# Bulk Renormalization Group Flows and Boundary States in Conformal Field Theories

## TL;DR

This paper introduces a variational approach using smeared boundary states to approximate ground states in conformal field theories deformed by relevant operators, providing insights into phase diagrams and universal amplitudes.

## Contribution

It proposes a novel method employing smeared boundary states for analyzing RG flows and boundary states in CFTs, with explicit results for 2D minimal models.

## Key findings

- Provides a criterion for boundary state selection based on bulk operators.
- Derives a phase diagram near the RG fixed point of the CFT.
- Establishes bounds on the universal free energy amplitude.

## Abstract

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.