The TT‾T\overline T deformation of quantum field theory as random geometry

We revisit the results of Zamolodchikov and others on the deformation of two-dimensional quantum field theory by the determinant $\det T$ of the stress tensor, commonly referred to as $T\overline T$. Infinitesimally this is equivalent to a random coordinate transformation, with a local action which is, however, a total derivative and therefore gives a contribution only from boundaries or nontrivial topology. We discuss in detail the examples of a torus, a finite cylinder, a disk and a more general simply connected domain. In all cases the partition function evolves according to a linear diffusion-type equation, and the deformation may be viewed as a kind of random walk in moduli space. We also discuss possible generalizations to higher dimensions.Comment: 32 pages. Final published version! Solution for t>0 clarifie

Logarithmic conformal field theories as limits of ordinary CFTs and some physical applications

We describe an approach to logarithmic conformal field theories as limits of sequences of ordinary conformal field theories with varying central charge c. Logarithmic behaviour arises from degeneracies in the spectrum of scaling dimensions at certain values of c. The theories we consider are all invariant under some internal symmetry group, and logarithmic behaviour occurs when the decomposition of the physical observables into irreducible operators becomes singular. Examples considered are quenched random magnets using the replica formalism, self-avoiding walks as the n->0 of the O(n) model, and percolation as the limit Q->1 of the Potts model. In these cases we identify logarithmic operators and pay particular attention to how the c->0 paradox is resolved and how the b-parameter is evaluated. We also show how this approach gives information on logarithmic behaviour in the extended Ising model, uniform spanning trees and the O(-2) model. Most of our results apply to general dimensionality. We also consider massive logarithmic theories and, in two dimensions, derive sum rules for the effective central charge and the b-parameter.Comment: 37 pages. v2: minor corrections and additions. Submitted to Special Issue of J. Phys. A on Logarithmic CF

Quantum Network Models and Classical Localization Problems

A review is given of quantum network models in class C which, on a suitable 2d lattice, describe the spin quantum Hall plateau transition. On a general class of graphs, however, many observables of such models can be mapped to those of a classical walk in a random environment, thus relating questions of quantum and classical localization. In many cases it is possible to make rigorous statements about the latter through the relation to associated percolation problems, in both two and three dimensions.Comment: 23 pages. To appear in '50 years of Anderson Localization', E Abrahams, ed. (World Scientific)

Linking numbers for self-avoiding walks and percolation: application to the spin quantum Hall transition

Non-local twist operators are introduced for the O(n) and Q-state Potts models in two dimensions which, in the limits n -> 0 (resp. Q -> 1) count the numbers of self-avoiding loops (resp. percolation clusters) surrounding a given point. This yields many results, for example the distribution of the number of percolation clusters which must be crossed to connect a given point to an infinitely distant boundary. These twist operators correspond to (1,2) in the Kac classification of conformal field theory, so that their higher-point correlations, which describe linking numbers around multiple points, may be computed exactly. As an application we compute the exact value \sqrt 3/2 for the dimensionless conductivity at the spin Hall transition, as well as the shape dependence of the mean conductance in an arbitrary simply connected geometry with two extended edge contacts.Comment: 4 pages, 3 figures; final version as will appear in PR

The Legacy of Ken Wilson

This is a brief account of the legacy of Ken Wilson in statistical physics, high energy physics, computing and education.Comment: Written version of a talk given at the Ken Wilson Memorial Session, StatPhys 25, Seoul, July 2013. To appear in the conference proceedings in J. Stat. Mec

Crossing Formulae for Critical Percolation in an Annulus

An exact formula is given for the probability that there exists a spanning cluster between opposite boundaries of an annulus, in the scaling limit of critical percolation. The entire distribution function for the number of distinct spanning clusters is also given. These results are found using Coulomb gas methods. Their forms are compared with the expectations of conformal field theory.Comment: 7 pages, 1 figure; v.2,3: minor corrections; v.4: published versio

Bulk Renormalization Group Flows and Boundary States in Conformal Field Theories

We propose using smeared boundary states $e^{-\tau H}|\cal B\rangle$ as variational approximations to the ground state of a conformal field theory deformed by relevant bulk operators. This is motivated by recent studies of quantum quenches in CFTs and of the entanglement spectrum in massive theories. It gives a simple criterion for choosing which boundary state should correspond to which combination of bulk operators, and leads to a rudimentary phase diagram of the theory in the vicinity of the RG fixed point corresponding to the CFT, as well as rigorous upper bounds on the universal amplitude of the free energy. In the case of the 2d minimal models explicit formulae are available. As a side result we show that the matrix elements of bulk operators between smeared Ishibashi states are simply given by the fusion rules of the CFT.Comment: 17 pages, 3 figures. v3: Reference to related work added; analysis of minimal models clarified; reformatted to conform with SciPost submission guidelines. v4: discussion of tricritical Ising expanded; minor improvements and added references suggested by referee