Conformal symmetry reduces a critical point to a list of operators

From scale invariance to conformal symmetry

A critical system is unchanged by translations, rotations and a uniform rescaling $x\to\lambda x$ . In practice it is also unchanged by transformations that rescale by a different amount at each point while preserving angles. These conformal transformations add special conformal transformations to the list.

Under a rescaling a local operator picks up a power of the factor: $\mathcal O(\lambda x)\to\lambda^{-\Delta}\mathcal O(x)$ . The number $\Delta$ is the scaling dimension. Operators that transform this way and are not derivatives of others are called primaries; derivatives of a primary are its descendants.

Source: S. Rychkov · D. Simmons-Duffin

What symmetry fixes

Conformal symmetry fixes the two-point function of a scalar primary completely, once it is normalised. The power $2\Delta$ in it is the $\eta$ of the previous chapter: for the Ising spin $2\Delta_\sigma=1/4$ .

Three-point functions are fixed up to one number per triple of operators, the OPE coefficient $\lambda_{123}$ . Four-point functions are not fixed: they depend on two cross-ratios in general, or one on a line. That freedom, constrained by consistency, is where the bootstrap works.

\langle\mathcal O(x)\mathcal O(0)\rangle=\frac{1}{|x|^{2\Delta}},\qquad \langle\mathcal O_1\mathcal O_2\mathcal O_3\rangle=\frac{\lambda_{123}}{|x_{12}|^{\Delta_1+\Delta_2-\Delta_3}|x_{13}|^{\Delta_1+\Delta_3-\Delta_2}|x_{23}|^{\Delta_2+\Delta_3-\Delta_1}}

Here

x_{ij}=x_i-x_j

. The coefficient

\lambda_{123}

is the only information left in a three-point function.

Source: D. Simmons-Duffin · S. Rychkov

The operator product expansion

When two operators approach each other, their product can be replaced by a sum of single operators at one point, with coefficients that grow or shrink as powers of the separation. Symmetry fixes the descendants in each term; only the primaries and their coefficients $\lambda$ are data.

Applied twice, the OPE reduces any four-point function to a sum over exchanged primaries, each weighted by $\lambda^2$ and multiplied by a known function called a conformal block. That sum is the subject of the next chapter.

\mathcal O_1(x)\,\mathcal O_2(0)=\sum_{k}\lambda_{12k}\,|x|^{\Delta_k-\Delta_1-\Delta_2}\bigl[\mathcal O_k(0)+\text{descendants}\bigr]

The sum runs over primaries. In a unitary theory the squared coefficients that appear in a four-point function of identical operators are non-negative.

Source: D. Simmons-Duffin

Worked example: reading dimensions from a ring of 64 spins

On a critical ring of $N$ sites each operator becomes an energy level: the gap above the ground state is $2\pi v\Delta/N$ , and the momentum in units of $2\pi/N$ is the spin. For the transverse-field chain $v=2J$ .

For $N=64$ the exactly computed gaps, multiplied by $N/2\pi v$ , are $0.12501$ , $0.9999$ , $1.1246$ twice and $1.9986$ . They are the spin operator $\sigma$ with $\Delta=1/8$ , the energy operator $\varepsilon$ with $\Delta=1$ , the two first derivatives of $\sigma$ at $9/8$ with spin $\pm1$ , and level-two states at $2$ .

At 256 sites the same numbers are $0.125$ , $0.99999$ , $1.12498$ and $1.99991$ . The corrections fall roughly as $1/N^2$ . The ground-state energy gives the central charge $c=0.500002$ , against the exact $1/2$ .

Source: G. B. Mbeng · D. Simmons-Duffin · J. Cardy · H. W. J. Blöte

Try it in the Lab

Find the Ising operators in the spectrum

Open the second chart at the critical preset. Predict which dashed lines the orange points will sit on.
Identify the spin operator, the energy operator and their first descendants by their energy and momentum.
Switch to sixteen spins and note how far each level moves from its dashed line.

Exercises with explained answers

Two-point functions of the Ising spin decay as $|x|^{-1/4}$ . What is $\Delta_\sigma$ ?
Show the answer
The power is $2\Delta_\sigma$ , so $\Delta_\sigma=1/8$ .
A ring of 128 critical spins has its first odd-sector gap at $E_1-E_0=0.012272J$ . What dimension does this give, with $v=2J$ ?
Show the answer
$\Delta=N(E_1-E_0)/(2\pi v)=128\times0.012272/(4\pi)=0.1250$ , the Ising spin dimension.
Why can a three-point function of primaries contain only one free number?
Show the answer
Three points can be moved to $0$ , $1$ and infinity by conformal transformations, leaving no invariant combination of positions. The function is therefore fixed up to an overall constant, the OPE coefficient.

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.
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.