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arXiv:2401.06818 · Journal of High Energy Physics

A bootstrap study of minimal model deformations

António Antunes, Edoardo Lauria, Balt C. van Rees

Published Jan 1, 2023• 1 citation
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Abstract

For QFTs in AdS the boundary correlation functions remain conformal even if the bulk theory has a scale. This allows one to constrain RG flows with numerical conformal bootstrap methods. We apply this idea to flows between two-dimensional CFTs, focusing on deformations of the tricritical and ordinary Ising model. We provide non-perturbative constraints for the boundary correlation functions of these flows and compare them with conformal perturbation theory in the vicinity of the fixed points. We also reproduce a completely general constraint on the sign of the $T\bar T$ deformation in two dimensions.

DOI
10.3204/pubdb-2023-05857

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For QFTs in AdS the boundary correlation functions remain conformal even if the bulk theory has a scale. This allows one to constrain RG flows with numerical conformal bootstrap methods. We apply this idea to flows between two-dimensional CFTs, focusing on deformations of the tricritical and ordinary Ising model. We provide non-perturbative constraints for the boundary correlation functions of these flows and compare them with conformal perturbation theory in the vicinity of the fixed points. We also reproduce a completely general constraint on the sign of the TTˉT\bar TTTˉ deformation in two dimensions.
A bstract For QFTs in AdS the boundary correlation functions remain conformal even if the bulk theory has a scale. This allows one to constrain RG flows with numerical conformal bootstrap methods. We apply this idea to flows between two-dimensional CFTs, focusing on deformations of the tricritical and ordinary Ising model. We provide non-perturbative constraints for the boundary correlation functions of these flows and compare them with conformal perturbation theory in the vicinity of the fixed points. We also reproduce a completely general constraint on the sign of the TT‾T\overline{T }TT deformation in two dimensions.
Abstract We consider perturbations of 2D CFTs by multiple relevant operators. The massive phases of such perturbations can be labeled by conformal boundary conditions. Cardy’s variational ansatz approximates the vacuum state of the perturbed theory by a smeared conformal boundary state. In this paper we study the limitations and propose generalisations of this ansatz using both analytic and numerical insights based on TCSA. In particular we analyse the stability of Cardy’s ansatz states with respect to boundary relevant perturbations using bulk-boundary OPE coefficients. We show that certain transitions between the massive phases arise from a pair of boundary RG flows. The RG flows start from the conformal boundary on the transition surface and end on those that lie on the two sides of it. As an example we work out the details of the phase diagram for the Ising field theory and for the tricritical Ising model perturbed by the leading thermal and magnetic fields. For the latter we find a pair of novel transition lines that correspond to pairs of RG flows. Although the mass gap remains finite at the transition lines, several one-point functions change their behaviour. We discuss how these lines fit into the standard phase diagram of the tricritical Ising model. We show that each line extends to a two-dimensional surface ξσ,c in a three coupling space when we add perturbations by the subleading magnetic field. Close to this surface we locate symmetry breaking critical lines leading to the critical Ising model. Near the critical lines we find first order phase transition lines describing two-phase coexistence regions as predicted in Landau theory. The surface ξσ,c is determined from the CFT data using Cardy’s ansatz and its properties are checked using TCSA numerics.
The boundary entropy log(g) of a critical one-dimensional quantum system (or two-dimensional conformal field theory) is known to decrease under renormalization group (RG) flow of the boundary theory. We study instead the behavior of the boundary entropy as the bulk theory flows between two nearby critical points. We use conformal perturbation theory to calculate the change in g due to a slightly relevant bulk perturbation and find that it has no preferred sign. The boundary entropy log(g) can therefore increase during appropriate bulk flows. This is demonstrated explicitly in flows between minimal models. We discuss the applications of this result to D-branes in string theory and to impurity problems in condensed matter.
We develop the theory of Hamiltonian Truncation (HT) to systematically study RG flows that require the renormalization of coupling constants. This is a necessary step towards making HT a fully general method for QFT calculations. We apply this theory to a number of QFTs defined as relevant deformations of d=1+1 CFTs. We investigated three examples of increasing complexity: The deformed Ising, Tricritical-Ising, and non-unitary minimal model M(3,7). The first two examples provide a crosscheck of our methodologies against well established characteristics of these theories. The M(3,7) CFT deformed by its \mathbb{Z}_2 ℤ 2 -even operators shows an intricate phase diagram that we clarify. At a boundary of this phase diagram we show that this theory flows, in the IR, to the M(3,5) CFT.
The boundary entropy log(g) of a critical one-dimensional quantum system (or two-dimensional conformal field theory) is known to decrease under renormalization group (RG) flow of the boundary theory. We study instead the behavior of the boundary entropy as the bulk theory flows between two nearby critical points. We use conformal perturbation theory to calculate the change in g due to a slightly relevant bulk perturbation and find that it has no preferred sign. The boundary entropy log(g) can therefore increase during appropriate bulk flows. This is demonstrated explicitly in flows between minimal models. We discuss the applications of this result to D-branes in string theory and to impurity problems in condensed matter.
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.