# Critical Three-Dimensional Ising Model on Spheriods from the Conformal
Bootstrap

Daniel Berkowitz  Yale University
Department of Physics
<EMAIL_ADDRESS>George T. Fleming  Yale University
Department of Physics
<EMAIL_ADDRESS>
###### Abstract

We construct a conformal map from $\mathbb{R}^{3}$ to a three-dimensional
spheriod, which includes $\mathbb{S}^{3}$, a double-cover of the 3-ball, and
$\mathbb{R}\times\mathbb{S}^{2}$ as limiting cases. Using the data of the
critical three-dimensional Ising model on $\mathbb{R}^{3}$ that was computed
using the conformal bootstrap method, we numerically estimate the fourth-order
Binder cumulant of the critical three-dimensional $\phi^{4}$ theory on
$\mathbb{S}^{3}$. We expect this estimate will enable an interesting
comparison between the conformal bootstrap and future calculations of critical
$\phi^{4}$ theory on $\mathbb{S}^{3}$ using the Quantum Finite Element (QFE)
method.

††preprint: APS/123-QED

##  I INTRODUCTION

The last decade has seen major advances in the general study of conformal
field theories beyond the special cases of two-dimensional spacetimes or
maximal supersymmetry through the widespread development of the conformal
boostrap [1]. Particularly notable is the success of the conformal boostrap in
constraining the data (scaling dimensions and OPE coefficients) of the three-
dimensional critical Ising model CFT [2, 3, 4, 5], surpassing the previous
best results using Markov chain Monte Carlo (MCMC) on regular cubic
discretizations of a three-dimensional torus [6, 7, 8]. This has also driven
the development of improved methods for constructing three-dimensional
conformal blocks, the basis functions of the conformal group analogous to
spherical harmonics for the rotation group [3, 9, 10]. These ingredients allow
for an accurate estimate of the four-point functions [11] of the critical
Ising model.

In order to keep up with developments in the conformal bootstrap, a new
approach for MCMC-based calculations optimized for the study of conformal
fixed points in quantum field theories, called Quantum Finite Elements (QFE),
has recently been developed [12, 13, 14, 15, 16, 17, 18, 19]. QFE is a general
framework for MCMC calculations on static curved manifolds. Relevant to the
study of CFTs, there is no conformal map from flat Euclidean space
$\mathbb{R}^{3}$, where conformal bootstrap calculations are performed, to the
torus $\mathbb{T}^{3}$, where traditional MCMC calculations on cubic grids are
performed, making direct comparison in the infinite volume limit difficult for
most of the CFT data except a few leading scaling dimensions determined via
finite size scaling. As we will show, conformal maps from $\mathbb{R}^{3}$ to
spheroids, including the special cases of the sphere $\mathbb{S}^{3}$; the
double cover of the 3-ball; and the cylinder $\mathbb{R}\times\mathbb{S}^{2}$,
do exist and MCMC calculations can be performed on these manifolds using QFE.
In particular, observables can be constructed in QFE to directly calculate all
the CFT data that appears in the conformal block expansion: scaling dimensions
and OPE coefficients.

As we shall see, one class of observables where the accuracy of QFE
calculations are expected to surpass those of the conformal bootstrap is in
the calculation of moments of the average magnetization

$\displaystyle M=\int d^{3}x\sqrt{g(x)}\phi(x),\quad$ (1) $\displaystyle
m_{n}=\left\langle\int
d^{3}x\sqrt{g(x)}\left(\phi(x)-M\right)^{n}\right\rangle.$

In QFE and traditional calculations on cubic lattices, moments of average
magnetization can be computed very accurately due to the increased statistics
of averaging over the volume.

We will first define the manifolds we are working with and obtain a general
Weyl factor which will allow us to define a conformal map between a 3D
spheroid and $\mathbb{R}^{3}$ and discuss the limiting cases of
$\mathbb{S}^{3}$, $\mathbb{R}\crossproduct\mathbb{S}^{2}$, and the double-
cover of 3-ball in detail. We will show how to map the approximate four-point
function computed in the conformal bootstrap on $\mathbb{R}^{3}$ for the
critical 3D Ising model to the four-point function on a spheroid. Then, we
integrate the two-point function and apply Monte-Carlo integration to the
four-point function and obtain estimates for the fourth-order Binder cumulant
of the critical 3D Ising model on $\mathbb{S}^{3}$. Finally, we will discuss
how this result can be compared with future lattice calculations, including
quantum finite elements (QFE).

##  II Conformal Invariance of a 3D Spheroid

###  II.1 Conformal Invariance of $\mathbb{S}^{3}$

In this section we will furnish a conformal mapping between a general 3D
spheroid and $\mathbb{R}^{3}$. We will later use the resultant Weyl factor to
obtain an estimate of the fourth-order Binder cumulant for the critical 3D
Ising model on $\mathbb{S}^{3}$. Our spheroid can be defined as a set of
points embedded in $\mathbb{R}^{4}$ which satisfies the following relation in
Cartesian coordinates

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}}+\frac{z^{2}}{a^{2}}+\frac{w^{2}}{b^{2}}=1\quad(a,b>0).$
(2)

When $b\rightarrow 0$ our spheroid approaches the 3D analogue of a disc or a
2-ball, which is a 3-ball. Technically speaking this 3-ball will be the
superposition of two 3-balls superimposed onto each other at their boundaries.
This will be discussed in more detail later. In the opposite limit when
$b\rightarrow\infty$ our spheroid can be understood to approach
$\mathbb{R}\crossproduct\mathbb{S}^{2}$. In an earlier calculation $\phi^{4}$
theory at its Wilson-Fisher fixed point was analyzed [19] on
$\mathbb{R}\crossproduct\mathbb{S}^{2}$ using the QFE. When $b\rightarrow a$
the spheroid approaches $\mathbb{S}^{3}$, which is the next manifold we wish
to study $\phi^{4}$ theory on using the QFE.

A set of equations which satisfy (2) and encapsulates all of the cases we just
outlined and everything else in-between are

$\displaystyle x=a\sin\psi\sin\theta\cos\phi,$ (3) $\displaystyle
y=a\sin\psi\sin\theta\sin\phi,$ $\displaystyle z=a\sin\psi\cos\theta,$
$\displaystyle w=b\cos\psi,$

where $\psi$ and $\theta$ range from 0 to $\pi$, and $\phi$ ranges from 0 to
$2\pi$. Intuitively this set of coordinates can be deduced by noticing that as
one goes up a dimension from the circle, $\mathbb{S}^{1}$, to the sphere,
$\mathbb{S}^{2}$, that an extra parameter, $\theta$, is introduced which
ranges from 0 to $\pi$ in the following manner

$\displaystyle x_{\mathbb{S}^{2}}=\sin\theta x_{\mathbb{S}^{1}},$ (4)
$\displaystyle y_{\mathbb{S}^{2}}=\sin\theta y_{\mathbb{S}^{1}},$
$\displaystyle z_{\mathbb{S}^{2}}=\cos\theta.$

For the coordinates originally associated with the lower dimensional sphere, a
factor of $\sin\theta_{i}$, where $\theta_{i}$ ranges from 0 to $\pi$ is
included and represents a new degree of freedom present on the higher
dimensional sphere. The new independent Cartesian coordinate which
differentiates the higher dimensional embedding space from the one dimension
lower space is parameterized by $\cos\theta_{i}$. We see that carrying out
this construction from the $\mathbb{S}^{2}$, parameterized using standard
spherical coordinates, to the $\mathbb{S}^{3}$ results in (3). Extending this
to arbitrary dimensions results in the following parameterization for a
n-sphere with radius $r$,

$\displaystyle x_{1}$ $\displaystyle=r\cos\left(\theta_{1}\right)$ (5)
$\displaystyle x_{2}$
$\displaystyle=r\sin\left(\theta_{1}\right)\cos\left(\theta_{2}\right)$
$\displaystyle x_{3}$
$\displaystyle=r\sin\left(\theta_{1}\right)\sin\left(\theta_{2}\right)\cos\left(\theta_{3}\right)$
$\displaystyle\vdots$ $\displaystyle x_{n-1}$
$\displaystyle=r\sin\left(\theta_{1}\right)\cdots\sin\left(\theta_{n-2}\right)\cos\left(\phi\right)$
$\displaystyle x_{n}$
$\displaystyle=r\sin\left(\theta_{1}\right)\cdots\sin\left(\theta_{n-2}\right)\sin\left(\phi\right).$

Because the n-sphere and in turn the n-spheroid can be parameterized in a
convenient coordinate system the results in this paper can be generalized to n
dimensions.

Using (3) we obtain the following metric representation for a 3D spheroid

$\displaystyle ds_{sph}^{2}=$ (6)
$\displaystyle\left(b^{2}\sin^{2}\psi+a^{2}\cos^{2}\psi\right)d\psi^{2}+a^{2}\sin^{2}{\psi}(d\theta^{2}+\sin^{2}{\theta}d\phi^{2}).$

Using the following change in coordinates inspired by [20],
$w=\int\sqrt{b^{2}\sin^{2}\psi+a^{2}\cos^{2}\psi}d\psi$, we can rewrite (6) as

$ds_{sph}^{2}=dw^{2}+f(w)^{2}(d\theta^{2}+\sin^{2}{\theta}d\phi^{2}),$ (7)

where $f(w)=a\sin\psi$. This metric can be related conformally to the
following metric on $\mathbb{R}^{3}$

$\begin{aligned}
ds_{flat}^{2}=(dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}{\theta}d\phi^{2})\end{aligned},$
(8)

by introducing this functional dependency, $r=e^{g(w)}$, which results in

$\displaystyle ds_{flat}^{2}=$ (9) $\displaystyle
e^{2g(w)}\left(\frac{dg}{dw}\right)^{2}\left(dw^{2}+\left(\frac{1}{\frac{dg}{dw}}\right)^{2}\left(d\theta^{2}+\sin^{2}{\theta}d\phi^{2}\right)\right).$

We can now set

$\left(\frac{1}{\frac{dg}{dw}}\right)=a\sin\psi(w)$ (10)

and obtain $g(w)=\int\frac{1}{a}\csc\psi dw$. Going back to the original
coordinate transformation we applied to (6), we can rewrite $g(w)$ as

$\displaystyle g(w)$
$\displaystyle=\int\frac{1}{a}\csc\psi\sqrt{b^{2}\sin^{2}\psi+a^{2}\cos^{2}\psi}d\psi$
(11) $\displaystyle=\int\sqrt{\frac{b^{2}}{a^{2}}+\cot^{2}\psi}d\psi.$

By doing so we recover the following metric which is conformal to (7)

$ds_{flat}^{2}=\frac{e^{2g(w)}}{a^{2}\sin^{2}\psi}\left(dw^{2}+f(w)^{2}(d\theta^{2}+\sin^{2}{\theta}d\phi^{2})\right).$
(12)

By comparing (12) to (7) we can deduce that the Weyl factor of our 3D spheroid
is

$\Omega_{spheroid}\left(x_{i}\right)=a\sin\psi{e^{-\int\sqrt{\frac{b^{2}}{a^{2}}+\cot^{2}\psi}d\psi}}.$
(13)

For the case of $\mathbb{S}^{3}$ when $b=a=1$ this reduces to

$\Omega_{\mathbb{S}^{3}}\left(x_{i}\right)=2\cos^{2}\frac{\psi}{2}.$ (14)

Going back to how we parameterized $r$ in (8) and setting $b=a=1$ we obtain
the following mapping between a point on $\mathbb{S}^{3}$ and a point in
$\mathbb{R}^{3}$

$\displaystyle z=\tan\left(\frac{\psi}{2}\right)\cos{\theta},$ (15)
$\displaystyle y=\tan\left(\frac{\psi}{2}\right)\sin{\theta}\sin{\phi},$
$\displaystyle x=\tan\left(\frac{\psi}{2}\right)\sin{\theta}\cos{\phi}.$

From a geometric perspective our conformal mapping of $\mathbb{S}^{3}$ to
$\mathbb{R}^{3}$ is the exact higher dimensional analogue of the standard
stereographic projection commonly performed from $\mathbb{S}^{2}$ to
$\mathbb{R}^{2}$. For the general case this mapping can be accomplished by
placing a $\mathbb{S}^{n}$ on $\mathbb{R}^{n}$ with its south pole centered on
the origin of $\mathbb{R}^{n}$ and drawing lines from the north pole which
intersect both $\mathbb{S}^{n}$ and $\mathbb{R}^{n}$. Each one of those lines
are oriented by a set of n angles and their intersection with $\mathbb{S}^{n}$
and $\mathbb{R}^{n}$ provides a one to one mapping between $\mathbb{S}^{n}$
and $\mathbb{R}^{n}$. The north pole itself cannot be mapped using only a
single cover because in the limiting case the line that would intersect the
north pole becomes parallel to $\mathbb{R}^{n}$ and hence never intersects it.
That is why $\mathbb{S}^{n}$ can be thought of as a one point compactification
of $\mathbb{R}^{n}$. A picture of this stereographic projection is provided in
Fig. 1.

(a) Figure 1: Illustration of the conformal mapping between an infinite plane
and $\mathbb{S}^{2}$ with the polar angle shifted in the following manner,
$\theta\rightarrow\pi-\theta$, compared to what we have in (14). This imagine
originally appeared in [20]

###  II.2 Conformal Invariance of $R\crossproduct\mathbb{S}^{2}$

As we previously mentioned, in the limit, $b\rightarrow\infty$, (2) gives the
equation for a 3D cylinder which is topologically equivalent to
$R\crossproduct\mathbb{S}^{2}$. One way of seeing this is to examine what
happens to (2) as $b\rightarrow\infty$ and $-\infty<w<\infty$. In the limit
of, $b\rightarrow\infty$, and $w$ being finite, (2) reduces to

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}}+\frac{z^{2}}{a^{2}}=1.$ (16)

Despite the disappearance of $w$ the manifolds which (16) admits are still
embedded in $\mathbb{R}^{4}$. Thus $w$ can be parameterized independently of
x, y, and z in an arbitrary fashion. Because we wish to recover
$R\crossproduct\mathbb{S}^{2}$ we will set $w=t$, where $t$ ranges from
$\left(-\infty,\infty\right)$ and parameterize x, y, and z using standard
spherical coordinates. Doing so results in the following metric

$\displaystyle ds_{3-cyl}^{2}=dx^{2}+dy^{2}+dz^{2}+dw^{2},$ (17)
$\displaystyle
ds_{3-cyl}^{2}=dt^{2}+a^{2}(d\theta^{2}+\sin^{2}{\theta}d\phi^{2}),$

where $a$ is the radius of $\mathbb{S}^{2}$. Using (5), this metric can be
generalized to $R\crossproduct\mathbb{S}^{n}$

$\displaystyle ds_{n-cyl}^{2}=dt^{2}+d\Omega^{2}_{n-1},$ (18)

where $d\Omega^{2}_{n-1}$ is the metric of $\mathbb{S}^{n-1}$ which can be
obtained for any n using (5). As the reader can verify
$dt^{2}+a^{2}(d\theta^{2}+\sin^{2}{\theta}d\phi^{2})$ is conformally related
to $(dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}{\theta}d\phi^{2})$ through the
following Weyl factor

$\displaystyle\Omega_{3-cyl}^{2}=e^{-2t/a},$ (19)

generated by defining $r$ as

$\displaystyle r=ae^{\frac{t}{a}}.$ (20)

It should be mentioned that setting $b$ equal to a positive, real, finite
number does not result in the geometry of a finite length
$\mathbb{R}\crossproduct\mathbb{S}^{2}$. Rather the resultant geometry is a 3D
ellipsoid. Only in the limit as $b\rightarrow\infty$ is
$\mathbb{R}\crossproduct\mathbb{S}^{2}$ realized.

Because $\mathbb{R}\crossproduct\mathbb{S}^{2}$ isn’t a compact manifold using
integration to find the Binder cumulant (34) isn’t trivial. To obtain an
estimate of the Binder cumulant for this non-compact geometry one can perform
a single point compactification of $\mathbb{R}\crossproduct\mathbb{S}^{2}$
which results in $\mathbb{S}^{1}\crossproduct\mathbb{S}^{2}$ and define a
lattice field theory on that manifold. Once a lattice field theory is defined
on $\mathbb{S}^{1}\crossproduct\mathbb{S}^{2}$ one can perform a Monte Carlo
simulation to compute an estimate of the Binder cumulant on that compactified
geometry as was done in [19]. When this compactification is done the manifold
is defined by two radii $r1$ and $r2$, where $r1$ denotes the radius of
$\mathbb{S}^{1}$ and, $r2$ is the radius of $\mathbb{S}^{2}$. In the limit as
$r1\rightarrow\infty$, $\mathbb{S}^{1}\crossproduct\mathbb{S}^{2}$ approaches
$\mathbb{R}\crossproduct\mathbb{S}^{2}$. Therefore if one has a method, such
as the QFE, which allows them to formulate lattice field theories on curved
manifolds they can study the critical 3D Ising model on
$\mathbb{S}^{1}\crossproduct\mathbb{S}^{2}$ as $r1\rightarrow\infty$ and
observe what the Binder cumulant approaches.

###  II.3 Conformal Invariance of 3-Ball

Before we move on to calculating the Binder cumulant on $\mathbb{S}^{3}$ it is
prudent to talk about how one would do the corresponding calculation for the
3-ball. The 3-ball is imminently related to $\mathbb{S}^{3}$. One can
construct $\mathbb{S}^{3}$ by superimposing two 3-balls and defining an
equivalence class which identifies all of the points which make up the shared
boundaries of this superimposed ball. In other words, the points which make up
the boundary of the two superimposed 3-balls, which are two 2-spheres, are
identified as a single point. This identification of the boundary with a
single point can be realized by projecting out the boundary of this
superimposed 3-ball into a higher dimensional space in which the boundary
points of the 2-spheres merge at the north pole; thus resulting in a newly
formed $\mathbb{S}^{3}$.

Mathematically we see a hint of this construction by setting, $b=0$, and
obtaining the following r from our earlier coordinate transformation

$\displaystyle r=\lim_{b\to
0}e^{\int\sqrt{\frac{b}{a}+\cot^{2}\psi}d\psi}=\sin\psi.$ (21)

As it can be seen when $b\to 0$ our $r$ doesn’t cover the whole plane
$\mathbb{R}^{3}$ because $r$ only ranges from 0 to 1. This is because these
coordinates only map points located on the lower hemisphere and excludes
points located on our 3-ball’s upper hemisphere. This is an artifact of
$\mathbb{S}^{3}$ being a construction of two superimposed 3-balls with their
shared boundaries being glued together at a single point. The total mapping
can be realized by noticing that we could have defined our functional
dependency right above (9) as $r=e^{-g(w)}$. Such a functional dependency
would have allowed us to obtain a different Weyl factor, $\Omega(x_{i})$,
which nonetheless yields the exact same Binder cumulant for the case of
$\mathbb{S}^{3}$ as the one we previously calculated. However it would also
result in the following radius

$\displaystyle r=\lim_{b\to
0}e^{-\int\sqrt{\frac{b}{a}+\cot^{2}\psi}d\psi}=\csc\psi.$ (22)

which ranges from 1 to $\infty$. Thus in order to compute the two and four-
point functions for the superimposed 3-ball one must differentiate between
points on the northern hemisphere and points on the southern hemisphere
because they both have different radial coordinates, (21) and (22). The
resulting calculation involves group averaging the four-point function over
all of the different combination of points that can be on one hemisphere and
the opposite hemisphere. More information of this group averaging for the
2-ball which can be extrapolated to the 3-ball can be found in [20].

Preliminary results obtained using the five operators reported in table 1 of
[21] and setting $\frac{b}{a}=10^{-5}$ yields, $U_{4}=0.38703\pm.0.00570$,
which suggests that the Binder cumulant for these two superimposed 3-balls is
similar to the Binder cumulant for $\mathbb{S}^{3}$ as we show in section IV.
This similarity between the Binder cumulant of the superimposed 3-ball and
$\mathbb{S}^{3}$ are in concord with the similarity found in [20] between the
Binder cumulant of the superimposed 2-ball(disc) and $\mathbb{S}^{2}$.

For the related case of the interior of a single, non-superimposed sphere that
was considered in [22] one has to take into account the boundary. This is done
by studying the boundary conformal field theory(BCFT)[23, 24] which has its
own scaling dimensions and operators associated with it. For the case with a
free boundary on $\mathbb{S}^{2}$ the scaling dimension of the relevant
operator was found to be [25, 26, 27, 28] $\Delta_{\tilde{\sigma}}=1.276(2)$.
Thus in computing the Binder cumulant of a 3-ball with a boundary one must
differentiate between pairs of points which are either both on the boundary,
inside the bulk or where one point is inside the bulk and the other is on the
boundary. The need to differentiate the location of points on the two
aforementioned manifolds is a similarity that the two-superimposed 3-balls and
the single 3-ball with boundary share with each other.

Using the known values of the scaling dimensions on the boundary of a 3-ball
and its interior allows one in theory to use the formalism we presented in
this paper to compute an estimate of the fourth-order Binder cumulant. An
estimate of the fourth-order Binder cumulant for the critical 3D Ising model
on a 3-ball with a boundary computed by integrating its two and four-point
functions can be compared with the estimate obtained in [22]. Comparing this
hypothetical estimate to what was obtained in [22] can increase our
understanding of how to study BCFTs via numerical simulations. In the future
we plan to do the calculation we outlined in this section in the limit when,
$\frac{b}{a}=0$, and perform a comparison of the value of the Binder cumulant
obtained from direct integration to [22]. Furthermore in the future the QFE
can be extended to apply to quantum field theories formulated on curved
manifolds with boundaries.

##  III The Four-Point Function for the Critical 3d Ising Model

The four-point function for the critical 3D Ising model whose form is
restricted by conformal symmetry is given below

$\left\langle\phi(x_{1})\phi(x_{2})\phi(x_{3})\phi(x_{4})\right\rangle_{flat}=\frac{g(u,v)}{\left|x_{1}-x_{2}\right|^{2\Delta_{\sigma}}\left|x_{3}-x_{4}\right|^{2\Delta_{\sigma}}},$
(23)

where $x_{i}$ is a point in $\mathbb{R}^{3}$ and $u$, and $v$ are the
following conformally invariant cross ratios

$\displaystyle
u=\frac{\left(x_{12}^{2}x_{34}^{2}\right)}{\left(x_{13}^{2}x_{24}^{2}\right)},$
(24) $\displaystyle
v=\frac{\left(x_{32}^{2}x_{14}^{2}\right)}{\left(x_{13}^{2}x_{24}^{2}\right)}.$

For our four-point function (23), $\Delta_{\sigma}$ is the scaling dimension
of $\phi(x_{i})$ and its value [29] estimated by the conformal bootstrap is
$\Delta_{\sigma}=0.5181489(10)$. The numerator of (23), $g(u,v)$, can be
expressed as a OPE in terms of conformal blocks

$g(r,\eta)=1+\sum_{\mathscr{O}\in\sigma\times\sigma}C_{\sigma\sigma\mathscr{O}}^{2}g_{\Delta_{\mathscr{O}},\ell_{\mathscr{O}}}(r,\eta),$
(25)

where

$\displaystyle
r=\sqrt{\frac{z\bar{z}}{\left(\sqrt{1-z}+1\right)^{2}\left(\sqrt{1-\bar{z}}+1\right)^{2}}}$
(26)
$\displaystyle\eta=\frac{\frac{z}{\left(\sqrt{1-z}+1\right)^{2}}+\frac{\bar{z}}{\left(\sqrt{1-\bar{z}}+1\right)^{2}}}{2\left(\frac{z\bar{z}}{\left(\sqrt{1-z}+1\right)^{2}\left(\sqrt{1-\bar{z}}+1\right)^{2}}\right)^{\frac{1}{2}}}$
$\displaystyle u=z\bar{z}$ $\displaystyle v=(1-z)(1-\bar{z}).$

More information on the physical meaning behind these coordinates (r, $\eta$,
$z$, $\bar{z}$) can be found in [11, 3].

The sum in (25) is over the operators (excluding the unit operator) that are
present in the $\phi(x_{i})\times\phi(x_{j})$ OPE of dimension
$\Delta_{\mathscr{O}}$ and spin $\ell_{\mathscr{O}}$. The scale dimensions and
spins for these operators are provided in table 2 of [30]. To evaluate (25) we
use following recursion relation which was first reported in [31]

$\displaystyle(4r)^{\Delta}h_{\Delta,\ell}(r,\eta)\equiv
g_{\Delta,\ell}(r,\eta)$ (27) $\displaystyle
h_{\Delta,\ell}(r,\eta)=h_{\ell}^{(\infty)}(r,\eta)+\sum_{k}\frac{c_{1}r^{n_{1}}}{\Delta-\Delta_{1}}h_{\Delta_{1}+n_{1},\ell_{1}}(r,\eta)$
$\displaystyle+\sum_{k}\frac{c_{2}r^{n_{2}}}{\Delta-\Delta_{2}}h_{\Delta_{2}+n_{2},\ell_{2}}(r,\eta)+\sum_{k}\frac{c_{3}r^{n_{3}}}{\Delta-\Delta_{3}}h_{\Delta_{3}+n_{3},\ell_{3}}(r,\eta)$

where information on what $h_{\ell}^{(\infty)}(r,\eta)$, $c_{i}$, $n_{i}$,
$\ell_{i}$, and $\Delta_{i}$ are is described in [3].

This recursion relation converges quickly and is easy to evaluate using a
computer algebra system like Mathematica. For our purposes we evaluate
$h_{\Delta,\ell}$ to 12th order in $r$ where $h_{\ell}^{(\infty)}$ is not
expanded in terms of r. The result of using this recursion relation (27) to
compute $h(r,\eta)$ up to nth order should be a nth order polynomial in r
whose coefficients include $h_{\ell}^{(\infty)}(r,\eta)$. In the appendix,
there will be a Mathematica code that evaluates (25) as a sum over the
operators present in table 1 of [5] up to any order in $r$. The code can be
easily modified to include the eleven terms which appear in table 2 of [30].

##  IV Binder Cumulant Estimate

Using the Weyl factor,
$\Omega\left(x_{i}\right)=2\cos^{2}\left(\frac{\psi_{i}}{2}\right)$, we
obtained earlier we can map the two and four-point functions of the critical
3D Ising model from $\mathbb{R}^{3}$ to $\mathbb{S}^{3}$ as is shown below in

$\displaystyle\left\langle\phi\left(x_{1}\right)\phi\left(x_{2}\right)\right\rangle_{g_{uv}}=\frac{1}{\Omega\left(x_{1}\right)^{\Delta_{\sigma}}}\frac{1}{\Omega\left(x_{2}\right)^{\Delta_{\sigma}}}\left\langle\phi\left(x_{1}\right)\phi\left(x_{2}\right)\right\rangle_{\text{flat
}}$ (28)
$\displaystyle\left\langle\phi\left(x_{1}\right)\phi\left(x_{2}\right)\phi\left(x_{3}\right)\phi\left(x_{4}\right)\right\rangle_{g_{uv}}$
$\displaystyle=\frac{1}{\Omega\left(x_{1}\right)^{\Delta_{\sigma}}}\ldots\frac{1}{\Omega\left(x_{4}\right)^{\Delta_{\sigma}}}\left\langle\phi\left(x_{1}\right)\ldots\phi\left(x_{4}\right)\right\rangle_{\text{flat
}}.$

We will use these variables in (15) to construct our two and four-point
functions

As it can be seen below conformal symmetry greatly restricts the form of the
following two-point function

$\left\langle\phi\left(x_{1}\right)\phi\left(x_{2}\right)\right\rangle_{flat}=\frac{1}{x_{12}^{2\Delta}},$
(29)

where $x_{ij}$ =$|x_{i}-x_{j}|$. The two quantities we need to find by
integrating our two and four-point functions over $\mathbb{S}^{3}$ in order to
obtain our fourth-order Binder cumulant are the following magnetization
densities

$\begin{array}[]{l}\left\langle\sigma^{2}\right\rangle=\rho^{2}\int\mathrm{d}S_{1}\mathrm{~{}d}S_{2}\left\langle\phi\left(x_{1}\right)\phi\left(x_{2}\right)\right\rangle_{g_{uv}},\\\
\left\langle\sigma^{4}\right\rangle=\rho^{4}\int\mathrm{d}S_{1}\cdots\mathrm{d}S_{4}\left\langle\phi\left(x_{1}\right)\phi\left(x_{2}\right)\phi\left(x_{3}\right)\phi\left(x_{4}\right)\right\rangle_{g_{uv}},\end{array}$
(30)

where $\rho$ is the areal density of the spins, and $\mathrm{d}S_{i}$
represents the number of spins in an infinitesimal area. For the
$\mathbb{S}^{3}$, $\rho$, and $\mathrm{d}S_{i}$ can respectively be expressed
as $\frac{1}{2\pi^{2}}$ and
$\sin^{2}{\psi_{i}}\sin{\theta_{i}}d\psi_{i}d\theta_{i}d\phi_{i}$. We can
reduce the computational cost of integrating our two and four-point functions
by taking advantage of the SO(4) symmetry of $\mathbb{S}^{3}$. Because SO(4)
has six generators, which corresponds to six independent rotations, we can
rotate our $\mathbb{S}^{3}$ in such a way that some of the angles that we
would normally need to integrate over are fixed; thus reducing the
computational cost of our multidimensional Monte Carlo integration. This can
be seen because SO(4) is locally isomorphic to $SO(3)\otimes SO(3)$. Thus
there are six independent rotations for $\mathbb{S}^{3}$ three for each SO(3).
This can be seen by noticing that the Lie Algebra of SO(4) can be represented
as two copies of the Lie Algebra of SO(3).

The key for efficiently evaluating these integrals (30) is to use the SO(4)
symmetry of $\mathbb{S}^{3}$. For the two-point function we can naively
evaluate a 6th dimensional integral over these points on its surface,
$(\psi_{1},\theta_{1},\phi_{1})$ and $(\psi_{2},\theta_{2},\phi_{2})$. By
rotating $\mathbb{S}^{3}$ we can set the first point to be $(0,0,0)$ and the
second point to be $(\psi_{2},0,0)$. This results in the following integral
for the 2nd order magnetization density.

$\left\langle\sigma^{2}\right\rangle=\int_{0}^{\pi}\frac{\left(\left(2\pi^{2}\right)(4\pi)\sin^{2}\left(\psi_{2}\right)\right)\left(\frac{1}{2\pi^{2}}\right)^{2}}{\left(2\cos^{2}\left(\frac{\psi_{2}}{2}\right)\right){}\left(\frac{\sin^{2}\left(\psi_{2}\right)}{\left(1+\cos\left(\psi_{2}\right)\right){}^{2}}\right)^{0.518149}}\,d\psi_{2}$
(31)

which yields from Mathematica’s INTEGRATE function

$\left\langle\sigma^{2}\right\rangle=0.847359$ (32)

The four-point function can be ultimately expressed using the following
coordinates $(\psi_{1},\theta_{1},\phi_{1})$,
$(\psi_{2},\theta_{2},\phi_{2})$, $(\psi_{3},\theta_{3},\phi_{3})$,
$(\psi_{4},\theta_{4},\phi_{4})$. Using the SO(4) group we can reduce the
dimensionality of our integral for $\left\langle\sigma^{4}\right\rangle$ from
twelve to six by fixing the following coordinates
$(0,0,0),(\psi_{2},0,0),(\psi_{3},\theta_{3},0),(\psi_{4},\theta_{4},\phi_{4})$.
Using Mathematica NINTEGRATE, we preformed 10,000 Monte Carlo evaluations and
obtained the following estimate of the fourth-order magnetization and its
associated statistical error

$\left\langle\sigma^{4}\right\rangle=1.59083\pm 0.00016.$ (33)

We now have all that we need to compute an estimate of the fourth-order Binder
cumulant.

$U_{4}=\frac{3}{2}\left(1-\frac{1}{3}\frac{\left\langle\sigma^{4}\right\rangle}{\left\langle\sigma^{2}\right\rangle^{2}}\right)$
(34) $U_{4}=0.39220\pm 0.00011.$ (35)

For now we exclude sources of error orientating from uncertainty inherent to
the CFT data obtained through the bootstrap.

This result must be understood within the context of the OPE representation of
the four-point function. Infinitely many operators of varying scaling
dimension and spin exist which must be summed in order to obtain an exact
expression for the four-point function of the critical 3D Ising model. The
finite number of operators whose CFT data has been obtained from the bootstrap
are the leading order operators which contribute the most to the four-point
function. However, because we only took into account data pertaining to eleven
of those operators a systematic error will be present in our calculation as a
result of us not being able to integrate the exact four-point function. The
remainder of the operators posses a higher spin and/or scaling dimension. Thus
their inclusion would allow us to more accurately compute the four point
function when the points are very close to each other.

The range of this systematic error can be estimated by taking the difference
between the Binder cumulant computed using the eleven operators listed in
table 2 of [30] and the resultant cumultant one obtains if they use only ten
of the operators. There is no definitive answer for which ten operators one
should include. Thus we performed this estimate using two similar, but
different methodologies. The first methodology involves computing a sequence
of Binder cumulants as operators are added to the OPE in the order of their
scaling dimension. This means first computing the Binder cumulant while only
including the operator with the lowest scaling dimension, $\epsilon$, in the
OPE representation of the four-point function and than including the operator
with the next lowest scaling dimension, $T_{\mu,\nu}$, until ten of the eleven
operators are included in the four-point function. The next methodology is
similar except operators are included sequentially into the OPE in order of
increasing spin.

In terms of increasing scaling dimension and spin, one respectively obtains
the following Binder cumulants and potential estimates for the systemic error

$\displaystyle U_{Scaling}=0.39216\pm 0.00011$ (36) $\displaystyle\Delta
U_{4-Scaling}=U_{4}-U_{Scaling}=0.00004,$

$\displaystyle U_{Spin}=0.39165\pm 0.00011$ (37) $\displaystyle\Delta
U_{4-Spin}=U_{4}-U_{Spin}=0.00141.$

One way to interpret the magnitude of the systematic errors that we obtained
using bootstrap data is to compare it to the magnitude of the systematic error
generated by doing the analogous calculation using the QFE. A direct
comparison of such nature cannot be done at the moment because we haven’t had
a chance to apply the QFE to $\phi^{4}$ theory on $\mathbb{S}^{3}$. However it
is reasonable to expect that the relative error that we will obtain when we do
the aforementioned calculation will be similar to the relative error obtained
for the analogous calculation [32, 33] on $\mathbb{S}^{2}$ which has already
been done.

Below are the statistical(58) and systematic(90) errors for an estimate of the
fourth-order Binder cumulant of $\phi^{4}$ theory at its Wilson-Fisher
conformal fixed point on $\mathbb{S}^{2}$ computed using the QFE (Monte Carlo
Values) [32, 33] and direct integration (Analytic CFT Values).

$\begin{array}[]{ll}\text{ Monte Carlo Values: }&U_{4,cr}=0.85020(58)(90)\\\
\text{ Analytic CFT Values: }&U_{4}^{*}=0.8510207(63).\end{array}$ (38)

If we compute the relative error of the above QFE result we obtain

$\delta U_{4,cr}=\frac{9\crossproduct 10^{-4}}{0.85020}\approx 10^{-3}.$ (39)

This is a reasonable estimate of the systematic error that we expect our QFE
calculation on $\mathbb{S}^{3}$ to yield. Therefore in order to interpret the
magnitude of the relative error that the bootstrap presently yields for the
fourth-Binder cumulant of the critical 3D Ising model we should compare it to
(39).

$\delta U_{4,scaling-dim}\approx 9\crossproduct 10^{-5},$ (40) $\delta
U_{4,spin}\approx 3.6\crossproduct 10^{-3}.$ (41)

If the systematic error in our calculation is closer to (40) that would
suggest that the current CFT data that we have from the bootstrap is enough to
compute an accurate estimate of the four-point function relative to the
implementation of the QFE that was used in [18]. However if the systematic
error is much closer to (41), that would indicate that the current bootstrap
results aren’t enough to match the accuracy of the QFE and that additional
data on higher order operators is needed so that the accuracy of the two
calculations can be in agreement with each other.

To demonstrate the convergence of the Binder cumulant as we add terms to the
four-point function we show in figure 2 the Binder cumulant as a function of
these operators for both methodologies. We used Monte-Carlo integration and
1,000 iterations to compute each Binder cumulant. We are confident that our
range is representative of the systematic error because it is evident that as
we include operators in the OPE that our results are converging to a
definitive value. The difference between preceding cumulants have a tendency
to shrink as we include more operators, thus the inclusion of more operators
will allow us to further minimize the systematic error. Thus in order to
compute a more accurate estimate of the fourth-Binder cumulant using
integration we need additional bootstrap results for higher order operators
and to include more Monte Carlo iterations.

Figure 2b shows an interesting phenomenon. Excluding when only the operator
with both the lowest scaling dimension and spin is included, the inclusion of
an operator with higher spin results in the value of $U_{4}$ jumping. When the
highest operator included in the four-point function has spin 2 the values of
$U_{4}$ varies little as additional spin 2 operators are included. It is only
when spin 4 operators are included that we see such a jump and again see that
the value of $U_{4}$ varies very little when additional spin 4 operators are
included. We see a similar jump when we include a spin 6 operator. The jump
though decreases in magnitude as higher and higher spin operators are
included. This suggests that the value of the Binder cumulant approaches some
definitive number as we increase the operators in the OPE representation of
the four point function. The origin of this phenomenon deserves to be
investigated.

As a check that our procedure for evaluating the four-point function on
$\mathbb{S}^{3}$ is correct we calculated the Binder cumulant for the free
theory. The correlation functions for a free CFT are given in [34] and its
Binder cumulant should be zero. Using Monte Carlo integration, 10,000
iterations, and accuracy goal 15, we calculated

$U_{4}=-1.9176\times 10^{-6}\pm 4.7357\times 10^{-5}.$ (42)

Our result is very comfortably within the range of the expected result of 0
for the Binder cumulant. We hope to check our results for the Binder cumulant
of the critical 3D Ising model on $\mathbb{S}^{3}$ using the QFE in the near
future.

##  V Concluding Remarks

Using the data of the critical 3D Ising model computed using the conformal
bootstrap method, we integrated the approximate two and four-point functions
to obtain an estimate of the fourth-order Binder cumulant. We also showed how
this approach could be used to estimate the Binder cumulants for the 3-ball
and other 3D spheroids. Our approach is an extension of the work by Deng and
Blote [20] to three dimensions and we showed how it could be extended further
to higher dimensional spheroids. The immediate application of our result is to
compare this estimate of the Binder cumulant with one computed in an upcoming
calculation of $\phi^{4}$ theory on $\mathbb{S}^{3}$ using quantum finite
elements (QFE). A favorable comparison of the two methods would give us
further confidence that QFE is a correct framework for computing non-
perturbative quantum field theories on curved manifolds.

(a) Binder cumulant plot in order of scaling dimension

(b) Binder cumulant plot in order of increasing spin

Figure 2: Plots of the Binder cumultant as we include operators into the OPE
of the four-point function in order of their scaling dimension and spin. We
count the unit operator as the first term included. We started with the 3rd
operator plot in order of scaling dimension. The statistical error bars don’t
show in the spin graph because of the large difference in values of the Binder
cumulant when only spin zero operator are included versus when spin $2>$
operators are included. The statistical errors for all of the points in our
spin graph are around $\approx.0004$.

###### Acknowledgements.

Both authors thank Richard Brower for his immense contributions towards the
development of the QFE. In addition we thank him for the intellectually
stimulating conversations that led us down the road to doing the calculation
we presented in this manuscript. We acknowledge support from the United States
Department of Energy through grant number DE-SC0019061.

##  Appendix A Appendixes

Using the following code and the CFT data reported for the five operators
included in table 1 of [5] we performed 100,000 Monte Carlo evaluations with
our accuracy goal set to 5 and calculated
$\left\langle\sigma^{4}\right\rangle$ to be $1.591463$ with an error of
$0.000050$. This allows us to obtain the following estimate of the fourth-
order Binder cumulant

$U_{4}=0.391765\pm 0.000035.$ (43)

The code below can be easily modified to include the additional CFT data [30]
that we used to compute (35).

This code defines the $h_{\Delta,\ell}(r,\eta)$ which we wish to calculate
through the recursion relation given in (27). The only input for the user is
”n”, which is the order one wishes to compute the recursion relation up to.

    H[\[CapitalDelta]\[CapitalDelta]_,
      LL_] := {n = 12;,
       recursion[a_] := Normal[Series[a /. hh -> h, {r, 0, n}]],
       Nest[recursion, {h[\[CapitalDelta]_,
            L_] := ((LegendreP[L, \[Eta]]/(
              Sqrt[1 - rr^2] Sqrt[-4 \[Eta]^2 rr^2 + (1 + rr^2)^2]) +
              Sum[((-((
                    2^(1 - 4 k)
                      k ((2 k)!)^2 Pochhammer[1 + L,
                      2 k])/((k!)^4 Pochhammer[1/2 + L, 2 k])))*(r)^(2*
                      k)/(\[CapitalDelta] - (1 - L - 2*k)))*
                hh[1 - L, L + 2*k], {k, 1, n/2}]
              +
              Sum[((-((
                    k (1/2 - k + L) Pochhammer[1/2, k]^2 Pochhammer[
                      1/4 (3 - 2 k + 2 L),
                      k]^2)/((1/2 + k + L) (k!)^2 Pochhammer[
                      1/4 (1 - 2 k + 2 L), k]^2)))*(r)^(2*
                      k)/(\[CapitalDelta] - ((3/2) - k)))*
                hh[(3/2) + k, L], {k, 1, n/2}] +
              Sum[((-((
                    2^(1 - 4 k)
                      k ((2 k)!)^2 Pochhammer[1 - 2 k + L,
                      2 k])/((k!)^4 Pochhammer[3/2 - 2 k + L,
                      2 k])))*(r)^(2*
                      k)/(\[CapitalDelta] - (2 + L - 2*k)))*
                hh[2 + L, L - 2*k], {k, 1,
                L/2}])), \[CapitalDelta] = \
    \[CapitalDelta]\[CapitalDelta];,
          L = LL;, (((LegendreP[L, \[Eta]]/(
              Sqrt[1 - rr^2] Sqrt[-4 \[Eta]^2 rr^2 + (1 + rr^2)^2]) +
              Sum[((-((
                    2^(1 - 4 k)
                      k ((2 k)!)^2 Pochhammer[1 + L,
                      2*k])/((k!)^4 Pochhammer[1/2 + L, 2*k])))*(r)^(2*
                      k)/(\[CapitalDelta] - (1 - L - 2*k)))*
                hh[1 - L, L + 2*k], {k, 1, n/2}]
              +
              Sum[((-((
                    k (1/2 - k + L) Pochhammer[1/2, k]^2 Pochhammer[
                      1/4 (3 - 2 k + 2 L),
                      k]^2)/((1/2 + k + L) (k!)^2 Pochhammer[
                      1/4 (1 - 2 k + 2 L), k]^2)))*(r)^(2*
                      k)/(\[CapitalDelta] - ((3/2) - k)))*
                hh[(3/2) + k, L], {k, 1, n/2}] +
              Sum[((-((
                    2^(1 - 4 k)
                      k ((2 k)!)^2 Pochhammer[1 - 2 k + L,
                      2 k])/((k!)^4 Pochhammer[3/2 - 2 k + L,
                      2 k])))*(r)^(2*
                      k)/(\[CapitalDelta] - (2 + L - 2*k)))*
                hh[2 + L, L - 2*k], {k, 1, L/2}])))}[[4]], n/2]}[[3]]

This code is the numerator that appears in the four-point function for the
critical 3D Ising model and is computed in accordance with (25)

    G = (1 + (H[1.412625, 0]*(((4*r)^(1.412625))*(1.0518537)^2)) + (H[
           3.82966, 0] (((4*r)^(3.82966))*(0.053029)^2)) + (H[3,
           2]*(((4*r)^(3)) ((0.5181489/Sqrt[0.946539])^2))) + (H[5.509,
           2] (((4*r)^(5.509))*(0.0172)^2)) + (H[5.02274,
           4] (((4*r)^(5.02274))*(0.1319)^2)));

This last piece of code plots the four-point function, showing that our
computation of the four-point function agrees with [11].

    {FourPointFunction = (G/(1 +
                Abs[z]^1.0362978 + (Abs[z]^1.0362978/
                   Abs[1 - z])^1.0362978) /. rr -> r /.
            r -> Abs[
              z/(1 + Sqrt[1 - z])^2] /. \[Eta] -> (z/(1 + Sqrt[1 - z])^2 +
                Conjugate[z]/(1 + Sqrt[1 - Conjugate[z]])^2)/(2*
               Abs[z/(1 + Sqrt[1 - z])^2]) /. z -> x + I*y);,
      Plot3D[FourPointFunction, {x, -1, .5}, {y, -1, 1},
       RegionFunction -> Function[{x, y}, Sqrt[x^2 + y^2] < 1 && x < 1/2],
        PlotRange -> All]}[[2]]

## References

  * Zamolodchikov and Zamolodchikov [1996] A. Zamolodchikov and A. Zamolodchikov, Conformal bootstrap in liouville field theory, Nuclear Physics B 477, 577 (1996).
  * El-Showk  _et al._ [2012] S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, and A. Vichi, Solving the 3d ising model with the conformal bootstrap, Physical Review D 86, 025022 (2012).
  * Kos  _et al._ [2014a] F. Kos, D. Poland, and D. Simmons-Duffin, Bootstrapping mixed correlators in the 3d ising model, Journal of High Energy Physics 2014, 109 (2014a).
  * Kos  _et al._ [2016a] F. Kos, D. Poland, D. Simmons-Duffin, and A. Vichi, Precision Islands in the Ising and $O(N)$ Models, JHEP 08, 036, arXiv:1603.04436 [hep-th] .
  * Komargodski and Simmons-Duffin [2017a] Z. Komargodski and D. Simmons-Duffin, The random-bond ising model in 2.01 and 3 dimensions, Journal of Physics A: Mathematical and Theoretical 50, 154001 (2017a).
  * Hasenbusch [2010] M. Hasenbusch, Finite size scaling study of lattice models in the three-dimensional ising universality class, Physical Review B 82, 10.1103/physrevb.82.174433 (2010).
  * Caselle  _et al._ [2015] M. Caselle, G. Costagliola, and N. Magnoli, Numerical determination of the operator-product-expansion coefficients in the 3D Ising model from off-critical correlators, Phys. Rev. D 91, 061901 (2015), arXiv:1501.04065 [hep-th] .
  * Costagliola [2016] G. Costagliola, Operator product expansion coefficients of the 3D Ising model with a trapping potential, Phys. Rev. D 93, 066008 (2016), arXiv:1511.02921 [hep-th] .
  * Penedones  _et al._ [2016] J. Penedones, E. Trevisani, and M. Yamazaki, Recursion relations for conformal blocks, Journal of High Energy Physics 2016, 1 (2016).
  * Hogervorst [2016] M. Hogervorst, Dimensional reduction for conformal blocks, Journal of High Energy Physics 2016, 1 (2016).
  * Rychkov  _et al._ [2017] S. Rychkov, D. Simmons-Duffin, and B. Zan, Non-gaussianity of the critical 3d ising model, SciPost Physics 2, 1 (2017).
  * Brower  _et al._ [2013] R. C. Brower, G. T. Fleming, and H. Neuberger, Lattice Radial Quantization: 3D Ising, Phys. Lett. B 721, 299 (2013), arXiv:1212.6190 [hep-lat] .
  * Brower  _et al._ [2012] R. C. Brower, G. T. Fleming, and H. Neuberger, Radial Quantization for Conformal Field Theories on the Lattice, PoS LATTICE2012, 061 (2012), arXiv:1212.1757 [hep-lat] .
  * Brower  _et al._ [2014] R. C. Brower, M. Cheng, and G. T. Fleming, Improved Lattice Radial Quantization, PoS LATTICE2013, 335 (2014), arXiv:1407.7597 [hep-lat] .
  * Brower  _et al._ [2015] R. C. Brower, M. Cheng, and G. T. Fleming, Quantum Finite Elements: 2D Ising CFT on a Spherical Manifold, PoS LATTICE2014, 318 (2015).
  * Brower  _et al._ [2016] R. C. Brower, G. Fleming, A. Gasbarro, T. Raben, C.-I. Tan, and E. Weinberg, Quantum Finite Elements for Lattice Field Theory, PoS LATTICE2015, 296 (2016), arXiv:1601.01367 [hep-lat] .
  * Brower  _et al._ [2017] R. C. Brower, E. S. Weinberg, G. T. Fleming, A. D. Gasbarro, T. G. Raben, and C.-I. Tan, Lattice Dirac Fermions on a Simplicial Riemannian Manifold, Phys. Rev. D 95, 114510 (2017), arXiv:1610.08587 [hep-lat] .
  * Brower  _et al._ [2018] R. C. Brower, M. Cheng, E. S. Weinberg, G. T. Fleming, A. D. Gasbarro, T. G. Raben, and C.-I. Tan, Lattice $\phi^{4}$ field theory on Riemann manifolds: Numerical tests for the 2-d Ising CFT on $\mathbb{S}^{2}$, Phys. Rev. D 98, 014502 (2018), arXiv:1803.08512 [hep-lat] .
  * Brower  _et al._ [2020] R. C. Brower, G. T. Fleming, A. D. Gasbarro, D. Howarth, T. G. Raben, C.-I. Tan, and E. S. Weinberg, Radial lattice quantization of 3d $\phi^{4}$ field theory, arXiv preprint arXiv:2006.15636 [hep-lat] (2020).
  * Deng and Blote [2003] Y. Deng and H. W. Blote, Conformal invariance and the ising model on a spheroid, Physical Review E 67, 036107 (2003).
  * Komargodski and Simmons-Duffin [2017b] Z. Komargodski and D. Simmons-Duffin, The Random-Bond Ising Model in 2.01 and 3 Dimensions, J. Phys. A 50, 154001 (2017b), arXiv:1603.04444 [hep-th] .
  * Cosme  _et al._ [2015] C. Cosme, J. V. P. Lopes, and J. Penedones, Conformal symmetry of the critical 3d ising model inside a sphere, Journal of High Energy Physics 2015, 1 (2015).
  * Cardy [1984] J. L. Cardy, Conformal invariance and surface critical behavior, Nuclear Physics B 240, 514 (1984).
  * Cardy [1996] J. Cardy,  _Scaling and renormalization in statistical physics_ , Vol. 5 (Cambridge university press, 1996).
  * Diehl and Shpot [1998] H. Diehl and M. Shpot, Massive field-theory approach to surface critical behavior in three-dimensional systems, Nuclear Physics B 528, 595 (1998).
  * Deng  _et al._ [2005] Y. Deng, H. W. Blöte, and M. Nightingale, Surface and bulk transitions in three-dimensional o (n) models, Physical Review E 72, 016128 (2005).
  * Hasenbusch [2011] M. Hasenbusch, Thermodynamic casimir force: A monte carlo study of the crossover between the ordinary and the normal surface universality class, Physical Review B 83, 134425 (2011).
  * Gliozzi  _et al._ [2015] F. Gliozzi, P. Liendo, M. Meineri, and A. Rago, Boundary and interface cfts from the conformal bootstrap, Journal of High Energy Physics 2015, 36 (2015).
  * Kos  _et al._ [2016b] F. Kos, D. Poland, D. Simmons-Duffin, and A. Vichi, Precision islands in the ising and o (n) models, Journal of High Energy Physics 2016, 1 (2016b).
  * Simmons-Duffin [2017] D. Simmons-Duffin, The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT, JHEP 03, 086, arXiv:1612.08471 [hep-th] .
  * Kos  _et al._ [2014b] F. Kos, D. Poland, and D. Simmons-Duffin, Bootstrapping mixed correlators in the 3d ising model, Journal of High Energy Physics 2014, 109 (2014b).
  * Mohamed  _et al._ [2018] M. S. Mohamed, A. N. Hirani, and R. Samtaney, Numerical convergence of discrete exterior calculus on arbitrary surface meshes, International Journal for Computational Methods in Engineering Science and Mechanics 19, 194 (2018).
  * Gasbarro [2018] A. D. Gasbarro,  _Studies of Conformal Behaviorin Strongly Interacting Quantum Field Theories_ , Ph.D. thesis, Yale University (2018).
  * Guerrieri  _et al._ [2016] A. L. Guerrieri, A. C. Petkou, and C. Wen, The free $\sigma$ cfts, Journal of High Energy Physics 2016, 1 (2016).