Properties of RG interfaces for 2D boundary flows

TL;DR

<p>We address the problem of characterising renormalisation group interfaces (RG interfaces) for 2D boundary flows by formulating conjectures that RG operators are conformal primaries and whose OPEs encode both the UV perturbation and possible IR leading irrelevant operators. The paper develops a multi-pronged approach: perturbative analysis near nearby fixed points, a Cardy-inspired variational method for boundary flows, and numerical checks via boundary TCSA in the tricritical Ising model. Key contributions include explicit calculations supporting the conjectures, a systematic variational framework for identifying IR endpoints and RG operators, and a detailed account of how RG interfaces transport states and operators across fixed points, including Gaiotto’s pairing and scaling eigenvectors. The results offer a coherent picture linking RG interface data to the space of boundary RG flows and provide tools that may generalise to bulk flows, with potential applications to constraining possible IR endpoints of given UV perturbations.</p>

Abstract

We consider RG interfaces for boundary RG flows in two-dimensional QFTs. Such interfaces are particular boundary condition changing operators linking the UV and IR conformal boundary conditions. We refer to them as RG operators. In this paper we study their general properties putting forward a number of conjectures. We conjecture that an RG operator is always a conformal primary such that the OPE of this operator with its conjugate must contain the perturbing UV operator when taken in one order and the leading irrelevant operator (when it exists) along which the flow enters the IR fixed point, when taken in the other order. We support our conjectures by perturbative calculations for flows between nearby fixed points, by a non-perturbative variational method inspired by the variational method proposed by J.~Cardy for massive RG flows, and by numerical results obtained using boundary TCSA. The variational method has a merit of its own as it can be used as a first approximation in charting the global structure of the space of boundary RG flows. We also discuss the role of the RG operators in the transport of states and local operators. Some of our considerations can be generalised to two-dimensional bulk flows, clarifying some conceptual issues related to the RG interface put forward by D.~Gaiotto for bulk $φ_{1,3}$ flows.

Figures (12)

• Figure 1: The boundary condition changing operator linking perturbed and unperturbed boundary conditions
• Figure 2: Norm squared of the perturbed theory vacuum represented as a short-distance limit of a two-point function of the RG operator
• Figure 3: A renormalised local operator in the perturbed theory can be obtained by surrounding a local operator in the UV theory by two interface operators, taking the limit when all 3 operators are at the same point, and subtracting divergences.
• Figure 4: A renormalised local bulk operator in the perturbed bulk theory can be obtained by surrounding a local operator in the UV theory by the deformation interface put on a circle of radius $\epsilon$ which is sent to zero and subtracting divergences.
• Figure 5: Two point function of the boundary condition changing operator is equivalent to perturbing the boundary condition on an interval