Elsevier

Physics Letters B

Volume 430, Issues 3–4, 2 July 1998, Pages 264-273
Physics Letters B

Truncated conformal space at c=1, nonlinear integral equation and quantization rules for multi-soliton states

https://doi.org/10.1016/S0370-2693(98)00543-7Get rights and content

Abstract

We develop truncated conformal space (TCS) technique for perturbations of c=1 conformal field theories. We use it to give the first numerical evidence of the validity of the non-linear integral equation (NLIE) derived from light-cone lattice regularization at intermediate scales. A controversy on the quantization of Bethe states is solved by this numerical comparison and by using the locality principle at the ultraviolet fixed point. It turns out that the correct quantization for pure hole states is the one with half-integer quantum numbers originally proposed by Fioravanti et al. [Phys. Lett. B 390 (1997) 243]. Once the correct rule is imposed, the agreement between TCS and NLIE for pure hole states turns out to be impressive.

Introduction

The scaling functions of 1+1 dimensional integrable models on the cylinder have proven to be a very useful non-perturbative tool of investigation of their finite size effects, in particular of the renormalization flow properties of the vacuum as well as of the excited states. Various methods have been proposed for the calculation of these important quantities, as the truncated conformal space (TCS) method [1]or the thermodynamic Bethe Ansatz (TBA) 2, 3. A very promising method is the one introduced some years ago by Destri and de Vega 4, 5(similar methods were independently introduced in condensed matter physics by other authors [6]). It consists in defining a lattice model and give evidence that its continuum limit reproduces the sine-Gordon QFT (sG) [7], then deriving a non-linear integral equation (NLIE) which is basically the continuum limit of the Bethe equations. There is one such equation for each excitation in the spectrum of the physical states. The problem is to give a general rule to write down the NLIE corresponding to a state characterized by a certain distribution of Bethe roots and holes. This problem, namely to find the NLIE for excited states, was first addressed, in a QFT context, in [8]which discussed the case of pure hole excitations. The general setup of excited states was described in detail later in [9]. The solution of the NLIE, the so-called counting function, is a central object in Bethe Ansatz approach to integrable QFT from which it is possible to reconstruct the eigenvalues of all the local conserved currents of the theory put on a cylinder of circumference L.

The aim of this paper is to present a numerical comparison between the NLIE and TCS. We develop a TCS for c=1 CFT perturbed by its cosβφ operator, that defines the sG model as perturbation of a CFT. We compare TCS data against scaling functions computed from numerical integration of NLIE and find a very good agreement, especially in the attractive regime, but also in the repulsive one. This agreement, however, only shows up for a specific choice of the quantization rule for the Bethe Ansatz states which is different from the one reported in [9]and instead agrees with the one made in Ref. [8]. This leads to the observation that the NLIE produces in fact more states than those present in the sG Hilbert space. It is known that the relation among the Hilbert space of c=1 CFT, the sG model and the massive Thirring (mTh) model is a quite delicate issue [10]. It is out of the scope of the present letter to investigate this very important question. We intend to return to it in a more extensive paper where a more detailed analysis of the results presented here will be given [11]. It is important to keep in mind that the NLIE (and in general Bethe Ansatz methods) are only able, at present, to reproduce states with even topological charge (i.e. even number of solitons minus antisolitons) in sG model.

Section snippets

The NLIE and its properties

The Lagrangian of sG theory is

L=dx12μΦ∂μΦ+M2β2:cosβΦ:.Often, it is convenient to use the parameter p=β28π−β2 or the 6-vertex anisotropy γ=πp+1.1

We do not reproduce here the deduction of the NLIE from the lattice Bethe equations. Although there are subtle remarks to do on all the derivations presented so far 5, 9, this

UV behaviour of multi-soliton states

We consider sG theory as the perturbation of a c=1 massless free boson Φ compactified on circle of radius R by a potential V=g2:(eiβΦ+e−iβΦ):, where g=M2β2and R=β. The interaction term has conformal dimensions ΔVV±=β2=pp+1 and becomes marginal when β2=8π which corresponds to p=∞. With the convention

Truncated conformal space at c=1

In this section we describe results obtained using the truncated conformal space (TCS) method which support the validity of the NLIE describing the vacuum and pure hole states.

The TCS method was originally created to describe perturbations of Virasoro minimal models in finite spatial volume [1]. We have developed an extension of the method to study perturbations of a c=1 compactified boson, more closely the perturbation corresponding to sG theory.

The perturbation keeps the winding number m and

TCS and NLIE in the attractive regime

We have made the comparison of the numerical data from TCS and the NLIE predictions at several values of the parameter p. For illustration, we present the case of p=27. For this and all other values of the coupling constant, we obtained a spectacular agreement between the results obtained by the two methods, up to deviations of order 10−4−10−3 which are smaller than the size of the points in the plots. The deviation grows with the volume L, exactly as expected for truncation errors. By studying

Conclusions

This letter should be seen as a preliminary report of the work done by our group to better investigate and clarify some of the properties of the NLIE in sG theory. We have given the first numerical check against truncated conformal space data – the only checks so far have been calculations of the conformal spectrum at small scale (kink limit) and comparison with the factorized scattering theory at large scale. The results presented here are the first evidence of the validity of NLIE in the

Acknowledgements

We are indebted to C. Destri, V.A. Fateev and E. Quattrini for useful discussions. This work was supported in part by NATO Grant CRG 950751, by European Union TMR Network FMRX-CT96-0012 and by INFN Iniziativa Specifica TO12. G.T. has been partially supported by the FKFP 0125/1997 and OTKA T016251 Hungarian funds.

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