Solving the 3d Ising Model with the Conformal Bootstrap II. $$c$$c-Minimization and Precise Critical Exponents

By S. El-Showk, M. Paulos, David Poland, S. Rychkov, David Simmons-Duffin, A. Vichi. 2012

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We use the conformal bootstrap to perform a precision study of the operator spectrum of the critical 3d Ising model. We conjecture that the 3d Ising spectrum minimizes the central charge $$c$$c in the space of unitary solutions to crossing symmetry. Because extremal solutions to crossing symmetry are uniquely determined, we are able to precisely reconstruct the first several $$\mathbb {Z}_2$$Z2-even operator dimensions and their OPE coefficients. We observe that a sharp transition in the operator spectrum occurs at the 3d Ising dimension $$\Delta _\sigma = 0.518154(15)$$Δσ=0.518154(15), and find strong numerical evidence that operators decouple from the spectrum as one approaches the 3d Ising point. We compare this behavior to the analogous situation in 2d, where the disappearance of operators can be understood in terms of degenerate Virasoro representations.

Subfields: Nuclear and High Energy Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/3954b013c8217489b8e92ee4f4d50ee68f49fd04

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Citations in the corpus

OPE Convergence in Conformal Field Theory (2012, background): “For reasons which are not yet fully understood, this simple-minded approach gives pretty strong, sometimes optimal [9, 16], bounds on the operator spectrum and also on the OPE coefficients.”
Critical exponents of the 3d Ising and related models from conformal bootstrap (2014, methodology): “To be more specific, following [15] we make the change of variables z = (a + √ b)/2, z̄ = (a − √ b)/2 and Taylor expand around a = 1 and b = 0.”
Modular bootstrap revisited (2016, neutral): “When the degeneracy bound at the gap ∆gap is saturated (for c−1 12 < ∆gap ≤ ∆mod), the entire spectrum of the CFT is in fact determined [25, 26], for reasons we explain below.”
Bootstrapping the O(N ) vector models (2013, methodology): “The results obtained so far have been particularly striking in 3D, where it was found in [12] that the CFT described by the critical 3D Ising model occupies a special place in the space allowed by crossing symmetry and unitarity.”
Convexity and liberation at large spin (2012, methodology): “Sometimes these numerical bounds seem to be saturated by actual CFTs, as has been demonstrated, for example, in [26].”
Infinite Chiral Symmetry in Four Dimensions (2013, methodology): “In large part, this progress has been inspired by the development of numerical techniques for extracting constraints on the defining data of a CFT using unitarity and crossing symmetry [3,4].”
Analyticity in spin in conformal theories (2017, methodology): “The modern numerical bootstrap [1] has been applied very successfully to numerous models in various dimensions (far too many to even attempt to review here) yielding for instance precise critical exponents for the three-dimensional Ising model [2, 3].”
The lightcone bootstrap and the spectrum of the 3d Ising CFT (2016, methodology): “13) and working order-by-order in y, we find λ12[12]n(h)λ43[12]n(h) ∼ ∑′”
Causality constraints in conformal field theory (2015, background): “In odd dimensions, the blocks are not known in closed form, but they can be computed efficiently by recursion relations [14].”
Bootstrapping the 3d Ising twist defect (2013, methodology): “We have thus obtained a conventional crossing equation with positive and equal coefficients on both sides, directly analogous to those used in higher dimensions [2, 11].”
Semiclassical Virasoro blocks from AdS3 gravity (2015, background): “They feature prominently in many applications of CFT, including in the conformal bootstrap program [10, 11] and in the study of the emergence of bulk locality from CFT [12–14].”
Projectors, shadows, and conformal blocks (2012, methodology): “More recently, an improved understanding of scalar conformal blocks in 3d led to novel determinations of operator dimensions in the 3d Ising model [20], with precision comparable to the best perturbative calculations and Monte-Carlo simulations.”
Conformal invariance in the long-range Ising model (2015, methodology): “In 3d, the most dramatic evidence for the conformal invariance of the SRI critical point comes from its recent exploration using the conformal bootstrap methods [14–19, 13, 20– 22].”