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 \U0001d7191,3 flows.
Highlights
We find this quite an appealing feature of RG interfaces in 2D that they must respect the infinite dimensional conformal symmetry while at the same time they must carry information about the RG flow that produced them
In [7] a conformal interface between neighbouring Aseries minimal models was constructed that was conjectured to be the RG interface for the RG flow triggered by the φ1,3 primary that was shown to link the two CFTs [8]
The difficulty stems from the fact that a conformal interface by virtue of the folding trick [11] is equivalent to a conformal boundary condition in a tensor product of the two theories
Summary
We consider some formal aspects of RG operators starting with mappings of states between the perturbed and unperturbed theories. We see that when renormalisation effects are properly taken into account the operator ψ[0,λ](0) maps the perturbed theory energy eigenstates into the eigenstates of the perturbed Hamiltonian acting in the unperturbed state space. This property is rather formal and does not give a recipe for constructing these states in H0. Fixing the normalisation of ψ[0,λ] to be such that this field has a finite twopoint function at the fixed point gives a representation of the vacuum state as an element in the anti-dual state space (H0) Another approach to constructing the perturbed theory energy eigenstates is by using the Gell-Mann-Low formula based on the adiabatic switching of the interaction. At the level of perturbation theory such constructions, that use the interface operator, should be equivalent to the usual Rayleigh-Schrodinger perturbation theory for the Hamiltonian eigenvalues and eigenvectors
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