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Physical review. B, Condensed matter

Phase boundaries near critical end points. II. General spherical models

Márcia C. Barbosa, Michael Ellis Fisher

Published May 1, 1991• 25 citations

Abstract

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 ${\mathit{Ds}}^{2}$-\ensuremath{\Vert}U\ensuremath{\Vert}${\mathit{s}}^{4}$+${\mathit{Vs}}^{6}$, 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.

DOI: 10.1103/physrevb.43.10635

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arXiv:2305.08245•Physica A Statistical Mechanics and its Applications•May 26, 2023•1 citation

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

{\mathit{Ds}}^{2}

-\ensuremath{\Vert}U\ensuremath{\Vert}

{\mathit{s}}^{4}

+

{\mathit{Vs}}^{6}

, 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.

We investigate the behavior of the correlation function and the susceptibility of the random-field spherical model at the critical point. In particular we calculate the critical exponents \ensuremath{\eta} and \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{} describing the divergences of the susceptibility and its disconnected part, respectively. In the case of short-range interactions we obtain \ensuremath{\eta}=\ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=0. For power-law interactions

{\mathit{U}}_{\mathit{i}\mathit{j}}

\ensuremath{\sim}${\mathit{r}}_{\mathit{i}\mathit{j}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\sigma}}}$ we find \ensuremath{\eta}=1/2\ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=D+2-\ensuremath{\sigma} (for DD+2), where D is the spatial dimension. The Schwartz-Soffer exponent inequality \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\le}2\ensuremath{\eta} is satisfied and becomes an equality independent of the functional form of the interaction.