Subdimensional Disorder and Logarithmic Defect

TL;DR

This work analyzes quenched disorder localized on a p-dimensional subspace within a d-dimensional CFT, showing that bulk operators remain ordinary primaries while defect operators organize into logarithmic multiplets, i.e., logarithmic defects. Using a Gaussian disorder and replica method in a free scalar theory, the authors identify a half-line of exactly marginal fixed points at codimension two ($p=d-2$) and demonstrate logarithmic correlation functions on the defect, highlighting logDOE structures. They develop a model-independent framework for logarithmic defect CFTs, including logarithmic defect operator expansions, defect conformal blocks, and bulk–defect bootstrap constraints, and instantiate this framework with explicit DOE data and correlators in the free-theory example. The results illuminate how subdimensional disorder can yield conformal defects with rich logarithmic structure, motivate RG monotones for disordered defect flows, and suggest broad generalizations to interacting CFTs and numerical explorations. Overall, the paper proposes a systematic, symmetry-driven program to bootstrap and understand logarithmic defects arising from subdimensional disorder in CFTs.

Abstract

We study quenched disorder localized on a $p$-dimensional subspacetime in a $d$-dimensional conformal field theory. Motivated by the logarithmic behavior often associated with disorder, we introduce a defect setup in which bulk local operators transform in ordinary conformal representations, while defect local operators assemble into logarithmic multiplets. We refer to such objects as logarithmic defects and investigate their model-independent properties dictated solely by conformal symmetry and its representation theory, including correlation functions, logarithmic defect operator expansions, and conformal blocks. As a concrete example, we analyze the free scalar theory with a generalized pinning defect subject to random coupling fluctuations, and we identify a half-line of fixed points describing the corresponding logarithmic conformal defects. Along the way, we propose a candidate monotone governing defect renormalization group flows induced by subdimensional disorder. We comment on various generalizations and the broader program of bootstrapping logarithmic defects.

Figures (5)

• Figure 1: Illustration of the defect setup studied in this paper. In the UV CFT, the correlation functions show usual power law, whereas in the IR defect CFT they develop logarithmic behavior due to subdimensional disorder.
• Figure 2: Sample diagrams contributing to the averaged defect free energy. There are no contributions at order $O(n)$.
• Figure 3: Schematic picture of logarithmic defect bootstrap equation.
• Figure 4: Feynman diagrams contributing to the bulk one-point function \ref{['eq:bulk_one_pt']}
• Figure 5: Contour deformation of the integral \ref{['eq:contor_integral']}.