A d-dimensional quantum spherical model with a long-range interaction (decreasing at long distances r as r−d−σ, 0 <σ ≤ 2) is considered close to its quantum critical point. The low-temperature behaviors of the entropy, the internal energy and the specific heat capacity are studied for σ/2 < d < 3σ/2 in the context of the finite-size scaling (FSS) theory. For a σ-dimensional system and a d-dimensional system with short-range interaction (σ = 2) the results for the corresponding critical amplitudes as functions of the dimensionality of the system are presented.
We propose a new class of universal amplitude ratios which involve the first terms of the short distance expansion of the correlators of a statistical model in the vicinity of a critical point. We will describe the critical system with a conformal field theory (UV fixed point) perturbed by an appropriate relevant operator. In two dimensions the exact knowledge of the UV fixed point allows for accurate predictions of the ratios and in many nontrivial integrable perturbations they can even be evaluated exactly. In three-dimensional O ( N ) scalar systems, feasible extensions of some existing results should allow perturbative expansions for the ratios to be obtained. By construction these universal ratios are a perfect tool to explore the short distance properties of the underlying quantum field theory even in regimes where the correlation length and one-point functions are not accessible in experiments or simulations.
We investigate a quantum version of the spherical model which is obtained from the classical Berlin-Kac spherical model by a simple canonical quantization scheme. We find a complete solution of the model for short-range as well as for long-range interactions. At finite temperatures the critical behavior is the same as in the classical spherical model whereas at zero temperature we find a quantum phase transition characterized by new critical exponents. Based on a functional-integral representation of the partition function the free energy of the model is shown to be equivalent to that of the nonlinear \ensuremath{\sigma} model in the limit of infinite order-parameter dimensionality. \textcopyright{} 1996 The American Physical Society.
We study d-dimensional ferromagnetic spherical models with power-law, J(R)\ensuremath{\sim}1/${\mathit{R}}^{\mathit{d}+\mathrm{\ensuremath{\sigma}}}$ (\ensuremath{\sigma}2), or short-range (\ensuremath{\sigma}\ensuremath{\simeq}2) interactions and on-site terms of the form
Ds2-\ensuremath{\Vert}U\ensuremath{\Vert}
s4+
Vs6, and establish that the phase diagrams for all d and \ensuremath{\sigma} with \ensuremath{\sigma}d2\ensuremath{\sigma}, may exhibit critical end points terminating nonclassical critical lines; tricritical points may arise only for 3/2\ensuremath{\sigma}\ensuremath{\le}d. The phase boundary ${\mathit{D}}_{\mathrm{\ensuremath{\sigma}}}$(T,h), where h is the magnetic field, between the noncritical, spectator phase and the ordering and critical phase is shown to exhibit singularities at the end point with amplitudes obeying the universal ratio laws advanced on phenomenological grounds in Part I.
The phase diagram in the vicinity of a Lifshitz point is analyzed using scaling arguments and renormalization-group calculations. Feynman-graph techniques in $d=4+\frac{1}{2}m\ensuremath{-}\ensuremath{\epsilon}$ dimensions are used to calculate the universal amplitude ratio governing the shape of the two branches of the critical line which intersect at an
m-fold Lifshitz point. For
m<6 the amplitude ratio is found to be $\frac{{A}_{H}(m)}{{A}_{F}(m)}=1\ensuremath{-}\frac{2}{m}+O(\ensuremath{\epsilon})$, where
AH(AF) is the amplitude associated with the helical (ferromagnetic) branch, so that to zeroth order in $\ensuremath{\epsilon}$, this ratio is independent of the number
n of components of the order parameter. The calculation indicates that along the remainder (
6< m< 8) of the critical dimensionality line, $\ensuremath{\epsilon}=0$, no helical long-range order is expected. We also calculate
AFAH for a spherical model of a
d-dimensional system exhibiting an
m-fold Lifshitz point. Along the critical dimensionality line $\ensuremath{\epsilon}=0$, the spherical model result reduces to that of the renormalization-group calculation.