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The conformal bootstrap at finite temperature

Iliesiu, Luca, Koloğlu, Murat, Mahajan, Raghu, Perlmutter, Eric, Simmons-Duffin, David·2018·Journal of High Energy Physics
DOI 10.1007/jhep10(2018)070
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Abstract

A bstract: We initiate an approach to constraining conformal field theory (CFT) data at finite temperature using methods inspired by the conformal bootstrap for vacuum correlation functions. We focus on thermal one- and two-point functions of local operators on the plane. The KMS condition for thermal two-point functions is cast as a crossing equation. By studying the analyticity properties of thermal two-point functions, we derive a “thermal inversion formula” whose output is the set of thermal one-point functions for all operators appearing in a given OPE. This involves identifying a kinematic regime which is the analog of the Regge regime for four-point functions. We demonstrate the effectiveness of the inversion formula by recovering the spectrum and thermal one-point functions in mean field theory, and computing thermal one-point functions for all higher-spin currents in the critical O ( N ) model at leading order in 1 /N . Furthermore, we develop a systematic perturbation theory for thermal data in the large spin, low-twist spectrum of any CFT. We explain how the inversion formula and KMS condition may be combined to algorithmically constrain CFTs at finite temperature. Throughout, we draw analogies to the bootstrap for vacuum four-point functions. Finally, we discuss future directions for the thermal conformal bootstrap program, emphasizing applications to various types of CFTs, including those with holographic duals.

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Figures

Figure 1 The OPE on S 1 β × R d -1 is valid if the two operators lie inside a sphere. The largest possible sphere has diameter β , wrapping entirely around the S 1 such that it is tangent to itself. Here, we illustrate such a sphere (blue) in d = 2.
Figure 1 The OPE on S 1 β × R d -1 is valid if the two operators lie inside a sphere. The largest possible sphere has diameter β , wrapping entirely around the S 1 such that it is tangent to itself. Here, we illustrate such a sphere (blue) in d = 2.CC-BY · From the PDF
Figure 2 Lightlike trajectories moving in the x L direction and around the thermal circle. One trajectory is z = 0 and the other is ¯ z = 0. Poles in the Lorentzian inversion formula come from the neighborhood of these trajectories.
Figure 2 Lightlike trajectories moving in the x L direction and around the thermal circle. One trajectory is z = 0 and the other is ¯ z = 0. Poles in the Lorentzian inversion formula come from the neighborhood of these trajectories.CC-BY · From the PDF
Image 3
CC-BY · From the PDF
Figure 4 For fixed r ∈ (0 , 1), the s -channel OPE (expansion around z = ¯ z = 0) implies that the thermal two-point function g ( z, ¯ z ) is analytic in an annulus in the w plane between radii r and 1 /r (shaded blue). The t -channel OPE (expansion around z = ¯ z = 1), together with symmetry under w ↔ -w , implies analyticity in the red-shaded regions, except for cuts running along ( -∞ , -1 /r ) , ( -r, 0) , (0 , r ) , (1 /r, ∞ ) (indicated with zig-zags). In this section, we argue for analyticity everywhere in the upper and lower half planes.
Figure 4 For fixed r ∈ (0 , 1), the s -channel OPE (expansion around z = ¯ z = 0) implies that the thermal two-point function g ( z, ¯ z ) is analytic in an annulus in the w plane between radii r and 1 /r (shaded blue). The t -channel OPE (expansion around z = ¯ z = 1), together with symmetry under w ↔ -w , implies analyticity in the red-shaded regions, except for cuts running along ( -∞ , -1 /r ) , ( -r, 0) , (0 , r ) , (1 /r, ∞ ) (indicated with zig-zags). In this section, we argue for analyticity everywhere in the upper and lower half planes.CC-BY · From the PDF
Figure 5 An illustration of the relation between s - and t -channels in the 〈 φφ 〉 β correlator. The two channels are related by moving the external operators around the thermal circle (gray). A single term in the t -channel OPE O ∈ φ × φ inverts to the sum over the [ φφ ] n families in the s -channel. Alternatively, the sum over the [ φφ ] n families in the s -channel reproduces the O term in the t -channel.
Figure 5 An illustration of the relation between s - and t -channels in the 〈 φφ 〉 β correlator. The two channels are related by moving the external operators around the thermal circle (gray). A single term in the t -channel OPE O ∈ φ × φ inverts to the sum over the [ φφ ] n families in the s -channel. Alternatively, the sum over the [ φφ ] n families in the s -channel reproduces the O term in the t -channel.CC-BY · From the PDF
Figure 6 Large-spin diagrams illustrating the contribution to the anomalous dimensions of [ φφ ] n from the exchange of O .
Figure 6 Large-spin diagrams illustrating the contribution to the anomalous dimensions of [ φφ ] n from the exchange of O .CC-BY · From the PDF
Figure 7 The diagram on the right can be thought of as the product of the two subdiagrams. Reading it from left to right, it's comprised of the contribution of O to the anomalous dimensions of [ φφ ] n , and the thermal coefficients of the [ φφ ] n families proportional to b 1 . Accordingly, it should be interpreted as the asymptotic piece with δ ( O ) [ φφ ] n ( ¯ h ) × a ( 1 ) [ φφ ] n ( ¯ h ) in the sum over [ φφ ] 0 .
Figure 7 The diagram on the right can be thought of as the product of the two subdiagrams. Reading it from left to right, it's comprised of the contribution of O to the anomalous dimensions of [ φφ ] n , and the thermal coefficients of the [ φφ ] n families proportional to b 1 . Accordingly, it should be interpreted as the asymptotic piece with δ ( O ) [ φφ ] n ( ¯ h ) × a ( 1 ) [ φφ ] n ( ¯ h ) in the sum over [ φφ ] 0 .CC-BY · From the PDF
Figure 8 The t -channel sum over the asymptotic parts represented by the diagram on the left inverts to the s -channel process on the right. Accordingly, the inversion should produce poles for the [ OO ] m families. The diagram on the right can itself be deciphered by reading it from right to left; first the external φ operators form [ φφ ] n , which mixes with [ OO ] m via exchange of a φ , then the [ OO ] m receive expectation values proportional to b 1 .
Figure 8 The t -channel sum over the asymptotic parts represented by the diagram on the left inverts to the s -channel process on the right. Accordingly, the inversion should produce poles for the [ OO ] m families. The diagram on the right can itself be deciphered by reading it from right to left; first the external φ operators form [ φφ ] n , which mixes with [ OO ] m via exchange of a φ , then the [ OO ] m receive expectation values proportional to b 1 .CC-BY · From the PDF
Figure 9 The t -channel diagram denotes a sum over the asymptotics δ ( O ) [ φφ ] n ( ¯ h ) × a ( O ′ ) [ φφ ] n ( ¯ h ). This inverts to poles for the [ OOO ′ ] m families in the s -channel. When O ′ = 1 we omit the line by convention and recover figure 8.
Figure 9 The t -channel diagram denotes a sum over the asymptotics δ ( O ) [ φφ ] n ( ¯ h ) × a ( O ′ ) [ φφ ] n ( ¯ h ). This inverts to poles for the [ OOO ′ ] m families in the s -channel. When O ′ = 1 we omit the line by convention and recover figure 8.CC-BY · From the PDF
Figure 10 Higher order terms δ m [ φφ ] log m ¯ z that sum up to shift the [ φφ ] poles in a (∆ , J ) are produced by sums over multi-twist families in the t -channel.
Figure 10 Higher order terms δ m [ φφ ] log m ¯ z that sum up to shift the [ φφ ] poles in a (∆ , J ) are produced by sums over multi-twist families in the t -channel.CC-BY · From the PDF
Figure 11 Poles in a (∆ , J ) for other double-twist families [ OO ] in 〈 φφ 〉 shift by anomalous dimensions through sums over multi-twist families [ φφ O ′ O ′ ] in the t -channel.
Figure 11 Poles in a (∆ , J ) for other double-twist families [ OO ] in 〈 φφ 〉 shift by anomalous dimensions through sums over multi-twist families [ φφ O ′ O ′ ] in the t -channel.CC-BY · From the PDF
Image 12
CC-BY · From the PDF
Figure 12 The function g ∆ max ( β ) in the free boson theory in 3d, plotted for the values ∆ max = 1 , 2 , . . . , 12 (colors), and ∆ max = ∞ (black dotted line). The value of b T in the free theory (gray horizontal line) is -3 ζ (3) 2 π ≈ -0 . 574.
Figure 12 The function g ∆ max ( β ) in the free boson theory in 3d, plotted for the values ∆ max = 1 , 2 , . . . , 12 (colors), and ∆ max = ∞ (black dotted line). The value of b T in the free theory (gray horizontal line) is -3 ζ (3) 2 π ≈ -0 . 574.CC-BY · From the PDF
Figure 13 The function g ∆ max ( β ) in the 3d Ising model, estimated using known operators only, and plotted for the values ∆ max = 1 , 2 , . . . , 10. The value of b T determined from Monte Carlo simulations is -0 . 459 (gray horizontal line, [58-60]).
Figure 13 The function g ∆ max ( β ) in the 3d Ising model, estimated using known operators only, and plotted for the values ∆ max = 1 , 2 , . . . , 10. The value of b T determined from Monte Carlo simulations is -0 . 459 (gray horizontal line, [58-60]).CC-BY · From the PDF

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Cited by

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Metadata

Authors
Iliesiu, L., Koloğlu, M., Mahajan, R., Perlmutter, E., & Simmons-Duffin, D.
Venue
Journal of High Energy Physics
Publisher
Springer Science and Business Media LLC
Published
2018-10
Volume
2018
Issue
10
ISSN
1029-8479
DOI
10.1007/jhep10(2018)070
Cited by
133
References
4
Language
en
Licence

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Semantic Scholar’s data is described in: Kinney, R., et al. (2023). The Semantic Scholar Open Data Platform. arXiv:2301.10140.