Some new exact critical-point amplitudes

The scaling properties of the free energy and some of universal amplitudes of a group of models belonging to the universality class of the quantum nonlinear sigma model and the O(n) quantum φ4 model in the limit n → ∞ as well as the quantum spherical model, with nearest-neighbor and long-range interactions (decreasing at long distances r as 1/rd+σ) is presented. For temperature driven phase transitions quantum effects are unimportant near critical points with Tc > 0. However, if the systems depends on another ”non thermal critical parameter” g, at rather low (as compared to characteristic excitations in the system) temperatures, the leading T dependence of all observables is specified by the properties of the zero-temperature (or quantum) critical point, say at gc. The dimensional crossover rule asserts that the critical singularities with respect to g of a d-dimensional quantum system at T = 0 and around gc are formally equivalent to those of a classical system with dimensionality d+ z (z is the dynamical critical exponent) and critical temperature Tc > 0. This makes it possible to investigate low-temperature effects (considering an effective system with d infinite spatial and z finite temporal dimensions) in the framework of the theory of finite-size scaling. A compendium of some universal quantities concerning O(n)-models at n → ∞ in the context of the finite-size scaling is presented. Casimir amplitudes in critical quantum systems Let us consider a critical quantum system with a film geometry L × ∞d−1 × Lτ , where Lτ = h̄/(kBT ) is the “finite-size” in the temporal (imaginary time) direction and let us suppose that periodic boundary conditions are imposed across the finite space dimensionality L (in the remainder we will set h̄ = kB = 1). The confinement of critical fluctuations of an order parameter field induces longranged force between the boundary of the plates [1, 2]. This is known as “statisticalmechanical Casimir force”. The Casimir force in statistical-mechanical systems is characterized by the excess free energy due to the finite-size contributions to the free energy of the bulk system. In the case it is defined as FCasimir(T, g, L|d) = − ∂f (T, g, L|d) ∂L , (1) where f (T, g, L|d) is the excess free energy f (T, g, L|d) = f(T, g, L|d)− Lf(T, g,∞|d). (2) 1 Here f(T, g, L|d) is the full free energy per unit area and per kBT , and f(T, g,∞|d) is the corresponding bulk free energy density. Then, near the quantum critical point gc, where the phase transition is governed by the non thermal parameter g, one could state that ( see, [3]) 1 L f (T, g, L|d) = (TLτ )LX ex(x1, ρ|d), (3) with scaling variables x1 = L δg, and ρ = L/Lτ . (4) Here ν is the usual critical exponent of the bulk model, δg ∼ g − gc, and X ex is the universal scaling function of the excess free energy. According to the definition (1), one gets F d Casimir(T, g, L) = (TLτ )L X Casimir(x1, ρ|d), (5) where X Casimir(x1, ρ|d) is the universal scaling functions of the Casimir force. It follows from Eq. (5) that depending on the scaling variable ρ one can define Casimir amplitudes ∆Casimir (ρ|d) := X Casimir (0, ρ|d) . (6) In addition to the “usual” excess free energy and Casimir amplitudes, denoted by the superscript “u”, one can define, in a full analogy with what it has been done above, “temporal excess free energy density” f ex t , f ex t (T, g, |d) = f(T, g,∞|d)− f(0, g,∞|d). (7) If the quantum parameter g is in the vicinity of gc, then one expects f ex t (T, g|d) = TL τ X ex (