FR37 · An interactive course
Conformal field theory and the bootstrap: what consistency alone can prove
At a critical point a magnet forgets its microscopic details. What is left can be described by a short list of numbers, and those numbers must fit together in a way that is surprisingly rigid. This course shows how that rigidity alone rules out most candidate theories, and how the two-dimensional Ising model appears at the edge of what remains.
From a critical spin chain to a published bound
The first chapter shows that a critical chain looks the same at every scale, using rings we can solve exactly. The second turns that into a list of operators and reads their dimensions from energy levels. The third writes one four-point function in two ways and asks that both agree, and the fourth checks the idea on free fields, which solve it exactly. The fifth uses a single inequality to rule out whole spectra, and the sixth audits what such a bound does and does not prove. The seventh finds the Ising model at a kink in the two-dimensional bound, and the last chapter reads the published table and figure that the Lab reproduces.
You need power laws, Taylor series and vectors. Conformal blocks, operator expansions and linear programming are introduced when they are first used. FR37
01 · L1
Why a critical chain looks the same at every scale
At a critical point spin correlations fall as a power of distance, so small and large rings show the same rescaled pattern. An exactly solvable chain shows it.
02 · L2
Conformal symmetry reduces a critical point to a list of operators
Conformal symmetry fixes two- and three-point functions, so a critical theory reduces to a list of operator dimensions, spins and OPE coefficients.
03 · L2
One four-point function expanded two ways gives the crossing equation
A four-point function can be expanded around either pair of points. Requiring the two sums to agree gives the bootstrap equation.
04 · L2
Free fields solve the crossing equation exactly
Generalized free fields solve crossing exactly, with every OPE coefficient positive. They give the bootstrap a test case and mark where bounds can end.
05 · L3
How one linear functional rules out a whole spectrum
Apply one linear functional to the crossing equation. If it is positive on every allowed operator, no spectrum with that gap can exist.
06 · L3
What a numerical bootstrap bound proves, and what it does not
A bound is only as strong as its assumptions, sampling and arithmetic. This chapter audits one exclusion step by step and separates evidence from proof.
07 · L4
The kink in the two-dimensional bound, and the Ising model at it
In two dimensions the scalar gap bound bends sharply at the Ising model. This chapter measures the kink and tests which truncations would fake it.
08 · L4
How to read Table 1 and Figure 27 of the TASI bootstrap lectures
A guide to the 2D bootstrap bounds in the TASI lectures: what each axis, curve and dot means, and how three table entries compare with the Lab.
Experiments along the way
The bounds in these experiments come from a linear programme that checks positivity on sampled dimensions. Each one is numerical evidence, labelled as such, and the published values it is compared with are shown separately.
Sources and what we recomputed
- D. Simmons-Duffin, TASI Lectures on the Conformal Bootstrap, arXiv:1602.07982v1 (2016)
Version 1. Eqs. (157)-(158): the four-point function as a sum over conformal multiplets, one conformal block each. Eqs. (164)-(168): the Casimir equation for blocks and their closed form in two dimensions, built from the one-variable function k. Eqs. (194)-(199), Sections 10.5-10.7: the crossing equation for identical scalars as a sum of vectors, the derivative functionals and the assumptions behind the scalar-gap bound. Fig. 26: 2D crossing vectors. Fig. 27 and Table 1: 2D bounds for derivative cutoffs 6 to 28 with the kink at the Ising point. Fig. 28: the 3D bound with its kink.
- D. Mazáč, Analytic bounds and emergence of AdS2 physics from the conformal bootstrap, arXiv:1611.10060v1; JHEP 04 (2017) 146
Version 1. Eqs. (2.12)-(2.17): the four-point function of identical operators on a line, the cross-ratio, the one-variable block and the crossing equation. Eq. (2.18) and Fig. 2: the numerical gap bound, which the text says converges to twice the external dimension plus one as derivatives are added. Eqs. (2.19)-(2.23): the generalized free fermion that saturates it and the zeros of the extremal functional. Fig. 1: the action of a typical extremal functional.
- D. Gaiotto, D. Mazáč and M. F. Paulos, Bootstrapping the 3d Ising twist defect, arXiv:1310.5078v2; JHEP 03 (2014) 100
Version 2. Section 4.1: one-dimensional generalized free fields. Eqs. (4.17) and (4.20) give the exchanged dimensions and OPE coefficients of the fermionic and bosonic solutions; the course resums them against the closed-form correlators.
- S. Rychkov, EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions, arXiv:1601.05000v2 (2016)
Version 2. Section 1.1: fixed points and scale invariance at a critical point. Section 1.3.4: how primary operators transform. Section 2.2: two- and three-point functions of primary scalars. Sections 3.1.3-3.1.4: the state-operator correspondence and the cylinder picture. Section 3.3: the operator product expansion. Section 4.3.1, Eqs. (4.14)-(4.16): the minimal models and the central charge and spin and energy dimensions of the 2D Ising model.
- G. B. Mbeng, A. Russomanno and G. E. Santoro, The quantum Ising chain for beginners, arXiv:2009.09208v2 (2020)
Version 2. Sections 1-3: the Jordan-Wigner mapping of the chain to free fermions. Eq. (55): the quasiparticle energies of the uniform chain, of which the transverse-field Ising ring is the fully anisotropic case. Eqs. (247)-(249): the ground-state correlation functions as determinants. The course uses these for the periodic transverse-field ring and checks them against exact diagonalisation of eight sites.
- H. W. J. Blöte, J. L. Cardy and M. P. Nightingale, Conformal invariance, the central charge, and universal finite-size amplitudes at criticality, Phys. Rev. Lett. 56, 742 (1986)
Read as an abstract only; the article itself was not opened, so no equation or number in this course is taken from it. The abstract states that the amplitude of the finite-size correction to the free energy of a critical strip is linearly related to the central charge. The coefficient the course uses comes from Cardy's lectures below.
- J. Cardy, Conformal Field Theory and Statistical Mechanics, Les Houches lectures, arXiv:0807.3472v1 (2008)
Version 1. Section 5.1, Eq. (19) and the text after it: on a cylinder of circumference L the Hamiltonian is set by the scaling dimensions and the central charge, so the ground-state energy carries the universal term with the central charge and the excitation gaps give the scaling dimensions. The lectures set the velocity to one; the course restores the chain velocity.
- S. Rychkov and A. Vichi, Universal constraints on conformal operator dimensions, arXiv:0905.2211v2; Phys. Rev. D 80, 045006 (2009)
Version 2. Eq. (2.8): the crossing sum rule written with conformal blocks. Section 3, Eqs. (3.1)-(3.2): the positivity argument, a linear functional of derivatives at the symmetric point that rules out a spectrum. Table 2: the two-dimensional bounds of that paper.
- D. Mazáč and M. F. Paulos, The analytic functional bootstrap. Part II, arXiv:1803.10233v2; JHEP 05 (2019) 088
Version 2. Section 4: extremal functionals in one dimension and the optimal gap bound equal to twice the external dimension plus one, saturated by the generalized free fermion.
- D. Poland, S. Rychkov and A. Vichi, The Conformal Bootstrap: Theory, Numerical Techniques, and Applications, arXiv:1805.04405v3; Rev. Mod. Phys. 91, 015002 (2019)
Version 3. Section IV.A: linear programming on a discretised set of dimensions, which the review calls obsolete but pedagogical. Section IV.B: semidefinite programming, which imposes positivity for all dimensions through polynomial approximations, a spin cutoff, and solvers run above machine precision. Fig. 5: vectors that can or cannot sum to zero with positive coefficients, and the separating plane. Fig. 7: the 3D single-correlator bound with the Ising model near its kink. Figs. 12, 14 and 15: the 3D Ising island from mixed correlators. Table II: the stable operators of the 3D Ising model. The islands and the table are reading material only.
- S. Rychkov and N. Su, New developments in the numerical conformal bootstrap, arXiv:2311.15844v3 (2024)
Version 3. Section 2: the solver SDPB uses arbitrary-precision arithmetic because the matrices are poorly conditioned. Section 3: the standard method, with derivatives at the symmetric point, polynomial approximations of block derivatives and a semidefinite feasibility problem. Footnote 18 in Section 4.2: approximations that remain not fully rigorous even in published error bars. Section 6.3.1: rigorous and non-rigorous error bars compared.