Exact Critical Exponents of the Staircase Model

The staircase model is a recently discovered [5] one-parameter family of integrable two-dimensional continuum field theories. We analyze the novel critical behavior of this model, seen as a perturbation of a minimal conformal theory Mp: the leading thermodynamic singularities are simultaneously governed by all fixed points Mp,Mp−1, . . . ,M3. The exponents of the magnetic susceptibility and the specific heat are obtained exactly. Various corrections to scaling are discussed, among them a new type specific to crossover phenomena between critical fixed points. PACS numbers: 5.70 Jk, 64.60 Kw, 64.60 Fr ∗ Electronic mail: iff299@DJUKFA11, [email protected] In the past few years, there has been renewed interest in exploring the consequences of integrability in two-dimensional statistical systems. At a critical point, an infinite number of integrals of motion appears if the theory is conformally invariant; one has even a partial classification of such universality classes [1]. For some perturbations away from criticality, a subset of these integrals survives and makes the theory solvable even at finite correlation length ξ. Two types of such systems with a generic (p− 1)critical point (p = 3, 4, . . .) described by the minimal conformal theory Mp have been known: (a) lattice models [2], whose manifold of integrability is parametrized about the fixed point Mp by a relevant temperature-like thermodynamic parameter as well as marginal and irrelevant parameters governing the lattice effects, and (b) exact factorizable scattering matrices for certain relevant perturbations of the conformal fixed points Mp [3]. These define massive continuum field theories that describe universal scaling behavior. The thermodynamic Bethe ansatz [4] is a way to calculate the universal ground state energy of the associated Hamiltonian on a circle of circumference R, E0(R, ξ) = 2π R f(R/ξ) . (1) The scaling function f(ρ) shows a simple crossover from the thermodynamic regime R ≫ ξ to the conformal regime R ≪ ξ; its ultraviolet limit is determined by the central charge c of the asymptotic conformal theory: limρ→0 f(ρ) = −c/12. The “staircase model” is a one-parameter family M(θ0) (θ0 > 0) of factorizable scattering theories with a new and more intricate scaling behavior, discovered very recently by Al.B. Zamolodchikov [5]. These theories contain a single type of massive particles that are characterized by the purely elastic S-matrix S(θ, θ0) = (sinh θ− i cosh 2θ0)/(sinh θ+ i cosh 2θ0), written in terms of the Lorentzinvariant rapidity difference θ. The scaling function f(ρ, θ0) shows a staircase pattern that interpolates between all central charges cp (see fig. 1). Hence the theory