Precision islands in the Ising and O(N ) models

Abstract

Abstract

We make precise determinations of the leading scaling dimensions and operator product expansion (OPE) coefficients in the 3d Ising, O(2), and O(3) models from the conformal bootstrap with mixed correlators. We improve on previous studies by scanning over possible relative values of the leading OPE coefficients, which incorporates the physical information that there is only a single operator at a given scaling dimension. The scaling dimensions and OPE coefficients obtained for the 3d Ising model, ([Delta] _[sigma] , [Delta] , [lambda]_σσ , [lambda]) = (0.5181489(10), 1.412625(10), 1.0518537(41), 1.532435(19) , give the most precise determinations of these quantities to date.

Published for SISSA by Springer

Received: June 6, 2016 Accepted: July 6, 2016 Published: August 4, 2016

Precision islands in the Ising and O(N) models

Filip Kos,a David Poland,a,b David Simmons-Du nb and Alessandro Vichic

aDepartment of Physics, Yale University,

New Haven, CT 06520, U.S.A.

bSchool of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey 08540, U.S.A.

cTheory Division, CERN,

Geneva, Switzerland

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , [email protected]

Abstract: We make precise determinations of the leading scaling dimensions and operator product expansion (OPE) coe cients in the 3d Ising, O(2), and O(3) models from the conformal bootstrap with mixed correlators. We improve on previous studies by scanning over possible relative values of the leading OPE coe cients, which incorporates the physical information that there is only a single operator at a given scaling dimension. The scaling dimensions and OPE coe cients obtained for the 3d Ising model, ( , [epsilon1], [epsilon1], [epsilon1][epsilon1][epsilon1]) = 0.5181489(10), 1.412625(10), 1.0518537(41), 1.532435(19)

, give the

JHEP08(2016)036

most precise determinations of these quantities to date.

Keywords: Conformal and W Symmetry, Nonperturbative E ects, Global Symmetries

ArXiv ePrint: 1603.04436

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP08(2016)036

Web End =10.1007/JHEP08(2016)036

Contents

1 Introduction 1

2 Bootstrap constraints 32.1 Ising model 32.2 O(N) models 5

3 Results 6

4 Conclusions 10

A Implementation details 11

1 Introduction

The conformal bootstrap [1, 2] in d > 2 has recently seen an explosion of exciting and nontrivial results, opening the door to the possibility of a precise numerical classi cation of non-perturbative conformal eld theories (CFTs) with a small number of relevant operators. Such a classi cation would lead to a revolution in our understanding of quantum eld theory, with direct relevance to critical phenomena in statistical and condensed matter systems, proposals for physics beyond the standard model, and quantum gravity.

One of the most striking successes has been in its application to the 3d Ising model, initiated in [3, 4]. In [5] we found that the conformal bootstrap applied to a system of correlators [notdef][angbracketleft][angbracketright], [angbracketleft][epsilon1][epsilon1][angbracketright], [angbracketleft][epsilon1][epsilon1][epsilon1][epsilon1][angbracketright][notdef] containing the leading Z2-odd scalar and leading Z2-

even scalar [epsilon1] led to a small isolated allowed region for the scaling dimensions ( , [epsilon1]).

In [6] this approach was pushed further using the semide nite program solver SDPB, leading to extremely precise determinations of the scaling dimensions and associated critical exponents.1

In [9] we found that this approach could also be extended to obtain rigorous isolated regions for the whole sequence of 3d O(N) vector models, building on the earlier results of [10, 11]. While the resulting \O(N) archipelago" is not yet as precise as in the case of the Ising model, it serves as a concrete example of how the bootstrap can lead to a numerical classi cation | if we can isolate every CFT in this manner and make the islands su ciently small, then we have a precise and predictive framework for understanding the space of nonperturbative conformal xed points. If the methods can be made more e cient, it is clear that this approach may lead to solutions of longstanding problems such as determining the conformal windows of 3d QED and 4d QCD.

1A complementary approach to solving the 3d Ising model with the conformal bootstrap was also developed in [7, 8].

{ 1 {

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Ising: Scaling Dimensions

Monte Carlo

1.41265

1.41264

1.41263

1.41262

1.41261

1.41260 0.518146 0.518148 0.518150 0.518152

Bootstrap

0.51808 0.51810 0.51812 0.51814 0.51816 0.51818

Figure 1. Determination of the leading scaling dimensions in the 3d Ising model from the mixed correlator bootstrap after scanning over the ratio of OPE coe cients [epsilon1][epsilon1][epsilon1]/ [epsilon1] and projecting to the ( , [epsilon1]) plane (blue region). Here we assume that and [epsilon1] are the only relevant Z2-odd and Z2-even scalars, respectively. In this plot we compare to the previous best Monte Carlo determinations [17] (dashed rectangle). This region is computed at = 43.

Compared to previous mixed-correlator studies [5, 6, 9] (see also [12{14]), the novelty of the present work is the idea of disallowing degeneracies in the CFT spectrum by making exclusion plots in the space of OPE coe cients and dimensions simultaneously. For example, in the 3d Ising model, by scanning over possible values of the ratio [epsilon1][epsilon1][epsilon1]/ [epsilon1], we can impose that there is a unique [epsilon1] operator. This leads to a three-dimensional island in ( , [epsilon1], [epsilon1][epsilon1][epsilon1]/ [epsilon1]) space whose projection to the ( , [epsilon1]) plane is much smaller than the island obtained without doing the scan. For each point in this island, we also bound the OPE coe cient magnitude [epsilon1]. The result is a new determination of the leading scaling dimensions ( , [epsilon1]) = 0.5181489(10), 1.412625(10)

, shown in gure 1, as well as precise determinations of the leading OPE coe cients ( [epsilon1], [epsilon1][epsilon1][epsilon1]) =

1.0518537(41), 1.532435(19)

. These scaling dimensions translate to the critical exponents ( , ) = 0.0362978(20), 0.629971(4)

.

We repeat this procedure for 3d CFTs with O(2) and O(3) global symmetry, focusing on the bootstrap constraints from the correlators [notdef][angbracketleft][angbracketright], [angbracketleft]ss[angbracketright], [angbracketleft]ssss[angbracketright][notdef] containing

the leading vector i and singlet s. We again nd that scanning over the ratio of OPE coe cients sss/ s leads to a reduction in the size of the islands corresponding to the

O(2) and O(3) vector models. The results are summarized in gure 2. In studying the O(2) model, we are partially motivated by the present 8 discrepancy between measure

ments of the heat-capacity critical exponent in 4He performed aboard the space shuttle

{ 2 {

1.4130

1.4129

1.4128

1.4127

1.4126

1.4125

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O

(N

): Scaling Dimensions

1.65s

1.60

1.55

O

(2

)

(3

)

O

1.50

1.45

1.40

1.35

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Ising

0.516 0.518 0.520 0.522 0.524

Figure 2. Allowed islands from the mixed correlator bootstrap for the O(2) and O(3) models after scanning over the ratio of OPE coe cients sss/ s and projecting to the ( , s) plane (blue regions). Here we assume that and s are the only relevant scalar operators in their O(N) representations. These islands are computed at = 35. The Ising island is marked with a cross because it is too small to see on the plot.

STS-52 [15] and the precise analysis of Monte Carlo simulations and high-temperature expansions performed in [16]. While our new O(2) island is not quite small enough to resolve this issue de nitively, our results have some tension with the reported 4He measurement and currently favor the combined Monte Carlo and high-temperature expansion determination.

This paper is organized as follows. In section 2 we review the bootstrap equations relevant for the 3d Ising and O(N) vector models and explain the scan over relative OPE coe cients employed in this work. In section 3 we describe our results, and in section 4 we give a brief discussion. Details of our numerical implementation are given in appendix A.

2 Bootstrap constraints

2.1 Ising model

We will be studying the conformal bootstrap constraints for 3d CFTs with either a Z2 or

O(N) global symmetry. In the case of a Z2 symmetry, relevant for the 3d Ising model, we consider all 4-point functions containing the leading Z2-odd scalar and leading Z2-even scalar [epsilon1]. The resulting system of bootstrap equations for [notdef][angbracketleft][angbracketright], [angbracketleft][epsilon1][epsilon1][angbracketright], [angbracketleft][epsilon1][epsilon1][epsilon1][epsilon1][angbracketright][notdef]

was presented in detail in [5]. Here we summarize the results. The crossing symmetry

{ 3 {

conditions for these correlators can be expressed as a set of 5 sum rules:

0 =

XO +

O [epsilon1][epsilon1]O

[vector]V+, ,[lscript] O

[epsilon1][epsilon1]O

!

+

XO

2[epsilon1]O[vector]V, ,[lscript] , (2.1)

where [vector]V, ,[lscript] is a 5-vector and [vector]V+, ,[lscript] is a 5-vector of 2 [notdef] 2 matrices. The detailed form of

[vector]V, describing the contributions of parity even or odd operators O in terms of conformal

blocks, is given in [5].

In [5, 6] we numerically computed the allowed region for ( , [epsilon1]) by assuming that and [epsilon1] are the only relevant dimensions at which scalar operators appear and searching for a functional [vector]

satisfying the conditions

1 1

[vector]

[notdef]

[vector]V+,0,0 11!

> 0 , for the identity operator ,

[vector] [vector]V+, ,[lscript] [followsequal] 0 , for Z2-even operators with even spin , [vector]

[notdef]

[vector]V, ,[lscript] 0 , for Z2-odd operators in the spectrum . (2.2)

If such a functional can be found, then the assumed values of ( , [epsilon1]) are incompatible with unitarity or re ection positivity. In [5, 6] we found that this leads to an isolated allowed island in operator dimension space compatible with known values in the 3d Ising model, with a size dependent on the size of the search space for the functional. One can additionally incorporate the constraint [epsilon1] = [epsilon1] by only requiring positivity for the combination

[notdef]

[vector] [vector]V+, [epsilon1],0 + [vector]V, ,0 1 0 0 0

! [followsequal]

0 , (2.3)

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reducing the size of the island somewhat further.

However, as noted in [5], the condition (2.3) is still stronger than necessary. In particular it allows for solutions of crossing containing terms of the form

Xi

i [epsilon1][epsilon1]i

[vector]V+, [epsilon1],0 + [vector]V, ,0 1 0 0 0

!

i

[epsilon1][epsilon1]i

!, (2.4)

represent an arbitrary number of (not necessarily aligned) two-component vectors. If instead we assume that and [epsilon1] are isolated and that there are no other contributions at their scaling dimensions, then we can replace (2.3) with the weaker condition

cos sin

[vector] [notdef]

[vector]V+, [epsilon1],0 + [vector]V, ,0

where i [epsilon1][epsilon1]i

1 0 0 0

! cos

sin

!

0 , (2.5)

for some unknown angle tan1( [epsilon1][epsilon1][epsilon1]/ [epsilon1]). By scanning over the possible values of

and taking the union of the resulting allowed regions (an idea rst explored in [18]), we can e ectively allow our functional to depend on this unknown ratio and arrive at a smaller allowed region, forbidding solutions to crossing of the uninteresting form (2.4).

{ 4 {

In addition, for any given allowed point in the ( , [epsilon1], ) space, we can compute a lower and upper bound on the norm [epsilon1]

p 2[epsilon1] + 2[epsilon1][epsilon1][epsilon1] of the OPE coe cient vector. This is obtained by substituting the conditions (2.2) with the optimization problem:

Maximize 1 1

[vector]

[notdef] [vector]V+,0,0

1 1 subject to

N =

cos sin

[vector] [notdef]

[vector]V+, [epsilon1],0 + [vector]V, ,0

1 0 0 0

! cos

sin

!,

[vector] [vector]V+, ,[lscript] [followsequal] 0 , for Z2-even operators with even spin , [vector]

[notdef]

[vector]V, ,[lscript] 0 , for Z2-odd operators in the spectrum . (2.6)

By choosing N = [notdef]1 we can obtain the sought upper and lower bounds:

N 2[epsilon1] 1 1

[vector]

[notdef]

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[vector]V+,0,0

1 1

. (2.7)

2.2 O(N) models

Similarly, when there is an O(N) symmetry, we can consider all 4-point functions containing the leading O(N) vector i and leading O(N) singlet s. The resulting system of bootstrap equations for [notdef][angbracketleft][angbracketright], [angbracketleft]ss[angbracketright], [angbracketleft]ssss[angbracketright][notdef] was studied in [9], leading to a set of 7 sum rules of

the form

0 =

XOS,[lscript] +

OS ssOS

[vector]VS, ,[lscript] OS

ssOS

!

+

XOT ,[lscript]+ 2OT [vector]VT, ,[lscript]

XOV ,[lscript][notdef]

2sOV

[vector]VV, ,[lscript] , (2.8)

+

XOA,[lscript]

2OA

[vector]VA, ,[lscript] +

where [vector]VT , [vector]VA, [vector]VV are 7-dimensional vectors corresponding to di erent choices of correlators and tensor structures and [vector]VS is a 7-vector of 2 [notdef] 2 matrices. The functions

[vector]VS, [vector]VT , [vector]VA, [vector]VV

describe the contributions from singlets OS, symmetric tensors OT , anti-symmetric tensors OA, and vectors OV , and are de ned in detail in [9].

To rule out an assumption on the spectrum, we will look for a functional satisfying the generic conditions

1 1

[vector]

[notdef]

[vector]VS,0,0 11!

0 , for the identity operator ,

[vector] [vector]VT, ,[lscript] 0 , for traceless symetric tensors with [lscript] even , [vector]

[notdef]

[vector]VA, ,[lscript] 0 , for antisymmetric tensors with [lscript] odd , [vector]

[notdef]

[vector]VV, ,[lscript] 0 , for O(N) vectors with any [lscript] , [vector]

[notdef]

[vector]VS, ,[lscript] [followsequal] 0 , for singlets with [lscript] even , (2.9)

where we take these constraints to hold for scalar singlets and vectors with 3, sym

metric tensors with 1, and all operators with spin satisfying the unitarity bound

{ 5 {

[lscript] + 1. Similar to the previous section, we will additionally allow for the contributions

of the isolated operators i and s by imposing the condition

cos N sin N

[vector] [notdef]

[vector]VS, s,0 + [vector]VV, ,0

1 0 0 0

! cos

N sin N

!

0 (2.10)

and scanning over the unknown angle N tan1( sss/ ss).

Similar to the previous section, for any allowed point in ( , , ) space, we can compute a lower and upper bound on the norm s

q 2s + 2sss. This is obtained by

substituting the conditions (2.9) with:

Maximize

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1 1

[vector]

[notdef] [vector]VS,0,0

1 1 subject to

N =

cos N sin N

[vector] [notdef]

[vector]VS, s,0 + [vector]V, s,0

1 0 0 0

! cos

N sin N

!,

[vector] [vector]VT, ,[lscript] 0 , for traceless symetric tensors with [lscript] even , [vector]

[notdef]

[vector]VA, ,[lscript] 0 , for antisymmetric tensors with [lscript] odd , [vector]

[notdef]

[vector]VV, ,[lscript] 0 , for O(N) vectors with any [lscript] , [vector]

[notdef]

[vector]VS, ,[lscript] [followsequal] 0 , for singlets with [lscript] even . (2.11)

3 Results

As shown in gures 1 and 3,2 we have used this procedure to determine the scaling dimensions and OPE coe cient ratio in the 3d Ising model to high precision at = 43,3 giving

= 0.5181489(10) , (3.1) [epsilon1] = 1.412625(10) , (3.2) [epsilon1][epsilon1][epsilon1]/ [epsilon1] = 1.456889(50) . (3.3)

We have also computed bounds on the magnitude of the leading OPE coe cients [epsilon1] at

= 27 over this allowed region, with the result shown in gure 4. These determinations yield the values

[epsilon1] = 1.0518537(41) , (3.4) [epsilon1][epsilon1][epsilon1] = 1.532435(19) . (3.5)

2In the plots in this work we show smooth curves that have been t to the computed points. The precise shape of the boundary is subject to an error which is at least an order of magnitude smaller than the quoted error bars.

3The functional [vector]

we search for is given as a linear combination of derivatives. The parameter limits the highest order derivative that can appear in the functional [vector]

. See [9] for the exact de nition of the

parameter .

{ 6 {

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Figure 3. Determination of the leading scaling dimensions ( , [epsilon1]) and the OPE coe cient ratio [epsilon1][epsilon1][epsilon1]/ [epsilon1] in the 3d Ising model from the mixed correlator bootstrap (blue region). This region is computed at = 43.

1.53248

1.53246

1.53244

1.53242

1.5324

1.53238 1.051842 1.051848 1.051854 1.05186 1.051866

Figure 4. Determination of the leading OPE coe cients in the 3d Ising model from the conformal bootstrap (blue region). This region was obtained by computing upper and lower bounds on the OPE coe cient magnitude at = 27, for points in the allowed region of gure 3.

{ 7 {

Ising: OPE Coefficients

O

(2

): Scaling Dimensions

1.520s

1.518

1.516

1.514

1.512

1.510

1.508

1.506 0.5185 0.5190 0.5195 0.5200 0.5205 0.5210

Figure 5. Allowed islands from the mixed correlator bootstrap for N = 2 after scanning over the OPE coe cient ratio sss/ s and projecting to the ( , s) plane (blue regions). Here we assumed that and s are the only relevant operators in their O(N) representations. These islands are computed at = 19, 27, 35. The green rectangle shows the Monte Carlo plus high-temperature expansion determination (MC+HT) from [16], while the horizontal lines show the 1 (solid) and 3 (dashed) con dence intervals from experiment [15].

Our determination of [epsilon1][epsilon1][epsilon1] is consistent with the estimate 1.45 [notdef] 0.3 obtained via Monte

Carlo methods in [21].4 An application of [epsilon1][epsilon1][epsilon1] is in calculating the properties of the 3d Ising model in the presence of quenched disorder in the interaction strength of neighboring spins [23].

In gure 2 we show similar islands for the leading vector and singlet operators in the O(2) and O(3) models, all computed at = 35. We show the zoom in of these regions as well as the regions at = 19, 27 in gures 5 and 6. Once the angle N has been computed at = 35, we determine the OPE coe cients ( s, sss) by bounding the magnitude s at = 27. The nal error in the OPE coe cients comes mostly from the angle, which is why we use a lower value of for the magnitude.

For the O(2) model, the resulting dimensions and OPE coe cients are

= 0.51926(32) , (3.6) s = 1.5117(25) , (3.7) sss/ s = 1.205(9) , (3.8)

s = 0.68726(65) , (3.9) sss = 0.8286(60) . (3.10)

4We disagree slightly with the determination in [22].

{ 8 {

19 =

=

27 =

35

4He 1

+HT

4He 3

MC

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O

(3

): Scaling Dimensions

1.610s

1.605

1.600

1.595

1.590

1.585

1.580 0.5180 0.5185 0.5190 0.5195 0.5200 0.5205 0.5210

Figure 6. Allowed islands from the mixed correlator bootstrap for N = 3 after scanning over the OPE coe cient ratio sss/ s and projecting to the ( , s) plane (blue regions). Here we assumed that and s are the only relevant operators in their O(N) representations. These islands are computed at = 19, 27, 35. The green rectangle shows the best previous determinations (MC+HT) from the Monte Carlo plus high-temperature expansion study in [19] and the more recent Monte Carlo simulations in [20].

A similar computation for the O(3) model gives

= 0.51928(62) , (3.11) s = 1.5957(55) , (3.12) sss/ s = 0.953(25) , (3.13)

s = 0.5244(11) , (3.14) sss = 0.499(12) . (3.15)

In the O(2) plot we compare to both the Monte Carlo plus high-temperature expansion determinations of [16] and the re-analysis of the experimental 4He data of [15], currently in

8 tension. Our result is easily compatible with [16] while it has started to exclude the

lower part of the 3 allowed region reported in [15]. Based on a na ve extrapolation to a higher derivative cuto , it seems plausible that the bootstrap result will eventually fully exclude the reported result of [15]. If this occurs, we would attribute the discrepancy to the fact that the t performed in [15] has a sizable sensitivity to which subleading contributions to the heat capacity are included, as can be seen in table II of [15]. It is therefore plausible to us that the experimental uncertainty in the extraction of the critical exponent should be larger than the reported error bars.

{ 9 {

JHEP08(2016)036

=

19 =

27 =

35

MC

+HT

Finally, we would like to emphasize that in all of our determinations, there is no additional error from truncations of the spectrum. For contributions at each spin, the formulation in terms of semide nite programming imposes positivity on operators of arbitrarily large dimension. As described in [9], the set of included spins is truncated, but in all cases we have chosen a su cient number of spins such that the functional exhibits an asymptotic behavior at large spin satisfying the positivity conditions described in section 2.

4 Conclusions

In this work we imposed the uniqueness of the relevant singlet operator appearing in the conformal block decomposition of [angbracketleft][angbracketright], [angbracketleft]ss[angbracketright], and [angbracketleft]ssss[angbracketright] in the Ising and O(N) mod

els.5 The absence of degeneracies is a natural restriction to impose on the CFT spectrum. It requires a modi ed numerical approach because the standard mixed correlator analysis used in previous works [5, 6, 9, 12{14] secretly allows for more general solutions of crossing symmetry that violate this assumption.

We implement this new constraint by scanning over the ratio of OPE coe cients sss/ s. By forbidding uninteresting solutions of crossing we further restricted the allowed region in the ( , s) plane. This results in a new precise determination of Ising critical exponents ( , ) = 0.0362978(20), 0.629971(4)

, almost two orders of magnitude better than the best Monte Carlo estimate [17]. We also improved on our previous determinations for O(2) and O(3), yielding exponents ( , )O(2) = 0.03852(64), 0.6719(11)

, although Monte Carlo results remain more precise in these cases. (The bootstrap however allows much more precise determinations of OPE coe cients.) Nevertheless, for O(2), we saw indications that the conformal bootstrap disfavors the commonly-quoted exponent extracted from experimental 4He data in the analysis of [15].

For the sake of completeness we also report qualitative results of attempts to reduce the size of the allowed regions by imposing additional assumptions. One natural ingredient not exploited so far is the constraint that the energy momentum tensor appears with the same central charge in all correlators. Enforcing this also requires imposing a gap between T = 3 and the dimension of the next spin two operator, T[prime] = 3 + . The net e ect is a non-negligible shrinking of the size of the O(2) island, but unfortunately it only carves out the upper right region of the island, leaving the rest essentially untouched. The e ect is also independent of the value of the gap as long as 0.2 1. Finally, we found

that the lower left endpoint of the O(2) island is controlled by the gap between s and the dimension of the next singlet scalar s[prime] ; however only when we assume s[prime] > 3.7 do we start changing the size of the O(2) island. This is not surprising since the expected value from Monte Carlo is s[prime] = 3.785(20) [16]. In order to keep the discussion general we decided not to push further in this direction.

As a byproduct of our analysis, we also obtained precise determinations of the OPE coe cients ( s, sss). While the latter is here computed for the rst time, the former

5To unify the discussion we use the O(N) notation to denote operator dimension and OPE coe cients, with the obvious dictionary to translate to the Ising model: ! , s ! [epsilon1].

( , )O(3) = 0.0386(12), 0.7121(28)

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and

{ 10 {

was already estimated for the Ising model in [4], using again a bootstrap approach. There the value [notdef] s[notdef] = 1.05183(86) was extracted, which should be compared with the result

in (3.4). The two determinations are fully compatible, despite the methods used to obtain the two estimates being somewhat di erent, both in the theoretical and numerical approach to the conformal bootstrap. The present work uses mixed correlators, translates the crossing constraints into a semide nite programming problem, and rules out unfeasible points in the CFT parameter space. The work of [4], instead, used a linear programming algorithm to solve the crossing equations directly under the assumption that the 3d Ising model is the 3d CFT which locally minimizes the central charge. The agreement of these methods is a further triumph of the numerical bootstrap.

The new ingredient studied in this work represents a further step in the numerical development of the conformal bootstrap. It not only further reduces the size of the allowed parameter space, but it also provides rigorous information on OPE coe cients. Such information is for example very important for predicting o -critical correlators, as shown in the recent application of these results [24]. It will be interesting to investigate the e ect of scanning over relative OPE coe cients in other situations where the bootstrap seems to be successful, both in known theories such as N = 4 supersymmetric Yang-Mills

theory [25], the 6d (2,0) SCFTs [26], and the conformal window of QCD [27], as well as in studies of the mysterious features that have appeared in the 4d N = 1 [28, 29] and 3d

fermion [30] bootstrap that may signal the existence of new islands in the ocean of CFTs.

Acknowledgments

We thank Zohar Komargodski, Slava Rychkov, and Ettore Vicari for comments and discussions. The work of DSD is supported by DOE grant number DE-SC0009988 and a WilliamD. Loughlin Membership at the Institute for Advanced Study. The work of DP and FK is supported by NSF grant 1350180. DP is additionally supported by a Martin A. and Helen Chooljian Founders' Circle Membership at the Institute for Advanced Study. The computations in this paper were run on the Bulldog computing clusters supported by the facilities and sta of the Yale University Faculty of Arts and Sciences High Performance Computing Center, on the Hyperion computing cluster supported by the School of Natural Sciences Computing Sta at the Institute for Advanced Study, and on the CERN cluster.

A Implementation details

Using the techniques described in the main text, we can set up a semide nite program to determine whether a triple ( , [epsilon1], ) is allowed. (In this discussion, we focus on the Ising model for simplicity.) Our choices and parameters for solving the semide nite program are identical to those quoted in [9]. To actually determine ( , [epsilon1], ) in the Ising model, we must make a 3d exclusion plot at successively larger values of . We proceed as follows:

We rst choose a relatively small value = 0. (For us, 0 = 11.) Since we roughly

know the 2d projection of the 3d Island from previous work [6], we begin by choosing

{ 11 {

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some points ( , [epsilon1]) in the 2d island and performing a 1d scan over . If we're lucky,

this gives at least one point p0 in the 3d island.

By scanning over a 3d grid near p0, we determine the rough shape S 0 of the 3d

island.

The island shrinks in an approximately self-similar way as is increased. Once we

know the shape S 0, we nd an a ne transformation T 0 : S 0 ! [1, 1]3 such that

S =19 becomes approximately spherical, with large volume in [1, 1]3. T 0 gives a

useful set of coordinates for a neighborhood of S 0. These coordinates are much better than ( , [epsilon1], ), because S 0 is extremely elongated and at in ( , [epsilon1], )

space ( gure 3). It is helpful to choose T 0 so that the plane [epsilon1] = 0 is parallel to

two of the axes in [1, 1]3. This ensures that a grid-based scan over [1, 1]3 involves

only a small number of values of [epsilon1] , which means we must compute fewer tables

of conformal blocks. This is the 3d generalization of the trick mentioned in [5].

Now that we have a better reference frame for S 0, our job is easier. We increase

0 ! 1 and determine a point p1 2 S 1 using a rough scan. We then determine

the boundary of S 1 by performing a binary search in the radial direction away from p1, in the T 0 coordinates. For the angular directions, we choose the vertices and edge-midpoints of an icosahedron centered at p1, oriented so that [epsilon1] takes as

few values as possible during the search. To get a higher resolution picture of S 1, we can pick a few more points in the interior and perform radial binary searches away from those points as well. Once we know S 1 we choose a new T 1 : S 1 ! [1, 1]3.

We now iterate the previous step to increase 1 ! 2 ! 3 . . . . After a few itera

tions, we can predict the point the islands are shrinking towards, removing the need for a scan at each stage. We take 0 = 11, 1 = 19, 2 = 27, 3 = 35, 4 = 43.

As an example, in the 3d Ising model, the inverse map T 1 =27 is given by

0

B

@

[epsilon1]

JHEP08(2016)036

1

C

A

= T 1 =27

0

B

@

x

y

z

1

C

A

0

B

@

2.76988363 [notdef] 106 6.95457153 [notdef] 107 9.83371791 [notdef] 106

= 2.76988363 [notdef] 106 6.95457153 [notdef] 107 9.39012428 [notdef] 105 2.66723434 [notdef] 105 2.70007022 [notdef] 106 5.48817612 [notdef] 105

1

C

A

0

B

@

x

y

z

1

C

A

+

0

B

@

0.51814922 1.41261837 0.96924816

1

C

A

, (A.1)

where (x, y, z) 2 [1, 1]3. Note that [epsilon1] is a function of z alone, which is helpful for

reducing the number of tables of conformal blocks needed for scans. The images of the 3d islands [notdef]S =27, S =35, S =43[notdef] under T =27 are shown in gure 7.

{ 12 {

x

-0.5 0.0 0.5

Figure 7. Images of the 3d islands [notdef]S =27, S =35, S =43[notdef] under the map T =27, where T 1 =27 is

given in (A.1).

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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