11institutetext: Department of Physics, University of Pisa and INFN,
Largo Pontecorvo 3, I-56127 Pisa, Italy22institutetext: Department of Physics,
University of Crete, Heraklion GR-70013, Greece33institutetext: Institute of
Theoretical and Computational Physics (ITCP), Department of Physics,
University of Crete, 70013 Heraklion, Greece44institutetext: CPHT, CNRS, École
Polytechnique, Institut Polytechnique de Paris,
Route de Saclay, 91128 Palaiseau, France

# Spectrum continuity and level repulsion: the Ising CFT from infinitesimal to
finite $\boldsymbol{\varepsilon}$

Johan Henriksson 1,2,3 Stefanos R. Kousvos 4 & Marten Reehorst
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Using numerical conformal bootstrap technology we perform a non-perturbative
study of the Ising CFT and its spectrum from infinitesimal to finite values of
$\varepsilon=4-d$. Exploiting the recent navigator bootstrap method in
conjunction with the extremal functional method, we test various qualitative
and quantitative features of the $\varepsilon$-expansion. We follow the
scaling dimensions of numerous operators from the perturbatively controlled
regime to finite coupling. We do this for $\mathbb{Z}_{2}$-even operators up
to spin 12 and for $\mathbb{Z}_{2}$-odd operators up to spin 6 and find a good
matching with perturbation theory. In the finite coupling regime we observe
two operators whose dimensions approach each other and then repel, a
phenomenon known as level repulsion and which can be analyzed via operator
mixing. Our work improves on previous studies in both increased precision and
the number of operators studied, and is the first to observe level repulsion
in the conformal bootstrap.

##  1 Introduction

The past decades have brought various advances in both perturbative and non-
pertubative methods for studying conformal field theories. On the perturbative
side, there is the $\varepsilon$-expansion which started with Wilson and
Fisher Wilson:1971dc 111See also Wilson:1973jj ; Kleinert2001 ;
Pelissetto:2000ek  for an incomplete list of useful references. and recently
led to the computation of anomalous dimensions up to order ${\varepsilon}^{7}$
and ${\varepsilon}^{8}$ Schnetz:2016fhy ; SchnetzUnp . On the other hand there
is the non-perturbative numerical conformal bootstrap, whose flagship
application has been the most precise determination of the critical exponents
in the 3d Ising model Kos:2016ysd . With these tools in hand, the time is ripe
to perform a precision study on the relation between the perturbative and non-
perturbative descriptions. We also probe the conjecture that a continuous
family of conformal field theories (CFTs) exists for any dimension between
$3\leqslant d\leqslant 4$ connecting the 3d Ising CFT to the Gaussian theory
in 4d.222This assumption has been implicit in a lot of work, see e.g.
LeGuillou:1987ph , and was formulated as “spectrum continuity” in
Hogervorst:2015akt . The family is further conjectured to exist all the way
down to $d=2$ where it is believed to merge with the 2d Ising model originally
solved in Onsager:1943jn .

In addition, we will test for the existence of inherently finite-coupling
effects such as level repulsion, known from quantum mechanics as the von
Neumann–Wigner non-crossing rule vonNeumann1929  or avoided crossing. In the
context of CFTs, level repulsion concerning eigenvalues of the dilatation
operator was discussed in Korchemsky:2015cyx , who noted that if two operators
have dimensions that get close to each other, when varying a coupling, their
dimensions can be approximated by the eigenvalues of a two-state “Hamiltonian”

$\mathbb{D}=\begin{pmatrix}\Delta_{1}(g)&x(g)\\\
x(g)&\Delta_{2}(g)\end{pmatrix}.$ (1)

The approximation holds for small $x$, in the region where
$|\Delta_{1}-\Delta_{2}|$ is comparable to $x$. An explicit example satisfying
these assumptions was given in the case of the large $N$ expansion of
$\mathcal{N}=4$ super-Yang–Mills (SYM) theory. One of the goals of our paper
is to investigate if (1) is a good model also in the Ising CFT, where the
mentioned assumptions are not guaranteed to hold. In our context, the role of
$g$ is played by the spacetime dimension $d$, or equivalently
${\varepsilon}=4-d$.

We investigate the above questions by studying the Ising CFT in fractional
dimensions via the conformal bootstrap. Although the $\varepsilon$-expansion
predicts that the conjectured family of CFTs is non-unitary for non-integer
$d$ Hogervorst:2015akt , it was found that the numerical conformal bootstrap
is in practice insensitive to this non-unitarity El-Showk:2013nia ;
Cappelli:2018vir . This is likely because the operators responsible for the
non-unitarity occur at large scaling dimensions Hogervorst:2015akt , to which
the bootstrap is less sensitive.

We find a continuous family of allowed islands in parameter space for
spacetime dimensions in the range $3\leqslant d\leqslant 4$ (as well as below
$d=3$). Inside these islands we extract the approximate solution to crossing
via the extremal functional method (EFM) El-Showk:2012vjm ; El-Showk:2014dwa ;
Komargodski:2016auf ; Simmons-Duffin:2016wlq ; El-Showk:2016mxr . We then
compare the resulting spectrum against known perturbative results obtained in
the $\varepsilon$-expansion for $\phi^{4}$ theory. In addition to the
anomalous dimensions related to the usual critical exponents (such as
$\beta=\frac{\Delta_{\phi}}{d-\Delta_{\phi^{2}}}$) we also compare against
perturbative predictions for a much larger part of the spectrum. These
predictions were computed in Kehrein:1992fn ; Kehrein:1994ff ;
Hogervorst:2015akt ; Henriksson:2022rnm .

Let us briefly summarize the numerical conformal bootstrap method. For an
overview of results see the review papers Poland:2018epd  and Poland:2022qrs .
The flagship application of the numerical conformal bootstrap is to find
bounds with rigorous error bars on conformal data – operator dimensions and
operator product expansion (OPE) coefficients – that enter in the four-point
function of primary operators. These bounds are found by systematically
probing parameter space for self consistency (defined as crossing symmetry
plus unitarity). The demand of self consistency can be phrased in terms of the
existence, or not, of a functional with specific properties (positive action
on all operator contributions). If a functional exists for a specific choice
of conformal data, one concludes that this set of data cannot correspond to a
self consistent CFT, as defined above. In addition, one may find complementary
non-rigorous approximations of the operator spectrum by looking for minima of
the aforementioned functional.

The Ising CFT has been subject to several previous non-perturbative numerical
bootstrap studies, with the goal of determining its operator spectrum and
other CFT-data, or as a benchmark target for new techniques. Important for our
work is Simmons-Duffin:2016wlq , where a precision estimate was given for
dimensions and OPE coefficients of numerous operators in the low dimensional
operators in the 3d Ising model, obtained via the EFM. That paper also
verified that the leading twist families behave as expected from analytic
computations in the light-cone limit Fitzpatrick:2012yx ; Komargodski:2012ek ;
Caron-Huot:2017vep . Part of this spectrum was further studied in
Reehorst:2021hmp  where rigorous error bars were provided for a larger set of
CFT data involving 13 quantities. Among other bootstrap approaches is the
truncation method of Gliozzi:2013ysa  which was applied to the 3d Ising model
in Gliozzi:2014jsa . In this approach instead of rigorously carving out
parameter space, a truncated set of crossing equations is solved by assuming
only a small finite number of exchanged operators. Another similar approach is
to minimize the error that arises from truncating the crossing equation. This
was recently implement in Kantor:2021jpz ; Kantor:2021kbx  using machine
learning, although it only explicitly investigated the case of $D=2$. See also
Laio:2022ayq  for an alternative of this idea using Monte Carlo. Finally we
should mention the recent hybrid numeric-analytic bootstrap proposed by
Su:2022xnj . Here the numerical bootstrap is supplemented by information on
the large spin operators coming from the analytic light-cone bootstrap. While
not strictly rigorous this was found to greatly improve the constraining power
of the numerical bootstrap.

Most pertinent to this study, El-Showk:2013nia ; Cappelli:2018vir  previously
used the numerical conformal bootstrap to study the Ising CFT at finite
fractional ${\varepsilon}$ by following the position of a kink in the
$(\Delta_{\sigma},\Delta_{\epsilon})$ exclusion plot across various values of
$d$ between 2 and 4. The approach we take is similar, we also work in
fractional values of $d$ but instead of following the position of a kink in
parameter space, we follow the position of an isolated island obtained using a
mixed correlator bootstrap setup.333Such a mixed correlator bootstrap setup
has previously been used to find islands at $3$, $3.25$, $3.5$ and $3.75$
space time dimensions Behan:2016dtz . We will study more values of $d$ at
higher derivative order and we will additionally be extracting the extremal
spectrum (which cannot reliably be done at a lower derivative order). We make
use of the recently discovered navigator method Reehorst:2021ykw  to
systematically locate small unitary islands in intermediate spacetime
dimensions $2.6\leqslant d<4$,444The original range considered was $3\leqslant
d<4$, however we extended the lower limit when we observed that the targeted
level repulsion effect would happen below rather than above three dimensions.
in steps of $0.1$. Each island is isolated in terms of the variables
($\Delta_{\sigma}$, $\Delta_{\epsilon}$, $\theta$). Here $\Delta_{\sigma}$ and
$\Delta_{\epsilon}$ are the scaling dimensions of the only relevant
$\mathbb{Z}_{2}$-odd and $\mathbb{Z}_{2}$-even operators and
$\theta=\arctan(\lambda_{\epsilon\epsilon\epsilon}/\lambda_{\sigma\sigma\epsilon})$
is the OPE coefficient angle. Within these islands we obtain the spectrum of
operators in the theory via the EFM.

In order to obtain a CFT spectrum by the EFM, one has to fix specific values
of for dimensions of the external operators and choose a parameter to
extremize. For the latter there are in principle infinitely many choice, e.g.
we can maximize any OPE coefficients or superposition of OPE coefficients. In
Simmons-Duffin:2016wlq  a sampling over $20$ points was performed and
minimization of the central charge was chosen as the direction of
extremization. This gave an estimate of all low-lying CFT data. The standard
deviation within this sample population was then interpreted as an estimate of
the error on these values. The rigorous bounds of Reehorst:2021hmp  were found
to closely match the intervals given by the error bars of the non-rigorous
estimates. Since the spread in the CFT data under the mentioned sampling was
found to be small, we avoid the computationally costly process of sampling and
study only a single spectrum spectrum for each dimension. In order to have a
unique well-defined selection criterion we examine only the spectrum at the
minimum of the navigator function.555To be precise, we minimize $C_{T}$ at the
external operator values corresponding to the navigator minimum. This point
can be seen as the “most allowed point” in the sense that it is the most
probable to remain un-excluded when increasing the number of constraints. We
hypothesize that the spectrum at this point gives a good estimate of the true
spectrum of the underlying CFT, see some comparisons in section D.1.

Indeed, we find that spectra close to the navigator minimum match very well
with the estimates from the $\varepsilon$-expansion for small $\varepsilon$.
Surprisingly, in some cases even order $\varepsilon^{1}$ un-resummed
predictions for scaling dimensions compare favourably with the non-
perturbative $d=3$ bootstrap results. For the case of $\mathbb{Z}_{2}$-even
scalars we find a clear example of level repulsion, involving the operators
with third and fourth smallest scaling dimensions. We also investigate which
operators present in the ${\varepsilon}$-expansion are first to go undetected
in the EFM. We find that whether an operator is detected or not is not just
determined by the dimension of the operator. Looking at e.g. the
$\mathbb{Z}_{2}$-even spin-$0$ sector we see that higher dimensional operators
with a larger OPE coefficient are detected while lower dimensional operators
with smaller OPE coefficients are overlooked.

To denote operators we will mostly follow the numerical bootstrap conventions
and denote $\mathbb{Z}_{2}$-even scalars by
$\epsilon,\epsilon^{\prime},\ldots$ and $\mathbb{Z}_{2}$-odd scalars by
$\sigma,\sigma^{\prime},\ldots$. However, when comparing the numerical data
with the $\varepsilon$-expansion we will also use the notation that is
conventional in that context, $\sigma=\phi$, $\epsilon=\phi^{2}$ etc.

##  2 Review

In this section we first briefly review the Ising CFT, focusing on previous
results relevant to our study. We then present a model for level repulsion in
the context of CFTs.

###  2.1 The Ising CFT

In integer dimensions, the Ising CFT can be defined in terms of the lattice
formulation of the Ising model. This lattice spin model is described by the
following Hamiltonian,

$H=J\sum_{\langle ij\rangle}\sigma_{i}\sigma_{j},$ (2)

where $\langle ij\rangle$ indicates that the sum runs over all nearest-
neighbor pairs of lattice points and $\sigma_{i}\in\\{-1,+1\\}$. We also take
$J<0$ so that the energy is minimized when the spins align.

At the continuous phase transition that is present in this model all
correlation functions become scale invariant at distances much larger than the
lattice spacing. In the continuum limit, these correlation functions are
believed to be described by a CFT.666In the special case of $d=2$ this has
been proven Chelkak2012 . For developments in $d=3$ and $d=4$ see Aizenman2014
and Aizenman:2019yuo  respectively. As can be seen from (2) this theory
possesses a $\mathbb{Z}_{2}$ flavor symmetry acting on the spin degrees of
freedom.

Taking the continuum limit one obtains a system described by the following
Hamiltonian density (see for instance Fisher1983 )

$\mathcal{H}=(\partial\phi)^{2}+m^{2}\phi^{2}+g\phi^{4}\,.$ (3)

This quantum field theory can be studied in $d=4-\varepsilon$ dimensions by
analytic continuation. In the limit $\varepsilon\rightarrow 0$, this theory
possesses an RG fixed point in the IR, at which $m^{2}=0$ and
$g=O(\varepsilon)$. The fixed point is of course scale invariant, since the
beta function vanishes there. Thus, under the assumption that the scale
invariance is enhanced to conformal invariance, this fixed point should again
be described by a $\mathbb{Z}_{2}$ symmetric CFT. Since this QFT describes the
continuum limit of the Ising model, its IR fixed-point is the Ising CFT.
Working at small $\varepsilon$ the coupling constant at the fixed-point is
infinitesimally small and provides the basis for a controlled perturbative
expansion.

An alternative to this perturbative understanding of the Ising CFT, is the
more axiomatic approach of the conformal bootstrap. In this approach conformal
symmetry is imposed and the Ising CFT is obtained as the only unitary
interacting CFT with $\mathbb{Z}_{2}$ symmetry that has $\Delta_{\sigma}$ and
$\Delta_{\epsilon}$ values close to those expected from other descriptions of
the Ising CFT.777This last condition is required because the numerical
conformal bootstrap cannot exclude solutions at large values of
$\Delta_{\sigma}$ where an additional peninsula always exists. An example of a
theory populating this peninsula is the supersymmetric Ising model Fei:2016sgs
; Rong:2018okz ; Atanasov:2018kqw ; Atanasov:2022bpi .

####  2.1.1 The ${\varepsilon}$-expansion

In this work, we will be comparing with various results obtained using the
$\varepsilon$-expansion. The techniques used to obtain these can roughly be
placed in three groups.

The first is the usual Feynman diagram expansion. This method provides the
data to highest order in $\varepsilon$. Calculations up to six loops for
generic multi-scalar theories have been carried out in Bednyakov:2021ojn ,
building on the calculations of Kompaniets:2017yct . Seven loop results
specifically for $\Delta_{\phi}$, $\Delta_{\phi^{2}}$ and $\Delta_{\phi^{4}}$
can be found in Schnetz:2016fhy . The scaling dimensions for $\phi$,
$\phi^{2}$ and $\phi^{4}$ that are used in this work were obtained with these
methods.888A result for $\Delta_{\phi}$ has recently been obtained to order
$\varepsilon^{8}$ SchnetzUnp .

The second group consists of calculations that work systematically for generic
operators (i.e. generic spin and global symmetry representation) at one loop
Kehrein:1992fn ; Kehrein:1994ff . The fact that one can systematically compute
corrections to anomalous dimensions of any operator at one loop, without
computing explicit diagrams, can be explained by the observation that the beta
function at one loop may be completely determined via knowledge of the OPE
coefficients at zeroth order Cardy:1996xt . The free theory OPE coefficients
themselves are computable via Wick contractions, see Hogervorst:2015akt  for
an application of this method to the one-loop dilatation problem. The result
of such methods are lists of primary operators constructed out of powers of
the order parameter field $\phi$, with the insertion of gradients, together
with results for their one-loop anomalous dimension Kehrein:1994ff ;
Henriksson:2022rnm .

The third method combines the $\varepsilon$ expansion with the analytic
bootstrap, which typically gives the CFT data for certain families of
operators, referred to as twist families. The most precise results concern the
leading twist operators. A general recipe for computing CFT data in $\phi^{4}$
theories with arbitrary global symmetries using this approach has been
presented in Henriksson:2020fqi  (also for large $N$). While less systematic
than directly computing diagrams, the method has various advantages. For
example, one obtains simultaneously the anomalous dimensions for an entire
family of operators, i.e. for arbitrary spins with twist and global symmetry
representation fixed. Another advantage is that the computation of OPE
coefficients is often not much more complicated than the computation of
scaling dimensions.999See for instance Gopakumar:2016cpb ; Alday:2017zzv ;
Carmi:2020ekr .

####  2.1.2 The Ising CFT in 3d

One way of obtaining quantitative predictions for conformal data in $d=3$ is
to continue the $\varepsilon$-expansion data to $\varepsilon=1$. Typically, in
order to obtain somewhat reliable estimates for exponents, one has to perform
resummations. These estimates are a posteriori found to be in good agreement
with other non-perturbative methods such as the conformal bootstrap (see e.g.
figure 3). While this is commonly believed to be the case for leading operator
dimensions, in this work we find that, remarkably, this also holds for many
operators higher up in the spectrum, even when we only have access to the
first-order anomalous dimensions in the ${\varepsilon}$-expansion.

On the conformal bootstrap side, state of the art results include Kos:2016ysd
, which confined $\Delta_{\sigma}$, $\Delta_{\epsilon}$ and
$\theta=\arctan(\lambda_{\epsilon\epsilon\epsilon}/\lambda_{\sigma\sigma\epsilon})$
to lie in a small isolated three dimensional island. These islands were so
small, that they provided the most precise determination of the critical
exponents describing the Ising critical point. In Simmons-Duffin:2016wlq ,
using the extremal functional method of El-Showk:2012vjm ; El-Showk:2016mxr ,
the author was able to estimate the operator data for many low dimensional
operators. A large part of these operators were observed to fall on so called
double twist trajectories, as predicted analytically in Komargodski:2012ek ;
Fitzpatrick:2012yx .101010In perturbation theory, such operators were observed
to exist much earlier Parisi:1973xn ; Callan:1973pu ; Kehrein:1995ia ;
Derkachov:1996ph . More recently, Reehorst:2021hmp  provided rigorous error
bars for the determination of some of the operators studied in Simmons-
Duffin:2016wlq .

####  2.1.3 Bootstrapping in non-integer dimensions

An important idea that we test in this paper is that of “spectrum continuity,”
i.e. that the spectrum, as well as other CFT-data such as OPE coefficients,
should change continuously as we follow the fixed-point across spacetime
dimensions $d$. While success of the $\varepsilon$-expansion for observables
like the critical exponents suggests that such continuation is possible, there
are some potential caveats. The most important in the context of this paper is
the fact that by a careful study of the $\varepsilon$-expansion, the Ising
model was found to be non-unitary in non-integer dimensions Hogervorst:2015akt
. This is manifest in several ways, such as negative-norm states, complex OPE
coefficients and complex anomalous dimensions. The continuation of the
spectrum across dimensions also raises questions about representation theory
of the Lorentz (rotation) group $SO(d)$ for non-integer $d$. Since the theory
also respects spacetime parity, we can combine Lorentz and parity to $O(d)$,
and the representation theory for that group can formally be continued across
$d$, see the discussion in Binder:2019zqc  for the $O(N)$ global symmetry
group.111111For $O(N)$ models, continuing in $N$ is in this respect similar to
continuing in $d$. Alternatively, the $O(N)$ model at non-integer $N$ can be
seen as a loop gas model. See for instance Grans-Samuelsson:2021uor ;
Gorbenko:2020xya  for representation theory of the 2d $O(N)$ model, and
Liu:2012ca  for a Monte Carlo study of the 3d $O(N)$ model.

Despite the violation of unitarity just discussed, in any practical
application so far the conformal bootstrap has been insensitive to it. For
example, El-Showk:2013nia  and Cappelli:2018vir  found that the kinks in
exclusion plots persist in fractional dimensions and moreover, the positions
of these kinks are in excellent agreement with perturbative predictions
LeGuillou:1987ph . Thus, at the derivative orders currently being studied, the
bootstrap appears to be essentially insensitive to any non-unitarity of this
type. In other words, the numerical conformal bootstrap is unable to exclude
these theories from parameter space even though they are non-unitary. The same
effect was found to persist also when a mixed correlator setup is used to find
isolated islands Behan:2016dtz .121212Below $d=2$, on the other hand, the
picture changes drastically, and the sequence of clear kinks for $d\geqslant
2$ degenerates into two features that then move away from each other
Golden:2014oqa .

A related observation was made in the study of a vector model with $O(500)$
symmetry in $d=5$ Li:2016wdp . This theory is also expected to be non-unitary
and yet imposing unitarity an isolated island was found. Note that this non-
unitarity is of a different type. Here the non-unitarity comes from large-$N$
instanton corrections to the scaling dimensions, i.e. contributions
proportional to $e^{-\frac{a}{1/N}}$ which multiply a complex number, where
$a$ is some $d$ dependent number Giombi:2019upv . Also in this case, the
bootstrap was insensitive to the non-unitarity.

Lastly, a more recent example is Sirois:2022vth . In this case the author
studied the $O(N)$ model for various non-integer values of $N$, which are know
to be non-unitary Binder:2019zqc . However, the bootstrap was apparently yet
again insensitive to the violation of unitarity in the range $N>1$.131313For
$N<1$, there is a clear manifestation of unitary violation in terms of
negative squared OPE coefficients involving low-lying operators even in the
free theory. In this region, indeed the bootstrap approach of Sirois:2022vth
was able to detect this unitarity violation. This trend continues in the
present work, where also do not see any noticeable effect due to the violation
of unitarity.

###  2.2 Level repulsion

In this section we will give some more details on level repulsion or avoided
crossing, which is a generic feature of Hermitian matrices, discussed in the
context of quantum mechanics by von Neumann and Wigner vonNeumann1929 . In the
context of conformal field theories, the closest equivalent to the Hamiltonian
of quantum mechanics is the dilatation operator, and the spectrum of energy
levels corresponds to the spectrum of scaling dimensions.141414Level repulsion
does not happen when states belong to different symmetry irreps. In what
follows, we assume that all operators discussed are primaries in the same
representation of the Lorentz and global symmetry group. Level repulsion would
be manifest as observing the scaling dimensions of two operators, as we vary
${\varepsilon}$, to first approach each other and then repel without crossing.

As mentioned in the introduction, level repulsion in the contexts of CFT was
discussed in Korchemsky:2015cyx  with examples from $\mathcal{N}=4$ SYM,
extremal spectra at the bootstrap bound and QCD baryon
distributions.151515Moreover, in Behan:2017mwi , it was shown that on a one-
dimensional conformal manifold, i.e. in the context of an exactly marginal
coupling, avoided crossing can be inferred from the differential equation that
governs the evolution of the CFT-data.

Here we will make a simplified argument for how a matrix like (1) would arise
to a reasonable approximation in a general setup. Although level repulsion
happens at finite coupling – for us manifested as finite ${\varepsilon}$ – it
is instructive to start by considering the situation in perturbation theory.
For concreteness, consider the case of perturbing the free theory with a
collection of operators,

$\mathcal{H}=\mathcal{H}_{0}+\sum_{i}g_{i}{\mathcal{O}}_{i}$ (4)

and focus on the case where there is a non-trivial conformal fixed-point,
$g_{i}=g^{*}_{i}$. Here, in principle, the sum runs over all operators in the
theory. Moreover, we assume that the couplings depend on a single underlying
parameter ${\varepsilon}$, so that $g^{*}_{i}\to 0$ as ${\varepsilon}\to
0$.161616For us, ${\varepsilon}$ will be the dimensional parameter
${\varepsilon}=4-d$, but for instance in the existence of a conformal
manifold, ${\varepsilon}$ could be the coordinate along a one-parameter curve
inside the conformal manifold. The perturbative spectrum is given by the
eigenvalues of the matrix of partial derivatives of the beta functions

$\omega=\left.\left(\frac{\partial\beta_{i}}{\partial
g_{j}}\right)\right|_{g^{*}}.$ (5)

More precisely,

$\Delta_{i}=d-\lambda_{i},\qquad\omega\vec{v}_{i}=\lambda_{i}\vec{v}_{i}$ (6)

In perturbation theory, typically $\omega$ is an upper block-triagonal
matrix,171717See e.g. Brown:1979pq . where the blocks are formed out of
operators with the same dimension in the free theory,

$\omega=\begin{pmatrix}\cline{1-1}\cr\vrule\lx@intercol\hfil\omega_{n_{1}\times
n_{1}}\hfil\lx@intercol&\vrule\lx@intercol\hfil
A\hfil\lx@intercol&A^{\prime}\\\
\cline{1-2}\cr&\vrule\lx@intercol\hfil\omega_{n_{2}\times
n_{2}}\hfil\lx@intercol&\vrule\lx@intercol\hfil
A^{\prime\prime}\hfil\lx@intercol&\ddots\\\ \cline{2-3}\cr
0&&\vrule\lx@intercol\hfil\omega_{n_{3}\times
n_{3}}\hfil\lx@intercol&\vrule\lx@intercol\hfil\hfil\lx@intercol&\\\
\cline{3-4}\cr&&&\vrule\lx<EMAIL_ADDRESS>(7)

Here $A$, $A^{\prime}$ etc. denote matrices with entries that are generically
non-zero. The zeros in this matrix may get non-perturbative
corrections.181818Such contributions may for instance be generated by
instanton contributions of the form $e^{-a/\varepsilon}$, where $a$ is a
number independent $\varepsilon$.  Neglecting such non-perturbative
corrections, the set of eigenvalues of (7) is the collection of the
eigenvalues of each of the diagonal blocks. The precise form of $\omega$ is
scheme dependent, however its eigenvalues (at the critical point) are not.

To a reasonable approximation, valid _a priori_ only at infinitesimal
${\varepsilon}$, the spectrum of the theory can then be written as the
eigenvalues of a matrix

$\mathbb{D}=\begin{pmatrix}\cline{1-3}\cr\Delta_{1}({\varepsilon})&&0&\vrule\lx@intercol\hfil\hfil\lx@intercol&&&\\\
&\ddots&&\vrule\lx@intercol\hfil\hfil\lx@intercol&X&&X^{\prime}\\\
0&&\Delta_{n_{1}}({\varepsilon})&\vrule\lx@intercol\hfil\hfil\lx@intercol&&&\\\
\cline{1-6}\cr&&&\vrule\lx@intercol\hfil\Delta_{n_{1}+1}({\varepsilon})\hfil\lx@intercol&&0&\vrule\lx@intercol\hfil\hfil\lx@intercol\\\
&X^{\dagger}&&\vrule\lx@intercol\hfil\hfil\lx@intercol&\ddots&&\vrule\lx@intercol\hfil
X^{\prime\prime}\hfil\lx@intercol\\\ &&&\vrule\lx@intercol\hfil
0\hfil\lx@intercol&&\Delta_{n_{1}+n_{2}}({\varepsilon})&\vrule\lx@intercol\hfil\hfil\lx@intercol\\\
\cline{4-7}\cr&X^{\prime\dagger}&&&X^{\prime\prime\dagger}&&\vrule\lx<EMAIL_ADDRESS>(8)

Here the diagonal elements are the eigenvalues of (7), computed block by
block, and we have accounted for some off-block-diagonal entries in the form
of matrices $X$, $X^{\prime}$, etc, which must vanish as ${\varepsilon}\to
0$.191919It may be that non-perturbative corrections also appear at the “zero
entries” inside the blocks of (8). In that case one can either perform a
further change of basis to accommodate them, or reduce the size of the blocks.
The conclusions for level repulsion below remain unchanged. They may get
contributions at higher order in perturbation theory, or be completely non-
perturbative.

As long as the off-diagonal terms $X$, $X^{\prime}$ etc. are small, the set of
eigenvalues of $\mathbb{D}$ in (8) is captured well by the diagonal entries,
which we take without loss of generality to be ordered by their values at
${\varepsilon}=0$, $\Delta_{i}(0)\leqslant\Delta_{i+1}(0)$ etc. This statement
breaks down when two consecutive diagonal entries in different blocks become
approximately equal, which is the case where there is an apparent crossing,
say $\Delta_{n_{1}}({\varepsilon})\approx\Delta_{n_{1}+1}({\varepsilon})$. In
the region where

$|\Delta_{n_{1}}({\varepsilon})-\Delta_{n_{1}+1}({\varepsilon})|\lesssim x$
(9)

we can no longer ignore the corresponding off-diagonal entries $x=X_{n_{1},1}$
and $X_{n_{1},1}^{*}$. Notice that by construction, this can only happen at
finite ${\varepsilon}$.

As long as no third operator dimension is also in the vicinity, we can
simplify the analysis further by looking at the eigenvalues of the following
submatrix,

$\begin{pmatrix}\Delta_{1}({\varepsilon})&x({\varepsilon})\\\
x({\varepsilon})&\Delta_{2}({\varepsilon})\end{pmatrix}.$ (10)

This is the matrix introduced in (1). For simplicity we consider the non-
diagonal terms to be real. The eigenvalues of (10) are

$\Delta_{\pm}=\frac{\Delta_{1}+\Delta_{2}}{2}\pm\sqrt{\left(\tfrac{\Delta_{1}-\Delta_{2}}{2}\right)^{2}+x^{2}}$
(11)

and the minimal distance, where $\Delta_{1}=\Delta_{2}$, is

$(\Delta_{+}-\Delta_{-})_{\mathrm{min}}=2x.$ (12)

Let us try to convert the preceding schematic discussion into to a testable
hypothesis. Our model (8) gives a good description of the spectrum of the
theory if the following two conditions hold:

  1. 1.

The off-block-diagonal terms in $X$, $X^{\prime}$ etc. are numerically small,
say $\lesssim x\ll 1$.

  2. 2.

Away from the regions where
$|\Delta_{i}({\varepsilon})-\Delta_{j}({\varepsilon})|\lesssim x$, the
diagonal entries $\Delta_{i}({\varepsilon})$, computed from perturbation
theory, approximate the spectrum well.

A theory in which these conditions hold can be said to be well approximated by
perturbation theory.202020It may also happen that $x$ is numerically small,
but the perturbative predictions of $\Delta_{i}({\varepsilon})$ fail to
reproduce the spectrum at finite ${\varepsilon}$. In that case we would still
expect level repulsion to occur, and expect the two operators involved in the
repulsion to have dimensions locally described by the eigenvalues (11), as
long as $\Delta_{i}$ and $x$ do not change too rapidly in ${\varepsilon}$. In
fact, a case where the discussion above can be put on a more firm footing is
in $\mathcal{N}=4$ SYM, at large number of colors $N$ Korchemsky:2015cyx .
There the diagonal elements of (8) are not computed from perturbation theory,
but can instead be taken to be the planar operator dimensions, known from
integrability Gromov:2009zb . In that case, the off-diagonal terms scale as
$O(1/N)$, leading for instance to level repulsion at order $1/N$ between the
Konishi operator and a sequence of double-trace operators with increasing
scaling dimension. In fact, by using holographic results for OPE coefficients
from Minahan:2014usa , the minimal separation in the case of mixing between
Konishi and the singlet double-trace operator
$[{\mathcal{O}}_{J},{\mathcal{O}}_{J}]$ at large $J$ was determined to leading
order in $1/N$ Korchemsky:2015cyx .

A concrete question that we ask in this paper is if we can numerically observe
the phenomenon of level repulsion, and then evaluate to what extent the two
above conditions hold. We stress that it is not obvious that they should apply
to our case. In fact, since strictly speaking we are considering a different
theory at each different dimensions $d$, we cannot exclude the possibility of
other, more exotic, scenarios to occur, such as operators annihilating and
going into the complex plane, or decoupling from the spectrum. Let us point
out that, to some extent, both these things do indeed happen in the Ising CFT;
there are operators with complex scaling dimensions, and operators that vanish
identically when one approaches integer spacetime dimensions
Hogervorst:2015akt . These effects, however, are expected to happen higher up
in the spectrum than what we are able to probe in this work.

##  3 Method

For the rest of this work we will make use of standardised numerical bootstrap
techniques, which will be briefly outlined below as well as in the appendices.
We study the same crossing equations as El-Showk:2012cjh ,212121These may also
be obtained in an automated fashion with Go:2019lke , which also works for a
large amount of supported groups beyond $\mathbb{Z}_{2}$. but instead of
studying only the case $d=3$ we will consider fractional dimensions between
$2.6$ and $4$. Such a comprehensive multi-correlator study in such a wide
region of dimensions has not been done before and is now possible due to
advancing numerical bootstrap techniques. Specifically we will be relying on a
a recent advancement in the numerical conformal bootstrap called the
_navigator_ Reehorst:2021ykw .

###  3.1 Obtaining the spectrum using the navigator and the EFM

The navigator, or navigator function, is a function that replaces the binary
‘allowed’ /‘disallowed’ information from the traditional bootstrap with some
continuous measure of success. I.e., it gives some measure of the distance to
an allowed point. Specifically, for the search space spanned by the dimension
of the relevant $\mathbb{Z}_{2}$-odd and $\mathbb{Z}_{2}$-even scalars
$\Delta_{\sigma}$ and $\Delta_{\epsilon}$ and the OPE angle
$\theta=\arctan(\lambda_{\epsilon\epsilon\epsilon}/\lambda_{\sigma\sigma\epsilon})$,
Reehorst:2021ykw  constructed a function
$\mathcal{N}(\Delta_{\sigma},\Delta_{\epsilon},\theta)$, with the following
properties:

  1. 1.

The function is bounded from above and exists in the whole search space.

  2. 2.

It differentiates between disallowed and allowed points by being positive on
the former and non-positive on the latter. (Being zero on the boundary by
continuity.)

  3. 3.

It is differentiable.

  4. 4.

It is cheaply computable by standard technology (it corresponds to an OPE
maximization using SDPB Simmons-Duffin:2015qma ; Landry:2019qug )

  5. 5.

Its derivatives are cheaply computable (see approx_objective provided with
SDPB).

These properties allow us to efficiently find isolated allowed regions (i.e.
islands) and the boundaries of these regions. It also allows us to find the
minimum of the navigator function which was conjectured to lie close in the
search space to the true underlying CFT.222222This can be seen for example by
considering how the island and the navigator function change when changing the
derivative order used in the numerics. Here we conjecture this to also extend
to the extremal spectrum. Specifically, we conjecture that the solution to the
(truncated) crossing equations, extracted by the extremal functional method
(EFM), at this minimum point, is a good predictor for the spectrum of the
underlying CFT.232323In the numerical conformal bootstrap a functional is
constructed from a certain number of derivatives at $z=\bar{z}=\frac{1}{2}$. A
parameter $\Lambda$ is usually used to parameterize the number of derivatives
in this basis of functionals. A larger value of $\Lambda$ leads to stronger
bounds. Bounds derived at any value of $\Lambda$ are rigorous and will hold
also for the solution to the full crossing equations. On the other hand a
solution extracted by the EFM method will only be a solution to the truncated
crossing equations given by the derivative expansion around the crossing-
symmetric point. This solution is expected to give a good approximation to the
low-dimensional operator spectrum of the solution to the full crossing
equations. Thus, we use the following procedure in order to estimate the
spectrum at all values of $\varepsilon$:

  1. 1.

Minimize the navigator function
$\mathcal{N}(\Delta_{\sigma},\Delta_{\epsilon},\theta)$. (Using the modified
Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm proposed in Reehorst:2021ykw
).

  2. 2.

Assuming a negative minimum is found, minimize the central charge $C_{T}$ in
order to obtain an extremal functional.

  3. 3.

Extract this extremal solution to the truncated crossing equations using for
example spectrum.py spectrum .

We choose to consistently present only the spectrum at the navigator minimum
rather than searching for many allowed points to minimize $C_{T}$ at and
averaging over the resulting spectra. The navigator minimum provides a clear
universal criterion to pick a point. Since the spread in the data found of
spectra obtained at different points is small, this should not qualitatively
affect our study. Given that the navigator point is the furthest from being
excluded at that given derivative order, we expect it to be a better predictor
of the true spectrum than any other point. We leave it for future work to
apply the methods of Reehorst:2021hmp  in order to turn our results into
rigorous bounds.

When constructing the navigator function we use the following assumptions:

$\displaystyle\Delta_{\epsilon^{\prime}}$ $\displaystyle\geqslant d\,,$ (13)
$\displaystyle\Delta_{\sigma^{\prime}}$ $\displaystyle\geqslant d\,,$ (14)
$\displaystyle\Delta_{T}$ $\displaystyle=d\,,$ (15)
$\displaystyle\Delta_{T^{\prime}}$ $\displaystyle\geqslant d+1\,.$ (16)

Here $\Delta_{\epsilon^{\prime}}$ denotes the dimension of the second lowest
$\mathbb{Z}_{2}$-even scalar, $\Delta_{\sigma^{\prime}}$ the second lowest
$\mathbb{Z}_{2}$-odd scalar and $\Delta_{T^{\prime}}$ the dimension of the
second lowest spin-2 $\mathbb{Z}_{2}$-even operator. We also assume the
existence of the stress tensor $T$. Finally we impose the stress-tensor ward
identity, i.e. we impose that the ratio of OPE coefficients with which the
stress tensor appears in the $\sigma\times\sigma$ and $\epsilon\times\epsilon$
OPEs is fixed as follows Osborn:1993cr :

$\lambda_{\sigma\sigma T}/\lambda_{\epsilon\epsilon
T}=\Delta_{\sigma}/\Delta_{\epsilon}.$ (17)

For the case of $d=3$ these assumptions are identical to the assumptions made
in Simmons-Duffin:2016wlq .

For more details on how these assumptions lead to the construction of the
navigator function $\mathcal{N}(\Delta_{\sigma},\Delta_{\epsilon},\theta)$ see
appendix B.

###  3.2 Initial search point

Given that we only wish to reach the minimum of the navigator function, and
not map out in detail the parameter space in the vicinity of the Ising CFT, we
started the navigator search as close as possible to the actual CFT. This
reduces the number of navigator function calls required to find the minimum
and increases numerically efficiency. To this end we used all previously
available information to make an optimal guess for the position of the allowed
island in various dimensions. In practice, we performed a Padé resummation of
the $\varepsilon^{7}$ results of Schnetz:2016fhy  such that in $d=3$ they
match the results of Kos:2016ysd .242424For the sake of clarity, we stress
that in Padé resummations throughout the rest of the work we do _not_ fix the
behavior at $d=3$. We then took the average between these, and the values
given by the interpolation of Cappelli:2018vir . Similarly, for the OPE
coefficients we used the results of Carmi:2020ekr  and Henriksson:2020jwk ,
and then required that the resummation matched the 3d value known from the
bootstrap.

###  3.3 Extremal functional method

When we use semi-definite programming to find a lower bound on $C_{T}$ at the
navigator minimum this provides an extremal functional El-Showk:2012vjm ; El-
Showk:2016mxr ; Simmons-Duffin:2016wlq . The zeros of the determinant of this
functional acting on the $\vec{V}$’s appearing in (37) should correspond to
the dimensions of the operators in the solution to the (truncated) crossing
equations at this point. We extract this spectrum using standard procedures
(see spectrum ). Due to finite numerical errors in the solution, the positions
of the double zeros of the functional will not really reach zero. Therefore it
is customary to look for the minima of the determinant instead. One caveat
should be noted: spectrum  excludes operators whose squared OPE coefficients
are negative. However, we found that there was often a numerical instability
due to which the OPE coefficients could not accurately be determined from the
primal solution. Therefore, the OPE coefficients of physical operators could
appear to be negative and wrongly be excluded. In order to not miss any
operators we instead report all minima whether spectrum.py finds a OPE
coefficient squared that is positive or not, see also Appendix D. Note that in
order to obtain an estimate of the OPE coefficients this method minimizes the
error in an over determined system. In principle all the equations in this
overdetermined system should approximately agree on the correct OPE
coefficients. However, we found that for some points this was not the case.
Picking some subset of equations and minimizing the error only for that subset
results in a large spread in the OPE coefficients found. The OPE coefficient
can easily change many magnitudes and even the sign depending on which
equations in the overdetermined system are included in the error
minimization.252525This phenomenon also occurs in $d=3$ and we therefore
believe it to be unrelated to actual negative squared OPEs appearing in the
solution.

##  4 Results

In this section we present the results from the numerical study of this paper,
and compare with perturbative data. For all $d$, a single negative navigator
minimum was found in the three-dimensional space parametrized by
($\Delta_{\sigma}$, $\Delta_{\epsilon}$, $\theta$). We present these in the
section below. Next we study the spectra extracted from these points, starting
with the $\mathbb{Z}_{2}$-even scalars where a clear example of level
repulsion is observed. We then briefly comment on the spectra in some other
interesting channels. Finally, we look at data for the leading two OPE
coefficients.

###  4.1 Navigator minima

The first step in the determination of the spectrum is to find a set of
allowed points. We performed this study at dimensions

$d\in\\{2.6,\,2.7,\,2.8,\,2.9,\,3.0,\,3.1,\,3.2,\,3.3,\,3.4,\,3.5,\,3.6,\,3.7,\,3.8,\,3.9,\,3.95\\}.$
(18)

At each of the points we find a local navigator minimum where the navigator is
negative. This gives a sequence of points

$\left(\Delta_{\sigma}|_{\mathrm{min}}(d),\Delta_{\epsilon}|_{\mathrm{min}}(d),\theta|_{\mathrm{min}}(d)\right),$
(19)

where
$\theta=\arctan(\lambda_{\epsilon\epsilon\epsilon}/\lambda_{\sigma\sigma\epsilon})$.
The complete collection of allowed points are reported in table 6 in appendix
C.2. In figure 1 we display the corresponding anomalous dimensions
$(\Delta_{\sigma}|_{\mathrm{min}}-\frac{d-2}{2},\Delta_{\epsilon}|_{\mathrm{min}}-(d-2))$.
This figure can be compared with the results of El-Showk:2013nia , who gave a
corresponding figure with kinks in the $\Delta_{\sigma},\Delta_{\epsilon}$
exclusion plots.

Figure 1: The final points of the navigator search, presented in the space of
anomalous dimensions. The point $d=3.95$ is excluded for clarity.

The points (19) provide estimates for the critical exponents using well-known
formulas, see e.g. Cardy:1996xt ; Henriksson:2022rnm .

###  4.2 $\mathbb{Z}_{2}$-even scalars

At each of the allowed points (19) we determined an estimate for the spectrum
using the extremal functional method, as described in appendix D. Here we will
discuss in detail the spectrum of $\mathbb{Z}_{2}$-even scalars, which is
shown in figure 2.

Figure 2: The dimensions of the $\mathbb{Z}_{2}$-even operators of spin 0. The
black dots indicate the spectrum found using the EFM at the navigator minimum.
The lines of various colors denote the leading order $\varepsilon$-expansion
prediction for operators of a certain type (see the main text).

In this figure, the black dots represent the numerical data corresponding to
the minima of the determinant of the extremal functional for
$\mathbb{Z}_{2}$-even scalars in each dimension. The lowest point displayed at
each $d$ is the value of the external dimension $\Delta_{\epsilon}$. Figure 2
also contains perturbative data for the first twelve $\mathbb{Z}_{2}$-even
scalar conformal primary operators, with their scaling dimensions truncated to
order ${\varepsilon}$. These estimates are shown as straight lines in the
figure. We use color coding to indicate how the operator is constructed in the
limit $d\to 4$. Specifically, we say that an operator is of “$\phi^{k}$ type”
if it is constructed out of $k$ fields $\phi$, with the potential insertion of
partial derivatives. The number of partial derivatives may be inferred from
the dimension of the operator at $d\rightarrow 4$.

Table 1: $\mathbb{Z}_{2}$-even scalars. The third column summarizes the known $\varepsilon$-expansion results while the fourth column refers to evaluations of various resummations at ${\varepsilon}=1$. Specifically, the evaluation of the hypergeometric–Meijer resummation for $O({\varepsilon}^{7})$ results, $\text{Pad\'{e}}_{[1,1]}$ for $O({\varepsilon}^{2})$ result (see (20)), and truncation for $O({\varepsilon})$ results. The fifth column contains the dimensions found in this work at $d=3$. The last column contains know bootstrap results at $d=3$ from previous works. The errors in bold refer to rigorous bounds.  $i$ | ${\mathcal{O}}$ | $\Delta_{\mathcal{O}}$ |  $\varepsilon$-expansion |  | This work | Bootstrap
---|---|---|---|---|---|---
1 | $\phi^{2}$ | $2-\frac{2{\varepsilon}}{3}$ | (${\varepsilon}^{7}$)Schnetz:2016fhy  | $1.4121(6)$ | Shalaby:2020xvv  |  | $1.41263$ | $1.412625(\boldsymbol{10})$ | Kos:2016ysd
2 | $\phi^{4}$ | $4-\frac{17{\varepsilon}^{2}}{27}$ | (${\varepsilon}^{7}$)Schnetz:2016fhy  | $3.8231(5)$ | Shalaby:2020xvv  |  | $3.82978$ | $3.82951(\boldsymbol{61})$ | Reehorst:2021hmp
3 | $\phi^{6}$ | $6+2{\varepsilon}$ | (${\varepsilon}^{2}$)Derkachov:1997gc  | $6.5918$ |  |  | $6.90348$ | $6.8956(43)$ | Simmons-Duffin:2016wlq
4 | $\square^{2}\phi^{4}$ | $8-\frac{8{\varepsilon}}{9}$ | (${\varepsilon}^{1}$)Kehrein:1994ff  | $7.1111$ |  |  | $7.29743$ | $7.2535(51)$ | Simmons-Duffin:2016wlq
5 | $\phi^{8}$ | $8+\frac{16{\varepsilon}}{3}$ | (${\varepsilon}^{2}$)Derkachov:1997gc  | $9.4135$ |  | 5 | $8.92059$ | $\sim 8.55$ | Simmons-Duffin:2016wlq
6 | $\square^{3}\phi^{4}$ | $10-\frac{5{\varepsilon}}{3}$ | (${\varepsilon}^{1}$)Henriksson:2022rnm  | $8.3333$ |  | 6 | $10.65903$ | $\sim 10.45$ | Simmons-Duffin:2016wlq
7 | $\square^{2}\phi^{6}$ | $10+\frac{2{\varepsilon}}{3}$ | (${\varepsilon}^{1}$)Kehrein:1994ff  | $10.6667$ |  | 7 | $12.21187$ | $\sim 11.6$ | Simmons-Duffin:2016wlq
8 | $\phi^{10}$ | $10+10{\varepsilon}$ | (${\varepsilon}^{2}$)Derkachov:1997gc  | $12.3126$ |  | 8 | $14.91822$ |  |

Studying figure 2 we can immediately make the following observations:

  * •

The numerical results capture correctly the existence of the first four
operators that match with the perturbative data in the limit ${\varepsilon}\to
0$. The only exceptions are the points $d=2.6$ and $d=2.7$, where the extremal
functional contains an additional zero at $\Delta=3.5379$ and $\Delta=4.3628$
respectively. These are likely to be spurious (i.e. they would disappear at a
higher derivative order). Apart from these extra zeros, the scaling dimensions
of the first four operators seem to follow continuous functions of $d$.

  * •

The numerical results completely miss the existence of the fifth operator,
which in perturbation theory is the operator $\phi^{8}$. However they do
appear to capture some features that may correspond to the next two operators.
In section 4.2.2 below we give a possible explanation for why $\phi^{8}$ is
the first operator to be missed.

  * •

The numerical data for the first four operators match well with estimates for
the first four operators from perturbation theory, and this agreement persists
also when ${\varepsilon}$ is not small. In figure 3 we compare the numerical
spectrum with perturbative estimates. They are done truncating the
perturbative series, and by working out Padé approximants of the form

Figure 3: Resummed perturbative estimates for the first four
$\mathbb{Z}_{2}$-even scalar operators. For the fourth lowest operator, there
are no results available beyond leading order in ${\varepsilon}$. The black
dots indicate the spectrum found using the EFM at the navigator minimum.

$\text{Pad\'{e}}_{[m,n]}({\varepsilon})=\frac{a_{0}+a_{1}{\varepsilon}+\ldots+a_{m}{\varepsilon}^{m}}{1+b_{1}{\varepsilon}+\ldots+b_{n}{\varepsilon}^{n}}.$
(20)

A list of perturbative data and estimates is given in table 7. Note that for
$\phi^{6}$ (like for all higher $\phi^{k}$ operators), both the
$O({\varepsilon})$ and $O({\varepsilon}^{2})$ terms in the anomalous
dimensions are numerically large, leading perhaps to somewhat worse
predictions.

As discussed in appendix D.1, the precise numerical values corresponding to
the operator dimensions are expected to change if the numerical precision is
increased. We believe that the qualitative picture involving the first few
operators will not change. We include the raw-data in an ancillary file; more
details are given in appendix C.

####  4.2.1 Level repulsion

Figure 2 shows that the scaling dimensions for the third and the fourth
operators approach each other and reach a minimum separation in the region
$2.6\lesssim d\lesssim 3$. The apparent behaviour of these two sequences of
values looks like the repulsion effect that we aimed to investigate in this
paper.262626Looking at just the order ${\varepsilon}$ dimensions corresponding
to the straight lines in figure 2, we were initially predicting that we would
see this effect above $d=3$. The fact that the strongest mixing happens
slightly below $d=3$ led us to extend our results to $d=2.6$.

In order to compare with the model (10) of a $2\times 2$ mixing matrix, we
would like to fit a curve that approximates the observed scaling dimensions in
the vicinity of the repulsion region. Since for constant $x$ and $\Delta_{i}$
linear in ${\varepsilon}$, the eigenvalues of (10) define a hyperbola in the
$(d,\Delta)$ plane, we fit the numerical data with a conic section. To this
end, we search for a $\mathcal{Q}(d,\Delta)=0$ where

$\mathcal{Q}(d,\Delta)=Ad^{2}+Bd\Delta+C\Delta^{2}+Dd+E\Delta+F$ (21)

We use the fitting procedure described in Bookstein1979  to perform a least
square fit of a conic section:

$\text{Minimize}\quad\tilde{\chi}^{2}\equiv\sum_{i}\mathcal{Q}(d_{i},\Delta_{i})^{2}\qquad\text{subject
to}\quad A^{2}+\frac{B^{2}}{2}+C^{2}=2$ (22)

The sum goes over the operators used for the fit, in our case selected in the
range $2.6\leqslant d\leqslant 2.9$ and $6<\Delta<8$. A list of these data
points is given in appendix C.3.

We find the values

$\displaystyle A$ $\displaystyle=-0.809771,$ $\displaystyle B$
$\displaystyle=0.200861,$ $\displaystyle C$ $\displaystyle=-1.150695,$ (23)
$\displaystyle D$ $\displaystyle=-3.068248,$ $\displaystyle E$
$\displaystyle=16.94456,$ $\displaystyle F$ $\displaystyle=-56.05891.$ (24)

The resulting conic section is a hyperbola seen in figure 4.

The minimal difference between the two branches of the fitted curve happens at
$d=2.777601$ and takes the value

$(\Delta_{{\mathcal{O}}_{4}}-\Delta_{{\mathcal{O}}_{3}})_{\mathrm{min}}=0.135915.$
(25)

Figure 4: Magnified version of figure 2, focusing on the level repulsion
effect. The blue curve is given by equation (26) and has been fitted using the
operator dimensions in the red box only.

To connect with (10), we rewrite $\mathcal{Q}(d,\Delta)=0$ describing the
hyperbola in a more suggestive form, to find

$\det\left(\Delta{\mathds{1}}-\begin{pmatrix}5.982656+0.930688\epsilon&0.067957\\\
0.067957&8.044623-0.756132\epsilon\end{pmatrix}\right)=0$ (26)

We observe, as seen in figure 4, that as ${\varepsilon}\to 0$ the two
eigenvalues approach the values $\Delta^{4d}\approx 6$ and $\Delta^{4d}\approx
8$ respectively to quite good precision. This surprisingly good agreement may
be somewhat accidental.

It useful at this point to make some remarks. From figure 3 we can draw a few
general insights about field theory in general, and the validity of the
$\varepsilon$-expansion specifically. We see that the two lowest lying
operators, namely $\phi^{2}$ and $\phi^{4}$, have their scaling dimensions
very well approximated by the perturbative predictions, especially after
resummations. Consequently, $\phi^{6}$ is observed to agree well with the
$\varepsilon$-expansion predictions very close to $d\rightarrow 4$ (e.g. for
$d=3.95$ and $d=3.9$). Furthermore, this agreement is improved once one
performs a resummation, e.g. we have good agreement down to $d=3.6\sim 3.5$.
However, we see that as we approach the mixing region the perturbative
prediction becomes progressively worse, as expected, since this mixing was not
taken into account when calculating this scaling dimension perturbatively.
This teaches a general lesson, one should be careful about taking
$\varepsilon$-expansion estimates at face value, at least ones that do not
take mixing between finitely separated (in scaling dimension) operators in
$d=4$. We stress, though, that this is not necessarily a breakdown of the
$\varepsilon$-expansion itself, the agreement could possibly be restored by
computing the off-diagonal elements $X$,$X^{\prime}$,… in (8). The calculation
of the mixing curve in figure 3 within the $\varepsilon$-expansion is an
interesting open problem.272727Since the Ising CFT is expected to be
(slightly) non-unitary at non-integer dimensions one could also consider non-
hermitian corrections to (8). This may lead to scaling dimensions that also
have an imaginary part, in addition to their real part. Hence, as a function
of the dimensionality $d$ we may expect the operator dimensions to move into
the complex plane. Such a scenario may be accommodated for by considering a
submatrix of the form
$\begin{pmatrix}\Delta_{1}({\varepsilon})&-x({\varepsilon})\\\
x({\varepsilon})&\Delta_{2}({\varepsilon})\end{pmatrix},$ (27) leading to the
following prediction for the scaling dimensions
$\Delta_{\pm}=\frac{\Delta_{1}+\Delta_{2}}{2}\pm\sqrt{\left(\tfrac{\Delta_{1}-\Delta_{2}}{2}\right)^{2}-x^{2}}\,.$
(28) The equation above describes an hyperbola with horizontal transverse
axis, i.e. a $90^{\circ}$ rotation compared to the best conic-section fit
shown in figure 4, with the dimensions moving of to the complex plane when
$\Delta_{1}$ closely approaches $\Delta_{2}$. Our fitting procedure is
agnostic to which scenario is correct since it searches the best fit in the
space of all conic-sections which also includes these types of hyperbolas as
well as the straight lines one would expect in a “crossing” scenario. The
scenario in (11) appears to be clearly numerically favored by a much better
fit. The fact that scenario leads to a much better fit can easily be seen by
eye, but can also be seen numerically by comparing the $\tilde{\chi}^{2}$
values of $0.000252652$ and $0.000420454$ respectively. Similarly a fit using
two straight lines, i.e. a degenerate conic-section, to account for a
“crossing” scenario has an even larger $\tilde{\chi}^{2}$ of $0.0055818$.
These comparisons should come with the qualification that large non-hermitian
corrections to (8) are essentially already excluded by the assumptions that go
into our numerical setup. In the sense that if large non-unitarities existed,
we wouldn’t find an island in the first place. Hence providing evidence that
to good approximation the low dimensional operators, to which the bootstrap is
most sensitive, are indeed unitary.

####  4.2.2 Missing operators

Let us now turn to the topic of operators existing perturbatively in
$4-{\varepsilon}$ dimensions that are not visible in our numerical results. In
the $\mathbb{Z}_{2}$-even scalar sector, the first such operator is $\phi^{8}$
with dimension

$\Delta_{\phi^{8}}=8-4{\varepsilon}+\frac{28}{3}{\varepsilon}-\frac{1198}{81}{\varepsilon}^{2}+O({\varepsilon}^{3})$
(29)

determined in Derkachov:1997gc .

Following the picture discussed in section D, one may discuss a few general
features of the extremal functional predictions. For each sector, i.e. for
each irrep of the global symmetry at a given spin, it has been empirically
observed that typically only the first few minima282828In terms of increasing
scaling dimension. correspond to actual physical operators, whereas subsequent
minima correspond to random/spurious operators. In general it has been
observed that the numerical conformal bootstrap becomes progressively less
constraining at higher values of the scaling dimensions. This has the
following intuitive understanding. The leading operators provide large
contributions to the sum rules, while operators higher up in the spectrum
provide smaller contributions. As a consequence the numerics do not fix the
latter contributions as tightly.

It is not unnatural that the first individual operator to not be correctly
captured by the EFM would be one that is less important in saturating the
crossing equations, in our case such an operator ${\mathcal{O}}$ would be
characterized (in the $\mathbb{Z}_{2}$-even case) by having numerically small
OPE coefficients $\lambda_{\phi\phi{\mathcal{O}}}$ and
$\lambda_{\phi^{2}\phi^{2}{\mathcal{O}}}$. To assess the situation for the
case at hand, we have gathered all known results for the OPE coefficients
involving the first few $\mathbb{Z}_{2}$-even scalar operators, presented in
table 2.

Table 2: OPE coefficients for $\mathbb{Z}_{2}$-even scalars. The value of Padayasi:2021sik  is a conjecture, see (30).  $i$ | ${\mathcal{O}}$ | $\lambda^{2}_{\phi\phi{\mathcal{O}}}$ |  | $\lambda^{2}_{\phi^{2}\phi^{2}{\mathcal{O}}}$ |
---|---|---|---|---|---
1 | $\phi^{2}$ | $2-\frac{2}{3}{\varepsilon}$ |  $({\varepsilon}^{4})$Carmi:2020ekr  | $8-8{\varepsilon}$ |  $({\varepsilon}^{1})$Henriksson:2020jwk
2 | $\phi^{4}$ | $\frac{\varepsilon^{2}}{54}-\frac{47{\varepsilon}^{3}}{1458}$ |  $({\varepsilon}^{3})$Carmi:2020ekr  | $6-\frac{20{\varepsilon}}{3}$ |  $({\varepsilon}^{1})$Bertucci:2022ptt
3 | $\phi^{6}$ | $\frac{5\varepsilon^{4}}{104796}$ |  $({\varepsilon}^{4})$Codello:2017qek  | $O({\varepsilon}^{2})$ |
4 | $\square^{2}\phi^{4}$ | $O({\varepsilon}^{4})$ | Alday:2017zzv  | $\frac{1}{6}-\frac{259\epsilon}{1296}$ |  $({\varepsilon}^{1})$Bertucci:2022ptt
5 | $\phi^{8}$ | $\frac{7{\varepsilon}^{6}}{37791360}$ |  $({\varepsilon}^{6})$Padayasi:2021sik * | $O({\varepsilon}^{4})$ |
6 | $\square^{3}\phi^{4}$ | $O({\varepsilon}^{4})$ |  | $\frac{1}{175}-\frac{71{\varepsilon}}{21000}$ |  $({\varepsilon}^{1})$Bertucci:2022ptt

Studying the values in table 2 we note that all operators of “$\phi^{4}$-type”
have $O(1)$ OPE coefficients $\lambda^{2}_{\phi^{2}\phi^{2}{\mathcal{O}}}$.
Therefore these operators can be seen as important in saturating crossing also
at small but finite ${\varepsilon}$. Operators with more than four fields are
suppressed in the $\phi^{2}\times\phi^{2}$ OPE by powers of ${\varepsilon}$,
unfortunately there are no numerical estimates for the coefficients in these
expansions.

In the $\phi\times\phi$ OPE, a bit more is known, and here all OPE
coefficients except that of $\phi^{2}$ are suppressed by powers of
${\varepsilon}$. One can think of these as extra powers of ${\varepsilon}$ as
powers of the coupling constant needed to create more fields in the operators
appearing in the OPE. Interestingly, we also note that operators of increasing
powers of $\phi$ have OPE coefficients $\lambda_{\phi\phi{\mathcal{O}}}^{2}$
that are not only suppressed by ${\varepsilon}$ powers, but also contain
numerically small constants at the leading non-zero term. In Padayasi:2021sik
, a formula was conjectured for the OPE coefficient of $\phi^{2k}$ in the
$\phi\times\phi$ OPE,292929The values at $k=2$ and $k=3$ have been
independently computed by other methods, see references in table 2.

$\displaystyle\lambda^{2}_{\phi\phi\phi^{2k}}$
$\displaystyle=\frac{10368k(2k-1)^{2}\Gamma(k)^{4}}{6^{4k}\Gamma(2k)}{\varepsilon}^{2k-2}\,,$
(30)

which evaluates to
$\frac{{\varepsilon}^{2}}{54},\frac{5{\varepsilon}^{4}}{104976},\frac{7{\varepsilon}^{6}}{37791360},\frac{{\varepsilon}^{8}}{952342272}$
for $k=2,3,4,5$, etc. We think that it would not be unreasonable for this
suppression to survive for finite values of $\varepsilon$.

Let us now compare with our results at finite ${\varepsilon}$ in figure 2.
Among the first six operators, the $\phi^{8}$ operator that is not seen in the
numerics is the one with the largest number of fields, which coincides with
being the operator with smallest OPE coefficients reported in table 2. Where
explicit formulas are available, the suppression holds both perturbatively in
${\varepsilon}$, and numerically in the size of the constants.

###  4.3 Other operator spectra

Having discussed the $\mathbb{Z}_{2}$-even scalars in detail, we now move on
to consider different symmetry irreps. In figures 5–10 we give the operator
spectra for $\mathbb{Z}_{2}$-even operators from spin-$2$ up to spin-$12$ with
a step of $2$. Since the $\mathbb{Z}_{2}$-even operators in our crossing
system only appear as the exchange of identical scalars, we cannot access any
$\mathbb{Z}_{2}$-even operator of odd spin in this work.303030However, these
operators would be exchanged in a system of correlators that includes two
different $\mathbb{Z}_{2}$-even external operators, for instance by adding
$\phi^{4}$ or the stress-tensor $T^{\mu\nu}$ to the system. Similarly, in
figures 11–17 we exhibit the spectra for $\mathbb{Z}_{2}$-odd operators from
spin $0$ to spin $6$ with a step of $1$. In each of the plots we display the
result of our computation together with order ${\varepsilon}$ anomalous
dimensions for operators in the ${\varepsilon}$-expansion. We display a subset
of this perturbative data is collected in tables 4 and 5.

Figure 5: $\mathbb{Z}_{2}$-even operators of spin 2. Figure 6:
$\mathbb{Z}_{2}$-even operators of spin 4. Figure 7: $\mathbb{Z}_{2}$-even
operators of spin 6. Figure 8: $\mathbb{Z}_{2}$-even operators of spin 8.
Figure 9: $\mathbb{Z}_{2}$-even operators of spin 10. Figure 10:
$\mathbb{Z}_{2}$-even operators of spin 12. Figure 11: $\mathbb{Z}_{2}$-odd
operators of spin 0. Figure 12: $\mathbb{Z}_{2}$-odd operators of spin 1.
Figure 13: $\mathbb{Z}_{2}$-odd operators of spin 2. Figure 14:
$\mathbb{Z}_{2}$-odd operators of spin 3. Figure 15: $\mathbb{Z}_{2}$-odd
operators of spin 4. Figure 16: $\mathbb{Z}_{2}$-odd operators of spin 5.
Figure 17: $\mathbb{Z}_{2}$-odd operators of spin 6.

In general, we find rather good agreement between some of the low-lying
operators in perturbation theory, and some sequences of points in the
numerical data. Whenever there is a single sequence of points that approach a
single line starting at $d=4-{\varepsilon}$, we can conclude that this set of
non-perturbative data would be an estimate of the scaling dimension of that
particular operator at finite ${\varepsilon}$. Also in a couple cases with
two-fold degeneracy at ${\varepsilon}\to 0$, the identification of operators
between numerics and perturbation theory seems unambiguous, namely the as the
dimension $8+O({\varepsilon})$ $\mathbb{Z}_{2}$-even operators at spins 2 and
4 (figures 5 and 6).

In a couple of our plots, odd spin 1 and spin 3 (figures 12 and 14), there are
persistent sequences of minima of the extremal functional that do not
correspond to operators in the ${\varepsilon}$-expansion. We have checked that
they are not sporadic features of a special navigator point but instead are
rather persistent features just like the dimensions that correspond to
physical operator.313131Figure 7 for $\mathbb{Z}_{2}$-even operators of spin 6
shows an interesting feature. Around $d\sim 3.6$, it looks like one of the
three operators with dimension $\Delta=10+O({\varepsilon})$ appears to deviate
from the perturbative line and migrate upwards as $d$ decreases. Then around
$d=3.1$ it looks like the same operator moves down again to appear below the
top yellow line at $d=3$. This behavour is interesting, since Simmons-
Duffin:2016wlq  reported only two operators in this region. However, a closer
analysis of several spectra before the final Navigator point shows that the
points $(3,8.78)$ and $(3.1,9.72)$ are unstable. However, we do not believe
that they are signals of new operators that were for some reason not
previously discovered in perturbation theory.323232The perturbative spectrum
reported in Kehrein:1994ff  has been independently investigated recently by
one of us Henriksson:2022rnm , and we have no reason to believe it to be
incorrect. Moreover, for the case of figure 14, it is not possible to
construct a $\mathbb{Z}_{2}$-odd spin-3 operator with dimension
$5+O({\varepsilon})$ out of fields and derivatives in the
${\varepsilon}$-expansion.

In summary, our results seem to confirm that the $\varepsilon$-expansion is
quite accurate at least in the Ising case. Also, at least for the operators
that we can track relatively well, there does not seem to be any operator that
decouples from the spectrum for some value of $d$. To be more precise, by
being able to track an operator well, we refer to the scenario were both
perturbation theory and the bootstrap overlap well at small $\varepsilon$, and
then the corresponding curve evolves smoothly. As a counterexample, in figure
9 close to $d=4$ we see a few cases of an extra operator close to $\Delta=12$,
however there is no corresponding prediction for any extra operator in the
$\varepsilon$-expansion, we thus conclude that these points are spurious.

###  4.4 OPE coefficients

Let us conclude this section by displaying the values of some OPE coefficients
in the theory. The data for OPE coefficients follow from the extremal crossing
solution, which we extract by spectrum.py spectrum . This method becomes quite
unstable for OPE coefficients higher up in the spectrum, even when there are
clear minima of the extremal functional. We therefore focus on the leading two
OPE coefficients. In the numerical problem, they correspond to the OPE
coefficients involving the leading two operators $\sigma$ and $\epsilon$,
which by the above considerations successfully matched with the operators
$\phi$ and $\phi^{2}$ in the ${\varepsilon}$-expansion. Therefore we have

Figure 18: OPE coefficient $\lambda_{\phi\phi\phi^{2}}$. Figure 19: OPE
coefficient $\lambda_{\phi^{2}\phi^{2}\phi^{2}}$.

$\lambda_{\sigma\sigma\epsilon}\leftrightarrow\lambda_{\phi\phi\phi^{2}},\qquad\lambda_{\epsilon\epsilon\epsilon}\leftrightarrow\lambda_{\phi^{2}\phi^{2}\phi^{2}}.$
(31)

In figures 18 and 19 we display our results for these OPE coefficients
extracted at the navigator minimum.333333We can only extract these up to an
overall sign since they appear squared in the crossing equations. Matching
with the ${\varepsilon}$-expansion fixes the sign to be positive. The plots
show good agreement with the perturbative values. For the first OPE
coefficients, we compare with a Padé approximant constructed using the order
${\varepsilon}^{4}$ result

$\lambda_{\phi\phi\phi^{2}}=\sqrt{2}-0.2357{\varepsilon}-0.1680{\varepsilon}^{2}+0.1036{\varepsilon}^{3}-0.2248{\varepsilon}^{4}+O({\varepsilon}^{5})$
(32)

from Carmi:2020ekr .343434The terms up to order ${\varepsilon}^{3}$ are known
on exact form and were computed in Gopakumar:2016cpb . The order
${\varepsilon}^{4}$ term is only known numerically. For the second OPE
coefficient, only the linear correction is known Henriksson:2020jwk ,

$\lambda_{\phi^{2}\phi^{2}\phi^{2}}=\sqrt{8}-\sqrt{2}{\varepsilon}+O({\varepsilon}^{2}).$
(33)

Finally, we can compare our estimates in $d=3$ with the rigorous values in
Kos:2016ysd ,

$\displaystyle(\lambda_{\sigma\sigma\epsilon},\lambda_{\epsilon\epsilon\epsilon})\big{|}_{\mathrm{min}}$
$\displaystyle=(1.0518527,\quad\ \ \,1.5324484),$ (34)
$\displaystyle(\lambda_{\sigma\sigma\epsilon},\lambda_{\epsilon\epsilon\epsilon})\big{|}_{\text{rigorous}}$
$\displaystyle=(1.0518537(41),1.532435(19)).$ (35)

We see that our values are inside the rigorous error bars.

##  5 Discussion and outlook

In this work we performed a comprehensive conformal bootstrap study on the
low-lying spectrum of the Ising CFT. This was done for a mixed correlator
system involving all possible correlators with $\sigma$ and $\epsilon$ as
externals. With this system we were able to access $\mathbb{Z}_{2}$-even
operators of even spin, and $\mathbb{Z}_{2}$-odd operators of any spin. We
explicitly presented results for $\mathbb{Z}_{2}$-even operators up to spin
$12$ and $\mathbb{Z}_{2}$-odd operators up to spin $6$ obtained using the
extremal functional method. The main observation of our work is that, in
absence of mixing effects, the $\varepsilon$-expansion produces results that
agree well with our non-perturbative results for various subleading operators.
This good agreement is somewhat surprising, since in many cases we only had
access to one order in the perturbative expansion. In the case of mixing, the
above statement holds away from the mixing region. In the mixing region, on
the other hand, we find that the operator dimensions match well with a fitted
hyperbola (26), which is expected to give a good local approximation of the
level repulsion caused by operator mixing. An interesting goal for the future
would be to reproduce the curve in figure 4 with an analytic computation, in
particular to give an estimate for the (half) minimal distance $x=0.067957$.

Another observation is that in the $\mathbb{Z}_{2}$-even spin-$0$ channel, we
were able to track the first four operators seemingly quite reliably. I.e. we
found good agreement as $d\rightarrow 4$, and then smooth continuous lines
with no apparent spurious operators in between (at least down to $d=3$). We
note similarly good agreement for some leading operators in the other
representations, and we can therefore see our work as an extension of
Cappelli:2018vir  to include operators with higher twist. This appears rather
promising for future studies, where with higher precision we may be able to
study operators even higher up in the spectrum.

There are a few obvious ways in which our results can be improved in the
future. If we include the operator $\phi^{4}$ $\sim$ $\epsilon^{\prime}$ as an
external operator in the system of correlators, we will now be able to access
$\mathbb{Z}_{2}$-even operators of odd spin. Such operators are inaccessible
in the $\sigma$, $\epsilon$ system, since they can only appear in the OPE of
non-identical external operators. We can also include external spinning
operators, such as the stress tensor, something that will allow us to access
space(time) parity odd operators. The aforementioned generalizations would
then give access the entire spectrum of the 3d Ising CFT, at least in
principle – in practice we expect to be able to study only a few low lying
operators in each sector. We find this particularly appealing, since it would
be one of the few, if any, non-integrable field theories where such an amount
of data would be known non-perturbatively.

In this work, as an initial proof of concept we showed evidence for level
repulsion phenomenon using the EFM to provide highly accurate estimates. It
would be interesting to turn these estimates into rigorous bounds and to
determine the relevant curves more precisely. This can be done by including
the values for $\Delta_{\epsilon^{\prime\prime}}$ and
$\Delta_{\epsilon^{\prime\prime\prime}}$ explicitly in the navigator function
Reehorst:2021hmp . Then one can minimize the distance between the two curves
as an efficient optimization problem.353535We acknowledge conversations with
Ning Su on this point. Enlarging the search space in such a way comes at the
cost of increased numerical run-times. However, we expect that with the
rapidly evolving bootstrap technology such numerics should be feasible in the
not so distant future. Such a study would be expected to conclusively settle
between the repulsion scenario (10) and the annihilation scenario (27).

The Ising CFT is of experimental relevance in spacetime dimensions $d=2$ and
$d=3$. The $d=2$ theory has been solved exactly, while the $d=3$ theory
remains an active area of research, also from a theoretical point of view.
This study focused on the region where the first strong example of level
repulsion takes place, but there are also questions to be answered about the
$d\rightarrow 2$ behaviour of the CFT. For example, the $d=2$ theory is to
some approximation accessible via the $d=4-\varepsilon$ expansion
LeGuillou:1987ph ; Kompaniets:2017yct  and as emphasized before, the
connection has been corroborated by bootstrap studies in intermediate
dimensions El-Showk:2013nia ; Cappelli:2018vir .363636 Another interesting
observation is that doing perturbation directly in $d=2$, if one goes to high
enough order and performs re-summations, one obtains values for critical
exponents surprisingly close to the exact result Serone:2018gjo . See also
Ballhausen:2003gx  for a study in intermediate dimensions using the non-
perturbative renormalization group. Hence, one could try to understand how a
theory (Ising in $d=4-\varepsilon$) that seems to not be exactly solvable,
continuously deforms into an integrable theory in $d=2$, see Cappelli:2018vir
; Li:2021uki  for work in this direction. It will be interesting for the
future to address this limit for a larger set of operator dimensions. To this
extent, we present an appetizer plot in figure 20, where we include some
preliminary results from performing our bootstrap study in $d=2.001$. In this
figure we show Padé approximants and the interpolating curves from
Cappelli:2018vir  for the first two operators, and our hyperbola for the
following two operators. Naïvely, the figure suggests that the third scalar
singlet decouples from the spectrum of the theory as we approach $d=2$. We
plan to investigate this phenomenon more carefully in future work.

Figure 20: Plot showing the collected state-of-the-art picture for the first
four $\mathbb{Z}_{2}$-even scalars, including our preliminary results in
$d=2.001$. The dots refer to results from the numerical bootstrap study, the
gray lines are Padé approximants computed from the ${\varepsilon}$-expansion
and values in 2d, the green dashed lines to previous fits from
Cappelli:2018vir , and blue lines to the hyperbola fit from the mixing region,
(26). Red markers denote the position of quasiprimaries in $d=2$. Note that
Cappelli:2018vir  has values for the first two operators for a more dense
sampling of values of $d$ near $d=2$.

There are other theories with an interpolating parameter, whose spectrum would
be interesting to study with respect to spectrum continuity and level
repulsion. An immediate target is the generalization to the $O(N)$ CFT, which
contains Ising at $N=1$. A first step in this direction was taken in
Sirois:2022vth , where a portion of the spectrum was probed as a function of
$N$ and the dimensionality $d$, focusing on the region $d\geqslant 3$. An
interesting open question in this context is the “Cardy–Hamber line” starting
at $N=d=2$, across which scaling dimensions are conjectured to be non-analytic
Cardy:1980at . A recent study using non-perturbative RG found no evidence of
non-analyticity Chlebicki:2020pvo , and it would be interesting to investigate
this with the bootstrap. With respect to level repulsion, the large $N$
expansion of the $O(N)$ model shows some interesting features of level
repulsion that may be interesting to study numerically Derkachov:1998js
.373737Specifically, the order $1/N$ anomalous dimensions of the third and the
fourth singlet scalars coincide at $d=17-\sqrt{193}=3.1076$. The apparent
contradiction with the non-crossing rule is resolved by taking higher-order
corrections into account. At order $1/N^{2}$, the anomalous dimension has only
been determined for fourth operator, and it has a pole exactly at the
coincident point. It would be interesting to study this effect non-
perturbatively. Also the level non-crossings in $\mathcal{N}=4$ SYM studied by
Korchemsky:2015cyx  and mentioned in section 2.2 would be interesting target
for a non-perturbative bootstrap study. Here the level repulsion between
Konishi and the first double-trace operator would be observable in a study at
large but finite $N_{c}$, and at finite coupling. With the recent localization
techniques for inserting the value of the coupling constant into the bootstrap
system Chester:2021aun , this problem looks tractable.

Another seemingly straightforward generalization of our work would be to study
the supersymmetric Ising CFT. In this case the CFT can be found as the fixed
point of the simplest GNY model one may write down Fei:2016sgs . In that case,
the supersymmetry appears to be emergent, even if one does not impose it from
the start. At the level of the $\varepsilon$-expansion, the signal for this is
the integer spacing of specific scaling dimensions. As was shown in
Rong:2018okz ; Atanasov:2018kqw  it is possible to get an island in parameter
space using exactly the same correlator system we used for this work. Using
the same system of crossing equations, the authors were also able obtain high
precision estimates for CFT data Atanasov:2022bpi . One potential complication
in this direction is that evanescent operators, i.e. the operators responsible
for violations of unitarity in fractional dimensions as discussed in
Hogervorst:2015akt , may appear for smaller values of the scaling dimensions
in fermionic theories than in purely bosonic theories Hogervorst:2015akt ;
Dugan:1990df ; Gracey:2008mf ; Ji:2018yaf . Hence, if these violations become
sizeable enough to be detected, this may have an effect on the bootstrap.

One may also study more complicated multi-scalar theories with various global
symmetries. A number of these are interesting experimentally, and others are
interesting due to field theoretical considerations.383838See Mukamel1975 ;
Mukamel1976 ; Mukamel1976b ; Bak1976 ; Pelissetto:2000ek  among others. For
example, once one allows for a global symmetry that is more complicated,
various phenomena can be observed, such as annihilation of fixed points, or
fixed points exchanging stability. Our present work thus serves also as a
proof of principle, showing that such theories may be reliably probed via the
bootstrap.

Let us finish by the outlook towards the bootstrap of conformal gauge
theories. Despite recent progress in bootstrapping gauge theories, e.g.
summarized in Poland:2022qrs , obtaining an isolated island in parameter space
corresponding to such theories still remains a very difficult task. The study
of such theories may be improved by applying the methodology used in this
paper, namely to attempt to track these theories from infinitesimal393939In
practice, in the bootstrap this would simply mean taking ${\varepsilon}$ to be
very small, such as e.g. $\varepsilon=0.01$. to finite and physically
interesting values of a control parameter such as $1/N$ or $\varepsilon$. This
procedure may provide us with a more clear (and non-perturbative) picture for
the spectrum of these complicated theories, which hopefully will give enough
information about the various gaps needed to reliably isolate these theories
in parameter space. See He:2021xvg  for some progress towards this
direction.404040See also Reehorst:2020phk ; Li:2020bnb  for discussions of the
manifestation of gauge-invariance in CFT spectra.

## Acknowledgements

JH has received funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement no. 758903). MR is supported by the Simons Foundation grant #488659
(Simons collaboration on the non-perturbative bootstrap). The research work of
SRK was initially supported by the Hellenic Foundation for Research and
Innovation (HFRI) under the HFRI PhD Fellowship grant (Fellowship Number:
1026). The research work of SRK subsequently 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 the
following people for useful conversations: Alessandro Vichi, Balt van Rees,
Benoit Sirois, Bernardo Zan, Connor Behan, Gabriel Cuomo, Hugh Osborn, Ning
Su, Slava Rychkov and Yuan Xin. Additionally, we thank an anonymous referee
whose comments helped improve this manuscript.

The numerical computations were performed on the Metropolis cluster of the
Crete Center for Quantum Complexity and Nanotechnology and on the Caltech High
Performance Cluster, and were partially supported by a grant from the Gordon
and Betty Moore Foundation.

##  Appendix A Numerical setup

For the numerics done in this work we set up various semi-definite programs
(SDP) using the programs qboot Go:2020ahx  and Simpleboot simpleboot . We
integrated these with a python code to run the modified BFGS algorithm of
Reehorst:2021ykw  to calculate the next point in the navigator optimization
tasks. To solve the SDPs the arbitrary precision solver SDPB Simmons-
Duffin:2015qma ; Landry:2019qug  is used. To extract the spectrum we used the
script spectrum.py spectrum  written originally for Komargodski:2016auf , see
also Simmons-Duffin:2016wlq  for more details. The script itself is an
implementation of the extremal functional idea of El-Showk:2012vjm ; El-
Showk:2016mxr . Note that this script discards operator for which it does not
extract a positive OPE coefficient squared. For this study it was useful to
also output these operators. Thus we used a modified version of spectrum.py.

The parameters we used when running SDPB can be found in table 3.

$\Lambda$ | 20 | 30
---|---|---
$\kappa$ | 30 | 30
order | 60 | 60
spins |  $\\{0,1,2,\ldots,49\\}\cup\\{55,56,$ $59,60,64,65,69,70,74,$ $75,79,80,84,85,89,90\\}$ |  $\\{0,1,2,\ldots,49\\}\cup\\{55,56,$ $59,60,64,65,69,70,74,$ $75,79,80,84,85,89,90\\}$
precision | 1100 | 1100
dualityGapThreshold | $10^{-30}$ | $10^{-30}$
primalErrorThreshold | $10^{-30}$ | $10^{-30}/10^{-60}$
dualErrorThreshold | $10^{-30}$ | $10^{-30}/10^{-60}$
initialMatrixScalePrimal | $10^{20}$ | $10^{20}/10^{40}$
initialMatrixScaleDual | $10^{20}$ | $10^{20}/10^{40}$
feasibleCenteringParameter | 0.1 | 0.1
infeasibleCenteringParameter | 0.3 | 0.3
stepLengthReduction | 0.7 | 0.7
maxComplementarity | $10^{100}$ | $10^{100}$
Table 3:  Parameters used to set up the SDPs, along with the SDPB parameters.
The definition of these parameters can be found in Simmons-Duffin:2015qma . In
particular cases where we report two values, e.g $10^{-30}/10^{-60}$, this
indicates that the bulk of our results were ran with the first value, however
tests were performed to confirm that different values of this parameter did
not change the results. In qboot we also use $n_{max}=500$. The parameters in
the $\Lambda=20$ column were used to explicitly check that our navigator
minima lie in islands.

##  Appendix B Construction of the navigator function.

We start from the crossing equation for a $\mathbb{Z}_{2}$ symmetric CFT
written in terms of sum rules:

$\vec{V}_{0,0}+p_{\epsilon}\vec{V}_{\epsilon}(\theta)+P_{T}\vec{V}_{T}+\sum_{(\Delta,\ell)\in
S_{+}}\operatorname{Tr}[P_{\Delta,\ell}\vec{V}_{+,\Delta,\ell}]+\sum_{(\Delta,\ell)\in
S_{-}}p_{\Delta,\ell}\vec{V}_{-,\Delta,\ell}=0\,,$ (36)

where

$\quad\vec{V}_{\epsilon}(\theta)=\text{Tr}\left[\begin{pmatrix}c_{\theta}^{2}&c_{\theta}s_{\theta}\\\
c_{\theta}s_{\theta}&s^{2}_{\theta}\end{pmatrix}\vec{V}_{+,\Delta_{\epsilon},0}+\begin{pmatrix}c_{\theta}^{2}&0\\\
0&0\end{pmatrix}\vec{V}_{-,\Delta_{\sigma},0}\right].$ (37)

Here we follow the notation of Reehorst:2021ykw .414141These equations were
first studied in El-Showk:2012cjh . The fact that scanning over the OPE angle
$\theta$ significantly improve the bootstrap bounds was first explored in
Kos:2016ysd . For the definitions of $P_{\Delta,\ell}$, $p_{\Delta,\ell}$,
$\vec{V}_{+,\Delta_{\epsilon},\ell}$,$\vec{V}_{-,\Delta_{\sigma},\ell}$,
$\vec{V}_{0,0}$, $p_{\epsilon}$ and $\vec{V}_{\epsilon}$ see equations (2.14)
to (2.17), (2.26) and (2.27) of that work. Here
$P_{T}=\frac{\lambda^{2}_{\sigma\sigma T}}{\Delta_{\sigma}^{2}}\sim
C_{T}^{-1}$ and $\vec{V}_{T}=\begin{pmatrix}\Delta_{\sigma}\\\
\Delta_{\epsilon}\end{pmatrix}\cdot\vec{V}_{+,3,2}\cdot\begin{pmatrix}\Delta_{\sigma}\\\
\Delta_{\epsilon}\end{pmatrix}$ see also (17). Finally, $c_{\theta}$ denotes
$\cos{\theta}$ and $s_{\theta}$ denotes $\sin{\theta}$. Our assumptions
restrict the allowed operators that can appear in the sum above to

$\displaystyle S_{+}=$ $\displaystyle\\{(\Delta,0):\Delta\geq
d\\}\cup\\{(\Delta,2):\Delta\geq
d+1\\}\cup\\{(\Delta,\ell):\ell=4,6,\ldots\text{ and }\Delta\geq
d+\ell-2\\}\,,$ (38) $\displaystyle S_{-}=$
$\displaystyle\\{(\Delta,0):\Delta\geq
d\\}\cup\\{(\Delta,\ell):\ell=1,2,3,\ldots\text{ and }\Delta\geq
d+\ell-2\\}\,.$ (39)

Depending on the choice of $p=(\Delta_{\sigma},\Delta_{\epsilon},\theta)$
equation (36) either does or does not have a solution. In order to efficiently
find and classify allowed regions and their boundary we construct a navigator
function $\mathcal{N}(\Delta_{\sigma},\Delta_{\epsilon},\theta)$. The key to
finding a navigator function that has all the properties we require is a so
called “navigator-improved” sum rule. By construction this sum rule has at
least one solution with $\lambda=\lambda_{*}>0$ for all points and looks like
this:

$\vec{V}_{0,0}+\lambda\vec{M}++p_{\epsilon}\vec{V}_{\epsilon}(\theta)+\sum_{(\Delta,\ell)\in
S_{+}}\operatorname{Tr}[P_{\Delta,\ell}\vec{V}_{+,\Delta,\ell}]+\sum_{(\Delta,\ell)\in
S_{-}}p_{\Delta,\ell}\vec{V}_{-,\Delta,\ell}=0\,.$ (40)

The vector $\vec{M}$ has to be chosen in order to guarantee the existence of a
solution with $\lambda=\lambda_{*}>0$. Our choice of $M$ will be motivated by
the observation that the generalized free field (GFF) would be a solution if
not for the fact that we explicitly forbid contributions from operators with
dimensions outside $S_{+}$ or $S_{-}$. Specifically, our assumptions forbid
contributions from the GFF operators $\phi_{1}^{2}$, $\phi_{2}^{2}$,
$\phi_{1}\phi_{2}$, $(T_{1\mu\nu})^{\prime}$ and $(T_{2\mu\nu})^{\prime}$
(where $\phi_{1}=\sigma$ and $\phi_{2}=\epsilon$). We can add these back to
the equation through the $\vec{M}$ term to guarantee the existence of a
solution with $\lambda=1$. This means taking

$\vec{M}=\sum_{(\Delta,\ell)\in
S^{\text{GFF}^{*}}_{+}}\operatorname{Tr}[P^{\text{GFF}}_{\Delta,\ell}\vec{V}_{+,\Delta,\ell}]+\sum_{(\Delta,\ell)\in
S^{\text{GFF}^{*}}_{-}}p^{\text{GFF}}_{\Delta,\ell}\vec{V}_{-,\Delta,\ell}=0.$
(41)

Here
$S^{\text{GFF}^{*}}_{+}=\\{(2\Delta_{\sigma},0),(2\Delta_{\epsilon},0),(2\Delta_{\sigma}+2,2)\\}$
and $S^{\text{GFF}^{*}}_{-}=\\{(\Delta_{\sigma}+\Delta_{\epsilon},0)\\}$. The
OPE coefficients $P^{\text{GFF}}_{\Delta,\ell}$ and
$p^{\text{GFF}}_{\Delta,\ell}$ have previously been computed in
Fitzpatrick:2011dm ; Fitzpatrick:2012yx ; Karateev:2018oml .

It is then easy to show that the navigator function

$\mathcal{N}(\bf{p})=\min\lambda\quad\text{ such that equation
\eqref{eq:crossingIsingWithM} has a solution}$ (42)

is bounded and has all the other properties we want as described in the main
text.

##  Appendix C Auxiliary data

###  C.1 Perturbative data

In tables 4 and 5, we display a collection of perturbative data for operators
with dimension $\Delta\leqslant 10$. Most of this data is available in
computer-readable format in Henriksson:2022rnm . The data for
$\mathbb{Z}_{2}$-even scalars is displayed in tables 1 and 2 in the main text.

$\ell$ | $i$ | ${\mathcal{O}}$ | $\Delta_{\mathcal{O}}$ | $\lambda^{2}_{\phi\phi{\mathcal{O}}}$ | $\lambda^{2}_{\phi^{2}\phi^{2}{\mathcal{O}}}$
---|---|---|---|---|---
2 | 1 | $T_{\mu\nu}$ | $4-\varepsilon$ |  | $\frac{d^{2}\Delta_{\phi}^{2}}{4(d-1)^{2}C_{T}}$ |  |  $\frac{d^{2}\Delta_{\phi^{2}}^{2}}{4(d-1)^{2}C_{T}}$
2 | ${\partial}^{2}\phi^{4}$ | $6-\frac{5\varepsilon}{9}$ | Kehrein:1994ff  | $\frac{\varepsilon^{2}}{1440}-\frac{\varepsilon^{3}}{1440}$ | Carmi:2020ekr ; Bertucci:2022ptt  |  $\frac{8}{5}-\frac{52\varepsilon}{27}$
3 | $\square{\partial}^{2}\phi^{4}$ | $8-\frac{10\varepsilon}{9}$ | Kehrein:1994ff  | $\frac{\varepsilon^{4}}{2592000}$ | Bertucci:2022ptt  | $\frac{1}{5}-\frac{1493\varepsilon}{5400}$
4 | ${\partial}^{2}\phi^{6}$ | $8+\frac{11\varepsilon}{9}$ | Kehrein:1994ff  |  |  | $O({\varepsilon}^{2})$
5 | $\square^{2}{\partial}^{2}\phi^{4}$ | $10-\frac{17\varepsilon}{9}$ | Henriksson:2022rnm  |  |  | $\frac{1}{999}$
6 | $\square^{2}{\partial}^{2}\phi^{4}$ | $10-\frac{16\varepsilon}{15}$ | Henriksson:2022rnm  |  |  | $\frac{4}{111}$
7 | $\square{\partial}^{2}\phi^{6}$ | $10+\frac{4\varepsilon}{9}$ | Kehrein:1994ff  |  |  | $O({\varepsilon}^{2})$
8 | ${\partial}^{2}\phi^{8}$ | $10+\frac{13\varepsilon}{3}$ | Kehrein:1994ff  |  |  |
4 | 1 | $C_{\mu\nu\rho\sigma}$ | $6-\varepsilon+\frac{7\varepsilon^{2}}{540}$ | ($\varepsilon^{4}$)Derkachov:1997pf  | $\frac{1}{35}-\frac{1217\varepsilon}{29400}$ | ($\varepsilon^{4}$)Alday:2017zzv  | $\frac{4}{35}-\frac{1831\varepsilon}{22050}$
2 | $({\partial}^{4}\phi^{4})_{1}$ | $8-\frac{14\varepsilon}{9}$ | Kehrein:1994ff  | $\frac{\varepsilon^{2}}{419580}$ | Alday:2017zzv ; Henriksson:2022rnm  | $\frac{125}{2331}$
3 | $({\partial}^{4}\phi^{4})_{2}$ | $8-\frac{11\varepsilon}{15}$ | Kehrein:1994ff  | $\frac{\varepsilon^{2}}{23976}$ | Alday:2017zzv ; Henriksson:2022rnm  | $\frac{8}{37}$
4 | $(\square{\partial}^{4}\phi^{4})_{1}$ | $10-\frac{\sqrt{697}+131}{90}\varepsilon$ | Henriksson:2022rnm  |  |  | $0.00201$
5 | $(\square{\partial}^{4}\phi^{4})_{2}$ | $10-\frac{-\sqrt{697}+131}{90}\varepsilon$ | Henriksson:2022rnm  |  |  | $0.03694$
6 | $({\partial}^{4}\phi^{6})_{1}$ | $10+\frac{13\varepsilon}{15}$ | Kehrein:1994ff  |  |  | $O({\varepsilon}^{2})$
7 | $({\partial}^{4}\phi^{6})_{2}$ | $10+0{\varepsilon}$ | Kehrein:1994ff  |  |  | $O({\varepsilon}^{2})$
6 | 1 | $\mathcal{J}_{6}$ | $8-\varepsilon+\frac{\varepsilon^{2}}{63}$ | ($\varepsilon^{4}$)Derkachov:1997pf  | $\frac{1}{462}-\frac{49807\varepsilon}{12806640}$ | ($\varepsilon^{4}$)Alday:2017zzv  | $\frac{2}{231}-\frac{23011\varepsilon}{3201660}$
2 | $({\partial}^{6}\phi^{4})_{1}$ | $10-1.50342\varepsilon$ | Kehrein:1994ff  | $1.67\cdot 10^{-7}\varepsilon^{2}$ | Alday:2017zzv ; Henriksson:2022rnm  | $O(1)$
3 | $({\partial}^{6}\phi^{4})_{2}$ | $10-1.32928\varepsilon$ | Kehrein:1994ff  | $0.76\cdot 10^{-7}\varepsilon^{2}$ | Alday:2017zzv ; Henriksson:2022rnm  | $O(1)$
4 | $({\partial}^{6}\phi^{4})_{3}$ | $10-0.82126\varepsilon$ | Kehrein:1994ff  | $27.5\cdot 10^{-7}\varepsilon^{2}$ | Alday:2017zzv ; Henriksson:2022rnm  | $O(1)$
8 | 1 | $\mathcal{J}_{8}$ | $10-\varepsilon+\frac{11\varepsilon^{2}}{648}$ | ($\varepsilon^{4}$)Derkachov:1997pf  | $\frac{1}{6435}+\ldots$ | ($\varepsilon^{4}$)Alday:2017zzv  | $\frac{4}{6435}+\ldots$
Table 4: Perturbative results for $\mathbb{Z}_{2}$-even operators with $\Delta\leqslant 10$ of even spin $\ell>0$. The OPE coefficients $\lambda^{2}_{\phi^{2}\phi^{2}{\mathcal{O}}}$ can all be extracted from the correlator given in Henriksson:2020jwk , see also Henriksson:2022rnm ; Bertucci:2022ptt  For $\lambda^{2}_{\phi^{2}\phi^{2}\mathcal{J}_{\ell}}$, with $\ell=4,6,\ldots$, order $\epsilon^{2}$ results are also available in Bertucci:2022ptt . For entries that are empty, it is not known to what order in ${\varepsilon}$ the OPE coefficients appear.  $\ell$ | $i$ | ${\mathcal{O}}$ | $\Delta_{\mathcal{O}}$ | $\lambda^{2}_{\phi\phi^{2}{\mathcal{O}}}$
---|---|---|---|---
0 | 1 | $\phi$ | $1-\frac{\epsilon}{2}+\frac{\epsilon^{2}}{108}$ | ($\epsilon^{8}$)SchnetzUnp  | $2-\frac{2\epsilon}{3}-\frac{34\epsilon^{2}}{81}$ | ($\epsilon^{4}$)Carmi:2020ekr
2 | $\phi^{5}$ | $5+\frac{5\epsilon}{6}-\frac{685\epsilon^{2}}{324}$ | ($\epsilon^{3}$)Zhang1982  | $\frac{5\epsilon^{2}}{108}$ | Codello:2017qek
3 | $\phi^{7}$ | $7+\frac{7\epsilon}{2}-\frac{959\epsilon^{2}}{108}$ | Derkachov:1997gc  |  |
4 | $\square^{2}\phi^{5}$ | $9-\frac{5\epsilon}{18}$ | Kehrein:1994ff  |  |
5 | $\phi^{9}$ | $9+\frac{15\epsilon}{2}-\frac{821\epsilon^{2}}{36}$ | Derkachov:1997gc  |  |
1 | 1 | $\square^{2}{\partial}\phi^{5}$ | $10-\frac{7\epsilon}{9}$ | Kehrein:1994ff  |  |
2 | 1 | ${\partial}^{2}\phi^{3}$ | $5-\frac{17{\varepsilon}}{18}+\frac{161\epsilon^{2}}{972}$ | Bertucci:2022ptt  | $\frac{1}{2}-\frac{199{\varepsilon}}{432}$ | Bertucci:2022ptt
2 | ${\partial}^{2}\phi^{5}$ | $7+\frac{{\varepsilon}}{6}$ | Kehrein:1994ff  |  |
3 | $\square{\partial}^{2}\phi^{5}$ | $9-\frac{{\varepsilon}}{2}$ | Kehrein:1994ff  |  |
4 | ${\partial}^{2}\phi^{7}$ | $9+\frac{47{\varepsilon}}{18}$ | Kehrein:1994ff  |  |
3 | 1 | ${\partial}^{3}\phi^{3}$ | $6-\frac{4{\varepsilon}}{3}+\frac{17{\varepsilon}^{2}}{2592}$ | Bertucci:2022ptt  | $\frac{2}{35}-\frac{911{\varepsilon}}{14700}$ | Bertucci:2022ptt
2 | ${\partial}^{3}\phi^{5}$ | $8-\frac{{\varepsilon}}{3}$ | Kehrein:1994ff  |  |
3 | $\square{\partial}^{3}\phi^{5}$ | $10-\frac{10{\varepsilon}}{9}$ | Kehrein:1994ff  |  |
4 | ${\partial}^{3}\phi^{7}$ | $10+2{\varepsilon}$ | Kehrein:1994ff  |  |
4 | 1 | ${\partial}^{4}\phi^{3}$ | $7-\frac{31\epsilon}{30}+\frac{3559{\varepsilon}^{2}}{20250}$ | Bertucci:2022ptt  | $\frac{1}{18}-\frac{13943{\varepsilon}}{194400}$ | Bertucci:2022ptt
2 | $({\partial}^{4}\phi^{5})_{1}$ | $9-\frac{17{\varepsilon}}{18}$ | Kehrein:1994ff  |  |
3 | $({\partial}^{4}\phi^{5})_{2}$ | $9-\frac{{\varepsilon}}{10}$ | Kehrein:1994ff  |  |
5 | 1 | ${\partial}^{5}\phi^{3}$ | $8-\frac{23{\varepsilon}}{18}+\frac{61{\varepsilon}^{2}}{1620}$ | Bertucci:2022ptt  | $\frac{2}{231}-\frac{378163{\varepsilon}}{28814940}$ | Bertucci:2022ptt
2 | $({\partial}^{5}\phi^{5})_{1}$ | $10-\frac{5{\varepsilon}}{3}$ | Kehrein:1994ff  |  |
3 | $({\partial}^{5}\phi^{5})_{2}$ | $10-\frac{7{\varepsilon}}{18}$ | Kehrein:1994ff  |  |
6 | 1 | $({\partial}^{6}\phi^{3})_{1}$ | $9-\frac{3{\varepsilon}}{2}$ | Kehrein:1992fn  |  |
2 | $({\partial}^{6}\phi^{3})_{2}$ | $9-\frac{15\epsilon}{14}+\frac{48799{\varepsilon}^{2}}{277830}$ | Bertucci:2022ptt  | $\frac{3}{572}-\frac{8019289{\varepsilon}}{961920960}$ | Bertucci:2022ptt
7 | 1 | ${\partial}^{7}\phi^{3}$ | $10-\frac{5{\varepsilon}}{4}+\frac{4297{\varepsilon}^{2}}{80640}$ | Bertucci:2022ptt  | $\frac{2}{2145}-\frac{2625239{\varepsilon}}{1545944400}$ | Bertucci:2022ptt
Table 5: Perturbative results for $\mathbb{Z}_{2}$-odd operators with
$\Delta\leqslant 10$.

###  C.2 Navigator minima

In table 6 we show the values for the navigator minima in the different
spacetime dimensions studied.

$d$ | $\Delta_{\sigma}|_{\mathrm{min}}$ | $\Delta_{\epsilon}|_{\mathrm{min}}$ | $\theta|_{\mathrm{min}}$
---|---|---|---
2.6 | 0.3440987308 | 1.219622499 | 0.8616022236
2.7 | 0.3860503791 | 1.265155812 | 0.8968882905
2.8 | 0.4291188478 | 1.312594686 | 0.9253317624
2.9 | 0.4731867209 | 1.361792983 | 0.9490037256
3. | 0.518148953 | 1.412627358 | 0.9692651416
3.1 | 0.5639124084 | 1.465006065 | 0.9870522563
3.2 | 0.6103917163 | 1.518841816 | 1.003021129
3.3 | 0.6575105206 | 1.574075519 | 1.017657414
3.4 | 0.7051992474 | 1.630668115 | 1.031330689
3.5 | 0.7533940071 | 1.688601476 | 1.044333198
3.6 | 0.8020356597 | 1.747882413 | 1.056907458
3.7 | 0.8510688458 | 1.808547471 | 1.069266184
3.8 | 0.9004410692 | 1.870674384 | 1.081610779
3.9 | 0.9501016191 | 1.934405213 | 1.094152721
3.95 | 0.9750242935 | 1.966945908 | 1.100574515
Table 6: Table of navigator minima.

###  C.3 Operators for fit of level repulsion plot

For the fit in section 4.2.1, we make use the points

$\displaystyle d$
$\displaystyle=2.6:\qquad\Delta\in\\{6.96815028,\,7.30348360\\},$
$\displaystyle d$
$\displaystyle=2.7:\qquad\Delta\in\\{7.04476307,\,7.20549735\\},$
$\displaystyle d$
$\displaystyle=2.8:\qquad\Delta\in\\{7.04351005,\,7.19424952\\},$ (43)
$\displaystyle d$
$\displaystyle=2.9:\qquad\Delta\in\\{6.98086197,\,7.24126426\\},$
$\displaystyle d$
$\displaystyle=3.0:\qquad\Delta\in\\{6.90348236,\,7.29743289\\}.$

##  Appendix D Extremal functional method

A key step in our analysis is the use of the extremal functional method (EFM),
developed in Poland:2010wg ; El-Showk:2012vjm ; El-Showk:2016mxr . For
demonstration purposes we show what the action of such a functional looks like
on a sum rule. More specifically, in figure 21, we show the determinant of the
extremal functional acting on the $\mathbb{Z}_{2}$-even spin-$0$ sector in
$d=3$, using the parameters from the second column of table 3, at the point in
parameter space referred to in the caption of table 7.

Figure 21: Extremal functional $\Lambda(\Delta)$ for $\mathbb{Z}_{2}$-even
scalars of dimensions $\Delta$ at $d=3.0$ dimensions.

From figure 21 we read off the following minima,

$\displaystyle\Delta\in$
$\displaystyle\\{3.82977589284,\,6.90348236615,\,7.29743289116,\ $
$\displaystyle
8.92058769366,\,10.6590253446,\,12.2118684512,\,14.9182183792,\ldots\\}.$ (44)

Alternatively, if we use the automatic reading of spectrum.py, we find

$\Delta\in\\{3.82977589284,\,6.90348236615,\,10.6590253446,\,12.2118684512,\ldots\\}.$
(45)

It is thus clear that this misses operators, in this case at $\Delta=7.2974$
and $\Delta=8.9206$. In fact spectrum.py is able to detect these minima, but
it discards them because solving the primal solution for the OPE coefficients
it finds a negative squared OPE coefficient.

We believe the inaccuracy in the OPE coefficients to be due to a numerical
instability. Hence, in order to be able to detect all operators, we read of
the minima of the determinants of the functionals in Mathematica. Or,
equivalently, using a modified version of spectrum.py that does not discard
these operators.

###  D.1 Systematic errors

For the case of $d=3$, we can compare with the values in Simmons-
Duffin:2016wlq . For $\mathbb{Z}_{2}$-even scalars above $\epsilon$, that work
found three stable operators, and some other visible features. In summary

$\Delta\in\\{3.82968(23),\,6.8956(43),\,7.2535(51),\,\sim 8.55,\,\sim
10.45,\,\sim 11.6\\}.$ (46)

We can see that for operators with $\tau\gtrsim 5$, our values are
consistently above those of Simmons-Duffin:2016wlq . It is likely that the
true operator dimensions are below those of both works. In table 7 we evaluate
the minima of the extremal functional for various values of $\Lambda$. One may
also compare with the values of Reehorst:2021hmp , which obtained
$\Delta_{\epsilon^{\prime}}=3.82951(61)$ with a rigorous error bar; our result
is $\Delta_{\epsilon^{\prime}}=3.82978$. See also, (34) and (35) for a
comparison of the leading OPE coefficients against values with rigorous error
bars.

| $\Delta_{\epsilon^{\prime}}$ | $\Delta_{\epsilon^{\prime\prime}}$ | $\Delta_{\epsilon^{\prime\prime\prime}}$ | $\Delta_{\epsilon^{\prime\prime\prime\prime}}$ | $\Delta_{\epsilon^{\prime\prime\prime\prime\prime}}$ | $\Delta_{\epsilon^{\prime\prime\prime\prime\prime\prime}}$
---|---|---|---|---|---|---
$\Lambda=15$ | $3.82994$ | $7.20245$ | $7.64939$ | $9.23416$ | $13.0288$ |
$\Lambda=21$ | $3.83012$ | $6.97177$ | $7.25821$ | $8.64518$ | $11.3409$ | $12.9575$
$\Lambda=30$ | $3.82978$ | $6.90348$ | $7.29743$ | $8.92059$ | $10.6590$ | $12.2119$
$\Lambda=43$ Simmons-Duffin:2016wlq  | $3.82968(23)$ | $6.8956(43)$ | $7.2535(51)$ | $\sim 8.55$ | $\sim 10.45$ |  $\sim 11.6$.
Table 7: Zeros of the functional for $\mathbb{Z}_{2}$-even scalars in $d=3$ at
different $\Lambda$. The first three lines were computed for the specific
point ($\Delta_{\phi}$ , $\Delta_{\phi^{2}}$ , $\theta$) $=$ ($0.518148953$ ,
$1.412627358$ , $0.9692651416$). The last line corresponds to the averaged
value of Simmons-Duffin:2016wlq .

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