PHYSICAL REVIEW D 101, 116020 (2020) Conformal bootstrap signatures of the tricritical Ising universality class Chethan N. Gowdigere , Jagannath Santara, and Sumedha School of Physical Sciences, National Institute of Science Education and Research, Jatni 752050, India and Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India (Received 28 November 2018; revised manuscript received 17 April 2020; accepted 12 June 2020; published 30 June 2020) We study the tricritical Ising universality class using conformal bootstrap techniques. By studying bootstrap constraints originating from multiple correlators on the conformal field theory (CFT) data of multiple operator product expansions (OPEs), we are able to determine the scaling dimension of the spin field Δσ in various noninteger dimensions 2 ≤ d ≤ 3. Here, Δσ is connected to the critical exponent η that governs the (tri)critical behavior of the two-point function via the relation η ¼ 2 − d þ 2Δσ. Our results for Δσ match with the exactly known values in two and three dimensions and are a conjecture for noninteger 3 dimensions. We also compare our CFT results for Δσ with ϵ-expansion results, available up to ϵ order. Our techniques can be naturally extended to study higher-order multicritical points. DOI: 10.1103/PhysRevD.101.116020 I. INTRODUCTION the restrictions imposed by conformal invariance and found that the conformal field theory corresponding to The consequences of the conformal hypothesis which the Ising universality class sits on the boundary of the posits conformal invariance to the behavior of physical allowed region at a kinklike point in the space of scaling systems at criticality, in addition to scale invariance, are dimensions of the only two relevant operators. most far reaching in two dimensions, where the conformal Furthermore, remarkably, they could extend this analysis symmetry is the infinite-dimensional Virasoro algebra. The to all noninteger dimensions between two and four, and seminal work of Belavin, Polyakov, and Zamoldchikov they showed that even here, the theory corresponding to the [1,2] resulted in the discovery of a whole class of two- Ising universality class is always located at a kinklike point dimensional conformal field theories (CFTs) viz. the [14]. Hitherto [15–20], only critical points have been Virasaro minimal models. Each of these models describes studied using conformal field theory methods. Other a universality class, and the exact knowledge of the scaling critical points (tricritical and multicritical) also have con- dimensions of the operators amounts to a derivation of formal symmetry, and here we use conformal field theory the critical exponents purely from conformal invariance. techniques to study tricritical points. In this paper, we look The Ising model, the tricritical Ising model, and the three- at the tricritical Ising point in two and higher dimensions and four-state Potts models in two dimensions were thus and show that just like the Ising critical point, the tricritical exactly solved 30 years ago. The infinitude of conformal Ising point can also be recognized by its special signatures symmetry made two dimensions rather special, and the use in the space of scaling dimensions of the appropriate of conformal field theories to study critical phenomena was operators. restricted to two dimensions alone. A tricritical point is a fixed point where three critical Recent times have seen a breakthrough in doing the same lines and a line of first-order transitions meet (or a point for three and other dimensions. Even though the conformal where three coexisting phases simultaneously become group is finite dimensional now, technical advances made critical) [21]. Quantum scalar field theory with a ϕ6 in the explicit computation of (global) conformal blocks interaction provides one realization of this universality (first in [3–5] and later in [6–8]) resulted in astounding class. Here, there is a line of Ising critical points which end progress (see [9] for a recent review) and has provided the in a higher-order critical point, namely, the tricritical point most precise values [10–13] for the critical exponents of which has a different set of critical exponents. These two the three-dimensional Ising model. Rychkov et al. analyzed critical behaviors have different upper critical dimensions: three for the tricritical Ising point and four for the Ising critical point [22]. Thus, field theory and renormalization Published by the American Physical Society under the terms of group based techniques such as the ϵ-expansion have had the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to limited success [23,24]. CFT techniques avoid the flow and the author(s) and the published article’s title, journal citation, only study the fixed point, which is where the extra scaling and DOI. Funded by SCOAP3. and conformal symmetries are present. Yet, so far, a CFT 2470-0010=2020=101(11)=116020(9) 116020-1 Published by the American Physical Society GOWDIGERE, SANTARA, and SUMEDHA PHYS. REV. D 101, 116020 (2020) study of the tricritical point has not had as much success as hϕðx1Þϕðx2Þϕðx3Þϕðx4Þi ∼ Gðu; vÞ: ð2Þ the critical point, in dimensions other than two. The paper is organized as follows. We start with a brief Using OPEs one can evaluate the four-point function in review of CFT in Sec. II. We recall the prime principles of terms of the CFT data CFT, the essential details of the conformal bootstrap X program, and some key results obtained so far in the Gðu; vÞ¼ λ λ G ðu; vÞ; ð Þ bootstrap program that we need for our analysis of the 12O 34O O 3 O tricritical CFT. In Sec. III, we present the bootstrap analysis that allows us to compute the tricritical exponent η for all where the summation is over the primary operators that noninteger dimensions between two and three. In the same occur in both the ϕ1 × ϕ2 and the ϕ3 × ϕ4 OPEs and section, we compare our results with the best available GOðu; vÞ is the (global) conformal block. This evaluation ϵ-expansion results for η. In Sec. IV, we present a further of the four-point function can be done in two different bootstrap analysis pertaining to multiple universality ways, and both ways agree that classes and multicriticality. We conclude the paper in Sec. V with a summary of results and future directions. X X λ12Oλ34OGOðu; vÞ¼ λ14O0 λ23O0 GO0 ðu; vÞð4Þ O O0 II. BRIEF REVIEW OF CFT We first provide a quick overview of the prime principles puts a constraint on the CFT data, which is referred to as the of CFT: CFT data, the operator product expansion (OPE), bootstrap equation or the crossing symmetry constraint. unitarity constraints, and crossing symmetry constraints. The conformal bootstrap program employs Eq. (4) (every A CFT is specified (partially) by a list of local primary four-point function provides one equation) to search for operators a.k.a. scaling operators (each operator is a CFTs by successively constraining the CFT data. This primary under the global conformal symmetry and is equation was shown to be tractable numerically starting specified by its scaling dimension and spin). The scaling with the seminal work [10] and leading up to [28–32], the dimension of a local operator in a unitary CFT is bounded main tools for the results of this paper. from below depending on its spin. A product of two such The first important step in the numerical conformal local operators is expandable in terms of all the local bootstrap program [10] is to focus on a certain region of the operators [25,26]; this is known as the operator product two-dimensional space of conformal cross ratios, now expansion, known as the spacelike diamond, in which the conformal blocks GOðu; vÞ turn out to be real and analytic and hence X have Taylor expansions. Equation (4) is a vector equation in ϕ ðx Þϕ ðx Þ¼ λ Cðx ; ∂ ÞOðx Þ; ð Þ 1 1 2 2 12O 12 x2 2 1 a real analytic function space. This function space is O coordinatized as follows [10], and this is inspired by the knowledge of how Eq. (4) is solved in the free scalar CFT. Cðx ; ∂ Þ where 12 x2 is fully determined by conformal invari- There is a certain point where the convergence of the ance and λ12O is referred to as an OPE coefficient, which is infinite sums in (4) is fastest (for the free scalar CFT): 1 1 real in a unitary CFT. The set of local primary operators and z ¼ 2 ; z¯ ¼ 2, the crossing-symmetric point. The z, z¯ coor- the set of OPE coefficients are together referred to as CFT dinates for the space of conformal cross ratios are related to data, which completely specifies a local CFT. the original coordinates u, v by u ¼ zz;¯ v ¼ð1 − zÞð1 − z¯Þ. Critical exponents are encoded in the scaling dimensions Every element of the real analytic function space in which of only a few low-lying primary operators (the relevant the conformal blocks live is parametrized by the coeffi- ones). There are only two relevant operators in the Ising 1 1 cients in its Taylor expansion around this z ¼ 2 ; z¯ ¼ 2 CFT, while there are four relevant operators in the CFT of point, an infinite-tuple of real numbers. Thus, the values of the tricritical Ising point [27]. Of the four, one only needs a real analytic function and the values of its various mixed three to define the critical exponents of the tricritical Ising partial derivatives at this special point form the coordinates universality class. In two dimensions these exponents are for this function space. Now, one solves Eq. (4) by known exactly [1,2]; the CFT is the second Virasaro truncating to a finite number of coordinates, depending minimal model consisting of six primary operators (while on the accuracy needed.