User:Sylvain Ribault/Bootstat2021/Vichi

These are notes on Alessandro Vichi's lectures.

Favourite example is the Ising model. Can be addressed using the renormalization group. using blocking or decimation. Couplings $J,K$ for nearest, next to nearest neighbour, change with scale. We learn there is a fixed point with a $\mathbb {Z} _{2}$. One stable, one unstable diretion, preserving or breaking this symmetry.

Conformal bootstrap: goal is to study quantitatively the critical points. Needs basic ingredients: symmetries, nb unstable directions. Non-perturbative, rigorous.

Framework of quantum field theory. Partition function on the lattice:

$Z=\sum _{\sigma _{i}}e^{-\beta H}\to \int D[\phi ]e^{-S[\phi ]}$

Observables are correlation functions of lattice observables. Assumption: there is a scaling limit of these observables, when the lattice spacing goes to zero, where these correlation functions become QFT correlation functions of local operators.

Restrict to unitary CFTs i.e. reflection positive. This implies that scaling dimensions are bounded from below. Examples of non-unitary CFTs: minimal models, Q-Potts, percolation.

Conformal symmetry

Conformal symmetry in dimension 2 or higher

How does the action change under a change of coordinates?

$x^{\mu }\to x^{\mu }+\xi ^{\mu }(x)\implies \delta S=\int T^{\mu \nu }\partial _{\mu }\xi _{\nu }$

Translation invariance $\partial _{\mu }T^{\mu \nu }=0$. Rotations: tensor must be symmetric.

Scale invariance implies conformal invariance under some conditions such as unitarity in 2d. FOr us it is an assumption that the trace of the stress tensor is zero. For free we get invariance under more geometric transformations.

Two dimensions: local and global conformal transformations. Global: translations, rotations, dilations, special conformal.

Higher dimensions: translations, rotations, dilations, special conformal transformation. (Global definitions.) These conformal maps form the conformal group. Group isomorphic to $O(1,d+1)$. Conformal algebra with its commutation relations. Translation generator is a raising operator i.e. increases dilation generator by one. Special conformal is a lowering operator. Commutation of SCT with translations gives dilations an rotations, important for unitarity. Analogs in two dimensions of these various generators. Commutation of rotations with dilations: can be simult. diagonalized.

Representations of the conformal algebra

States classified by two quantum numbers: conformal dimension i.e. eigenvalue of dilation generator, and a representation of the rotation group $SO(d)$. 2d analogs: left and right conformal dimensions. What happens if we act with other generators? lower or raise dimensions, and also change representation of rotations. Keep acting with lowering operators: go to dimensions unbounded from below. Contradicts unitarity, QM stability. Also makes correlation functions grow with distance. We restrict to irreducible representations such that there is a state with a minimum dimension: primary states, killed by SCT generators. (In 2d these would be called quasi primary. Primaries are killed by all annihilation modes.)

Act on primary with raising operator: increase conformal dimension by one. Can increase or decrease spin, depending on whether we add an index or contract an index. (Nice 2d diagram whose coordinates are conformal dimension and spin.)

Correlation functions

Notation

$\left\langle O_{1}(x_{1})\cdots O_{n}(x_{n})\right\rangle$

Local operators are convenient objects but what matters are correlation functions themselves.

Conformal transformation

${\frac {\partial x'^{\mu }}{\partial x^{\nu }}}=\Omega (x)\Lambda _{\nu }^{\mu }$

(Infinitesimal condition for preserving angles.)

Transformations of operators under rotations and scale transformations. Written for primary operators. Definition of primary operators: commute with generators of SCT. Example: two scalar operators, i.e. operators invariant under rotations. Consider two-point correlation function $\left\langle \phi _{1}(x_{1})\phi _{2}(x_{2})\right\rangle$. Just like in QFT, must be a function of the distance. Scale invariance says

$\left\langle \phi _{1}(x_{1})\phi _{2}(x_{2})\right\rangle \propto x_{12}^{-\Delta _{1}-\Delta _{2}}$

SCT further imply it is zero unless the two dimensions are equal. We can assume two-point function vanishes unless the two operators are equal, and set the prefactor to one by rescaling operators. For three-point function, conformal symmetry fixes the coord dependence, but the constant prefactor can no longer be absorbed by rescaling operators.

Definition of the central charge from the two-point function of stress-energy tensor. Cannot rescale stress-energy tensor as we want its OPE with a primary to give conformal dimension? Tensor structures in 3pt functions. Example of scalar-scalar-T three-point function, we fix its overall factor by some specific conventions.

Any 3pt fct is fixed up to a finite number of constants. Higher correl fct: no longer true. Csd 4pt fct of scalars with all the same dimension.

$\left\langle \phi _{1}(x_{1})\cdots \phi _{4}(x_{4})\right\rangle =x_{12}^{-2\Delta }x_{34}^{-2\Delta }$

This ansatz would transform correctly but we can add invariant factors: fct of cross-ratios $u,v$. Or their combinations

$u=z{\bar {z}}\qquad v=(1-z)(1-{\bar {z}})$

$z,{\bar {z}}$ would be cplx cjg in Euclidean, real independent in Minkowskian.

Can use conformal invariance to set $x_{1}=0,x_{2}=z,x_{3}=1,x_{4}=\infty$. Conformal frame: radial coordinates for $\rho$ defined by $z={\frac {4\rho }{(1+\rho )^{2}}}$.

Principles of the conformal bootstrap

Radial quantization

Foliate space using concentric spheres. Define evolution operator as rescaling. Map to cylinder: $\tau =\log r$ where $r$ is radius. Then dilatation operator assumes role of Hamiltonian for the time $\tau$. Beware that metrics are related by Weyl transformation. Would need to be careful if we wanted to make precise relation with cylinder. Here we use cylinder for intuition.

Using map to cylinder we translate cylinder's time ordering into radial ordering of correlation functions. So states are defined on spheres, by inserting operators inside spheres. The origin corresponds to infinite negative time. Also import concept of conjugation. On cylinder, conjugation is:

$|\phi \rangle =\phi (\tau )|0\rangle \to \langle \phi |=\langle 0|\phi (-\tau )=(|\phi \rangle )^{*}$

On sphere we define inverted operators

$O(x)|0\rangle \to (O(x)|0\rangle )^{\dagger }=\langle 0|IO(x)I$

State-operator correspondence: to an operator at point $0$ we associate a state. This allows us to infer a lot of things starting with the OPE.

Operator product expansion (OPE)

Consider the state-operator correspondence for two primary operators $O_{1}(0),O_{2}(x)$. Inserting both operators creates a state, no longer primary, which we can decompose over a basis of our Hilbert space. This state should then correspond to an operator at point $0$. This leads to the OPE

$O_{2}(x)O_{1}(0)\sim \sum _{\Delta ,r}\lambda _{1,2,\Delta ,r}\left(O_{\Delta ,r}(0)+\cdots \right)$

where the dots are for descendants. We can repackage descendants and coefficients as a coefficient

$O_{2}(x)O_{1}(0)\sim \sum _{\Delta ,r;I}C_{1,2,\Delta ,r}^{I}(X,\partial )O_{\Delta ,r}^{I}(0)$

where derivatives are descendants, as the creation mode (momentum) takes derivatives. Conformal symmetry fixes the OPE coefficients $C^{I}$. They are related to the three-point function $\left\langle O_{1}O_{2}O_{\Delta ,r}^{I}\right\rangle$, which is fixed up to a constant coefficient. From now on we assume we know the OPE coefficients.

Case of two dimensions: examples of OPEs involving the stress-energy tensor. Fusion rules, which are not OPEs (not really explained). Fusion rules are not OPEs, they say which representations can appear but not what the coefficients are i.e. the strengths of interactions.

OPE of scalars:

$O_{1}(x_{1})O_{2}(x_{2})\sim \sum _{k}{\frac {\lambda _{O_{1}O_{2}O_{k}}}{x_{12}^{h_{1}+h_{2}-h_{k}}}}O_{k}$

where we only write scalar primaries. Selection rules: only some irreps of $SO(d)$ can appear. OPEs of scalars only yield integer spin representations. If there is a global symmetry i.e. a symmetry that commutes with conformal symmetry, then OPEs amount to tensor products of repres. of the global symmetry.

Radius of convergence of OPE: converges if no other operator is too close.

Conformal blocks

Go back to 4pt function of four identical scalars.

$\left\langle \phi (x_{1})\phi (x_{2})\phi (x_{3})\phi (x_{4})\right\rangle =\sum _{O_{k}\in \phi \times \phi }x_{12}^{\Delta _{\phi }-2\Delta }x_{34}^{\Delta _{\phi }-2\Delta }\lambda ^{2}\cdots$

where we apply OPEs in pairs $12$ and $34$. Compare with the expectation from conformal invariance: a function $g(u,v)$ of cross-ratios. Write this function as

$g(u,v)=\sum _{O_{k}\in \phi \times \phi }\lambda ^{2}g_{\Delta _{k},r_{k}}(u,v)$

The objects $g_{\Delta _{k},r_{k}}(u,v)$ sum up the contributions of an irrep. They are computable in principle, and they are called conformal blocks. In practice it is challenging to compute them. For Virasoro blocks a recursive formula was found by Al. Zamolodchikov in 1984 by analyzing singularities. In higher dimension, Dolan and Osborne computed blocks by brute force and also by solving a differential equation in the variables $z,{\bar {z}}$. Conformal blocks in even dimension have a simple form. Let us start with two dimensions, global blocks:

$g_{\Delta ,\ell }(z,{\bar {z}})=K_{\Delta +\ell }(z)K_{\Delta -\ell }({\bar {z}})+(z\leftrightarrow {\bar {z}})$

where

$K_{\beta }(x)=x^{\frac {\beta }{2}}F({\frac {\beta }{2}},{\frac {\beta }{2}},\beta ,x)$

Similar formulas of increasing complexity in any even dimension. But still an open problem to find simple expression in odd dimension. Alternative methods: recursion Kos, Poland, Simmons-Duffin 2014.

This leads to a formula

$g_{\Delta ,\ell }(r,\theta )=(4r)^{\Delta }\sum _{n=0}^{\infty }r^{n}\omega _{n}(\ell ,\Delta ,\eta )$

where $\omega _{n}$ is rational in $\Delta$. It can be computed very efficiently, its derivatives too. In practice we truncate the series to a given $n$. Under some conditions on $\Delta$, the denominator is positive.

There is a recursion on the dimension, relating blocks at $d$ and $d-2$.

Crossing symmetry

We defined blocks by taking OPEs in a specific way i.e. $12,34$. Doing $23,14$ is also possible. In the region where both OPEs converge, they have to agree, although they do not look the same, they are related by $2\leftrightarrow 4$ which amounts to $u\leftrightarrow v$. We get

$\sum _{\Delta ,\ell }\lambda _{\Delta ,\ell }^{2}\left(u^{-\Delta _{\phi }}g_{\Delta ,\ell }(u,v)-v^{-\Delta _{\phi }}g_{\Delta ,\ell }(v,u)\right)=0$

For more general fields we would have additional non-equivalent equations. The equation is not satisfied term by term.

Reflection positivity and unitarity

Boils down to states having positive norm squared. This is constraining. If we start with a primary field we can compute norms of descendants and they are not automatically of the same sign.

We obtain a constraint of the type

$\Delta \geq \Delta _{\text{min}}(d,r)$

and also reality of 3pt couplings. Unitarity bounds in three dimensions: irreps labelled by $J\in {\frac {1}{2}}\mathbb {N}$. Condition is $\Delta \geq J+1$ for $J>{\frac {1}{2}}$, bound is 1 and 0 for spins 1/2 and 0. Saturation of bound occurs when there is a descendant with zero norm.

Numerical bootstrap

A CFT is a collection of CFT data: local operators, called the spectrum, together with their three-point structure constants (aka OPE coefficients), such that unitarity is obeyed (bounds are obeyed), and crossing symmetry is obeyed for any four-point function. Crossing symmetry is a condition on the spectrum and OPE coefficients.

These are necessary conditions, maybe not sufficient. Cf gauge theory, nonlocal operators.

Principles

Which choice of data obey the conditions? Want to check a finite number of crossing equations. This can allow us to rule out our data, or to check consistency so far.

For example, start with a scalar of a given dimension, choose trial spectrum, study crossing symmetry of scalar 4pt function.

Check consistency of spectrum: look for linear functional on functions of $z,{\bar {z}}$; such that

$\alpha [F_{\Delta ,\ell }]\geq 0$

except for identity which should yield 1. If we find such a functional then we get a contradiction with crossing symmetry and unitarity, for any choice of OPE coefficients. Set of useful functionals: combinations of derivatives at the crossing symmetric point $z={\bar {z}}={\frac {1}{2}}$. Vectors of derivatives of $F_{\Delta ,\ell }$, labelled by $\Delta ,\ell$. Positions of these vectors in space: can we find functional i.e. hyperplane, such that all vectors are one the same side? If yes, spectrum is inconsistent. If no, spectrum is feasible. We can then check a smaller spectrum.

There are algorithmes (semi-definite programming) for doing this.

Application to the Ising model

We know we have $\mathbb {Z} _{2}$ symmetry, an even scalar deformation i.e. spinless operator $\epsilon$, and an odd scalar deformation $\sigma$. Let us choose fusion rules for the OPE $\sigma \times \sigma$. By parity $\sigma$ cannot appear. We expect the identity, $\epsilon$, and extra scalars and non-scalars with higher dimensions, respecting unitarity. The ansatz is the same for $\epsilon \times \epsilon$. The OPE $\sigma \times \epsilon$ is different by parity, it should start with $\sigma$.

We write a crossing equation for $\langle \sigma ^{4}\rangle$. Check consistency as fct of the two dimensions $\Delta _{\sigma },\Delta _{\epsilon }$. We find allowed and disallowed regions.

The assumptions that we made so far are consistent with many theories: any non-trivial unitary minimal model would do, and there is an infinite series of such models. With free fields we also have a continuous line, i.e. $V_{\alpha }\times V_{\alpha }=1+V_{2\alpha }$. There is an allowed region in the plane, mininmal models sit on the boundary. The Ising model is a kink of the boundary. We did not use Virasoro symmetry. This method can be extended to higher dimensions.

In 3d we again find a kink that corresponds to the Ising model. If we add crossing symmetry for more 4pt functions, and restrict the spectrum by assuming there are no more operators of dimension less than 3, the allowed region shrinks to a small region around Ising. We obtain dimensions of our two fields with about 6 significant digits.