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

### - Published as SciPost Phys. 3, 011 (2017)

• Other versions of this Submission (with Reports) exist:

### Submission summary

 As Contributors: John Cardy Arxiv Link: http://arxiv.org/abs/1706.01568v4 Date accepted: 2017-08-02 Date submitted: 2017-07-31 Submitted by: Cardy, John Submitted to: SciPost Physics Domain(s): Theoretical Subject area: Mathematical Physics

### 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.

### List of changes

In response to Anonymous Report 1:

- Comparison with work of Fateev: sentence added at end that this would be interesting to do.
- on p.3, discussion of correspondence between RG sinks and boundary states in Ising expanded.
- after eqs (5,6) it is made clear when I am specializing to 1+1 dimensions.
- in sec.2, T and Tbar are defined.
- after eq, 6 it is explained in the text what is the direction of quantization, rather than introducing a new figure.
- in eq (26) the rescaling of E_a is stated explicitly
- typo after eq (23) corrected.

In response to Paul Fendley:

1. It should work in principle for all RCFTS but e.g even the boundary states haven't been worked out in general. I have inserted wording that there is no obstacle in principle, as far as I know.

2. Are 1-point functions known for boundary states in integrable cases? I don't think so.

3. Thanks, this point now emphasized and reference to Huse added at this point.

4. Levin et al state in their first sentence: `the Casimir force between parallel plates is attractive'. They then look at other geometries, not relevant to this work.

5. Thanks, I have now included discussion using Affleck's identification of boundary states, which I agree is more intuitive.

6. I have added to the caption, hopefully making what is a rather stretched comparison (which came from a comment from G Vidal) more comprehensible.

7. I added references to my 1989 paper and also a good review by Petkova and Zuber.

8. Thanks, yes, corrected.