise predictions of the scaling exponents [6][7][8][9][10]. See [11] for a recent review including many other successful applications.
Supersymmetry is a symmetry between bosons and fermions [12][13][14]. Even though it was first proposed to solve the long-standing hierarchy problem in particle physics, it has also recently attracted a lot of attention in condensed matter physics due to the possibility of experimental realization [1,[15][16][17][18][19][20][21]. Typically in these models supersymmetry is realized as an emergent symmetry at the second-order phase transition point. At such a critical point, the symmetry is described by the so-called superconformal field theory (SCFT), where the usual conformal group is enlarged into the superconformal group. In this work, we develop the numerical bootstrap technique to study the 2 + 1 dimensional N = 1 superconformal field theories. This allows us to determine the critical exponents of various models.
In this work, we study the minimal superconformal field theory in 2+1 dimensions. We show another example where numerical bootstrap allows us to determine the scaling exponents to unprecedented precision. The Lagragian of the model is which σ → −σ and ψ → γ 0 ψ. When λ 1 = λ 2 , the model has N = 1 supersymmetry (SUSY) and the Lagrangian can be rewritten into a Wess-Zumino model with superpotential is a real superfield. When λ 1 = λ 2 , SUSY is broken. However, it is expected that the theory would still flow to the supersymmetric fixed point, and SUSY is realized as an emergent symmetry. It was argued in [1] that this fixed point might be realized as a quantum critical point at the boundary of a 3 + 1D topological superconductor. Emergent supersymmetry means that the critical point can be reached by fine-tuning a single physical parameter, which is crucial for experimental realization. This condition is equivalent to saying that the spectrum of the SCFT contains only one relevant scalar operator that is even under time-reversal parity (T-parity). The operator corresponds to the mass term m 2 σ 2 that can be added to the Lagragian (1.1), we need to tune its coefficicent to zero to reach the critical point. We find that when bootstrapping the N =1 Ising model, the physical condition of emergent supersymmetry plays an important role.
To bootstrap this theory, one needs to consider four point correctors σσσσ , and σσ , with being the superconformal descendant of σ ( is a conformal primary). The operator product expansion (OPE) coefficients in σ × σ, σ × and × are related to each other, and the relation is fixed by superysmmetry. This is a generalization of the "long multiplet bootstrap" idea [22] used in two-dimensional superconformal bootstrap. Notice that unlike SCFTs with a higher number of supersymmetry, N =1 SCFTs have no R symmetry and the scaling dimension of Φ can not be determined exactly by analytic methods. Imposing the emergent supersymmetry condition, we can determine ∆ σ to high precision, providing both an upper and lower bound for its value. Our numerical bootstrap work can be viewed as a numerical "proof" of the emergent supersymmetry of the model (1.1). Furthermore, if we assume that there are only two relevant operators that are T-parity odd in the spectrum, the allow region for (∆ σ , ∆ σ ) becomes an isolated island. This helps us determine the critical exponents η σ , η ψ , 1/ν and ω all together. We also calculate the value of C T , which appears in the two-point function for stress-energy tensor.
The model (1.1) belongs to a family of models called the Gross-Neveu models, whose Lagragian is given by a four fermion interaction [23] In D>2, the four fermion interaction is non-renormalizable. The action is the UV completion of the Gross-Neveu models in 2<D<4 [24,25], which are sometimes referred as the Gross-Neveu-Yukawa (GNY) model or the chiral Ising model. Here N counts the number of Majorana fermions. To determine the scaling exponents of the GNY model have been the has been a longtime goal of many theoretical work, either based on -expansion [26][27][28][29][30][31], large N method [32][33][34][35]. The N = 4 and N = 8 models are of special

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interest since they could also be studied using Monte Carlo simulation [27,[36][37][38][39]. Also, the N = 8 GNY model is known to describe the quantum critical point of the semimetal to charge density wave order transition on graphene [40]. It is a nice surprise that we can use the modern numerical bootstrap method to determine the critical exponents of N = 1 special case to high precision. This also allows us to perform a two-sided Padé re-sum the large N critical exponents of (1.4) calculated in [33][34][35], and get the results for the N = 4 and N = 8 GNY models. The result is summarised in table 1 and compared with other methods.

Bootstrap the minimal N = 1 SCFT
Conformal multiplets in N =1 superconformal field theories group themselves into supermultiplets. There are in total four types of multiplets, which we denote as B l + , B l − , F j + and F j − as in [41]. "B/F " tells us whether the super-primary field is bosonic or fermionic. A generic super-multiplet contains four conformal primaries, suppose the superconformal primary has spin l and scaling dimension ∆ 0 , there are two level-1 (super)descendant with ∆ = ∆ 0 + 1/2 and spin l ± 1. There is also a level-2 descendant with ∆ = ∆ 0 + 1 and spin l. The subscritp +/− denote the parity of a bosonic field within the super-multiplet. The super-multiplet B [l] Here we denote a conformal primary field with spin l, scaling dimension ∆ and even parity to be [l] + ∆ . In our convention, the super-field Σ, define in (1.3), is a B (l=0) − multiplet. The four point function of the superfield Σ, when expanded in the Grassmann variable θ, contains the four point functions involving not only the superconformal σ, but also super-descendants ψ and . For simplicity, in this work, we consider only bosonic fields σ and . In [6], the critical exponents of the threedimensional Ising model is determined to high precision by studying the set of crossing equation involving the set of four point functions { σσσσ , σσ , }. The crossing equations turn out to be The vectors V ± were calculated in [6]. For the reader's convenience, we will note their explicit expressions in appendix A. In previous attempts to determine the critical exponents of (1.1), either based on bootstrapping three-dimensional scalar operators [42], or

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fermions [43], the only SUSY constrain taken into account was the relation between the scaling dimension of operators, ∆ = ∆ σ + 1 and ∆ ψ = ∆ σ + 1/2, since they belong to the same supermultiplet Σ. Here we however show that OPE relations are much more powerful, therefore are essential in bootstrapping N =1 SCFTs. Suppose O + and O − are from the same superconformal multiplet, their OPE coefficients λ σσO + , λ O − and λ σ O − are proportional to each other. Their ratios are fixed by supersymmetry and can be determined by considering the θ-expansion of three-point functions of superfields ΣΣO . ΣΣO is determined by superconformal symmetry and can be worked out based on the results of [44]. Plug these OPE relations into (2.4), and collect blocks for O + and O − that belongs to the same multiplet, the crossing equations becomes l ∈ even ∆ and l (or j) are the scaling dimensions and spins of super-conformal primaries. As will be presented explicitly in appendix A, there are four different types of superconformal blocks Each corresponds to a type of super-multiplets appearing in Σ × Σ OPE. Schematically, the OPE looks like As mentioned in the introduction, we are going to assume that the N =1 Ising model has emergent supersymmetry. This is equivalent to requiring the spectrum to contain only one relevant T-parity even scalar. In terms of constraints on the super-multiplets, this amounts to imposing the following conditions − super multiplet (which we denote the superfield as Σ and it super-primary as σ ) to have scaling dimension bigger or equal to ∆ σ . We can use numerical bootstrap to carve out the region in (∆ σ , ∆ σ )-plane that allows an unitary N = 1 SCFT to exist. The result is shown in figure 1, where the maximum number of derivatives for conformal block approximation is Λ = 13. For technical details of numerical bootstrap, we refer to [45]. The sharp spike at ∆ σ ≈ 0.584 indicates an SCFT which we will identify as the 3D N = 1 minimal SCFT. The kink at ∆ σ ≈ 0.96 appears when the bound meets the line ∆ σ = ∆ σ . The identification of this kink to an previously known conformal field theory remains mysterious at the moment. We have checked that if we use higher Λ [45] JHEP06(2021)154 to do the numerics, this kink is pushed to ∆ σ → 1. One preliminary guess is that the kink is related to the existence of SCFT with higher number of supersymmetries, since the supermultiplets multiplets of SCFT with N > 1 may contain protected scalar with scaling dimension ∆ = 1. For example, the N = 2 flavor current multiplet, when viewed as N = 1 multiplets, branches into a conserved current multiplet, and a real scalar multiplet with the scaling dimension ∆ = 1.
If we further assume that Σ and Σ are the only two relevant scalar superfields in the spectrum, the allowed region becomes an isolated island. This is shown in figure 2, where we use the parameters S Λ=27 in [45] for the numerics. The bootstrap island can also be found if one considers only SUSY OPE relations involving scalar operators 1 [46,47]. Notice that the scaling dimension of Σ is higher than 2, so that the corresponding T-parity even descendant is irrelevant. This bootstrap island can be viewed as a non-perturbative numerical "proof" of the emergent supersymmetry of the model (1.1).

Scaling exponents and CFT data
From the island, it is easy to get ∆ σ = 0.584444 (30), corresponding to the critical exponents η σ = η ψ = 0.168888(60), 1/ν = 1.415556 (30). (3.1) Σ contains a super-primary with ∆ σ = 2.882(9) and also a super-descendant which is the lowest dimensional irrelevant T-parity even scalar operator. This helps us determine the critical exponent ω = 0.882 (9). model. We can also compare our results with previous bootstrap determinations of the critical exponents. In [42], it was observed that the SUSY line ∆ = ∆ σ + 1 intersects with the region at ∆ σ = 0.565, this provides a lower bound for η σ > 0.13. In [43], the allowed region touches the SUSY line ∆ ψ = ∆ σ + 1/2 at ∆ σ ≈ 0.582 (though one need to impose the condition ∆ σ ≥ 3). Our results show that N = 1 minimal model is indeed located in this region. We can also calculate the constant C T in the stress-tensor two-point function when it is normalized using the Ward identity. By bootstrapping the OPE coefficient λ 2 F − , with F ∆=5/2,j=3/2 − being the SUSY current multiplet, we get where C f.s.
The resumed result is presented in the first two rows of table 1, and compared with the results using other methods. Our η ψ and η σ are also compatible with various kinks observed by bootstrapping 3D fermions [50]. We focus on the N = 4 and N = 8 models due to possible realization using Monte Carlo simulation [27,[36][37][38][39]. What's more, the N = 8 model corresponds to the quantum critical point of the semimetal to charge density wave order transition on graphene [40].

JHEP06(2021)154 4 Discussion
A Monte Carlo simulation of this N = 1 super-Ising models would certainly be interesting. The Lagrangian (1.1) has an anomaly under time-reversal symmetry. One would possibly need to study such a system at the boundary of a 3 + 1 dimensional lattice, which can make the simulation time-consuming. Another interesting direction, as pointed out in [20], is to construct lattice models with non-local interaction. One important lesson that we have learned from this work is that long multiplet bootstrap [22], hence crossing equation involving super-descendant operators (which are conformal primaries), play important role in numerical bootstrap. In other words, it would be interesting to try imposing similar constraints when bootstrapping SCFT's with higher number of supercharges, extending the works of [51,52].

A Superconformal bootstrap equations
To derive the superconoformal bootstrap equations, we first follow the result [44] to consider the three-point function of two Σ's and a third superfield,

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Here η 1 is an auxiliary two component spinor, valued by commuting numbers. There exist four different t-structures B (l) Performing a series expansion of the three-point function in terms of the Grassmann variables θ 1 , θ 2 and θ 3 , we obtain the SUSY OPE relations. It is also necessary to take into account the fact that the operators from θ expansion are not yet properly normalized. One needs to fix this by expanding the two-point functions in Grassmann variables, and then read out the operators' normalization and scaling the OPE relations properly. The convention in this work is the same as [11]. The calculation is in the same spirit of [54], where N = 1 superconformal block in four dimensions was calculated.
A similar calculation can be done for F Notice an interesting feature in the calculation is that only one conformal primary of the super multiplet appears in the σ × σ OPE (or σ × OPE).
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