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.
At a critical end point a critical phase coexists with a noncritical phase \ensuremath{\alpha} and singularities arise beyond those on the associated critical line. New universal amplitude ratios are defined for the shape of the \ensuremath{\alpha}-phase boundary and for the noncritical surface tensions near end points. A postulated correspondence with wall/surface criticality leads to predictions for general dimension d that are exact at (and near) d=2 and for d\ensuremath{\ge}4 (where mean-field theory applies) and yield experimentally testable estimates for d=3.
Thermodynamic functions in the vicinity of an ordinary critical point are expected to obey asymptotic scaling laws as $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$ and the ordering field,
h, approach zero. However, the optimal scaling variables are the nonlinear scaling fields, ${g}_{t}=t+{b}_{t}{h}^{2}+{c}_{t}{t}^{2}+\ensuremath{\cdots}$ and ${g}_{h}=h(1+{c}_{h}t+\ensuremath{\cdots})$. The nonlinearities yield correction factors to the leading power-law (and scaling) variation of thermodynamic quantities,
L, of the form ($1+{a}_{L}t+{b}_{L}{t}^{2}+\ensuremath{\cdots}$), etc., where the correction amplitudes
aL,bL,…, are uniquely determined by the nonlinear scaling-field coefficients. It follows that "analytic" corrections to, e.g., the susceptibility, are directly related to those for the free energy and magnetization (in zero field). The term
bth2 also generates nonanalytic contributions such as an additive, energylike term, varying as ${|t|}^{1\ensuremath{-}\ensuremath{\alpha}}$ in the zero-field susceptibility, and factors like ($1+{c}_{L}{|h|}^{2\ensuremath{-}\frac{1}{\ensuremath{\Delta}}}$) on the critical isotherm,
t=0. Irrelevant scaling fields yield further, in general distinct, nonanalytic corrections, and cause shifts in
Tc and various amplitudes although "universal" ratios remain constant.
This article outlines a tutorial review of some recent progress in the theory of phase transitions and criticality. Topics mentioned are: |φ|6 field theories in (d=3)-dimensional space which correspond to the limits of tricriticality in spherical models; the desirability of finding real systems displaying critical endpoints, etc., in a magnetic field; scaling theories for first-order magnetic transitions in block and cylinder geometries, including spin-wave contributions, ϵ=4 - d expansions for finite-size effects at and near criticality; the universal correlation length amplitude for critical-point correlations in a finite cylinder and its evaluation in d = 2 dimensions on the basis of conformal covariance; the predictions of conformal covariance for other finite-size and surface critical-point properties; the implications, stemming from field theory, of conformal covariance and unitarity for general criticality in two dimensions, including infinitely many sets of exact critical exponents for Ising-like, three-state Potts-like, … systems; the discovery of new, exactly soluble two-dimensional, “restricted solid-on-solid” models which display all the predicted conformally covariant multicritical points and a second infinite set as well!
Phase transitions are conventionally defined by nonanalyticities of thermodynamic potentials in the thermodynamic limit. In this Letter, we show that the singularity is not the definition of criticality but its asymptotic outcome: criticality is already written in the microcanonical entropy derivatives at any finite size as intrinsic morphological structures -- inflection points and extrema. The singularity is then the endpoint of a sharpening process that evolves with increasing system size. Combining microcanonical inflection-point analysis (MIPA) with the Berlin-Kac spherical model -- for which the microcanonical density of states is known in closed form at every finite
N -- we systematically identify these structures in the energy profiles of entropy derivatives that encode the transition. An inflection point in the inverse temperature
βN(ε)=∂εSN and a pronounced peak in its derivative
γN(ε)=∂ε2SN define a well-controlled pseudocritical trajectory whose controlled sharpening and drift culminate in the macroscopic cusp at the critical energy
εc in the thermodynamic limit. This establishes an intrinsic, order-parameter-free notion of criticality that precedes its singular asymptotic representation.