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Physical review. B, Condensed matter

Critical correlations and susceptibilities in the random-field spherical model

Thomas Vojta, Michael Schreiber

Published Jul 1, 1994• 12 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.50.1272

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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 Uij{\mathit{U}}_{\mathit{i}\mathit{j}}Uij​\ensuremath{\sim}rij\ensuremath−\ensuremathσ{\mathit{r}}_{\mathit{i}\mathit{j}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\sigma}}}rij\ensuremath−\ensuremathσ​ 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.
We investigate the influence of spatial correlations between the values of the random field on the critical behavior of random-field lattice models and derive a generalized version of the Schwartz-Soffer inequality for the averages of the susceptibility and its disconnected part. At the critical point this leads to a modification of the Schwartz-Soffer exponent inequality for the critical exponents \ensuremath{\eta} and \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{} describing the divergences of the susceptibility and its disconnected part, respectively. It now reads \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\le}2\ensuremath{\eta}-2y where 2y describes the divergence of the random-field correlation function in Fourier space. As an example we exactly calculate the susceptibility and its disconnected part for the random-field spherical model. We find that in this case the inequalities actually occur as equalities.
In extensive Monte Carlo simulations the phase transition of the random-field Ising model in three dimensions is investigated. The values of the critical exponents are determined via finite-size scaling. For a Gaussian distribution of the random fields it is found that the correlation length \ensuremath{\xi} diverges with an exponent \ensuremath{\nu}=1.1\ifmmode\pm\else\textpm\fi{}0.2 at the critical temperature and that \ensuremath{\chi}\ensuremath{\sim}${\ensuremath{\xi}}^{2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}}$ with \ensuremath{\eta}=0.50\ifmmode\pm\else\textpm\fi{}0.05 for the connected susceptibility and ${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$\ensuremath{\sim}${\ensuremath{\xi}}^{4\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}$ with \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=1.03\ifmmode\pm\else\textpm\fi{}0.05 for the disconnected susceptibility. Together with the amplitude ratio A=${\mathrm{lim}}_{\mathit{T}\ensuremath{\rightarrow}\mathit{T}\mathit{c}}$${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$/${\mathrm{\ensuremath{\chi}}}^{2}$(hr{\mathit{h}}_{\mathit{r}}hr​/T)2{)}^{2})2 being close to one this gives further support for a two-exponent scaling scenario implying \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\eta}. The magnetization behaves discontinuously at the transition, i.e., \ensuremath{\beta}=0. However, no divergence for the specific heat and in particular no latent heat is found. Also the probability distribution of the magnetization does not show a multipeak structure that would be characteristic for the phase-coexistence at first-order phase-transition points.
The behavior of the spherical model of a ferromagnet with an interaction energy between the magnetic spins which varies with distance as $\frac{1}{{r}^{d+\ensuremath{\sigma}}}$ (where ddd is the dimensionality of the lattice and $\ensuremath{\sigma}>0$) is analyzed. It is shown that the model exhibits a ferromagnetic transition in one and two dimensions, providing $0<\ensuremath{\sigma}<d$. (The usual spherical model with nearest-neighbor interactions does not have a transition in one and two dimensions.) The critical-point behavior is investigated. It is found that the singularities in the specific heat and susceptibility are dependent on $\ensuremath{\sigma}$ and ddd, but the behavior of the magnetization is independent of $\ensuremath{\sigma}$ and ddd. In three dimensions the susceptibility diverges as ${(T\ensuremath{-}{T}_{c})}^{\ensuremath{-}\ensuremath{\gamma}}$, where $\ensuremath{\gamma}=1$ for $0<\ensuremath{\sigma}<\frac{3}{2}$, $\ensuremath{\gamma}=\frac{\ensuremath{\sigma}}{(3\ensuremath{-}\ensuremath{\sigma})}$ for $\frac{3}{2}<\ensuremath{\sigma}<2$ and $\ensuremath{\gamma}=2$ for $\ensuremath{\sigma}>2$. The asymptotic form of the spin-spin correlation function $\ensuremath{\Gamma}(\mathrm{r})$ is studied in the neighborhood of the critical temperature Tc{T}_{c}Tc​. At T=TcT={T}_{c}T=Tc​, $\ensuremath{\Gamma}(\mathrm{r})$ decays for large rrr as $\frac{1}{{r}^{d\ensuremath{-}\ensuremath{\sigma}}}$. Several two-dimensional models with long-range interactions falling off as 1r2\frac{1}{{r}^{2}}r21​ in certain directions only are also investigated.
We present an analytic real-space renormalization group calculation for the random-field Ising model. We apply the Migdal-Kadanoff approximation for the renormalization of a cubic cell in dimensions d , introducing a new field partitioning scheme which allows us to treat the random-field fluctuations in a coherent manner. Our scheme leads naturally to a lower critical dimensionality and allows us to calculate a complete set of three independent exponents in arbitrary dimension. In three dimensions the magnetization exponent and the Schwartz-Soffer inequality is almost satisfied as an equality. We expand analytically in . Further, we show that and the magnitude of the inequality go to zero exponentially with . We calculate the crossover exponent, from pure to the random-field system and find surprisingly good agreement with experimental values. We find that satisfies the Schwartz-Soffer inequality: , the susceptibility exponent of the pure system. We expand in and find that the magnitude of the inequality varies exponentially in . Finally we find that dimensional reduction is satisfied to first order in , with the reduced dimension .
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{\mathit{Ds}}^{2}Ds2-\ensuremath{\Vert}U\ensuremath{\Vert}s4{\mathit{s}}^{4}s4+Vs6{\mathit{Vs}}^{6}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.
Random-field models are considered with long-range exchange, J(r) approximately 1/r d+ sigma , and/or long-range correlated random fields, (h(r)h(r')) approximately 1/ mod r-r' mod d- rho . A general scaling analysis is presented, in which the existence of three independent critical exponents is emphasized. Four different universality classes are identified, according to whether the fixed point controlling the critical behaviour is of short-range or long-range character for the exchange and/or the random-field correlations. The lines in the rho - sigma plane at which the long-range couplings first become relevant are expressed in terms of the exponents eta and eta of the corresponding short-range problem: long-range exchange is relevant for sigma <2- eta , long-range field correlations for rho >2 eta - eta . The scaling results are confirmed and amplified by explicit calculations of three independent exponents of the n-component spin model, correct to order 1/n. The results reveal errors in published expansions around the upper critical dimension.