The Hybrid Bootstrap

Ning Su
Department of Physics, University of Pisa, I-56127 Pisa, Italy

###### Abstract

Finding a method to combine the numerical bootstrap with the analytic
lightcone bootstrap is an important goal to advance the conformal bootstrap
program. We propose a hybrid bootstrap method to do just that. The numerical
and analytic bootstrap approaches are sensitive to different regions of the
spectrum and complement each other. When they are effectively combined, the
hybrid bootstrap enjoys the best of both worlds and the prediction for the
actual CFT can be significantly improved. In this work, we discuss the general
strategy to perform such a hybrid bootstrap, and we make a partial
implementation of the strategy for 3D Ising CFT $\\{\sigma,\epsilon\\}$
system. Even at relatively low derivative order $\Lambda=19$, the hybrid
bootstrap predicts very precise values for the scaling dimension
$\Delta_{\sigma},\Delta_{\epsilon}$ that are within the previous $\Lambda=43$
rigorous error bars.

###### Contents

  1. 1 Introduction
  2. 2 Review of the numerical bootstrap and analytic bootstrap
    1. 2.1 Numerical bootstrap
    2. 2.2 Analytical bootstrap
  3. 3 The hybrid bootstrap
    1. 3.1 General strategy
    2. 3.2 Alternative approaches
    3. 3.3 Bootstrapping the 3D Ising CFT
    4. 3.4 Why it works? How to determine $L_{0}$?
    5. 3.5 How to improve further?
    6. 3.6 A preliminary attempt on estimation of the error bar
    7. 3.7 Balance out multiple analytics with different accuracy
  4. 4 Conclusion and outlook
  5. A Code availability
  6. B Technical details
  7. C Collection of results

##  1 Introduction

The conformal bootstrap program has achieved much progress, both numerically
and analytically, in the past few years. On the numerical side, critical
exponents of various CFTs have been determined precisely with rigorous error
bars111See [1] for a thorough review. See [2, 3, 4, 5, 6] for some recent
computations, where rigorous error bars are determined in various CFTs.. On
the analytic side, tools have been developed to solve bootstrap equations in
the lightcone limit $z\ll 1-\bar{z}\ll 1$[7, 8, 9, 10, 11, 12, 13, 14, 15,
16]222See also [17] for a review of the analytic bootstrap.. The result is
that the conformal data of the leading twist operators can be accurately
approximated using the data of low lying operators.

The numerical and analytic approaches are sensitive to different regions. In
the numerical bootstrap, we project the conformal block function
$G_{\Delta,\ell}\left(z,\bar{z}\right)$ to a finite dimension vector by
expanding it around the crossing symmetric point $z=\bar{z}=1/2$ as
$G^{(m,n)}=\partial_{z}^{m}\partial_{\bar{z}}^{n}G_{\Delta,\ell}\left(z,\bar{z}\right)|{}_{z=\bar{z}=1/2}$.
However, in the lightcone limit, we know
$G_{\Delta,\ell}\left(z,\bar{z}\right)\propto z^{\Delta/2}\log(\bar{z})$. It
would be difficult to capture the $\text{log}(\bar{z})$ divergence using the
Taylor expansion at $\bar{z}=1/2$. On the other hand, the analytic bootstrap
predicts the large spin operators have to exist in order to match the
$\text{log}(\bar{z})$ divergence in the lightcone limit. This is just the
shortcomings of the numerical bootstrap, because these large spin operators
are not visible to the numerical methods : using the extreme functional method
(EFM)[18], the numerically determined spectrum only has accurate information
for the small spin operators even if we start with a large spin set in the
setup.

To better improve the prediction from the numerical bootstrap, we should make
the numerics converge not just around $z=\bar{z}=1/2$, but also more uniformly
in other regions of $z,\bar{z}$. To achieve that, a natural idea is to inject
the information from the analytic lightcone bootstrap, which is valid around
the lightcone limit $z\ll 1-\bar{z}\ll 1$, into the numerical bootstrap. In
this work, we propose a hybrid bootstrap method to realize this
idea333Alternative approaches might be to follow [19] or [20].. A rough idea
about the hybrid method was actually already suggested in [13]. But in the
past, there were two difficulties in the implementation of the hybrid
bootstrap. One is that the lightcone analytics is parameterized by a few
variables that one has to scan over, but numerically scanning over a higher
dimensional parameter space was forbiddingly expensive444The scan problem was
partially solved in [4], where the surface cutting algorithm was invented to
handle the space of the OPE coefficient. But this algorithm is not suitable
for the generic parameter space that involves the scaling dimensions.. The
second difficulty was that the lightcone analytics is not exact, which
contradicts the previous paradigm of the numerical bootstrap : the bootstrap
numerics works in the “feasibility mode”, i.e. we make an assumption about the
CFT spectrum, then the numerics rigorously rule out the spectrum if it
contradicts with the bootstrap equation. In this feasibility mode, the
bootstrap numerics can’t distinguish a “less wrong parameter” from “a more
wrong parameter” — they are all just infeasible parameters. Therefore it was
not obvious how to incorporate non-rigorous information in a controllable way.
Fortunately, recently the navigator function [21] was developed, where the
binary information “feasible/infeasible” was replaced by a continuous measure
of success. With this technique, both difficulties can be effectively handled.
In this paper, we will investigate the hybrid bootstrap method based on the
navigator function.

Figure 1: In the hybrid numerics, the accuracy of the small spin is improved,
while the large spin operators matches with the analytics.

A summary of the content and the structure of this paper is the following. In
Section 2, we briefly review the numerical bootstrap using the navigator
technique, and lightcone analytic bootstrap in the language of the inversion
formula. We compute a few ingredients which are needed for the later sections.
In Section 3.1 and 3.2, we discuss the general strategy of the hybrid
bootstrap. We also show a few alternative approaches and explain why we choose
our main approach. The key idea is that the numerical bootstrap is accurate
for low twist operators with small spins, while the analytic bootstrap work
better for the large spins, therefore we should trust each approach in the
region where it has an advantage and glue the spectrum in some intermediate
range of the spin. The gluing process effectively single out an unique point
in the theory space whose leading twist operators match with the analytics as
much as possible. At that point, the errors for a wide range of spins are
improved, as the schematic illustration in Figure 1). There are many technical
aspects when we actually try to implement this strategy. For example, how to
decide the intermediate region and how to match the spectrum. From Section 3.3
to 3.7, we demonstrate how to use the hybrid bootstrap in the example of 3D
Ising $\\{\sigma,\epsilon\\}$ system and address those technical issues. We
will see the hybrid bootstrap yields very accurate predictions for the Ising
CFT data, even at a low derivative order.

##  2 Review of the numerical bootstrap and analytic bootstrap

###  2.1 Numerical bootstrap

The bootstrap equation of the 3D Ising $\\{\sigma,\epsilon\\}$ system is

$\displaystyle\sum_{{\cal O}^{+}}\begin{pmatrix}f_{\sigma\sigma{\cal
O}}&f_{\epsilon\epsilon{\cal
O}}\end{pmatrix}\vec{V}^{(+)}_{\Delta,\ell}\begin{pmatrix}f_{\sigma\sigma{\cal
O}}\\\ f_{\epsilon\epsilon{\cal O}}\end{pmatrix}+\sum_{{\cal
O}^{-}}f_{\sigma\epsilon{\cal O}}^{2}\vec{V}^{(-)}_{\Delta,\ell}$
$\displaystyle=$ $\displaystyle 0,$ (1)

where $\vec{V}^{(+/-)}_{\Delta,\ell}$ is the crossing vectors defined in
(3.12) of [22].

To use the navigator method, we take the parameter space to be
$(\Delta_{\sigma},\Delta_{\epsilon},\theta)$ with
$\theta=\text{tan}^{-1}(f_{\epsilon\epsilon\epsilon}/f_{\sigma\sigma\epsilon})$,
and on each point of the parameter space, we define a semi-definite program
(SDP) as:

$\displaystyle objective=\vec{\alpha}.(\vec{V}^{(+)}_{\Delta=0,\ell=0}),$
$\displaystyle conditions:$
$\displaystyle\vec{\alpha}\cdot(\vec{V}^{(\theta)})\geq 0,$
$\displaystyle\vec{\alpha}\cdot(\vec{V}^{(+)}_{\Delta,\ell=0})\geq 0\text{ for
}\Delta\geq 3,$
$\displaystyle\vec{\alpha}\cdot(\vec{V}^{(-)}_{\Delta,\ell=0})\geq 0\text{ for
}\Delta\geq 3,$
$\displaystyle\vec{\alpha}\cdot(\vec{V}^{(+)}_{\Delta,\ell})\geq 0\text{ for
}\Delta\geq\Delta_{unitary}\text{ and }\ell=2,4...,$
$\displaystyle\vec{\alpha}\cdot(\vec{V}^{(-)}_{\Delta,\ell})\geq 0\text{ for
}\Delta\geq\Delta_{unitary}\text{ and }\ell=1,2,...,$ (2)

where

$\displaystyle\vec{V}^{(\theta)}=\vec{V}^{(+)}_{\Delta_{\epsilon},0}+\vec{V}^{(-)}_{\Delta_{\sigma},0}\otimes\left(\begin{array}[]{cc}1&0\\\
0&0\end{array}\right).$ (5)

and $\alpha$ is a linear functional acting on the crossing vectors. For the
normalization of the SDP, we use the $\Sigma$-navigator normalization [21].
Specifically the normalization is defined by summing the crossing vectors in
each channel at 20 equally spaced $\Delta_{i}$ between the first gap and the
gap+6 (for above SDP, the gap is $\Delta_{unitary}$ for $\ell>0$ and 3 for
$\ell=0$).

The SDP can be solved using the sdpb program [23] and the optimal value of the
objective defines a navigator function ${\cal
N}(\Delta_{\sigma},\Delta_{\epsilon},\theta)$. The positive/negative sign of
the navigator function indicates the parameter
$(\Delta_{\sigma},\Delta_{\epsilon},\theta)$ is infeasible/feasible. We can
use Algorithm 2 of [21] to maximize/minimize certain parameters in the
feasible region or use Algorithm 1 of [21] to minimize the navigator value
${\cal N}(\Delta_{\sigma},\Delta_{\epsilon},\theta)$. After the
maximizing/minimizing procedure, we obtain an unique point in the parameter
space. In this paper, we may refer to this unique point as the optimal point,
or simply the result of the optimization. As shown in [21], the result of the
navigator minimization gives a good prediction of the actual CFT. Throughout
this paper, we refer this computation as

######  Numerics Setup 1.

Minimize the navigator function ${\cal
N}(\Delta_{\sigma},\Delta_{\epsilon},\theta)$ subject to the SDP condition of
(2).

In Figure 2, we present the $(\Delta_{\sigma},\Delta_{\epsilon})$ from the
navigator minimization for $\Lambda=11,19,27,35,43$. The technical details of
the computation (such as the choice of various parameters) are presented in
the Appendix B, while the full result of all computations in this paper are
presented in the Appendix C.

Figure 2: The locations of the minimum navigator at $\Lambda=11,19,27,35,43$
from the computation of the Numerics Setup 1. The light blue curve on the
right figure is the boundary of the $\Lambda=43$ Ising island of [2].

The $\Lambda=43$ result gives the best prediction of the actual CFT:

$\displaystyle\Delta_{\sigma}\approx
0.518148884,\qquad\Delta_{\epsilon}\approx 1.41262383,\qquad\qquad\qquad$
$\displaystyle f_{\epsilon\epsilon\epsilon}\approx 1.05185442,\qquad
f_{\sigma\sigma\epsilon}\approx 1.53243407,\qquad f_{T}\approx 1.25885570$ (6)

where
$\Delta_{\sigma},\Delta_{\epsilon},f_{\epsilon\epsilon\epsilon}/f_{\sigma\sigma\epsilon}$
is obtained from the minimal navigator point while the rest of the data555The
convention of the OPE coefficient and the block in this paper is defined in
the following way. We refer the block convention of the n-th row of the Table
1 of [1] as “Convention n”. We say the OPE coefficient is in Convention n if
the OPE coefficient is defined by $G(u,v)=\sum_{\cal O}f^{2}g_{\Delta_{{\cal
O}},\ell_{\cal O}}(u,v)$ where $G$ is the four-point function and $g$ is the
block in Convention n. The block and OPE coefficient are in the Convention 1
throughout this paper except Section 2.2. For Section 2.2, the block and OPE
coefficient is in the Convention 6, because this convention is more convenient
for the analytic computation. When using the result of Section 2.2 in the
numerics, one has to first convert the convention accordingly. is extracted
using the extremal functional method (EFM)[18]. The $f_{T}$ is defined by
$f_{T}=f_{\sigma\sigma T}/\Delta_{\sigma}=f_{\epsilon\epsilon
T}/\Delta_{\epsilon}$ using the Ward identity666The data from our numerics has
a small difference between $f_{\sigma\sigma
T}/\Delta_{\sigma},f_{\epsilon\epsilon T}/\Delta_{\epsilon}$. What we present
in (2.1) is the mean value..

The spectrum from the EFM at the minimal navigator point has a good quality.
In the ${\mathbb{Z}}_{2}$ even channel, we observed there are fake operators
with dimensions exactly at the unitarity bound, due to the sharing effect [13,
16]. The dimension and OPE coefficient of our data for $[\sigma\sigma]_{0}$ is
obtained by a cleanup procedure
$\Delta_{\text{eff}}=(\sum_{\mathcal{O}}f_{\mathcal{O}}^{2}\Delta_{\mathcal{O}})/(\sum_{\mathcal{O}}f_{\mathcal{O}}^{2})$
and $f_{\text{eff}}=\sqrt{\sum_{\mathcal{O}}f_{\mathcal{O}}^{2}}$, where
$\sum_{\cal O}$ is a summation over all operators with twist smaller than 1.1
in each spin. After the cleanup, both the dimension and the OPE coefficient
matches much better with the analytics in Section 2.2. On the other hand, the
situation in the ${\mathbb{Z}}_{2}$-odd channel is more complicated. We
present the raw data for operators in ${\mathbb{Z}}_{2}$-odd channel with
twist $\tau<2$ in Figure 3. There are some operators below
$[\sigma\epsilon]_{0}$ with OPE coefficient roughly two orders of magnitude
smaller than the OPE coefficient of $[\sigma\epsilon]_{0}$. Whether those
operators are physical operators or fake operators due to the sharing effect,
is not clear. In fact, the $[\sigma\epsilon]_{0}$ shouldn’t be the lowest
twist family in the ${\mathbb{Z}}_{2}$ odd channel. The lowest should be
$[\sigma T]_{0}$, because $\tau_{\sigma}>\tau_{T}$777We thank Johan Henriksson
for pointing out this fact and sharing some results of their ongoing work[24].
So the blue circles in Figure 3 could be related to physical
operators888However according to [25], $[\sigma T]$ shouldn’t exist for
$\ell<6$ in the $4-\epsilon$ expansion.. Another observation is that if we
process the raw data in the same way as the ${\mathbb{Z}}_{2}$ even channel,
the result is further away from the analytics in Section 2.2. Therefore we
choose to clean up the sharing effect for $[\sigma\sigma]_{0}$, while for
$[\sigma\epsilon]_{0}$ we use the raw data. In later sections, when we refer
to “$\Lambda=43$ EFM spectrum”, we always mean the data obtained in this way.

Figure 3: The spectrum in ${\mathbb{Z}}_{2}$-odd channel with twist $\tau<2$.
The red circles are the operators in the $[\sigma\epsilon]_{0}$ family, while
the blue circles are the rest. The size of the circle is proportional to
$1/|\text{log}(\lambda_{\sigma\epsilon[\sigma\epsilon]_{0}})|$.

###  2.2 Analytical bootstrap

In [13], it was demonstrated how to solve the Ising bootstrap equation in the
lightcone limit $z\ll 1-\bar{z}\ll 1$. The conclusion is that the conformal
data of the leading twist $[\sigma\sigma]_{0,\ell},[\sigma\epsilon]_{0,\ell}$
at large spin can be effectively approximated using only five parameters
$\\{\Delta_{\sigma},\Delta_{\epsilon},f_{\epsilon\epsilon\epsilon},f_{\sigma\sigma\epsilon},f_{T}\\}$,
where $f_{T}=f_{\sigma\sigma T}/\Delta_{\sigma}=f_{\epsilon\epsilon
T}/\Delta_{\epsilon}$. Alternatively, one can use the inversion formula [15,
26] to do similar computations [16]. The difference is that, certain non-
perturbative term that is relevant to small $\ell$ is omitted in the lightcone
bootstrap[14], while the inversion formula approach has the full contribution
of an isolated operator, thus is more accurate for small $\ell$. On the other
hand, the result of [13] is given in term of simple $\Gamma$ function and can
be evaluated efficiently, while the formalism of [16] contains hypergeometric
function ${}_{4}F_{3}$ or ${}_{3}F_{2}$ and is computationally very expensive.

In this section we briefly review some of the results of [13, 16]. We then
derive an improved version for $[\sigma\sigma]_{0,\ell}$, which will be used
later. Our goal is actually slightly different from [27, 13, 16, 28]. For the
purpose of hybrid bootstrap, the analytics has to simultaneously meet (1),
computational efficiency; (2), accuracy at large spin; (3), dependence only on
a few parameters (at most
$\\{\Delta_{\sigma},\Delta_{\epsilon},f_{\epsilon\epsilon\epsilon},f_{\sigma\sigma\epsilon},f_{T}\\}$).
On the other hand, the accuracy at small spins doesn’t matter too much for us.
The reason will be clear later.

Let’s consider the correlator $\langle\sigma\sigma\sigma\sigma\rangle$.
Following the formalism and notations of [16], we define a generating function
that capturing all the information in the $\sigma\times\sigma$ OPE:

$\displaystyle
C(z,\bar{h})=\sum_{\mathcal{O}\in\sigma\times\sigma}\lambda_{\sigma\sigma\mathcal{O}}^{2}(\bar{h}_{\mathcal{O}})z^{h_{\mathcal{O}}}$
(7)

where $h=\frac{\Delta-\ell}{2},\bar{h}=\frac{\Delta+\ell}{2}$ and the
$\lambda$ is related to the OPE coefficient that we use in the numerical
bootstrap by
$f_{\sigma\sigma\mathcal{O}}^{2}(\bar{h}_{\mathcal{O}})=(1-\frac{\partial
h_{\mathcal{O}}(\bar{h})}{\partial\bar{h}}){}^{-1}\lambda_{\sigma\sigma\mathcal{O}}^{2}(\bar{h}_{\mathcal{O}}).$

The inversion formula prediction that the contribution of a single operator
$\mathcal{O}^{\prime}$ to the t-channel generating function
$C_{\mathcal{O}^{\prime}}^{t}\left(z,\bar{h}\right)$ at small $z$ is

$\displaystyle C_{\mathcal{O}^{\prime}}^{t}(z,\bar{h})\approx
2\sin(\pi(h_{\mathcal{O}^{\prime}}-2h_{\sigma})){}^{2}\sum_{p=0}^{\infty}\sum_{q=-p}^{p}\mathcal{A}_{p,q}^{h_{32},h_{14}}(h_{\mathcal{O}^{\prime}},\bar{h}_{\mathcal{O}^{\prime}})\frac{z^{h_{1}+h_{2}}k_{\bar{h}_{\mathcal{O}^{\prime}}+q}^{h_{32},h_{14}}(1-z)}{(1-z)^{h_{2}+h_{3}}}\kappa_{2\bar{h}}\Omega_{\bar{h},h_{\mathcal{O}^{\prime}+p},h_{2}+h_{3}}^{h_{1},h_{2},h_{3},h_{4}}.$
(8)

where the function $\mathcal{A},k,\kappa,\Omega$ is defined in [16]. At large
$\bar{h}=h_{5}$ the $\kappa\Omega$ in (8) can expand to asymptotic series
999This is essentially (G.1) of [16], except we flipped the $\Gamma$ function
for easier numerical implementation. :

$\displaystyle(\kappa\Omega)_{h_{5},h_{6},h_{2}+h_{3}}^{h_{1},h_{2},h_{3},h_{4}}=$
$\displaystyle\sum_{m=0}^{\infty}\frac{\Gamma(m-h_{2}-h_{3}+h_{6}+1)\Gamma(m-h_{1}-h_{4}+h_{6}+1)}{\pi^{2}}\times$
(9)
$\displaystyle\frac{\Gamma(-h_{1}+h_{2}+h_{5})\Gamma(-h_{3}+h_{4}+h_{5})\Gamma(-m+h_{1}+h_{3}+h_{5}-h_{6}-1)}{\Gamma(2h_{5}-1)\Gamma(m-h_{1}-h_{3}+h_{5}+h_{6}+1)}\times$
$\displaystyle\frac{(-1)^{m}\Gamma(2h_{6})\Gamma(m+h_{2}-h_{3}+h_{6})\Gamma(m-h_{1}+h_{4}+h_{6})}{2m!\Gamma(h_{2}-h_{3}+h_{6})\Gamma(-h_{1}+h_{4}+h_{6})\Gamma(m+2h_{6})}.$

If we have information of a set of operators $\\{\mathcal{O}^{\prime}\\}$, we
can use them to approximate the t-channel generating function as
$C^{t}(z,\bar{h})\approx\sum_{\mathcal{O}^{\prime}}\lambda_{\sigma\sigma\mathcal{O}^{\prime}}^{2}C_{\mathcal{O}^{\prime}}^{t}(z,\bar{h})$
and the full generating function $C(z,\bar{h})=2C^{t}(z,\bar{h})$. We can
extract the dimension and OPE coefficient of the leading twist family
$[\sigma\sigma]_{0}$ by matching our approximated generating function
$C(z,\bar{h})$ with the ansatz
$C(z,\bar{h})=C_{[\sigma\sigma]_{0}}(\bar{h})z^{2h_{\sigma}+\delta
h_{[\sigma\sigma]_{0}}(\bar{h})}+\text{...}=C_{[\sigma\sigma]_{0}}(\bar{h})z^{2h_{\sigma}}(1+\log(z)\delta
h_{[\sigma\sigma]_{0}}+\text{...})+\text{...}$. This matching is of course
can’t be exact and we will use the recipe in [13, 14] : if our approximated
generating function has the form of
$C(z,\bar{h})=z^{2h_{\sigma}}(A(\bar{h})+B(\bar{h})\log(z))$, we extract the
OPE coefficient and dimension as

$\displaystyle\delta
h_{[\sigma\sigma]_{0,\ell}}(\bar{h})=\frac{B(\bar{h})}{A(\bar{h})+\log(z_{0})B(\bar{h})},\qquad
f_{\sigma\sigma[\sigma\sigma]_{0,\ell}}^{2}(\bar{h})=A(\bar{h})$ (10)

for a choice of $z_{0}\in(0,1]$. Now we solve the equation

$\displaystyle\Delta_{[\sigma\sigma]_{0,\ell}}=2\Delta_{\sigma}+\ell+2\delta
h_{[\sigma\sigma]_{0,\ell}}(\bar{h})$ (11)

for $\Delta_{[\sigma\sigma]_{0,\ell}},\bar{h}_{[\sigma\sigma]_{0,\ell}}$ and
plug back to (10) to get $f_{\sigma\sigma[\sigma\sigma]_{0,\ell}}$.

If we take the operator set $\\{\mathcal{O}^{\prime}\\}$ to be
$\\{\epsilon,T\\}$ and $p=q=m=0$ in (8), (9), $z_{0}=1$ in (10), we recover
the lightcone bootstrap result (6.1) of [13]. We show the percentage
difference of this result to $\Lambda=43$ result in Figure 4 (the blue dots).
There is an almost constant $1.8\%$ discrepancy at larger spins, which doesn’t
meet our goal. Now let’s try to understand why.

Consider a generic set of operators $\\{\mathcal{O}^{\prime}\\}$. Expand
$\lambda_{\sigma\sigma[\sigma\sigma]_{0}}^{2},\delta h(\bar{h})$ in large
$\bar{h}$ limit and keep only the leading term for each operator
$\mathcal{O}^{\prime}$:

$\displaystyle\delta
h(\bar{h})\approx-\sum_{\mathcal{O}^{\prime}}\lambda_{\sigma\sigma\mathcal{O}^{\prime}}^{2}\frac{\Gamma(2h_{\sigma})^{2}\Gamma(2\bar{h}_{\mathcal{O}^{\prime}})}{\Gamma(2h_{\sigma}-h_{\mathcal{O}^{\prime}})^{2}\Gamma(\bar{h}_{\mathcal{O}^{\prime}})^{2}}\bar{h}^{-2h_{\mathcal{O}^{\prime}}}$
(12)
$\displaystyle\lambda_{\sigma\sigma[\sigma\sigma]_{0}}^{2}/\lambda_{\text{GFF}}^{2}\approx(1-\sum_{\mathcal{O}^{\prime}}\lambda_{\sigma\sigma\mathcal{O}^{\prime}}^{2}\frac{\Gamma(2\bar{h}_{\mathcal{O}^{\prime}})}{\Gamma(\bar{h}_{\mathcal{O}^{\prime}})^{2}}\frac{\Gamma(2h_{\sigma})^{2}}{\Gamma(2h_{\sigma}-h_{\mathcal{O}^{\prime}})^{2}}2H_{\bar{h}_{\mathcal{O}^{\prime}}-1}\bar{h}^{-2h_{\mathcal{O}^{\prime}}})$
$\displaystyle\lambda_{\text{GFF}}^{2}\approx\frac{\sqrt{\pi}}{\Gamma(2h_{\sigma})^{2}}2^{3-2\bar{h}}\bar{h}^{-\frac{3}{2}+4h_{\sigma}}$

where $H$ is the harmonic number. We see that at large spin, the leading
contributions come from operators with small twists. In
${\mathbb{Z}}_{2}$-even channel, $\\{\epsilon,T,[\sigma\sigma]_{0,\ell}\\}$
are the small twist operators with $\tau<1.5$. Therefore potentially they can
all affect the large spin result, so we should sum up the contribution from
$[\sigma\sigma]_{0,\ell}$. Following the double twist improvement (DTI)
procedure of [14]), a single operator in
$\mathcal{O}^{\prime}\in[\sigma\sigma]_{0,\ell}$ contribute to $C^{t}$ as

$\displaystyle
C^{t}(z,\bar{h})_{[\sigma\sigma]_{0,\ell}}\approx\frac{\sqrt{\pi}}{\Gamma(2h_{\sigma})^{2}}2^{4-2\bar{h}_{\mathcal{O}^{\prime}}}\bar{h}_{\mathcal{O}^{\prime}}^{-\frac{3}{2}+4h_{\sigma}}(\lambda_{\sigma\sigma[\sigma\sigma]_{0}}^{2}/\lambda_{\text{GFF}}^{2})(\delta
h_{\mathcal{O}^{\prime}})^{2}$ (13)
$\displaystyle\sum_{p=0}^{\infty}\sum_{q=-p}^{p}\hat{\mathcal{A}}_{p,q}^{0,0}(2h_{\sigma})\frac{z^{2h_{\sigma}}k_{\bar{h}_{\mathcal{O}^{\prime}}+q}^{0,0}(1-z)}{(1-z)^{2h_{\sigma}}}\kappa\Omega_{\bar{h},2h_{\sigma}+p,2h_{\sigma}}^{h_{\sigma}}$

Now we can expand
$(\lambda_{\sigma\sigma[\sigma\sigma]_{0}}^{2}/\lambda_{\text{GFF}}^{2})(\delta
h_{\mathcal{O}^{\prime}})^{2}$ to a linear combination of
$1/\bar{h}^{\tau}_{i}$. For each term, we use (4.41) of [13] to approximately
sum up the contribution from $\ell=\ell_{0},\ell_{0}+2,...\infty$. For
example, if we pick up a term proportional to
$1/\bar{h}^{2h_{\mathcal{O}^{\prime}_{k}}}$ from
$(\lambda_{\sigma\sigma[\sigma\sigma]_{0}}^{2}/\lambda_{\text{GFF}}^{2})$ and
a term proportional to
$(1/\bar{h}^{2h_{\mathcal{O}^{\prime}_{i}}})(1/\bar{h}^{2h_{\mathcal{O}^{\prime}_{j}}})$
from $(\delta h_{\mathcal{O}^{\prime}})^{2}$, the
summation101010Interestingly, this summation also contributes to the term
$z^{\tau_{tot}}$, which seems to be corresponding to a triple trace operator
$[\mathcal{O}_{i}\mathcal{O}_{j}\mathcal{O}_{k}]$. Similarly for
$[\sigma\epsilon]_{0}$, $\delta h_{[\sigma\epsilon]_{0}}$ is a linear
combination of $\bar{h}^{-h_{\sigma}},1/\bar{h}^{-h_{T}},...$ at large spin.
Taking $\bar{h}^{-(h_{\sigma}+h_{T})}$ term in $\delta
h_{[\sigma\epsilon]_{0}}^{2}$, with the help of (4.47) of [13] we can sum up
the double twist $[\sigma\epsilon]_{0}$ contribution to the generating
function of $\langle\sigma\epsilon\sigma\epsilon\rangle$. This summation
contribute to $z^{h_{\sigma}+h_{T}}$ of the generating function. This seems to
suggests that $[\sigma T]_{0}$ in principle should appear in the bootstrap of
the $\\{\sigma,\epsilon\\}$ system. contributes to the $z^{2h_{\sigma}}$ as

$\displaystyle
C^{t}(z,\bar{h})_{[\sigma\sigma]_{0,\ell=\ell_{0}\text{...}\infty}}\approx
4\pi^{2}D_{\text{tot}}\frac{\Gamma(\Delta_{\sigma}-\frac{\tau_{\text{tot}}}{2})^{2}}{\Gamma(\Delta_{\sigma})^{2}}\sum_{p=0}^{\infty}\sum_{q=-p}^{p}2^{2q}\hat{\mathcal{A}}_{p,q}^{0,0}(2h_{\sigma})z^{2h_{\sigma}}\kappa\Omega_{\bar{h},\Delta_{\sigma}+p,\Delta_{\sigma}}^{h_{\sigma}}$
(14)
$\displaystyle\left(\alpha_{0}^{\text{even}}\left[S_{-\Delta_{\sigma}+\frac{\tau_{\text{tot}}}{2}}\right](q+\Delta_{\sigma}+\ell_{0})\log(z)+\beta_{0}^{\text{even}}\left[S_{-\Delta_{\sigma}+\frac{\tau_{\text{tot}}}{2}}\right](q+\Delta_{\sigma}+\ell_{0})\right)$

where
$\tau_{\text{tot}}=\tau_{i}+\tau_{j}+\tau_{k},D_{\text{tot}}=D_{i}D_{j}D_{k}2H_{\bar{h}_{\mathcal{O}^{\prime}_{k}}-1}$
with
$D_{\mathcal{O}^{\prime}}=-\lambda_{\sigma\sigma\mathcal{O}^{\prime}}^{2}\frac{\Gamma(2h_{\sigma})^{2}\Gamma(2\bar{h}_{\mathcal{O}^{\prime}})}{\Gamma(2h_{\sigma}-h_{\mathcal{O}^{\prime}})^{2}\Gamma(\bar{h}_{\mathcal{O}^{\prime}})^{2}}$,
and $\alpha_{0}^{\text{even}},\beta_{0}^{\text{even}}$ are defined in (4.41)
of [13].

Now we can compute the individual contribution of $\\{\epsilon,T\\}$ using (8)
and sum up the double twist contribution using (13) to get an approximate
generating function. To further improve it, we may iterate the inversion
formula a few times, i.e. we take the spectrum extracted from the approximate
generating function and plug it into the RHS of the inversion
formula111111Similar procedure has been used in [13, 28].. We also have to
make a balance between our goals : in various infinite summations, we choose
cut-off parameters small enough to make the computation fast, but also large
enough so that the result is accurate. Specifically, our choice is the
following.

######  Analytics Setup 1.

We take cut-off parameters $p_{\text{max}}=1$ in (8), $m_{\text{max}}=1$ in
(9), $K_{\text{max}}=\ell_{\text{max}}=7$ in (4.40) of [13] and $z_{0}=0.05$.
To solve (11), we use GFF value as the initial value for $\Delta$ and iterate
(11) for 3 times. In first step, we take $\\{\epsilon,T\\}$ as individual
contributions using (8) and sum up double twist contribution from $\ell_{0}=4$
using (13). To do the double twist summation, we take the identity
contribution for
$\lambda_{\sigma\sigma[\sigma\sigma]_{0}}^{2}/\lambda_{\text{GFF}}^{2}$ and
the $\\{\epsilon,T\\}$ contributions for $\delta h(\bar{h})$ in (12). Using
this generating function, we compute the dimension and OPE coefficient of
$[\sigma\sigma]_{0,\ell}$ for spin $4,...,20$. In the second step. we take the
spectrum $[\sigma\sigma]_{0,\ell=4,...20}$ from the first step as individual
contributions and repeat the process with $\ell_{0}=22$.

The result only depends on
$\\{\Delta_{\sigma},\Delta_{\epsilon},f_{\sigma\sigma\epsilon},f_{T}\\}$. It’s
also accurate and fast enough for our purpose121212When implemented in
Mathematica, on a computer with 64 AMD EPYC 7532 32-Core Processor,
computations for $[\sigma\sigma]_{0,\ell=4,...,52}$ takes about 7 seconds..
The result is shown as red dots in Figure 4 by comparing with the $\Lambda=43$
spectrum of the Numerics Setup 1. Now we understand the $1.8\%$ discrepancy in
the red dots : eventually the discrepancy will disappear for very large spins,
but for the spin range of the figure, it has to be corrected by the DTI
procedure.

In this work, we will mainly use the analytics in [13] as our analytic input
in the hybrid bootstrap, and occasionally we also use Analytics Setup 1. We
won’t repeat the derivation of [13] here, but simply layout the notation:

######  Analytics Setup 2.

For twist family $[\sigma\sigma]_{0,\ell}$, the analytic predictions of the
conformal data
$\Delta^{(analytic)}_{[\sigma\sigma]_{0,\ell}}=2\Delta_{\sigma}+\ell+\delta^{(analytic)}_{[\sigma\sigma]_{0,\ell}}$,
$f^{(analytic)}_{\sigma\sigma[\sigma\sigma]_{0,\ell}}$,
$f^{(analytic)}_{\epsilon\epsilon[\sigma\sigma]_{0,\ell}}$ are given by (6.1),
(6.2), (6.31) of [13]. For $[\sigma\epsilon]_{0,\ell}$, the data
$\Delta^{(analytic)}_{[\sigma\epsilon]_{0,\ell}}=\Delta_{\sigma}+\Delta_{\epsilon}+\ell+\delta^{(analytic)}_{[\sigma\epsilon]_{0,\ell}}$,
$f^{(analytic)}_{\sigma\epsilon[\sigma\epsilon]_{0,\ell}}$ is given by (6.32),
(6.34) of [13].

Figure 4: Comparing the lightcone analytics with the $\Lambda=43$ EFM
spectrum. Horizontal axes : spin. Vertical axes :
$\delta^{(analytic)}/\delta^{(numerics)}-1$, where $\delta$ is the correction
to the GFF dimension. On the left: Red dots are from Analytics Setup 1; Blue
dots are from Analytics Setup 2. On the right : green/blue dots are for
odd/even spin, both from Analytics Setup 2. All the analytics are computed
using the values in (2.1).

The result of Analytics Setup 2 is shown in Figure 4 by comparing with the
$\Lambda=43$ EFM spectrum131313In this paper, when we want to show the
spectrum of the leading twist families, we often plot the percentage
difference respect to either the analytics or the $\Lambda=43$ EFM spectrum,
because otherwise it’s hard to see the tiny difference in the spectrum with
bare eyes.. One might notice they match well for the spin roughly from 10 to
25, but mismatch for smaller or larger spins. This is a key observation that
inspired the construction of the hybrid method which will be discussed in the
next subsection.

##  3 The hybrid bootstrap

###  3.1 General strategy

Let’s consider the leading twist family in a CFT. A general intuition is that
the bootstrap numerics is sensitive to operators with small $\ell$. On the
other hand, as the spin gets larger, the analytic lightcone bootstrap in
Section 2.2 is more accurate. So there must be some intermediate spin range
around $L_{0}$ where the analytics and numerics have similar accuracy. See the
schematic Figure 5. That’s why in Figure 4, the analytics and numerics don’t
precisely agree for small spin and large spin, but they match well for some
intermediate values of spin. Since they make up for each other’s shortcomings,
one might expect to find a hybrid method that takes the best of both sides. To
do that, we should trust the analytics for large spin region, trust the
numerics for small spin region and demand they have similar results in the
intermediate region.

Figure 5: A schematic figure to show the error in analytics v.s. numerics.

Since our analytic predictions are not exact, we need to find a relatively
robust way to use them. A natural idea is that, starting from a certain spin
$L_{0}$, we may demand the operator exist within a bounding band around the
analytic prediction. The width of the band should be smaller as the spin gets
larger, because we expect the analytics is more accurate and trustable. We
then demand there is a gap between this band and the next twist family.

Figure 6: A schematic figure to show the strategy of the hybrid bootstrap. The
red curve is the analytic prediction of the leading twist family. The blue
dashed line is the leading twist and the next twist family of the actual CFT.
We impose positivity on the shaded region and shrink the red shaded region as
much as possible in order to glue the analytics with the numerics.

See Figure 6 for a schematic illustration. Technically, those requirement
translate to the following positivity requirement:

$\displaystyle\vec{\alpha}\cdot(V_{\Delta,\ell})\geq 0\text{ for
}\Delta^{(\text{band})}_{\text{min}}\leq\Delta\leq\Delta^{\text{(band)}}_{\text{max}}\text{
and }\Delta\geq\Delta_{gap},$ (15)

where $V$ is the crossing vector. Such conditions can be implemented using the
“interval positivity”[29]. To demand that the leading twist is as close as
possible to the analytics, we may minimize the width of the band until the
feasible region of the parameter (such as the dimension of external operators)
shrinks to a point.

One may make sense of this process in the following way. The bootstrap
equation at finite $\Lambda$ is a truncated bootstrap equation. It admits many
solutions, located at different points of the feasible region. By demanding
the band is as narrow as possible until the feasible region shrinks to a
point, we single out an unique solution whose leading twist matches the most
with our analytics.

If we also have the analytic prediction for the OPE coefficient of the leading
twist, we may inject that information into numerics as well. To use the
information of the OPE coefficient, we have to sum the contribution of the
operators with the identity block (i.e.
$G_{0,0}(u,v)+f_{\mathcal{O}}^{2}G_{\Delta_{\mathcal{O}},\ell_{\mathcal{O}}}(u,v)+...$).
To make the process more robust, when plug in OPE coefficient, we may make it
slightly smaller than the analytic prediction, then demand it to be as close
as possible to the analytics until the feasible region shrink to a
point141414A useful observation is that, from the sharing effect in [13], we
see that the numerics care more about the total “effective” OPE coefficient
from a group of nearby operators, but care less about the specific $\Delta$ of
those operators. Therefore assuming our OPE coefficient and $\Delta$ is not
very accurate, it can still be corrected by another operator with a small OPE
coefficient in the band. Since we summed the major portion of the OPE
coefficient with the identity, we already demanded the existence of the
leading twist family with the spectrum data being compatible with the analytic
prediction.

If we know the information of higher twist families, we can continue this
process by imposing more bounding bands for those twist families. We may also
encounter a situation where we have major twist families with big OPE
coefficients and minor twist families with much smaller OPE coefficients. The
3D Ising ${\mathbb{Z}}_{2}$-odd channel is such an example: the leading twist
$[\sigma T]_{0}$ has very small OPE coefficients while the major twist
$[\sigma\epsilon]_{0}$ is above $[\sigma T]_{0}$ but have a much bigger OPE
coefficients. In this situation, the above strategy still works for the major
twist families as long as $L_{0}$ is not too small. We can simply ignore the
minor twist families, because for spin larger than $L_{0}$ the numerics are
not very sensitive and won’t be affected too much by the minor twist families.
Of course, if we have analytic predictions for the minor twists, we may plug
in those data and sum the contributions of the twist families with the
identity block to make the bootstrap more robust.

Now we still need to know concretely how to choose $L_{0}$ as well as the form
of the band. Careful analysis of the error on both the numerics and the
analytics could certainly help to determine the $L_{0}$. But even if we don’t
have the quantitative version of Figure 5, there are some simple and model
independent rules to determine $L_{0}$. We will discuss this in detail in
Section 3.4. For the form of the band, in fact any form that shrinks
sufficiently fast in large spins is fine. We have some simple and model
independent choices that work well.

In Section 3, 3.4, 3.5, we will implement above strategy to the Ising
$\\{\sigma,\epsilon\\}$ system and show concretely how and why it works. But
before that, in the next Section 3.2, let’s comment on a few alternative
approaches and their advantages and disadvantages. We will explain why we
choose our main approach in this section over the alternatives.

###  3.2 Alternative approaches

A natural idea is that why don’t we simply plug the analytic data of the
leading twist into the bootstrap equation and just minimize the navigator
function to get a prediction for the actual CFT? We certainly could do this
for the large spin region of Figure 5. But if we also plug in data for the
intermediate spin and even small spin, the feasible region in the parameter
space will disappear and the scale of the minimum navigator value (positive)
is usually much larger than the scale of the minimum navigator in Section 2.1,
indicating that we get an even more wrong solution. Now let’s assume we only
do it for the large spin region. This is in fact equivalent to our main
strategy with a large $L_{0}$ and the band shrinks to a line. The analytics is
indeed much more accurate than the numerics in the large spin region. However
the numerics is also exponentially insensitive to those large spin data,
therefore the constraining power is also loose. It is really the process of
“gluing” the numerics and analytics in the intermediate region generating a
strong constraining power. Let’s make an analogy to illustrate this argument a
bit more. In the loop diagram computation, one may compute $4-\epsilon$
expansion and the $2+\epsilon$ expansion, then use the 2-sided pedé
approximation to smoothly interpolate between 2d and 4d. If one only uses
$2+\epsilon$ expansion, usually the result won’t be very accurate for larger
$\epsilon$ due to a branch cut at 4d. Similarly, in the hybrid bootstrap the
numerics and the analytics work for different regions and we should smoothly
connect them in the intermediate region. Without demanding they match in the
intermediate region is just like that in the loop computation, we use
$2+\epsilon$ and $4-\epsilon$ expansion for $2<d<3$ and $3<d<4$ respectively
but don’t demand they smoothly match at $d=3$: then the $4-\epsilon$ expansion
won’t help with the overall accuracy for $2<d<3$. However this alternative
approach could still be very useful for another purpose. In Section 3.4, we
will use it to determine the intermediate region.

Another alternative approach is that we may make a very mild assumption of the
error of our analytics, then maximize/minimize the various parameters in the
feasible region. If our assumption is mild enough so that we are totally
confident about it’s validity, the feasible region we get from the hybrid
bootstrap will be almost a rigorous bound. This strategy actually could work.
Let’s use the Ising $\\{\sigma,\epsilon\\}$ system as an example. Based on
Figure 4, we can make a conservative assumption on the error of
$\Delta^{(analytic)}_{[\sigma\sigma]_{0}}$ from the Analytics Setup 2 (blue
dot), by comparing it with either the $\Lambda=43$ EFM spectrum, or the
Analytics Setup 1 (red dots). Our assumption is plotted as dashed lines in the
left of Figure 7.

Figure 7: On the left: The red dots (from Analytics Setup 1) and blue dots
(from Analytics Setup 2) are same as Figure 4. The two dashed lines are
$\text{Err}^{(upper)}_{[\sigma\sigma]_{0}}=0.03$,
$\text{Err}^{(lower)}_{[\sigma\sigma]_{0}}=-0.317285/\ell^{1.5}-0.00687528$.
On the right : The optimal points from maximizing/minimizing $\Delta_{\sigma}$
and minimizing the navigator function in the feasible region at $\Lambda=11$.
The “hybrid” points are from the computation of the Hybrid Setup 1. The “non-
hybrid” points are from the computation of the Numerics Setup 2. The blue line
is the boundary of the $\Lambda=11$ island from Figure 13 of [21].

We assume the actual CFT operators $\Delta_{[\sigma\sigma]_{0},\ell\geq 4}$
located between the two dashed lines. To be rigorous, we won’t put a bound for
$[\sigma\epsilon]_{0}$ family in the ${\mathbb{Z}}_{2}$ odd channel, because
there are operators (such as $[\sigma T]_{0}$) below the
$[\sigma\epsilon]_{0}$ family. Concretely, the strategy translate to the
following SDP that depends on the parameters
$\\{\Delta_{\sigma},\Delta_{\epsilon},f_{\epsilon\epsilon\epsilon}/f_{\sigma\sigma\epsilon}\\}$
:

$\displaystyle
objective=\vec{\alpha}\cdot(\vec{V}^{(+)}_{\Delta=0,\ell=0}+(\lambda_{\sigma\sigma\epsilon},\lambda_{\epsilon\epsilon\epsilon}).\vec{V}^{(\theta)}.(\lambda_{\sigma\sigma\epsilon},\lambda_{\epsilon\epsilon\epsilon})$
$\displaystyle\qquad\qquad\qquad+\lambda_{T}^{2}(\Delta_{\sigma},\Delta_{\epsilon}).\vec{V}^{(+)}_{\Delta=3,\ell=2}.(\Delta_{\sigma},\Delta_{\epsilon}))$
$\displaystyle conditions:$
$\displaystyle\vec{\alpha}\cdot(\vec{V}^{(+)}_{\Delta,\ell\geq 2})\geq 0\text{
for }\Delta^{(+)}_{upper}\geq\Delta\geq\Delta^{(+)}_{lower}\text{ or
}\Delta\geq\Delta^{(+)}_{gap},$
$\displaystyle\vec{\alpha}\cdot(\vec{V}^{-}_{\Delta,\ell\geq 1})\geq 0\text{
for }\Delta^{(-)}_{upper}\geq\Delta\geq\Delta^{(-)}_{lower}\text{ or
}\Delta\geq\Delta^{(-)}_{gap},$ and the conditions of (17). (16)
$\displaystyle\textit{conditions for ${\mathbb{Z}}_{2}$ even spin 0, 2 and
${\mathbb{Z}}_{2}$ odd spin 0, 1}:$
$\displaystyle\vec{\alpha}\cdot(\vec{V}^{(+)}_{\Delta,\ell=0})\geq 0\text{ for
}\Delta\geq 3.7,$
$\displaystyle\vec{\alpha}\cdot(\vec{V}^{(+)}_{\Delta,\ell=2})\geq 0\text{ for
}\Delta\geq 5,$
$\displaystyle\vec{\alpha}\cdot(\vec{V}^{(-)}_{\Delta,\ell=0})\geq 0\text{ for
}\Delta\geq 5,$
$\displaystyle\vec{\alpha}\cdot(\vec{V}^{(-)}_{\Delta,\ell=1})\geq 0\text{ for
}\Delta\geq 5.$ (17)

The normalization of the SDP is the same as Section 2.1. For
${\mathbb{Z}}_{2}$ even spin 0, 2 and ${\mathbb{Z}}_{2}$ odd spin 0, 1, we
manually set up some gaps. Those conditions will be used repeatedly in
different setups in later sections, so we label them separately. The full
setup of the problem is

######  Hybrid Setup 1.

The framework of the SDP is given by (16). In the conditions, we set

$\displaystyle\Delta^{(+)}_{\text{upper}/\text{lower}}=\Delta_{\text{GFF}}+\delta^{(analytic)}_{[\sigma\sigma]_{0,\ell}}(1+\text{Err}^{(upper/lower)}_{[\sigma\sigma]_{0}}),$
$\displaystyle\Delta^{(+)}_{\text{gap}}=\Delta_{[\sigma\sigma]_{0,\ell}}^{(analytic)}+1,$
$\displaystyle\Delta^{(-)}_{\text{upper}}=\Delta^{(-)}_{\text{lower}}=\Delta^{(-)}_{\text{gap}}=\Delta_{\text{unitary}}.$

where $\text{Err}^{(upper/lower)}_{[\sigma\sigma]_{0}}$ is the two dashed line
of Figure 7. We maximize/minimize the parameters (such as $\Delta_{\sigma}$)
while maintaining $objective\leq 0$.

The result for maximizing/minimizing $\Delta_{\sigma}$ at $\Lambda=11$ is
presented in Figure 7. To benchmark the result, we also run another pure
numerics (without using the analytic information). The setup are

######  Numerics Setup 2.

Minimize the navigator (i.e. objective) or maximize/minimize the parameters
(such as $\Delta_{\sigma}$) of the SDP of (16) while maintaining
$objective\leq 0$. For both ${\mathbb{Z}}_{2}$-even and ${\mathbb{Z}}_{2}$-odd
channel, we set
$\Delta^{(-)}_{upper}=\Delta^{(-)}_{lower}=\Delta^{(-)}_{gap}=\Delta_{unitary}$.

The result for $\Lambda=11$ is presented in Figure 7. Comparing the Numerics
Setup 2, the error bar of Hybrid Setup 1 shrinks but not very much. At higher
$\Lambda$, the difference would be even less (the $\Lambda=19$ results are
given in the Appendix C). The effectiveness of the approach will really rely
on how small the analytic error is in the intermediate region. One might make
some compromise between this approach and our main approach in Section 3.1,
i.e. demand the numerics to be fairly close to the analytics but still leave
it with some finite size feasible region and use that as an estimation of
error bar. The confidence level of the error bar will be the same as the
confidence level of the error bar of the analytics.

If our analytics is very accurate compared to the numerics, both of the
alternatives can actually work.

###  3.3 Bootstrapping the 3D Ising CFT

In this section, we use the 3D Ising $\\{\sigma,\epsilon\\}$ system as an
example to implement our hybrid bootstrap strategy.

To demand the leading twist is as close as possible to the analytics, we use
the following prescription. For $\ell<L_{0}$, we require
$\Delta\geq\Delta_{unitary}$. For $\ell\geq L_{0}$, we sum up operators at
$\Delta_{analytic}$ with OPE coefficient $f=f_{analytic}(1-e^{a\ell})$ where
$a$ is a negative number. Meanwhile, we assume an operator can exist in the
interval $[\Delta_{GFF}+\delta_{lower},\Delta_{GFF}+\delta_{upper}]$, where
$\Delta_{GFF}$ is the GFF dimension and $\delta_{lower/upper}$ is given by
minimum/maximum of $\delta_{analytic}(1\pm e^{a\ell})$. Then we minimize the
parameter $a$ until the navigator function is zero. Note that in general the
error bar of a CFT data from lightcone analytics does not decay exponentially.
Instead, most like the decay behavior is a linear combination of the power
function. Here we choose the exponential function because after $L_{0}$ the
numerics is exponentially less sensitive, so we can just trust analytical
prediction. The $1\pm e^{a\ell}$ form is just a model independent choice to
glue the numerics with the analytics. There are more quantitative arguments in
Section 3.4 to show why the exponential function is a good choice.

Technically the above strategy translates to the following SDP problem that
smoothly depending on the parameters
$\\{\Delta_{\sigma},\Delta_{\epsilon},f_{T},f_{\sigma\sigma\epsilon},f_{\epsilon\epsilon\epsilon},a\\}$:

$\displaystyle
objective=\vec{\alpha}.(\vec{V}^{(+)}_{\Delta=0,\ell=0}+(\lambda_{\sigma\sigma\epsilon},\lambda_{\epsilon\epsilon\epsilon}).\vec{V}^{(\theta)}.(\lambda_{\sigma\sigma\epsilon},\lambda_{\epsilon\epsilon\epsilon})+\lambda_{T}^{2}(\Delta_{\sigma},\Delta_{\epsilon}).\vec{V}^{(+)}_{\Delta=3,\ell=2}.(\Delta_{\sigma},\Delta_{\epsilon})+$
$\displaystyle\sum_{\ell_{max}\geq\ell_{\mathcal{O}}\geq
L_{0},\mathcal{O}=[\sigma\sigma]_{0,\ell}}(f^{*}_{\sigma\sigma\mathcal{O}},f^{*}_{\epsilon\epsilon\mathcal{O}})\cdot\vec{V}^{(+)}_{\Delta_{\mathcal{O}},\ell_{\mathcal{O}}}\cdot(f^{*}_{\sigma\sigma\mathcal{O}},f^{*}_{\epsilon\epsilon\mathcal{O}})+\sum_{\ell_{max}\geq\ell_{\mathcal{O}}\geq
L_{0},\mathcal{O}=[\sigma\epsilon]_{0,\ell}}{f^{*}_{\sigma\epsilon\mathcal{O}}}^{2}\vec{V}^{(-)}_{\Delta_{\mathcal{O}},\ell_{\mathcal{O}}};$
$\displaystyle conditions:$
$\displaystyle\vec{\alpha}\cdot(\vec{V}^{(+)}_{\Delta,\ell\geq 2})\geq 0\text{
for
}\Delta^{(+)}_{\text{upper}}\geq\Delta\geq\Delta^{(+)}_{\text{lower}}\text{ or
}\Delta\geq\Delta^{(+)}_{\text{gap}},$
$\displaystyle\vec{\alpha}\cdot(\vec{V}^{(-)}_{\Delta,\ell\geq 1})\geq 0\text{
for
}\Delta^{(-)}_{\text{upper}}\geq\Delta\geq\Delta^{(-)}_{\text{lower}}\text{ or
}\Delta\geq\Delta^{(-)}_{\text{gap}},$ and the conditions of (17). (18)

The normalization of the SDP is defined similarly as in Section 2.1. The full
setup of the hybrid bootstrap is 151515Due to the proof in [30], the leading
twist in the singlet channel actually enjoys the convexity starting from spin
2, therefore we can demand twist gaps increase linearly between spin 2 and
$L_{0}$, which is a stronger condition. The effect of this condition on the
result is small though. In this paper, we will not use this condition. We
thank Johan Henriksson for discussion on the convexity of in the singlet
channel and suggest this stronger condition.

######  Hybrid Setup 2.

The framework of the SDP is given by (18). In the conditions, we take

$\displaystyle\Delta^{(+/-)}_{\text{upper}/\text{lower}}=\Delta^{(+/-)}_{\text{gap}}=\Delta_{\text{unitary}}\quad\text{for}\quad\ell<L_{0},$
$\displaystyle\Delta^{(+)}_{\text{upper}/\text{lower}}=\Delta_{\text{GFF}}+\delta^{(analytic)}_{[\sigma\sigma]_{0,\ell}}(1\pm
e^{a\ell})\quad\text{for}\quad\ell\geq L_{0},$
$\displaystyle\Delta^{(-)}_{\text{upper}/\text{lower}}=\Delta_{\text{GFF}}+\delta^{(analytic)}_{[\sigma\epsilon]_{0,\ell}}(1\pm
e^{a\ell})\quad\text{for}\quad\ell\geq L_{0},$
$\displaystyle\Delta^{(+)}_{\text{gap}}=\Delta_{[\sigma\sigma]_{0,\ell}}^{(analytic)}+1\quad\text{for}\quad\ell\geq
L_{0},$
$\displaystyle\Delta^{(-)}_{\text{gap}}=\Delta_{[\sigma\epsilon]_{0,\ell}}^{(analytic)}+1\quad\text{for}\quad\ell\geq
L_{0}.$

In the objective, we take $f^{*}=f_{analytic}(1-e^{a\ell})$ and
$\ell_{max}=100$161616Note $\ell_{max}$ only affects the summation in the
objective, while in the conditions the spin set is given in Appendix B. All
analytics are computed using Analytics Setup 2. We minimize the $a$ parameter
until the navigator (i.e. objective) is zero.

The different analytic information
($\Delta^{(analytic)}_{[\sigma\sigma]_{0,\ell}}$,
$f^{(analytic)}_{\sigma\sigma[\sigma\sigma]_{0,\ell}}$,
$f^{(analytic)}_{\epsilon\epsilon[\sigma\sigma]_{0,\ell}}$,
$\Delta^{(analytic)}_{[\sigma\epsilon]_{0,\ell}}$,
$f^{(analytic)}_{\sigma\epsilon[\sigma\epsilon]_{0,\ell}}$) we used in the
hybrid bootstrap may come with different accuracy. In a situation where one
analytic information is much more accurate than another, we might have to
choose the $a$ parameter differently in different places (say if
$\Delta^{(analytic)}_{[\sigma\sigma]_{0,\ell}}$ is much more accurate than
$\Delta^{(analytic)}_{[\sigma\epsilon]_{0,\ell}}$, then the magnitude of the
$a$ parameter for ${\mathbb{Z}}_{2}$-even channel should be larger than that
of ${\mathbb{Z}}_{2}$-odd channel, i.e. in intermediate region we demand the
matching for $\Delta^{(analytic)}_{[\sigma\sigma]_{0,\ell}}$ should be better
than for $\Delta^{(analytic)}_{[\sigma\epsilon]_{0,\ell}}$). This is a
complicated situation and a preliminary approach to determine different
bounding bands will be discussed in Section 3.7. In this section let’s simply
use same $a$ for all analytics. We also empirically fix some $L_{0}$. The
justification for those choices will be discussed in the next section.

Figure 8: Results of the computations for the Hybrid Setup 2 at various
$\Lambda$ and $L_{0}$. The light blue curve is the boundary of the
$\Lambda=43$ Ising island of [2] Figure 9: The path of the computation of
Hybrid Setup 2 with $\Lambda=11,L_{0}=6$. The greenish points are points from
early steps while the redish points are from later steps.

The result of Hybrid Setup 2 for various $\Lambda$ and $L_{0}$ in
$(\Delta_{\sigma},\Delta_{\epsilon})$ plane is plotted in Figure 8, while the
full results are in the Appendix C. We see even with relatively small
$\Lambda=11,19$, the result is already quite accurate, comparing to the
Numerics Setup 1 and 2. In Figure 9, we show a path for the minimization $a$
in $\Lambda=11,L_{0}=6$ case, starting from
$(\Delta_{\sigma},\Delta_{\epsilon},f_{\sigma\sigma\epsilon},f_{\epsilon\epsilon\epsilon},f_{T},a)=(0.5151,1.457,1,1.5,1.25,-0.1)$,
which is quite far from the actual CFT. The algorithm is a modification of the
Algorithm 2 of [21] (see Appendix B for more details). After 256 function
calls, it eventually reached
$(\Delta_{\sigma},\Delta_{\epsilon})=(0.51815015,1.4126784)$.

In the next two subsections, we will use only one analytic information (either
$\Delta_{[\sigma\epsilon]_{0}}$ or $\Delta_{[\sigma\sigma]_{0}}$) for the
hybrid bootstrap. The advantage is that with only one analytic information
it’s easier to analyze why and how the strategy works.

###  3.4 Why it works? How to determine $L_{0}$?

In this subsection, we will try to understand why the strategy works and
analyze $L_{0}$ more carefully. To make the situation simpler, we will only
use the analytics for $\Delta_{[\sigma\epsilon]_{0}}$. The full setup is:

######  Hybrid Setup 3.

The framework of the SDP is given by (18). In the conditions, we take

$\displaystyle\Delta^{(-)}_{\text{upper}/\text{lower}}=\Delta^{(-)}_{\text{gap}}=\Delta_{\text{unitary}}\quad\text{for}\quad\ell<L_{0},$
$\displaystyle\Delta^{(-)}_{\text{upper}/\text{lower}}=\Delta_{\text{GFF}}+\delta^{(analytic)}_{[\sigma\epsilon]_{0,\ell}}(1\pm
e^{a\ell})\quad\text{for}\quad\ell\geq L_{0},$
$\displaystyle\Delta^{(-)}_{\text{gap}}=\Delta_{[\sigma\epsilon]_{0,\ell}}^{(analytic)}+1\quad\text{for}\quad\ell\geq
L_{0},$
$\displaystyle\Delta^{(+)}_{\text{upper}}=\Delta^{(+)}_{\text{lower}}=\Delta^{(+)}_{\text{gap}}=\Delta_{\text{unitary}}\quad\text{for
all $\ell$}.$

In the objective, we take all $f^{*}=0$. All analytics are computed using
Analytics Setup 2. We minimize the $a$ parameter until the navigator (i.e.
objective) is zero.

The result for $\Lambda=19$ with various $L_{0}$ is plotted in Figure 10.

Figure 10: Results of the computations for the Hybrid Setup 3 at $\Lambda=19$
and various $L_{0}$. The light blue curve is the boundary of the $\Lambda=43$
Ising island of [2]

To understand how the numerics and analytics glued together, let’s make a
quantitative version of the Figure 5 for Hybrid Setup 3. In Figure 11, the red
dots are estimated errors of the analytics, obtained by comparing with the
$\Lambda=43$ EFM spectrum. The blue curve is the estimated error of the pure
numerics, obtained in the following way. We first set up a computation as
(18), but we take $L_{0}$ to be large enough so we can exactly plug in the
scaling dimension from the analytics, i.e.
$\Delta^{(+/-)}_{\text{upper}}=\Delta^{(+/-)}_{\text{lower}}=\Delta^{(analytic)}$.
We minimize the navigator function and find minimal navigator = $3.74\times
10^{-12}$ (this order of magnitude doesn’t sensitive to $L_{0}$). At the
minimal navigator point, we compute the derivative of the navigator function
respect to $\Delta^{(analytic)}$ at each spin, i.e.
$\partial_{\Delta_{\ell}}{\cal N}$ (the order of magnitude is also insensitive
to $L_{0}$ except when $\ell$ is close to $L_{0}$). Then
$\partial_{\Delta_{\ell}}{\cal N}/(3.74\times 10^{-12})$ should be the
estimation of the error bar at large $\ell$. For small spin, we can rigorously
bootstrap the error bar. The computation is the same as (2) except for a
certain spin $\ell_{0}$ we plug in $\Delta_{\ell_{0}}$ and assume next
operator has dimension $\geq\Delta_{\ell_{0}}+1$. We then maximize/minimize
$\Delta_{\ell_{0}}$ and the rigorous error bar is given by (max-min)/2. The
black dots are the results of such computations for $\ell=2,10$. With those
data, we fit the large spin log(error) with a degree 4 polynomial and demand
the fit to match with $\ell=2$ rigorous data. The result is the blue curve :
$-10.9349+0.242082\ell+0.0193981\ell^{2}-0.000138822\ell^{3}+4.02564\times
10^{-7}\ell^{4}$.

Figure 11: $\text{log}(\text{error}(\Delta_{[\sigma\epsilon]_{0}}))$ v.s.
spin. Red dots : estimated error of the analytics. Blue curve : estimated
error of the numerics. Black dot: rigorous error bar for the numerics.

From Figure 11, we see the errors of analytics and numerics intersecting
around $\ell=7$. However this error of the numerics is for the pure numeric
bootstrap (without injecting analytics), not the hybrid bootstrap. Once we
injecting the information of analytics at large spin, the numerical error for
the small spins will also be improved (as we have seen in Figure 8 and Figure
10), and the intermediate region should shift to the right. Therefore we may
say the small spin region is $L_{0}\leq 7$.

We expect the error in the hybrid bootstrap should schematically look like the
dashed curve of Figure 1 in Section 1. Let’s try to quantitatively confirm
this scenario in the example of the Hybrid Setup 3. We take the EFM spectrum
of the $L_{0}=10$ case and plot it on top of the $\Lambda=43$ EFM spectrum for
$\Delta_{[\sigma\epsilon]_{0}}$. The relative difference by comparing with the
Analytics Setup 2 is shown in the left of Figure 12171717We have to plot the
relative difference, because had we plot the absolute
$\Delta_{[\sigma\epsilon]_{0}}$ or $\tau_{[\sigma\epsilon]_{0}}$, we won’t be
able to see the tiny difference with bare eyes.. One may notice that indeed
for larger spin, the $\Delta_{[\sigma\epsilon]_{0}}$ is forced to be sit on
the analytic prediction $\Delta^{analytic}_{[\sigma\epsilon]_{0}}$. In fact
for $\ell=25,26$, our $\Lambda=19$ EFM spectrum is likely to be even more
precise than the $\Lambda=43$ EFM spectrum, because at $\ell=25,26$ the
$\Lambda=43$ EFM spectrum already start to fluctuating away from the analytic
prediction181818In this paper, for $\Lambda=19$ we used the spin set in (B),
so we only see operators with spin $\ell=...25,26,49,50$ appear in the EFM.
The EFM spectrum for the last two operators are
$\Delta_{[\sigma\epsilon]_{0,\ell=49}}=50.8677486,\Delta_{[\sigma\epsilon]_{0,\ell=50}}=51.9847355$,
sitting inside the narrow band around the analytic prediction. They are
certainly much more precise than the $\Lambda=43$ EFM spectrum.. The
logarithemic of the error (estimated by comparing with $\Lambda=43$ EFM
spectrum) is shown on the right of Figure 12. Indeed we see the small spin
accuracy is improved while the large spin operators match with the analytics,
confirming our intuition in the schematic Figure 1.

Figure 12:  In both figures, the blue dots are related to the EFM spectrum of
$\Delta_{[\sigma\epsilon]_{0}}$ from the Hybrid Setup 3 at
$\Lambda=19,L_{0}=10$. On the left: comparing the spectrum from the hybrid
setup with the $\Delta^{(analytic)}_{[\sigma\epsilon]_{0}}$ of the Analytics
Setup 2 computed at the values of (2.1). Horizontal axes : spin. Vertical axes
: $\delta^{(numerics)}/\delta^{(analytic)}-1$, where $\delta$ is the
correction to the GFF dimension. Blue dots : $\delta^{(numerics)}$ is from the
hybrid setup. Red dots: $\delta^{(numerics)}$ is from the $\Lambda=43$ EFM
spectrum (the same as the right of Figure 4). Dashed lines: $\pm e^{a\ell}$
with $a=-0.347263$ from the result of the hybrid computation. On the right:
The red dots and blue line is the same as Figure 11. The blue dots is
log(error) for the EFM spectrum $\Delta_{[\sigma\epsilon]_{0}}$ of the hybrid
setup, where the error is computed by comparing with the $\Lambda=43$ EFM
spectrum of Section 2.1.

Now let’s try to determine the large spin region. We do a simple test: if we
exactly plug in the information from analytics, how small we can go in $\ell$.
For the case in this section, the setup is

######  Hybrid Setup 4.

The SDP is the same as Hybrid Setup 3 with $a=-\infty$. We decrease $L_{0}$
until we can’t decrease it anymore without losing the feasible region. We call
it $L_{\text{min}}=\text{min}(L_{0})$ subject to $objective\leq 0$.

For the Hybrid Setup 3, $L_{\text{min}}=14$. It means up to $\ell=14$, the
numerics still couldn’t tell if the analytics is right or wrong. So we may say
$\ell\geq 14$ is the large spin and $8\leq\ell\leq 13$ is the intermediate
region. In practice, we found a simple choice for $L_{0}$ is a few spins
smaller than $L_{\text{min}}$. In the $\Lambda=11$ run of the Hybrid Setup 2,
we find $L_{\text{min}}=7$; and $L_{\text{min}}=15$ for $\Lambda=19$, so we
made those choices of $L_{0}$ in Figure 8.

One may wonder why $L_{0}=2,4$ seems to also work in Figure 10. It will be
clear by looking at the $a$ parameter at those points. In Figure 13, we plot
the $-e^{a\ell}$ on top of the left of Figure 4. We see that for
$L_{0}=2,...12$, the $a$ parameter is almost the same (roughly around
$-0.35$). Indeed, since for small spin, the physical operator is well inside
the $\pm e^{a\ell}$ bound, so even if we impose the interval posivity from
$\ell=2$, it will do no harm. Of course it may do harm if we also plug in OPE
coefficients, because such an operator will sit at a dimension that the
numerics can say it’s wrong. So we shouldn’t choose $L_{0}$ too small for the
runs of Figure 8. We see indeed the exponential decay function is a good
choice for gluing the numerics and analytics, although other choices may also
work, as long as they decay faster enough191919If the function doesn’t decay
fast enough, the risk is that when we shrink the size of the bounding band, it
may get stuck at the small spins, so the accurate information for the larger
spins won’t used efficiently..

Figure 13: The blue and green dots are the same as the right of Figure 4.
Orange curve : $-e^{a\ell}$ with $a=-0.361$ for $L_{0}=12$. Red curves :
$-e^{a\ell}$ with $a$ between -0.343 and -0.347 for $L_{0}=2,4,...10$.

###  3.5 How to improve further?

Figure 14: If we improve the numerics, the intermediate region will shift to
the right; while if we improve the analytics, the intermediate region will
shift to the left.

Let’s think about how to systematically improve the hybrid bootstrap result.
We can compute the numerics at higher $\Lambda$, so the error curve for the
numerics will move downward and the intermediate region shift to the right
(see Figure 14). We have already seen such an example in Figure 8. In this
section, we will try to push in the other direction : after improving the
analytics, we expect the intermediate region will shift to the left and the
result should be more accurate. Let’s use the analytics of
$[\sigma\sigma]_{0,\ell}$ as an example. The full setups for two different
versions of the analytics are

######  Hybrid Setup 5.

The framework of the SDP is given by (18). In the conditions, we take

$\displaystyle\Delta^{(+)}_{\text{upper}/\text{lower}}=\Delta^{(+)}_{\text{gap}}=\Delta_{\text{unitary}}\quad\text{for}\quad\ell<L_{0},$
$\displaystyle\Delta^{(+)}_{\text{upper}/\text{lower}}=\Delta_{\text{GFF}}+\delta^{(analytic)}_{[\sigma\sigma]_{0,\ell}}(1\pm
e^{a\ell})\quad\text{for}\quad\ell\geq L_{0},$
$\displaystyle\Delta^{(+)}_{\text{gap}}=\Delta_{[\sigma\sigma]_{0,\ell}}^{(analytic)}+1\quad\text{for}\quad\ell\geq
L_{0},$
$\displaystyle\Delta^{(-)}_{\text{upper}}=\Delta^{(-)}_{\text{lower}}=\Delta^{(-)}_{\text{gap}}=\Delta_{\text{unitary}}\quad\text{for
all $\ell$}.$

In the objective, we take all $f^{*}=0$. All analytics are computed using
Analytics Setup 2. We minimize the $a$ parameter until the navigator (i.e.
objective) is zero.

######  Hybrid Setup 6.

The setup is the same as Hybrid Setup 5, except we use Analytics Setup 1.

We first follow the logic of Section 3.4 to determine $L_{\text{min}}$. Set
$a=-\infty$ in the Hybrid Setup 5, we find $L_{\text{min}}=14$. For the Hybrid
Setup 6, our analytics is so accurate that we can actually exactly plug in
$\Delta^{(analytic)}_{[\sigma\sigma]_{0,\ell}}$ up to $L_{\text{min}}=6$! The
result of both setups for some $L_{0}<L_{\text{min}}$ are shown in Figure 15.
One can see indeed the Analytics Setup 1 significantly improve the accuracy of
the hybrid bootstrap result.

Figure 15: The results of the computations for the Hybrid Setup 5 and 6 at
$\Lambda=19$ are shown on the left and the right respectively. The light blue
curve is the boundary of the $\Lambda=43$ Ising island of [2]. Figure 16:
Comparing the EFM spectrum $\Delta_{[\sigma\sigma]_{0}}$ from the computation
of the Hybrid Setup 6 at $\Lambda=19,L_{0}=4$ with the
$\Delta^{(analytic)}_{[\sigma\sigma]_{0}}$ of the Analytics Setup 1 (computed
at the values of
$(\Delta_{\sigma},\Delta_{\epsilon},f_{\sigma\sigma\epsilon},f_{T})$ from the
hybrid setup). Horizontal axes : spin. Vertical axes :
$\delta^{(numerics)}/\delta^{(analytic)}-1$, where $\delta$ is the correction
to the GFF dimension. Blut dots: $\delta^{(numerics)}$ is from the hybrid
computation. Red dots : $\delta^{(numerics)}$ is from the $\Lambda=43$ EFM
spectrum (after compensating the sharing effect). Light red dots :
$\delta^{(numerics)}$ is from the raw $\Lambda=43$ EFM spectrum (without
compensating the sharing effect). Dashed lines: $\pm e^{a\ell}$ with
$a=-1.17306876127$ from the result of the hybrid computation.

Next we will look at the EFM spectrum of $\Delta_{[\sigma\sigma]_{0,\ell}}$
extracted at $L_{0}=4$ point of the Hybrid Setup 6. The result (relative to
Analytics Setup 1) is shown in Figure 16. We see that $\Lambda=19$ EFM
spectrum from Hybrid Setup 6 is even more accurate than the $\Lambda=43$ EFM
spectrum raw data (without averaging over both real operator and fake
operator)202020In fact, the evidence seems suggests blue dots of Figure 16 is
also more accurate than the red dots, because (1), if the red dots are more
close to the real operators, we expected the blue dots should sit at the lower
bound of the band, instead of the upper bound; (2), the red dots is obtained
through the averaging method aiming to compensate the sharing effect (see
Section 2.1). Since it’s just an empirical method, it’s hard to believe after
such a process the data could precisely located on the actual operators. The
data from Table 3 of [13] is close to the red dots.. Another interesting
feature is that the $\Lambda=19$ spectrum only have one operator near
$\Delta_{[\sigma\sigma]_{0,\ell}}$ in the bounding band212121In Figure 16, the
analytics is computed using the values from Table 10 and the blue dot is
precisely on the upper dashed line. Had we use the values of (2.1), the
difference is tiny and invisible in the figure., i.e. it doesn’t suffer from
the sharing effect222222It would be interesting to see how the pure bootstrap
numerics changes if we are able to remove the sharing effect, i.e. demand
there is only one operator near the $\Delta_{[\sigma\sigma]_{0,\ell}}$,
although such condition might be hard to implement.. On the other hand, the
spectrum for $\Delta_{[\sigma\epsilon]_{0,\ell}}$ doesn’t perfectly match the
analytics, as expected. This suggests we should throw more analytic
information into the numerics.

Lastly, we would like to comment on the gaps of (17). If we relax those gaps
to

$\displaystyle\vec{\alpha}\cdot(\vec{V}^{(+)}_{\Delta,\ell=0})\geq 0\text{ for
}\Delta\geq 3.1,$
$\displaystyle\vec{\alpha}\cdot(\vec{V}^{(+)}_{\Delta,\ell=2})\geq 0\text{ for
}\Delta\geq 4,$
$\displaystyle\vec{\alpha}\cdot(\vec{V}^{(-)}_{\Delta,\ell=0})\geq 0\text{ for
}\Delta\geq 3.1,$
$\displaystyle\vec{\alpha}\cdot(\vec{V}^{(-)}_{\Delta,\ell=1})\geq 0\text{ for
}\Delta\geq 3.1,$ (19)

we find surprisingly the computations of setups only use
$\Delta^{(analytic)}_{[\sigma\sigma]_{0}}$ will yield less accurate results,
but the computations of setups only use
$\Delta^{(analytic)}_{[\sigma\epsilon]_{0}}$ won’t change much. In
particularly, we checked that the Hybrid Setup 6 with condition (19) at
$\Lambda=19,L_{0}=4$ gives result
$(\Delta_{\sigma},\Delta_{\epsilon})=(0.51816320,1.41276289)$, while the
Hybrid Setup 3 with condition (19) at $\Lambda=19,L_{0}=4$ gives result
$(\Delta_{\sigma},\Delta_{\epsilon})=(0.51814895,1.41262720)$. We don’t have a
good explanation of the difference. But it does suggest again that we should
use both analytic information at the same time. We will discuss some
preliminary attempts in Section 3.7. Before doing that, we need to have a tool
to estimate the error.

###  3.6 A preliminary attempt on estimation of the error bar

A missing piece in our method is a satisfactory method to estimate the error
bar of the result. We don’t have a perfect method yet, but in this subsection
we will make a preliminary attempt.

The result we obtained in various hybrid setups in this paper will have some
errors. Let’s think about the source of errors. Taking the Hybrid Setup 6 as
an example, the result we obtained has errors, because in Figure 16 we pushed
the numerics too close to the analytics, which is not exact. By the moment the
feasible region shrinking to a point, we definitely have already “over pushed”
it and over passed the spin 4 operator ($[\sigma\sigma]_{0,\ell=4}$) of the
actual CFT. We should expect the actual operator for spin 4 in Figure 16 is
slightly above the blue dot 232323One may reason in this way: by the moment
the feasible region shrinking to a point, the numerics can almost tell the
analytics is wrong, because had the $a$ decrease even a tiny bit more, the
navigator function would be positive. We expect the main power that the
numerics can tell the analytics is marginally wrong comes from the spin 4
operators, because for spin 6 even if we exactly plug in the analytic value
the numerics won’t complain (i.e. $L_{\text{min}}=6$)..

But it’s not obvious how to estimate how much had we overpassed the dimension
of the actual operator. It’s also not obvious what is the consequence of this
over-push. It may not necessarily damage the prediction of
$(\Delta_{\sigma},\Delta_{\epsilon})$. Let’s consider the example of
$\Delta_{\sigma^{\prime}}$ bound, where $\sigma^{\prime}$ is the next spin 0
operator (after $\sigma$) appearing in the $\sigma\times\epsilon$ OPE. If we
assume $\Delta_{\epsilon}$ saturates the
$\langle\sigma\sigma\sigma\sigma\rangle$ single correlator bootstrap bound and
the first operator in the ${\mathbb{Z}}_{2}$-odd spin 1 channel has dimension
$\geq 2.3$, the plot of the bound on $\Delta_{\sigma^{\prime}}$ v.s.
$\Delta_{\sigma}$ looks like dark blue region of the Figure 6 of [31]. If we
maximize $\Delta_{\sigma^{\prime}}$ in the feasible region, although it would
overpass the real operator, the $\Delta_{\sigma}$ actually get more accurate.
But it doesn’t necessarily mean the spectrum at the optimal point of the
maximization is overall a more accurate spectrum. Therefore there is a risk
that the accurate result we obtain in this paper might partially due to
similar effect. With this in mind, let’s now try to estimate the errors.

We may expect, if the numerics is more sensitive to the bounding band, the
scale of $\partial_{a}{\cal N}$ should be larger. In fact, if the navigator
function around the feasible region of
$\\{\Delta_{\sigma},\Delta_{\epsilon},f_{T},f_{\sigma\sigma\epsilon},f_{\epsilon\epsilon\epsilon},\Delta_{[\sigma\sigma]_{0,\ell=4}}\\}$
is a quadratic function, we can simply use the gradient and Hessian at a local
point to reconstruct an approximate navigator function and estimate the size
of the feasible region based on that. In reality, the navigator function is
not a quadratic function. But let’s still try to estimate the error based on
the approximate quadratic navigator function. Hopefully the estimation may
scale with the actual error.

After each hybrid bootstrap computation, we use the procedure in Appendix C of
[21] to compute the Hessian 242424As a rough estimation, one may use the the
approximate Hessian at the end of Algorithm 2 of [21]. However we found the
precision of the approximate Hessian is not good enough for our purpose. For
example, the approximate Hessian is always positive definite, but in reality
the Hessian could be negative., and construct the approximate navigator
function as

$\displaystyle{\cal
N}^{(approx)}(x)=g(x_{f})(x-x_{f})+\frac{1}{2}(x-x_{f})^{T}B(x-x_{f}).$ (20)

where $g$ is the gradient, $B$ is the Hessian, and
$x=(\Delta_{\sigma},\Delta_{\epsilon},f_{T},f_{\sigma\sigma\epsilon},f_{\epsilon\epsilon\epsilon},a)$.
Now we can maximize/minimize various parameters subject to ${\cal
N}^{(approx)}(x)\leq 0$ and use it as an estimation of the error bars.

Let’s take the computation in Hybrid Setup 6 with $\Lambda=19,L_{0}=4$ as an
example. The gradient at the optimal point is

$\displaystyle(4.59\times 10^{-17},7.70\times 10^{-17},2.81\times
10^{-16},7.97\times 10^{-18},7.53\times 10^{-17},-8.05231\times 10^{-9})$

while the Hessian is given by

$\displaystyle\begin{pmatrix}$28828.7$&$-6254.82$&$-8598.62$&$427.614$&$-1345.41$&$-0.0429191$\\\
$-6254.82$&$1914.45$&$3752.10$&$-51.2511$&$905.168$&$0.0741709$\\\
$-8598.62$&$3752.10$&$8956.93$&$14.2040$&$2481.62$&$0.231068$\\\
$427.614$&$-51.2511$&$14.2040$&$9.71850$&$24.8149$&$0.00365034$\\\
$-1345.41$&$905.168$&$2481.62$&$24.8149$&$776.322$&$0.0795515$\\\
$-0.0429191$&$0.0741709$&$0.231068$&$0.00365034$&$0.0795515$&$0.0000548614$\end{pmatrix}.$

Maximizing/minimizing various parameters in the approximate navigator
function, we obtain the data in Table 1252525The error in Table 1 is defined
by $(\text{max}-\text{min})$. For the last column, we took the data from Table
2 of [13]. For $f_{T}$ in the last column, we estimate it by
$f_{T}=(f_{\epsilon\epsilon T}/\Delta_{\sigma}+f_{\sigma\sigma
T}/\Delta_{\epsilon})/2$ and convert to our convention, while the error is
derived from the error of $f_{\epsilon\epsilon T}$.. We found all the error
bars are compatible with the data in [13].

Parameter | value | min | max | error | value of [13]
---|---|---|---|---|---
$\Delta_{\sigma}$ | $0.51814904$ | $0.51814721$ | $0.51815085$ | $0.00000363$ | 0.5181489(10)
$\Delta_{\epsilon}$ | $1.41262750$ | $1.41260625$ | $1.41264878$ | $0.00004253$ | 1.412625(10)
$f_{\sigma\sigma\epsilon}$ | $1.05185297$ | $1.05186070$ | $1.05184522$ | $0.00001548$ | 1.0518537(41)
$f_{\epsilon\epsilon\epsilon}$ | $1.53244587$ | $1.53240724$ | $1.53248517$ | $0.00007793$ | 1.532435(19)
$f_{T}$ | $1.25885744$ | $1.25885555$ | $1.25885928$ | $0.00000373$ | 1.2588576(90)
Table 1: Estimation of the errors in Hybrid Setup 6 with $\Lambda=19,L_{0}=4$.

For various setups in this paper, the estimated errors
$\Delta_{\sigma,\max}-\Delta_{\sigma,\min}$,
$\Delta_{\epsilon,\max}-\Delta_{\epsilon,\min}$ and the total error
$\prod^{6}_{i=1}(x_{i,\text{max}}-x_{i,\text{min}})$ is presented in Table
2262626For some of the computations, the Hessian is not positive, therefore
the error would be $\infty$ based on the approximate navigator function.. We
find all error boxes are compatible with the data in [13]. We also observe
that this estimation can tell qualitatively : (1), Hybrid Setup 6 is better
than other setups; (2), in Hybrid Setup 2, $\Lambda=19$ computations are
better than the $\Lambda=11$ computation; (3), the results from Hybrid Setup 5
are the least accurate ones; (4), When $L_{0}$ is too large, usually the
result is not good. Indeed this method is still helpful. On the other hand, we
are not sure how robust the method is in other scenarios and other models.

Setup | $\Delta_{\sigma,\max}-\Delta_{\sigma,\min}$ | $\Delta_{\epsilon,\max}-\Delta_{\epsilon,\min}$ | $\prod_{i=0}^{6}\left(x_{i,\max}-x_{i,\min}\right)$
---|---|---|---
$\text{Hybrid Setup \ref{hybrid:fullsetup} $\Lambda$=11 }L_{0}\text{= 6}$ | $0.00016363$ | $0.00198749$ | $4.225\times 10^{-17}$
$\text{Hybrid Setup \ref{hybrid:fullsetup} $\Lambda$=19 }L_{0}\text{=10}$ | $0.00001632$ | $0.00019559$ | $6.392\times 10^{-24}$
$\text{Hybrid Setup \ref{hybrid:fullsetup} $\Lambda$=19 }L_{0}\text{=12}$ | $0.00001679$ | $0.00020124$ | $1.095\times 10^{-23}$
$\text{Hybrid Setup \ref{hybrid:fullsetup} $\Lambda$=19 }L_{0}\text{=14}$ | $\infty$ | $\infty$ | $\infty$
$\text{Hybrid Setup \ref{hybrid:se0} $\Lambda$=19 }L_{0}\text{= 2}$ | $0.00001939$ | $0.00023415$ | $2.872\times 10^{-23}$
$\text{Hybrid Setup \ref{hybrid:se0} $\Lambda$=19 }L_{0}\text{= 4}$ | $0.00001620$ | $0.00019514$ | $1.125\times 10^{-23}$
$\text{Hybrid Setup \ref{hybrid:se0} $\Lambda$=19 }L_{0}\text{= 6}$ | $0.00003387$ | $0.00040541$ | $4.811\times 10^{-22}$
$\text{Hybrid Setup \ref{hybrid:se0} $\Lambda$=19 }L_{0}\text{= 8}$ | $0.00001720$ | $0.00020704$ | $1.612\times 10^{-23}$
$\text{Hybrid Setup \ref{hybrid:se0} $\Lambda$=19 }L_{0}\text{=10}$ | $0.00002485$ | $0.00029873$ | $1.204\times 10^{-22}$
$\text{Hybrid Setup \ref{hybrid:se0} $\Lambda$=19 }L_{0}\text{=12}$ | $0.00003137$ | $0.00037567$ | $1.735\times 10^{-21}$
$\text{Hybrid Setup \ref{hybrid:ss0_v2} $\Lambda$=19 }L_{0}\text{= 4}$ | $\infty$ | $\infty$ | $\infty$
$\text{Hybrid Setup \ref{hybrid:ss0_v2} $\Lambda$=19 }L_{0}\text{= 8}$ | $\infty$ | $\infty$ | $\infty$
$\text{Hybrid Setup \ref{hybrid:ss0_v2} $\Lambda$=19 }L_{0}\text{=10}$ | $\infty$ | $\infty$ | $\infty$
$\text{Hybrid Setup \ref{hybrid:ss0_v2} $\Lambda$=19 }L_{0}\text{=12}$ | $\infty$ | $\infty$ | $\infty$
$\text{Hybrid Setup \ref{hybrid:ss0_v3} $\Lambda$=19 }L_{0}\text{= 4}$ | $0.00000363$ | $0.00004253$ | $2.453\times 10^{-28}$
Table 2: Estimation of errors for various setups in this paper.

It might be possible to find other methods to estimate the error. For example,
instead of using $\pm e^{a\ell}$ for the form of the bounding band, we can use
$\pm Ae^{a\ell}$. By varying $A$, we glue the numerics and the analytics
slightly differently and the result will be slightly different. The range
where the parameters fluctuate may be used as estimation of the error. This is
just like in the loop computation, different Padé approximation schemes give
slightly different results and one may use the range as an estimation of the
error. Similarly, we may change the value of $z_{0}$ in the analytics setup,
observe how much the result of the hybrid bootstrap fluctuate with $z_{0}$,
and use that as an error estimation. However we haven’t thoroughly
investigated those approaches.

###  3.7 Balance out multiple analytics with different accuracy

Although the Hybrid Setup 2 used all the information from Analytics Setup 2,
it seems the result isn’t as good as the Hybrid Setup 3, which only used
$\Delta^{analytic}_{[\sigma\epsilon]_{0}}$. After our analysis in Section 3.4
and 3.5, this shouldn’t be too surprising. Imagine in a hybrid bootstrap
setup, we have two analytic information with very different accuracy, say
$\Delta^{analytic}_{[\sigma\epsilon]_{0}}$ from Analytics Setup 2 and
$\Delta^{analytic}_{[\sigma\sigma]_{0}}$ from Analytics Setup 1. If we use the
same $a$ for the bounding bands for the both analytics, $a$ certainly can’t
decrease below the optimal value in Hybrid Setup 3 (around $-0.343$).
Therefore the very accurate information from
$\Delta^{analytic}_{[\sigma\sigma]_{0}}$, which allows $a\sim-1.17$, won’t be
used effectively. This situation remind us the Liebig’s barrel, or the law of
the minimum : a barrel’s capacity is limited by its shortest stave.

To overcome this issue, the natural idea is that we set different decay
functions for the bands that bound analytics with different accuracy. A main
difficulty is that we have to choose how they decay relative to each other as
we shrink the feasible region to a point. In Section 3.6 we found a method to
estimate the error (or an indicator that more or less scales with the error),
therefore we may choose a ratio between different decay factors such that the
estimated error is minimized. Since our method for the error is not precise,
the ratio may not be perfect. But let’s still try it as a preliminary attempt.
In this subsection, we will use both
$\Delta^{analytic}_{[\sigma\sigma]_{0}},\Delta^{analytic}_{[\sigma\epsilon]_{0}}$
from the Analytics Setup 2. One may notice from Figure 15 and 10, the result
from using $\Delta^{analytic}_{[\sigma\sigma]_{0}}$ only is less accurate than
the result from using $\Delta^{analytic}_{[\sigma\epsilon]_{0}}$ only and both
predict a $\Delta_{\sigma}$ that is too small. We want to see if we can get a
better result by using both analytics. The full setup is

######  Hybrid Setup 7.

The framework of the SDP is given by (18). In the conditions, we take

$\displaystyle\Delta^{(+)}_{\text{upper}/\text{lower}}=\Delta^{(+)}_{\text{gap}}=\Delta_{\text{unitary}}\quad\text{for}\quad\ell<L_{0},$
$\displaystyle\Delta^{(+)}_{\text{upper}/\text{lower}}=\Delta_{\text{GFF}}+\delta^{(analytic)}_{[\sigma\sigma]_{0,\ell}}(1\pm
e^{a_{e}\ell})\quad\text{for}\quad\ell\geq L_{0},$
$\displaystyle\Delta^{(+)}_{\text{gap}}=\Delta_{[\sigma\sigma]_{0,\ell}}^{(analytic)}+1\quad\text{for}\quad\ell\geq
L_{0},$
$\displaystyle\Delta^{(-)}_{\text{upper}/\text{lower}}=\Delta^{(-)}_{\text{gap}}=\Delta_{\text{unitary}}\quad\text{for}\quad\ell<L_{0},$
$\displaystyle\Delta^{(-)}_{\text{upper}/\text{lower}}=\Delta_{\text{GFF}}+\delta^{(analytic)}_{[\sigma\epsilon]_{0,\ell}}(1\pm
e^{a_{o}\ell})\quad\text{for}\quad\ell\geq L_{0},$
$\displaystyle\Delta^{(-)}_{\text{gap}}=\Delta_{[\sigma\epsilon]_{0,\ell}}^{(analytic)}+1\quad\text{for}\quad\ell\geq
L_{0}.$

In the objective, we take all $f^{*}=0$. All analytics are computed using
Analytics Setup 2. We fix the ratio $r=a_{e}/a_{o}$ and minimize the $a$
parameter until the navigator (i.e. objective) is zero. We change the $r$
until the estimated total error (using the method of Section 3.6) is
minimized.

Figure 17: Results of the computations for the Hybrid Setup 7 at $\Lambda=19$,
$L_{0}=8$ and various $r=a_{e}/a_{o}$ ratios. The red dots correspond to
ratios where the estimated errors are small. The light blue curve is the
boundary of the $\Lambda=43$ Ising island of [2].  Figure 18: The estimated
error for the Hybrid Setup 7 at $\Lambda=19$, $L_{0}=8$. Horizontal :
$r=a_{e}/a_{o}$. Vertical : $\text{log}_{10}(\text{error})$, where
$\text{error}=\prod_{i=0}^{6}(x_{i,\max}-x_{i,\min})$ is estimated using the
method of Section 3.6. The two vertical grids correspond to two local minimums
with $r=0.38,1.06875$. The horizontal grid is the $\text{log}(\text{error})$
for the Hybrid Setup 3, while the Hybrid Setup 5 has $\text{error}=\infty$.

The results for some $r$ are shown in Figure 17. We found surprisingly the
prediction fluctuate around the $\Lambda=43$ island as $r$ varies. The
estimated total error is also not smooth on $r$. So in practice, we scanned
many values of $r$. The estimated errors are shown in Figure 18. The logic to
choose this range of $r$ is the following. From Table 9 and 8, we have seen
$a_{e}=-0.3951$ from the Hybrid Setup 5 and $a_{o}=-0.3459$ from the Hybrid
Setup 3 when we only use one analytics. So a natural choice could be
$r=(-0.3951)/(-0.3459)=1.1422$272727The result for $r=1.1422$ is
$(0.518146278,0.51814627)$, outside the range of Figure 17. The estimated
error is $\infty$.. But since the Hybrid Setup 5 has estimated error $\infty$,
we should choose $r<1.1422$ to favor the analytics in ${\mathbb{Z}}_{2}$-odd
channel. Thus we scanned those values of $r$ in Figure 17 and 18. From those
result, we observe the prediction from various $r$ come cross a certain region
of the island several times, where the estimated errors are small. And for
most ratios, the result is better than both Hybrid Setup 5 and 3. So this
preliminary method seems to have worked for our setup. If we want to
incorporate more analytics, such as the OPE coefficients, it might be
difficult to find global minimum due to the non-smoothness of error function
on the ratios. On the other hand, as we seen in Figure 17 and 18, local
minimums of the error function also give good predictions. The validity and
robustness of this method is subject to future studies in other scenarios and
models. It would also be interesting to find other methods to balance out
multiple analytics282828We may naively think, why not just let $a_{e},a_{o}$
be free and minimize the total area of both bounding bands? Unfortunately this
won’t work. The reason is that the numerics is more sensitive to the band of
the more accurate analytics. By minimizing the total area, we more or less
“equalized” the sensitivity. But this will favor the less accurate analytics
instead of the more accurate one. In other words, the area of different
bounding bands can’t be compared directly. To carefully compensate for the
difference, it seems we have to use the information of the 2nd order
derivative of the navigator function. Our method of estimating error is of
this type..

##  4 Conclusion and outlook

In this work, we have found a hybrid bootstrap method to combine numerical
conformal bootstrap with the analytical lightcone bootstrap. We have explained
the logic of the hybrid bootstrap and the general strategy. Partial
implementation of the strategy in the examples in 3D Ising
$\\{\sigma,\epsilon\\}$ system have been worked out. We have observed that
once injecting the analytical information into the numerics, the prediction
for the actual CFT in the hybrid setups is significantly improved. We also
have made a preliminary attempt to estimate the errors. We studied a method to
use multiple analytic information at the same time, where we have found the
result is better than the result of just using one.

The best computation in this paper comes from combining a refined version of
$\Delta^{(analytic)}_{[\sigma\sigma]_{0}}$ with the $\Lambda=19$ numerics. The
estimated error box of our best hybrid run, as well as the error box of the
pure numerics (non-hybrid) at $\Lambda=19$ are presented in Figure 19. Even in
relatively low derivative order $\Lambda=19$ with some partial analytic
information of the leading twist operators, the prediction of hybrid bootstrap
is very accurate, clearly showing the potential of this method to advance the
conformal bootstrap computation.

Figure 19:  The star : the result of the Hybrid Setup 6 at
$\Lambda=19,L_{0}=4$. The dashed box : an estimation of the error for the star
(the data is from Table 1). The black box : the error box of the Numerics
Setup 2 at $\Lambda=19$ (the data is from Table 6).

More work is needed to really bring this method to its full fledged form. For
3D Ising CFT, we need to systematically improve the efficiency and accuracy of
the analytics, carefully balance the accuracy of various analytic information,
and do the computation at higher $\Lambda$. We also need to further
investigate methods for the error estimation and test their validity in more
scenarios. For most computations in this paper, we used the $\Lambda=43$
island of [2] as a benchmark. But as we go beyond $\Lambda=19$ in the hybrid
bootstrap, we will be in an uncharted territory and the validity of the error
estimation will be essential.

This hybrid bootstrap won’t give the rigorous error box as the traditional
numerical bootstrap does, so it can’t completely replace the rigorous method.
But by sacrificing some rigorousness, we might gain a lot in the accuracy of
the prediction for the actual CFT. Such a trade-off is certainly favorable in
many scenarios. If we can obtain a non-rigorous but still very trustable
spectrum, we will still be able to answer important questions in physics.

Our method can be applied to many other models. As we have shown, even some
partial analytic information could significantly improve the result. When
applying the hybrid bootstrap to other CFTs, one may start with the most
reliable analytics that one has. Quick applications could be super-Ising and
the O(N) CFTs, where some analytics were worked out in [16, 28]292929It’s
likely that one will have to further improve the analytics for large $\bar{h}$
before applying it to the hybrid bootstrap, similar to what we did in Section
2.2..

Besides the lightcone limit, the analytic information from other limits could
be useful as well. The numerical bootstrap is not sensitive to large $\Delta$.
If we have some estimation of the effective contribution from larger $\Delta$,
we probably can inject that information into the numerical bootstrap as well.
Another interesting limit is the large charge expansion for the $O(N)$ CFTs
[32]. To access operators with larger $O(N)$ charge in the numerical
bootstrap, we have to include more and more external operators with large
charges in the bootstrap equation. This looks like a very bad situation
because the number of bootstrap equations grows very fast with the number of
external operators. But as we have shown in this paper, if we inject the
analytics of the leading twist into the numerics, we probably don’t need very
high $\Lambda$ for every equation. Without increasing the total number of
components of the functional $\alpha$, we may redistribute those components to
other equations that have access to higher charge. In this way, we might see a
higher charge spectrum and hopefully could connect the large charge analytics
with the numerics. Beside the conformal bootstrap, it would be interesting to
see if a similar hybrid method works for other bootstrap methods. For example,
in the modular bootstrap there is an analogy of the lightcone analytics in
terms of the Cardy formula.

Another interesting direction to explore in the future is to see if the hybrid
bootstrap can help with bootstrapping a four-point function whose external
operator has a large dimension. CFTs from gauge theories are of this type. It
is known that in general numerical bootstrap works less well if the external
operator has a larger dimension. For example, the single correlator bound [33]
converges less well for larger $\Delta_{\sigma}$. We also see an example in
gauge theory [34], where the numerics for 2d scalar QED converge much better
than the 3d case. We don’t fully understand this phenomenon. But one
contributing factor might be, as the dimension of the external operator gets
larger, the leading twist family usually also has a larger twist that is
further away from the unitary bound, so the unitary bound we put in each
channel may not be sufficient. A relevant hint is that the EFM spectrum of the
super-Ising CFT [28] also suffers from the sharing effect, i.e. it has fake
operators on the unitary bound. Since the $\Delta_{\sigma}$ in the super-Ising
CFT is larger than $\Delta_{\sigma}$ in the Ising CFT, those fake operators
will further away from the true operators, thus could do more damage to the
convergence. On the other hand, in the hybrid bootstrap the bounding band is
imposed close to the true operator and the EFM spectrum in the band doesn’t
suffer from the sharing effect. If this fact (free from the sharing effect)
really contributes to the nice convergence of the hybrid bootstrap
computations in this work, we expect the hybrid bootstrap with leading twist
analytics is important for many bootstrap setups involving large external
operators. To confirm this scenario, we should implement the hybrid bootstrap
for the super-Ising CFT.

## Acknowledgements

We thank Johan Henriksson, Yinchen He, Brian McPeak, David Poland, Marten
Reehorst, Jongchen Rong, Slava Rychkov, David Simmons-Duffin, Benoit Sirois,
Alessandro Vichi, Balt van Rees for discussions. We thank Shai Chester, Johan
Henriksson, Jongchen Rong, Alessandro Vichi for their critical opinions on
this manuscript. We thank Walter Landry for very helpful advice on C++
parallel scientific programming. This project has received funding from the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement no. 758903). We thank
Yinchen He and David Simmons-Duffin for support on computational resources.
The computations in this paper were partially run on the Symmetry cluster of
Perimeter institute. The computations presented here were partially conducted
in the Resnick High Performance Computing Center, a facility supported by
Resnick Sustainability Institute at the California Institute of Technology.

##  Appendix A Code availability

During this project, we developed several programs to facilitate the
implementation. All the programs are open-source software and available
online.

To compute the conformal block we used a fork version of scalar_blocks with
the temporary name scalar_blocks_mod(https://gitlab.com/suning-git/scalar-
blocks-mod). This program computes the scalar block, convolves it and saves
the block polynomials to a binary format or the Mathematica format. One
difference between scalar_blocks_mod and original version is that for
correlators of the form $\langle\sigma\sigma\epsilon\epsilon\rangle$, certain
poles have zero residue and they are explicitly removed in scalar_blocks_mod.

To generate the SDP for sdpb, we used a fork version of sdp2input with the
temporary name sdp2input_mod(https://github.com/suning-
git/sdpb/tree/sdp2input_mod). This program depends on three inputs. The first
one is the block data from the scalar_blocks_mod. The second one is called a
parameter file, which contains the values for a set of user-defined variables.
The third one is called a SDP template file. This is a Mathematica file
similar to the input for original sdp2input, except that all the polynomials
from the blocks are not plugged in but just stubs, and it also allows user-
defined variables. This program parses the SDP template file, plugs in the
actual values for the block polynomials and user-defined variables, evaluates
elementary Mathematica expressions, and generates the final SDP for sdpb. The
typical runtime of sdp2input_mod for generating a SDP at $\Lambda=19,43$ in
Section 2.1 is about 7 seconds and 110 seconds respectively on a computer with
64 AMD EPYC 7532 32-Core Processor.

The workflow is managed using simpleboot 3.0 (https://gitlab.com/bootstrap
collaboration/simpleboot) as follows. Given a bootstrap problem, the SDP might
depend on various variables (such as the dimensions of the external operators
and the gaps in each channel). The simpleboot package generates a template
file for sdp2input_mod that contains those variables, but the value is not
plugged in. Such a template file is generated once for all in a setup (with
fixed $\Lambda$). In a specific sdpb run, simpleboot will compute the actual
values for the variables in the template and save it as the parameter file,
then call scalar_blocks_mod and sdp2input_mod to generate the SDP. This
workflow minimizes the operations in Mathematica and the most heavy
computation is done in C++. For the computations in this work, the evaluation
of the gradient of the navigator function requires generating the SDP multiple
times, thus the efficient workflow is essential. For users on a cluster
without Mathematica, one possible way to proceed is to use simpleboot to
generate a template file locally, then upload it to the cluster and write a
script to replace above management role of simpleboot (i.e. decide the
variables for the template file).

For the computation of the gradient of the navigator function, we used an
experimental version of approx_objective. The difference from the original
version is that this program will take the change of the SDP variable ‘‘$A$"
(defined in (2.14) of [23]) into consideration. However we later found this is
not really necessary for the computations in this paper, because if we fix the
non-polynomial part of the rational approximation of the conformal block
($\chi_{\ell}(\Delta)$ in (3.7) of [23]), the $A$ variable won’t change.

By the time this paper is published, all the programs are still in the testing
stage and there is very little documentation support. But we are committed to
make the method in this paper more accessible to the reader. Interested users
may come back to those links in a few months for the documentation and example
files.

##  Appendix B Technical details

For all computations, we used the following choices for the set of spins at
each value of $\Lambda$:

$\displaystyle S_{11}$ $\displaystyle=\\{0,\dots,20\\}\cup\\{49,52\\}\,,$
$\displaystyle S_{19}$ $\displaystyle=\\{0,\dots,26\\}\cup\\{49,52\\}\,,$
$\displaystyle S_{27}$
$\displaystyle=\\{0,\dots,31\\}\cup\\{33,34,37,38,41,42,45,46,49,50\\}\,,$
$\displaystyle S_{35}$
$\displaystyle=\\{0,\dots,44\\}\cup\\{47,48,51,52,55,56,59,60,63,64,67,68\\}\,,$
$\displaystyle S_{43}$
$\displaystyle=\\{0,\dots,64\\}\cup\\{67,68,71,72,75,76,79,80,83,84,87,88\\}\,.$
(21)

For $\Lambda=11,19,27,35,43$, the ‘‘--keptPoleOrder" parameter of
scalar_blocks_mod are $12,14,20,30,40$ and the ‘‘--order" parameter are 4
times ‘‘--keptPoleOrder". The sdpb parameters are given in table 3.

$\Lambda$ | 11,19,27,35 | 43
---|---|---
precision | 765 | 1024
dualityGapThreshold | $10^{-40}$ | $10^{-50}$
primalErrorThreshold | $10^{-40}$ | $10^{-50}$
dualErrorThreshold | $10^{-40}$ | $10^{-50}$
initialMatrixScalePrimal | $10^{20}$ | $10^{20}$
initialMatrixScaleDual | $10^{20}$ | $10^{20}$
feasibleCenteringParameter | 0.1 | 0.1
infeasibleCenteringParameter | 0.3 | 0.3
stepLengthReduction | 0.7 | 0.7
maxComplementarity | $10^{70}$ | $10^{70}$
Table 3: Parameters used for the sdpb.

To compute the gradient of the navigator function, we used a finite difference
of $10^{-40}$ in each argument of the navigator function. To maximize/minimize
a parameter in the navigator method, we used Algorithm 2 in [21] with
$g_{\text{tol}}=10^{-15}$. To minimum navigator, we used Algorithm 1 in [21]
with $g_{\text{tol}}=10^{-15}$. We made a small modification to Algorithm 1.
The standard BFGS line search is good for the most of the runs except the
$\Lambda=43$ run of Figure 2. The issue is that the gradient of the navigator
function for points outside the $\Lambda=43$ island is typically in the order
of $10^{-3}$, while inside the island it’s $10^{-16}$. If we use the standard
Broyden–Fletcher–Goldfarb–Shanno (BFGS) line search, at the first feasible
point, the line search step length will be very small because the gradient is
very small while the scale of the Hessian is still inferred based on the
points outside the island. The standard BFGS line search simply multiple the
step length by 4 in subsequent trials, but it will take too many steps
($\sim\text{log}_{4}(10^{16})=26.5$) to have the correct scale for the step
length. We fix this issue in the following way : if the first step with
$\alpha=1$ in the line search failed, we use the information at $\alpha=0,1$
to fit a quadratic polynomial in $\alpha$ and use this polynomial to do
extrapolation to determine the next step. This fix effectively reduce the
$\sim 26$ steps to 3 steps. We also made a small modification to Algorithm 2.
We add an option that if one start from an infeasible initial point, certain
parameters will be fixed until the first feasible point is found. The behavior
of algorithm before finding first feasible point is very similar to Algorithm
1 of [21]. For the run of Figure 9, the fixed parameter is $a$. The complete
code of both algorithms can be found in simpleboot.

##  Appendix C Collection of results

In the tables of this section, we listed the results of computations in this
paper.

$\Lambda$ | $\Delta_{\sigma}$ | $\Delta_{\epsilon}$ | $f_{\sigma\sigma\epsilon}/f_{\epsilon\epsilon\epsilon}$
---|---|---|---
$11$ | $0.518258316$ | $1.41269841$ | $1.45655643$
$19$ | $0.518134913$ | $1.41244118$ | $1.45641973$
$27$ | $0.518149340$ | $1.41261849$ | $1.45684755$
$35$ | $0.518148980$ | $1.41262326$ | $1.45688261$
$43$ | $0.518148886$ | $1.41262383$ | $1.45688799$
Table 4: Computations of the Numerics Setup 1. Setup | $\Delta_{\sigma}$ | $\Delta_{\epsilon}$ | $f_{\sigma\sigma\epsilon}$ | $f_{\epsilon\epsilon\epsilon}$ | $f_{T}$
---|---|---|---|---|---
$\left.\text{$\Lambda$=11 max(}\Delta_{\sigma}\right)$ | $0.51833135$ | $1.4147102$ | $1.0510383$ | $1.5356715$ | $1.2592846$
$\left.\text{$\Lambda$=11 min(}\Delta_{\sigma}\right)$ | $0.51775600$ | $1.4079531$ | $1.0536891$ | $1.5244978$ | $1.2579070$
$\left.\text{$\Lambda$=19 max(}\Delta_{\sigma}\right)$ | $0.51816152$ | $1.4127727$ | $1.0518006$ | $1.5327088$ | $1.2588684$
$\left.\text{$\Lambda$=19 min(}\Delta_{\sigma}\right)$ | $0.51812509$ | $1.4123433$ | $1.0519570$ | $1.5319139$ | $1.2588299$
Table 5: Computations of the Hybrid Setup 1. Setup | $\Delta_{\sigma}$ | $\Delta_{\epsilon}$ | $f_{\sigma\sigma\epsilon}$ | $f_{\epsilon\epsilon\epsilon}$ | $f_{T}$
---|---|---|---|---|---
$\left.\text{$\Lambda$=11 max(}\Delta_{\sigma}\right)$ | $0.51853855$ | $1.4172801$ | $1.0501657$ | $1.5413379$ | $1.2591992$
$\left.\text{$\Lambda$=11 min(}\Delta_{\sigma}\right)$ | $0.51772067$ | $1.4075014$ | $1.0538377$ | $1.5235323$ | $1.2579841$
$\Lambda$=11 min(nvg) | $0.51798601$ | $1.4106308$ | $1.0525454$ | $1.5281088$ | $1.2588351$
$\left.\text{$\Lambda$=19 max(}\Delta_{\sigma}\right)$ | $0.51816437$ | $1.4128073$ | $1.0517879$ | $1.5327764$ | $1.2588716$
$\left.\text{$\Lambda$=19 min(}\Delta_{\sigma}\right)$ | $0.51812208$ | $1.4123071$ | $1.0519696$ | $1.5318388$ | $1.2588280$
$\Lambda$=19 min(nvg) | $0.51813719$ | $1.4124867$ | $1.0519042$ | $1.5321763$ | $1.2588426$
Table 6: Computations of the Numerics Setup 2. ‘‘nvg" means the navigator function. Setups | $\Delta_{\sigma}$ | $\Delta_{\epsilon}$ | $f_{\sigma\sigma\epsilon}$ | $f_{\epsilon\epsilon\epsilon}$ | $f_{T}$ | $a$
---|---|---|---|---|---|---
$\Lambda=11,L_{0}=6$ | $0.51815015$ | $1.4126784$ | $1.0518300$ | $1.5327085$ | $1.2588618$ | $-0.55899947$
$\Lambda=19,L_{0}=10$ | $0.51815135$ | $1.4126552$ | $1.0518428$ | $1.5324985$ | $1.2588596$ | $-0.19533902$
$\Lambda=19,L_{0}=12$ | $0.51815078$ | $1.4126480$ | $1.0518455$ | $1.5324834$ | $1.2588589$ | $-0.19681209$
$\Lambda=19,L_{0}=14$ | $0.51814723$ | $1.4126063$ | $1.0518600$ | $1.5324020$ | $1.2588556$ | $-0.75148583$
Table 7: Computations of the Hybrid Setup 2. $L_{0}$ | $\Delta_{\sigma}$ | $\Delta_{\epsilon}$ | $f_{\sigma\sigma\epsilon}$ | $f_{\epsilon\epsilon\epsilon}$ | $f_{T}$ | $a$
---|---|---|---|---|---|---
$2$ | $0.51814914$ | $1.4126296$ | $1.0518523$ | $1.5324528$ | $1.2588570$ | $-0.34342837$
$4$ | $0.51814939$ | $1.4126327$ | $1.0518510$ | $1.5324596$ | $1.2588574$ | $-0.34375960$
$6$ | $0.51814853$ | $1.4126222$ | $1.0518549$ | $1.5324393$ | $1.2588564$ | $-0.34511252$
$8$ | $0.51814780$ | $1.4126136$ | $1.0518578$ | $1.5324206$ | $1.2588557$ | $-0.34595671$
$10$ | $0.51814901$ | $1.4126279$ | $1.0518529$ | $1.5324495$ | $1.2588568$ | $-0.34726307$
$12$ | $0.51815349$ | $1.4126802$ | $1.0518339$ | $1.5325445$ | $1.2588612$ | $-0.36100753$
Table 8: Computations of the Hybrid Setup 3. $L_{0}$ | $\Delta_{\sigma}$ | $\Delta_{\epsilon}$ | $f_{\sigma\sigma\epsilon}$ | $f_{\epsilon\epsilon\epsilon}$ | $f_{T}$ | $a$
---|---|---|---|---|---|---
$4$ | $0.51814534$ | $1.4125839$ | $1.0518688$ | $1.5323629$ | $1.2588538$ | $-0.39513721$
$8$ | $0.51814545$ | $1.4125852$ | $1.0518683$ | $1.5323654$ | $1.2588539$ | $-0.39519309$
$10$ | $0.51814544$ | $1.4125851$ | $1.0518683$ | $1.5323651$ | $1.2588539$ | $-0.39525413$
$12$ | $0.51814545$ | $1.4125852$ | $1.0518683$ | $1.5323653$ | $1.2588539$ | $-0.39525418$
Table 9: Computations of the Hybrid Setup 5. Setup | $\Delta_{\sigma}$ | $\Delta_{\epsilon}$ | $f_{\sigma\sigma\epsilon}$ | $f_{\epsilon\epsilon\epsilon}$ | $f_{T}$ | $a$
---|---|---|---|---|---|---
$\text{$\Lambda$=19 }L_{0}\text{=4}$ | $0.51814904$ | $1.4126275$ | $1.0518530$ | $1.5324459$ | $1.2588574$ | $-1.1730688$
Table 10: Computations of the Hybrid Setup 6. $r$ | $\Delta_{\sigma}$ | $\Delta_{\epsilon}$ | $f_{\sigma\sigma\epsilon}$ | $f_{\epsilon\epsilon\epsilon}$ | $f_{T}$ | $a_{o}$
---|---|---|---|---|---|---
$0$ | $0.51814848$ | $1.4126215$ | $1.0518548$ | $1.5324348$ | $1.2588571$ | $-0.34387402$
$0.2$ | $0.51814809$ | $1.4126169$ | $1.0518568$ | $1.5324276$ | $1.2588560$ | $-0.34326748$
$0.38$ | $0.51814855$ | $1.4126223$ | $1.0518546$ | $1.5324367$ | $1.2588571$ | $-0.34297320$
$0.4$ | $0.51814861$ | $1.4126231$ | $1.0518542$ | $1.5324375$ | $1.2588572$ | $-0.34281567$
$0.6$ | $0.51814891$ | $1.4126265$ | $1.0518528$ | $1.5324416$ | $1.2588574$ | $-0.34109519$
$0.8$ | $0.51814895$ | $1.4126270$ | $1.0518525$ | $1.5324415$ | $1.2588574$ | $-0.34014029$
$0.9$ | $0.51814889$ | $1.4126263$ | $1.0518528$ | $1.5324397$ | $1.2588574$ | $-0.33986723$
$0.975$ | $0.51815014$ | $1.4126410$ | $1.0518478$ | $1.5324710$ | $1.2588587$ | $-0.33875725$
$1$ | $0.51815029$ | $1.4126428$ | $1.0518472$ | $1.5324750$ | $1.2588589$ | $-0.33811048$
$1.06875$ | $0.51814889$ | $1.4126259$ | $1.0518534$ | $1.5324424$ | $1.2588575$ | $-0.33557516$
$1.1$ | $0.51814793$ | $1.4126145$ | $1.0518576$ | $1.5324205$ | $1.2588565$ | $-0.33393662$
$1.14216$ | $0.51814628$ | $1.4125946$ | $1.0518649$ | $1.5323818$ | $1.2588547$ | $-0.33053125$
Table 11: Computations of the Hybrid Setup 7.

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