Correlation functions in the two-dimensional Ising model in a magnetic field at T = Tc

Abstract

The one and two-particle form factors of the energy operator in the two-dimensional Ising model in a magnetic field at T = T c are exactly computed within the form factor bootstrap approach. Together with the matrix elements of the magnetisation operator already computed in ref. [7], they are used to write down the large distance expansion for the correlators of the two relevant fields of the model.The last years have seen important progresses in the non-perturbative study of twodimensional quantum field theories and related statistical mechanical models. If conformal symmetry provided us with an exact description of critical points and universality classes [1,2], the study of off-critical models turned out to be better approached within the framework of relativistic scattering theory. In fact, if the off-critical model under consideration is integrable (i.e. admits infinite conservation laws), it can be usually solved exploiting very general bootstrap techniques [3,4,5,6]. This circumstance appears to be particularly important in light of the fact that a large number of physically interesting two-dimensional systems can actually be described in terms of integrable models. A remarkable example is provided by the scaling limit of the two-dimensional Ising model in a magnetic field at T = T c (IMMF in the sequel) [4]. It can be formally described by the actionwhere A CF T denotes the action of the conformal minimal model M 3,4 and σ(x) the magnetisation operator of scaling dimension 2∆ σ = 1/8. The coupling constant h (magnetic field) has physical dimension h ∼ m 15/8 , m being a mass scale. Apart form the magnetisation operator, the only other relevant scaling field in the Ising model is the energy density ε(x) with scaling dimension 2∆ ε = 1.Zamolodchikov showed that the theory described by the action (1) possesses an infinite number of integrals of motion which can be used in order to determine the exact particle spectrum and S-matrix of the theory [4]. He found that the spectrum consists of eight massive particles A a (a = 1, 2, . . . , 8) whose masses stay in the following ratios with the mass m 1 of the lightest particle The interaction between these particles is described by a factorised, reflectionless S-matrix 1 characterised by the two-particle amplitudesThe set of numbers A ab and the multiplicity factors µ α are given in Table 1. We use the standard rapidity parameterisation of the on-shell momenta pIn ref.[7] the knowledge of the S-matrix (3) was exploited in order to approach the computation of the correlation functions of the model (1) within the form factor bootstrap method. The basic idea of this approach is to express the (euclidean) correlation functions (e.g. the two-point ones) as a spectral sum over a complete set of intermediate multiparticle statesand to exploit the fact that the form factors (FF)are exactly computable in integrable models once the S-matrix is known [8,6]. The spectral series (4) is manifestly a large distance expansion. Nevertheless, it has been observed in several...

First-order phase transitions and integrable field theory. The dilute q-state Potts model

“…We have shown how integrability allows an exact scattering description of the scaling limit and how this leads 7 An example of this mechanism is obtained specialising Eq. to the computation of correlation functions for the interesting operators through the form factor approach.…”

“…The situation is more subtle when Φ = σ i . In fact, although the number of terms in the primed sum vanishes as q → 1, now the sum contains contributions which diverge in the same limit 7 . In this way, the correlator Ω i |Θ(x)σ i (0)|Ω i q=1 receives a finite contribution even from excitations which are unphysical at q = 1.…”

“…For these models, integrability seems to emerge as a property of the scaling limit, after that the non-universal, lattice dependent features have been eliminated. Among the examples of this kind which have been studied in more detail, we mention the Ising model in a magnetic field at T = T c [1,6,7], the off-critical q-state Potts model with 0 < q ≤ 4 [8,9], and the off-critical Ashkin-Teller model [10].…”

Matrix elements of the operator in integrable quantum field theory

“…If the correspondence between primary operators and minimal form factor solutions is many times unambiguous, even the identification of primaries can become non-trivial in absence of internal symmetries: the Ising model without [11] and with a magnetic field [12,13,14] provides the simplest illustration of the two situations. It was shown in [12] for unitary theories that the scaling dimension of the operator determines a very restrictive upper bound on the asymptotic behavior of the form 1 factors.…”

“…Being non-linear in the operators the factorization equations single out specific solutions within the linear subspaces selected by the asymptotic bound and have been exploited for the identification of primary operators, in particular in the above mentioned case of the Ising model in a magnetic field [13,14].…”