Universal amplitude ratios in the two-dimensional Ising model

We use the results of integrable field theory to determine the universal amplitude ratios in the two-dimensional Ising model. In particular, the exact values of the ratios involving amplitudes computed at nonzero magnetic field are provided. Work supported by the European Union under contract FMRX-CT96-0012 Universality is one of the most fascinating concepts of statistical mechanics [1]. Briefly stated, it says that physical systems with different microscopic structure but having in common some basic internal symmetry exhibit the same critical behaviour in the vicinity of a phase transition point. The point is best illustrated considering the singular behaviour of the various thermodynamic quantities nearby the critical point. For a magnetic system exhibiting a second order phase transition the usual notation is C ≃ (A/α) τ , τ > 0 , h = 0 C ≃ (A/α) (−τ)−α′ , τ < 0 , h = 0 C ≃ (Ac/αc) |h|−αc , τ = 0 , h 6= 0 M ≃ B (−τ)β , τ < 0 , h = 0 χ ≃ Γ τ , τ > 0 , h = 0 χ ≃ Γ (−τ)−γ′ , τ < 0 , h = 0 χ ≃ Γc |h|−γc , τ = 0 , h 6= 0 h ≃ Dc M |M |δ−1 , τ = 0 , h 6= 0 ξ ≃ ξ0 τ , τ > 0 , h = 0 ξ ≃ ξ 0 (−τ)−ν ′ , τ < 0 , h = 0 ξ ≃ ξc |h|−νc , τ = 0 , h 6= 0 where τ = a (T − Tc) (a positive constant), h is the applied magnetic field and the limit towards the critical point τ = 0, h = 0 is understood. C, M , χ and ξ denote the specific heat, the magnetisation, the susceptibility and the correlation length, respectively. The critical exponents α, β,.. are the same for all systems within a given universality class and are related by the scaling and hyperscaling relations in d dimensions α = α , γ = γ , ν = ν ′ , γ = β(δ − 1) , α = 2− 2β − γ , 2− α = dν , αc = α/βδ , γc = 1− 1/δ , νc = ν/βδ . The critical amplitudes A, B,.., on the other hand, depend on the scale factors used for τ and h and are nonuniversal. However, universal ratios of amplitudes can be constructed in which any dependence on metric factors cancels out. Together with the critical exponents, these ratios further characterise the given universality class. The standard amplitude combinations considered in the literature are [2] A/A , Γ/Γ , ξ0/ξ ′ 0 , (1) RC = AΓ/B 2 , R ξ = A ξ0 , (2) Rχ = ΓDcB δ−1 , RA = AcD −(1+αc) c B −2/β , Q2 = (Γ/Γc)(ξc/ξ0) γ/ν . (3) 1 A substantial progress was made over the last years in the derivation of nonperturbative theoretical results in two-dimensional statistical mechanics and quantum field theory. The solution of conformal field theories (CFTs) [3, 4] provided an almost complete classification of universality classes for second order phase transitions in d = 2. In particular, it solved the problem of the exact determination of the critical exponents. The critical amplitudes, however, carry information about the scaling region outside the critical point and are not determined by CFT. In this respect, the Zamolodchikov’s observation that specific perturbations of the critical point lead to integrable off-critical theories [5] is of crucial importance. The integrable theories obtained in this way, regarded as quantum field theories in 1 + 1 dimensions, are characterisable through the determination of their exact S-matrix. A number of physical quantities can then be computed using different techniques [6]. In particular, the results provided by the thermodynamyc Bethe ansatz (TBA) [7] and the form factor approach [8, 9, 10, 11] enable the determination of amplitude ratios and have been used for this purpose in the problems of self-avoiding walks [12] and percolation [13] (see also [14]). It is the purpose of this note to illustrate the derivation of universal amplitude ratios from (integrable) field theory through the very basic example of the two-dimensional Ising model. Of course, the purely “thermal” ratios (1) and (2) are exactly known for this case since the seventies, when the correlation functions of the Ising model at h = 0 were first computed on the lattice [15, 16]. The possibility to determine the ratios (3), on the contrary, relies on the more recent realisation that the scaling limit of the Ising model at τ = 0 and h 6= 0 is an integrable theory [5]. We will regard the scaling limit of the two-dimensional Ising model as described by the euclidean field theory defined by the action S = SCFT − τ ∫