D-dimensional Spherical Model Research Articles

The imperfect Bose gas in d dimensions: critical behavior and Casimir forces

We consider the d-dimensional imperfect (mean-field) Bose gas confined in a slit-like geometry and subject to periodic boundary conditions. Within an exact analytical treatment we first extract the bulk critical properties of the system at Bose–Einstein condensation and identify the bulk universality class to be the one of the classical d-dimensional spherical model. Subsequently we consider finite slit width D and analyze the excess surface free energy and the related Casimir force acting between the slit boundaries. Above the bulk condensation temperature (T > Tc) the Casimir force decays exponentially as a function of D, with the bulk correlation length determining the relevant length scale. For T = Tc and for T < Tc its decay is algebraic. The magnitude of the Casimir forces at Tc and for T < Tc is governed by the universal Casimir amplitudes. We extract the relevant values for different d and compute the scaling functions describing the crossover between the critical and low-temperature asymptotics of the Casimir force. The scaling function is monotonic at any d∈(2,4) and becomes constant for d > 4 and T ≤ Tc.

Response of non-equilibrium systems with long-range initial correlations

The long-time dynamics of the d-dimensional spherical model with a non-conserved order parameter and quenched from an initial state with long-range correlations is studied through the exact calculation of the two-time autocorrelation and autoresponse functions. In the ageing regime, these are given in terms of non-trivial universal scaling functions of both time variables. At criticality, five distinct types of ageing are found, depending on the form of the initial correlations, while at low temperatures only a single type of ageing exists. The autocorrelation and autoresponse exponents are shown to be generically different and to depend on the initial conditions. The scaling form of the two-time response functions agrees with a recent prediction coming from conformal invariance.

Lattice two-point functions and conformal invariance

A new realization of the conformal algebra is studied which mimics the behaviour of a statistical system on a discrete albeit infinite lattice. The two-point function is found from the requirement that it transforms covariantly under this realization. The result is in agreement with explicit lattice calculations of the (1 + 1)-dimensional Ising model and the d-dimensional spherical model. A hard core is found which is not present in the continuum. For a semi-infinite lattice, profiles are also obtained.

Critical endpoints in the spherical model with rapidly decaying power-law interactions

We study the d-dimensional spherical model with a rapidly decaying power-law interaction J( R) ∼ 1/ R d + σ with σ > 2( d + 2) /3, introducing a potential W( s) = Ds 2 − | U | s 4 + Vs 6 at each site. The phase diagrams for different values of d and σ are shown to exhibit nonclassical critical lines terminating at critical endpoints. We show that, although these lines exhibit the usual spherical model leading order critical behavior independent of the specific range of σ, significant σ-dependent correction-to-scaling terms are present when the interaction decays fast (with σ ≈ 2).

A study of the asymptotic behavior of the D-dimensional Ising model

A general method is introduced for studying the asymptotic behavior of the D -dimensional Ising model. In our approach, the continuum representation of the Ising partition function is analyzed by using bifurcation theory in conjunction with well-known asymptotic methods of analysis. Following a suggestion of Gallavotti, it is shown rigorously that in the range H ( = J/kT) < 1 4 D the gaussian model yields the correct asymptotic behavior of the Ising model, and there is no phase transition in this range. For the range H > 1 4 D, the behavior of the Ising model is obtained bu “bifurcating” from the gaussian model, and the solution in this range is found to be very similar in structure to the one obtained for the corresponding D-dimensional spherical model. We find bifurcation in all dimensions, no phase transition in one-dimension, but a phase transition in all D ⩾ 2. For D = 2, the phase transition is characterized by a logarithmic singularity in the specific heat (in accordance with the exact behavior found by Onsager). For D = 3, the transition is characterized by an algebraic singularity in the specific heat (in particular, the singularity is (λ - λ c) -1 2 where λ is a parameter related to, but not identical with, the temperature). With regard to quantitative estimates, since in this paper the bifurcation analysis was carried through only to order ε 2, where ε is the bifurcation parameter, our estimates of the transition temperature in 2 and 3 dimensions are somewhat displaced from the accepted values. Finally, the possible relevance of our approach to other problems of interest in the theory of phase transitions is indicated.