Benchmarking the Ising Universality Class in3d 4dimensions Claudio BONANNO1a Andrea CAPPELLI1b

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Benchmarking the Ising Universality Class

in 3≤d < 4dimensions

Claudio BONANNO 1,a, Andrea CAPPELLI 1,b,

Mikhail KOMPANIETS 2,3,c, Satoshi OKUDA 4,d, Kay Jörg WIESE5,e

1INFN, Sezione di Firenze, Via G. Sansone 1, 50019 Sesto Fiorentino (FI), Italy

2Saint Petersburg State University, 7/9 Universitetskaya Embankment, St. Petersburg,

199034, Russia

3Bogoliubov Laboratory of Theoretical Physics, JINR, 6 Joliot-Curie, Dubna, 141980,

Russia

4Department of Physics, Rikkyo University Toshima, Tokyo 171-8501, Japan

5Laboratoire de Physique de l’Ećole Normale Supérieure, Université PSL, CNRS, Sorbonne

Université, Université Paris-Diderot, Sorbonne Paris Cité, 24 rue Lhomond, 75005 Paris,

France

Abstract

The Ising critical exponents η,νand ωare determined up to one-per-thousand

relative error in the whole range of dimensions 3≤d < 4, using numerical conformal-

bootstrap techniques. A detailed comparison is made with results by the resummed

epsilon expansion in varying dimension, the analytic bootstrap, Monte Carlo and non-

perturbative renormalization-group methods, ﬁnding very good overall agreement. Pre-

cise conformal ﬁeld theory data of scaling dimensions and structure constants are ob-

tained as functions of dimension, improving on earlier ﬁndings, and providing bench-

marks in 3≤d < 4.

aclaudio.bonanno@fi.infn.it

bandrea.cappelli@fi.infn.it

cm.kompaniets@spbu.ru

dokudas@rikkyo.ac.jp

ewiese@lpt.ens.fr

arXiv:2210.03051v3 [hep-th] 3 Apr 2023

1 Introduction

Many approaches to critical phenomena obtain results in continuous space dimension,

although physically relevant dimensions are integer. Most notable is the perturbative renor-

malization group in d= 4−dimensions [1–4]. This is not merely a technical issue: quantities

as functions of real dcan clarify features that are harder to see at discrete values. E.g., one

can follow the topology of the renormalization-group (RG) ﬂow as a function of dimension

and ﬁnd instances where the universality class changes at non-integer values. This proved

particularly useful for systems with long-range interactions [5–7] or disorder [8–13].

The recent very precise numerical conformal bootstrap [14–16] has been formulated in

continuous dimension [17, 18], in particular for the Ising model in its whole range 4> d ≥

2[19–21]. The interest lies in understanding how the strongly interacting Ising conformal

ﬁeld theory connects to a free scalar in d= 4 and to the integrable fully-solvable model

in d= 2 [22, 23]. Analytic bootstrap approaches which use the dimension as a tunable

parameter were also developed [24–32]. Initially, the non-unitarity of the theory in non-

integer dimensions [33] was thought to hamper the numerical methods involving positive

quantities. These concerns have been overcome by de facto never observing problems for the

quantities of interest, as explained later.

In this paper, we extend the numerical approach of Ref. [20] using a single correlator,

the SDPB [34] routine for determining the unitarity domain, and the Extremal Functional

Method [35, 36] for solving the bootstrap equations. We obtain improved results for the

scaling dimensions in 4> d ≥3by a denser scanning of the unitary region near the Ising

point, i.e., the kink. The latter gets parametrically sharper as dapproaches 4, allowing for

its better identiﬁcation. The conformal spectrum in dimensions 4> d ≥2.6has also been

obtained in Ref. [21] via the advanced navigator bootstrap technique [37]. We use these very

precise results in combination with ours to obtain a consistent description of the low-lying

spectrum.

The achieved precision allows us to perform a detailed comparison with state-of-the-art

epsilon expansion in two regimes: for dclose to 4, the series is directly compared to bootstrap

data, using the necessary ﬁner scale for the latter; for intermediate values between 4and

3(included), the divergent perturbative series is resummed using well-established methods

involving the Borel transform [38–41].

The analysis is done on the dimensions of the conformal ﬁelds σ, , 0, corresponding to

spin, energy and subleading energy. They determine the critical exponents η, ν, ω. The

precision of our bootstrap data is summarized by the (mostly) d-independent value of the

relative error Err(γ)/γ =O(10−3)for the anomalous dimensions γof the conformal ﬁelds σ

and . As the anomalous dimensions are very small for d≈4, the precision for the conformal

dimensions ∆σ,∆is even higher in this region. Regarding the subleading energy, the relative

error Err(∆0)/∆0stays at three digits, as explained later. Some of the structure constants

are determined with a higher O(10−4)accuracy.

We compare our data with recent results of the analytic bootstrap [27–32], Monte Carlo

simulations [42–44] and the non-perturbative RG [45, 46]. We ﬁnd that the data by all

methods agree very well. This is rather rewarding given the achieved precision. Besides

conﬁrming the high quality of conformal-bootstrap results, our analysis provides a reference

point for further analytic and numerical methods aiming at exploring critical phenomena in

varying dimensions.

The outline of this paper is the following. In Sec. 2 we summarize our bootstrap proto-

col [20] and present the results for the three main conformal dimensions mentioned above,

together with their polynomial ﬁts as a function of dimension and the estimation of errors. In

Sec. 3 we brieﬂy recall the properties of the epsilon expansion and resummation techniques.

We then compare its predictions with our bootstrap data and the results by other methods,

and authors. A detailed analysis of all issues is presented. In Sec. 4, we report the numerical

bootstrap data for scaling dimensions of structure constants and other conformal ﬁelds, and

compare them to the existing epsilon expansion. In the conclusions in Sec. 5 we discuss open

questions.

2 Conformal bootstrap in non-integer dimension

The aim of this section is to summarize our procedure for deriving conformal data of scaling

dimensions and structure constants, as a function of the space-time dimension 4> d ≥2.

We ﬁrst discuss the conformal dimensions of three main ﬁelds O=σ, , 0. Our goal is to

provide a polynomial description of ∆Oas a function of y= 4 −d, by performing a best ﬁt of

the data obtained at several values of d1. Our results are ﬁnally compared to those obtained

from the resummed epsilon expansion in Section 3.

2.1 Summary of numerical methods

The conformal dimensions and structure constants of the critical Ising model as a func-

tion of dare computed in the setup of Ref. [20], which we shortly summarize for the reader’s

convenience. We consider a single 4-point correlator hσ(x1)σ(x2)σ(x3)σ(x4)i, where σ(x)is

the primary scalar ﬁeld with lowest dimension, denoted ∆σ. We truncate the functional

bootstrap equation to 190 components2. The unitarity condition for this equation is deter-

mined through the SDPB algorithm [34], leading to a bound in the (∆σ,∆)plane; next, the

Extremal Functional Method (EFM) [35,36] is used to solve the equations on this boundary.

We use the generalization of these numerical methods to non-integer dimensions developed

in Ref. [20], and detailed in its Appendix A.

Our 1-correlator numerical bootstrap approach has been surpassed by more recent imple-

mentations [16,19,21,47,48], but we ﬁnd it convenient for determining the low-lying spectrum

with modest computing resources. The complete determination of the conformal data for

one value of drequires about 20 hours on 256 cores, corresponding to 5000 core hours. This

simple setting allows us to evaluate the spectrum for various dimensions d.

1Note that is the energy ﬁeld, the next-to-lowest scalar primary ﬁeld, not to be confused with the

deviation from four dimensions denoted by y.

2This corresponds to the standard bootstrap parameter Λ = 18, which counts the number of derivatives

in the approximation of the functional basis.

The ﬁrst crucial step is to locate the Ising critical point in parameter space. To this end,

we adopt the twofold strategy of Ref. [20], consisting in searching the kink on the unitarity

boundary in the (∆σ,∆)plane and, at the same time, minimizing the central charge c[15].

This procedure allow us to determine for each value of dan interval of values for ∆σ,∆and

c, that we take as the Ising conformal theory, accompanied by an estimate of the uncertainty.

This procedure is displayed in Fig. 1, where we show the identiﬁcation of the Ising point

for d= 3,3.25,3.5and 3.75. The gray area in the plots indicates the chosen errors for ∆σ,∆

and c, which are roughly determined by the mismatch between the positions of the minimum

and the kink. As a conservative choice, we consider an interval of four data points for each

value of d.

The precision is greater than in Ref. [20], because we perform a ﬁner scan of the ∆σvalues

around the kink. We observe that the kink and the minimum get sharper for d→4, as

shown by the four pairs of plots drawn on the same scale in Fig. 1; this is convenient in our

approach, since it leads to an increased precision when anomalous dimensions are smaller.

In Fig. 2, we show the point d= 3.875, not considered in the earlier work. It is necessary for

studying the region of d→4. Here the curves are so steep that magniﬁed scales are needed.

Once the Ising point is determined, we obtain the rest of the conformal data as follows.

The solution of the bootstrap equations gives a spectrum of conformal dimensions ∆Oand

structure constants fσσOas a function of ∆σ; they are divided into diﬀerent sets characterized

by the spin `= 0,2,4, . . . of the operator O. The estimation of ∆Oand fσσOis obtained by

taking the central value of such quantities for ∆σvarying in the interval previously identiﬁed

as the Ising point (grey areas in Figs. 1 and 2). The error is obtained from their dispersion.

It is interesting to point out that, although we largely improved the precision of our results

for 4>d>3with respect to Ref. [20], we observe no signs of trouble associated to non-

unitarity in our bootstrap spectrum. On general grounds, non-unitarity contributions are

expected to appear for non-integer values of ddue to the presence of negative-norm states [33].

However, these occur at very high order in the OPE expansion of the correlator hσσσσi, thus

we may argue that they have numerically negligible structure constants. As a matter of fact,

their presence does not seem to yield problems in solving the bootstrap equations with our

method. This conclusion was also reached by recent 3-correlator bootstrap studies of the

critical O(N)models [18] and the Ising model [21] in non-integer space dimensions using the

navigator method [37].

Figure 1: Determination of the Ising critical point for d= 3,3.25,3.5,3.75 (d= 3 data from

Ref. [20]). Left plots: Identiﬁcation of the kink; the blue points correspond to the solutions of the

bootstrap equations. Right plots: position of the cminimum. The grey shaded areas represent the

estimated errors on ∆σ,∆and c.

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BenchmarkingtheIsingUniversalityClassin3d<4dimensionsClaudioBONANNO1;a,AndreaCAPPELLI1;b,MikhailKOMPANIETS2;3;c,SatoshiOKUDA4;d,KayJörgWIESE5;e1INFN,SezionediFirenze,ViaG.Sansone1,50019SestoFiorentino(FI),Italy2SaintPetersburgStateUniversity,7/9UniversitetskayaEmbankment,St.Petersburg,199034,Russia...

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Benchmarking the Ising Universality Class in3d 4dimensions Claudio BONANNO1a Andrea CAPPELLI1b

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