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Bulk Renormalization Group Flows and Boundary States in Conformal Field Theories

John Cardy

TL;DR

This work introduces smeared boundary states $e^{-\tau H}|\mathcal{B}\rangle$ as variational approximations to the ground state of conformal field theories perturbed by relevant bulk operators, connecting RG flows near CFT fixed points to boundary-state data and producing a tractable phase-diagram and bounds on universal free-energy contributions. The approach yields explicit formulas in 2d minimal models, relates bulk-boundary couplings to modular data, and reveals a simple fusion-rule structure for matrix elements between smeared Ishibashi states, via the Verlinde formula. Applied to the Ising and tricritical Ising models, it reproduces qualitative features of RG flows and boundary sectors, though it fails to capture true massless flows and tends to predict first-order transitions between competing sinks. The results illuminate the relationship between boundary conformal data and ground states of gapped theories, and hint at topological aspects of bulk-boundary dynamics through fusion rules.

Abstract

We propose using smeared boundary states $e^{-τH}|\cal B\rangle$ 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.

Bulk Renormalization Group Flows and Boundary States in Conformal Field Theories

TL;DR

This work introduces smeared boundary states as variational approximations to the ground state of conformal field theories perturbed by relevant bulk operators, connecting RG flows near CFT fixed points to boundary-state data and producing a tractable phase-diagram and bounds on universal free-energy contributions. The approach yields explicit formulas in 2d minimal models, relates bulk-boundary couplings to modular data, and reveals a simple fusion-rule structure for matrix elements between smeared Ishibashi states, via the Verlinde formula. Applied to the Ising and tricritical Ising models, it reproduces qualitative features of RG flows and boundary sectors, though it fails to capture true massless flows and tends to predict first-order transitions between competing sinks. The results illuminate the relationship between boundary conformal data and ground states of gapped theories, and hint at topological aspects of bulk-boundary dynamics through fusion rules.

Abstract

We propose using smeared boundary states 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.

Paper Structure

This paper contains 10 sections, 53 equations, 3 figures.

Table of Contents

  1. Introduction
  2. General formalism.
  3. Trace of the energy-momentum tensor.
  4. 2d minimal models
  5. The Ising model.
  6. Logarithmic anomaly.
  7. The tricritical Ising model.
  8. Matrix elements between Ishibashi states and the fusion rules.
  9. Conclusion.
  10. Funding information.

Figures (3)

  • Figure 1: Path integral for smeared boundary state (left) and its lattice discretization (right) as a matrix product state. On the right, each vertical column of lattice sites represents a matrix. The horizontal lines represent contractions between these in the internal space, and the vertical dangling bonds label the physical degrees of freedom.
  • Figure 2: Landau-Ginzburg assignment of bulk operators in the $A_4$ Kac table.
  • Figure 3: Correspondence between boundary conditions in lattice models and Kac labels of conformal boundary states: in the $A_4$ model according to Ref. PB (upper labels), and in the Blume-Capel model, according to Ref. Affleck (lower labels).