User:Sylvain Ribault/Bootstat2021/Zirnbauer

Conformal field theory of the integer quantum Hall transitions: a status report.

The integer quantum Hall effect is observed in semi-conductors under strong magnetic field and low temperature. It is observed even in dirty samples, which still exhibit exact quantization. We ignore the Coulomb interaction, which is justified as disorder dominates. Varying the magnetic field, we observe transitions between plateaus. Scaling limits should be described by CFT, at the middle point between two plateaus. Hopefully, we can perturb the CFT and obtain a phenomenologically interesting theory.

Anderson transitions traditionally thought to be between metallic and insulator states. More recently, it was understood that in any dimension (2d, 3d) there are five transitions between strong topological insulators and band insulators. (With weak disorder, other cases arise.) The 2d integer quantum Hall transition is one of these transitions. It is the only CFT we are beginning to understand.

Hyperbolic symmetry

Singe-particle Green's fct encode observable quantities. Retarded, advanced Green's fct:

$G_{\pm }^{E}(x',x)=\det(\epsilon \mp i(E-H))\int D\phi \phi _{\pm }(x'){\bar {\phi }}_{\pm }(x)e^{\pm i\int {\bar {\phi }}_{\pm }(E\pm i\epsilon -H)\phi _{\pm }}$

We can have a number $r$ of replicas. Wegner's hyperbolic symmetry: in the limit $\epsilon \to 0^{+}$ we have the sesquilinear form

${\bar {\phi }}_{+}\phi _{+}-\phi _{-}{\bar {\phi }}_{-}+{\bar {\psi }}_{+}\psi _{+}+\psi _{-}{\bar {\psi }}_{-}$

which defines supergroup $U(r,r|2r)$.

Upper critical dimension

Local density of states is discontinuity of Green's fct across real axis and is positive:

$\mathbb {E} \left({\frac {G_{-}^{E}(x,x)-G_{+}^{E}(x,x)}{2\pi i}}\right)=\left\langle |\phi _{-}|^{2}+|\phi _{+}|^{2}\right\rangle \geq 0$

Get a quartic interaction from averaging over disorder. We obtain a bare Lagrangian (where we omit fermions)

$L=-i{\bar {\phi }}_{+}(\nabla ^{2}+E)\phi _{+}+i{\bar {\phi }}_{-}(\nabla ^{2}+E)\phi _{-}+\lambda ^{2}(|\phi _{+}|^{2}-|\phi _{-}|^{2})^{2}$

Can this be treated in mean field in the fashion of Landau for some dimensions? Consider orbits of symmetry group $SU(1,1)$. These orbits are hyperboloids. If the group was compact the field could localize at zero at high temperature, and the symmetry would be unbroken. Here, in an Anderson transition, the critical behaviour is not in the density of states, and the important role is not played by zero field. So the mean field scenario cannot apply.

In systems with a large local density of states, we can derive nonlinear sigma model, whose target is a group coset $U/K$ with $U=U(r,r|r+r)$. Conventional scenario of Anderson transition: in the metallic state (extended states), the symmetry $U$ is broken spontaneously. In the insulator state (localized states), it is restored. In this traditonal wisdom, there is no upper critical dimension.

Consider a system with a low density of states. We cannot derive the conventional sigma model. Low density means high disorder, short elastic mean free path, large $\lambda$. So the hyperbolic orbit that matters is the light cone $|\phi _{+}|^{2}=|\phi _{-}|^{2}$. The target space (after bosonization) becomes nilpotent, and the field obeys $Q^{2}=0$. Mobility along light ray, inertia transverse to it, leading to spontaneous symmetry breaking. The field gets stuck on a light ray, which is the target space of a WZW model $GL(r|r)$ with level $n=4$ and marginal deformation $\gamma =1$. This is not a Lie supergoup target, but a Riemannian symmetric superspace. The central charge of the CFT vanishes. The WZW field has dimension zero but is not the identity. The WZW field is identified with the local density of states. This CFT matches observations. The volume of the target is one, the volume of a Lie supergroup would be zero. Here, we have fewer indecomposable representations.

Integrating out the fermions, one gets Gaussian free field with background charge.

Prospects for conformal bootstrap

The vacuum of the theory is a constant function on an hyperboloid so it cannot be integrated.

Primary fields are vertex operators with scaling dimensions

$\Delta _{p,q}={\frac {1}{n}}(p(p-1)-q(q-1))+{\frac {1-\gamma }{n^{2}}}(p^{2}+q^{2})$

with $p\in \mathbb {N}$ and $q\in {\frac {1}{2}}+i\mathbb {R}$ so that dimension is positive. Actually we have $\gamma =1$.

Comparison with Liouville theory: three-point functions in Liouville theory exist because the zero-mode integral exists. In our case, due to the fermions, the similar integral diverges for three vertex operators. We need any correlation function to include a point contact.

A point contact is a Dirac delta in the WZW field, whereas vertex operators are exponentials.

We have been focusing on the $r=1$ theory, in principle we want to understand other values of $r$, all these models are supposed to be part of one larger CFT.