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Journal of Statistical Mechanics Theory and Experiment

A renormalization group for the truncated conformal space approach

Giovanni Feverati, K Graham, Paul A. Pearce, Gábor Zsolt Tóth, G.M.T. Watts

Published Mar 18, 2008• 43 citations
Publisher

Abstract

We discuss the errors introduced by level truncation in the study of boundary renormalization group (RG) flows using the truncated conformal space approach (TCSA). We show that the TCSA results can have the qualitative form of a sequence of RG flows between different conformal boundary conditions. In the case of a perturbation by the field ϕ (13) , we propose a RG equation for the coupling constant which predicts a fixed point at a finite value of the TCSA coupling constant and we compare the predictions with data obtained using thermodynamic Bethe ansatz equations.

DOI
10.1088/1742-5468/2008/03/p03011

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We discuss the errors introduced by level truncation in the study of boundary renormalization group (RG) flows using the truncated conformal space approach (TCSA). We show that the TCSA results can have the qualitative form of a sequence of RG flows between different conformal boundary conditions. In the case of a perturbation by the field ϕ (13) , we propose a RG equation for the coupling constant which predicts a fixed point at a finite value of the TCSA coupling constant and we compare the predictions with data obtained using thermodynamic Bethe ansatz equations.
We discuss the errors introduced by level truncation in the study of boundary renormalisation group flows by the Truncated Conformal Space Approach. We show that the TCSA results can have the qualitative form of a sequence of RG flows between different conformal boundary conditions. In the case of a perturbation by the field phi(13), we propose a renormalisation group equation for the coupling constant which predicts a fixed point at a finite value of the TCSA coupling constant and we compare the predictions with data obtained using TBA equations.
Abstract The Truncated conformal space approach (TCSA) is a numerical technique for finding finite size spectrum of Hamiltonians in quantum field theory described as perturbations of conformal field theories. The truncation errors of the method have been systematically studied near the UV fixed point (when the characteristic energy related to the coupling is less than the truncation cutoff) where a good theoretical understanding has been achieved. However numerically the method demonstrated a good agreement with other methods for much larger values of the coupling when the RG flow approaches a new fixed point in the infrared. In the present paper we investigate this regime for a number of boundary RG flows testing the leading exponent and truncation errors. We also study the flows beyond the first fixed point which have been observed numerically but yet lack a theoretical understanding. We show that while in some models such flows approximate reversed physical RG flows, in other models the spectrum approaches a stable regime that does not correspond to any local boundary condition. Furthermore we find that in general the flows beyond the first fixed point are very sensitive to modifications of the truncation scheme.
In this paper we continue the study of the truncated conformal space approach to perturbed conformal field theories, this time applied to bulk perturbations and focusing on the leading truncation-dependent corrections to the spectrum. We find expressions for the leading terms in the ground state energy divergence, the coupling constant renormalisation and the energy rescaling. We apply these methods to problems treated in two seminal papers and show how these RG improvements greatly increase the predictive power of the TCSA approach. One important outcome is that the TCSA spectrum of excitations is predicted not to converge for perturbations of conformal weight greater than 3/4, but the ratios of excitation energies should converge.
Renormalization group (RG) flows for systems subject to an external field can be derived either by fixing the field or by fixing the conjugate order parameter. We show how to implement these boundary conditions within the functional renormalization group approach. We derive and compare the corresponding flow equations for ϕ4ϕ^4ϕ4 theory and show that some scaling variables and scaling exponents associated with the linearized flow of the relevant three-point vertex in the vicinity of the Ising fixed point depend on the boundary conditions. We also discuss the effect of elasticity on Ising criticality. In this case two of the fixed points (namely the renormalized Ising fixed point and the spherical fixed point) obtained for constant strain are shifted to physically inaccessible regimes of parameter space when we perform the RG at constant stress.
<!--HTML-->I give an introduction the Truncated Conformal Space Approach, which was originally invented by Yurov and Zamolodchikov in the framework of the scaling Lee-Yang model. Since then it has become a very powerful tool to study not only finite volume spectra, but also form factors, resonances, phase transitions and renormalization group flows in one-dimensional quantum field theories, both integrable and non-integrable. Its scope has been extended to theories with boundaries and impurities, and also generalized to settings beyond perturbed conformal field theories. In this talk, beyond explaining the basics of the approach I also give a brief overview of the recently introduced renormalization methods to deal with cut-off dependence, which led to a major improvement in precision and also widened the potential scope of applications.
Recently, Gaiotto (2012 arXiv:1201.0767) considered conformal defects which produce an expansion of infrared local fields in terms of the ultraviolet (UV) ones for a given renormalization group flow. In this paper, we propose that for a boundary RG flow in two dimensions there exist boundary condition changing fields (RG defect fields) linking the UV and the infrared (IR) conformal boundary conditions which carry similar information on the expansion of boundary fields at the fixed points. We propose an expression for a pairing between IR and UV operators in terms of a four-point function with two insertions of the RG defect fields. For the boundary flows in minimal models triggered by ψ 13 perturbation, we make an explicit proposal for the RG defect fields. We check our conjecture by a number of calculations done for the example of ( p , 2) → ( p − 1, 1)⊕( p + 1, 1) flows.