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Physical review. B, Solid state

System Exhibiting a Critical Point of Order Four: Ising Planes with Variable Interplanar Interactions

Fredric Harbus, Alex Hankey, Eugene Stanley, T. S. Chang

Published Sep 1, 1973• 21 citations
Publisher

Abstract

The first phase diagram explicitly exhibiting intersecting lines of tricritical points is presented. Their point of intersection is a critical point of order 4. The system is a simple assembly of ferromagnetic Ising planes coupled with an arbitrary interplanar interaction, with Hamiltonian $\mathcal{H}=\ensuremath{-}J\left(\ensuremath{\Sigma}\stackrel{\mathrm{xy}}{〈\mathrm{ij}〉}{s}_{i}{s}_{j}+\mathcal{R}\ensuremath{\Sigma}\stackrel{z}{〈\mathrm{ij}〉}{s}_{i}{s}_{j}\right)\ensuremath{-}\ensuremath{\mu}H\ensuremath{\Sigma}\stackrel{}{i}{s}_{i}\ensuremath{-}\ensuremath{\mu}{H}^{\ensuremath{'}}{(\ensuremath{-}1)}^{\ensuremath{\eta}}\ensuremath{\Sigma}\stackrel{}{i}{s}_{i},$ where $J>0$, ${s}_{i}=\ifmmode\pm\else\textpm\fi{}1$, and $\ensuremath{\mu}$ is the magnetic moment per spin. The first sum is over nearestneighbor (nn) spins in an $x\ensuremath{-}y$ plane, the second sum over nn spins coupled along the $z$ direction, $H$ is the magnitude of a uniform magnetic field, and ${H}^{\ensuremath{'}}$ is a staggered magnetic field which acts oppositely on adjacent planes of constant $z$ ($\ensuremath{\eta}=0$ on even planes, +1 on odd planes). The Hamiltonian is invariant under $\mathcal{R}\ensuremath{\rightarrow}\ensuremath{-}\mathcal{R}$, $H\ensuremath{\rightarrow}{H}^{\ensuremath{'}}$, and ${s}_{i}\ensuremath{\rightarrow}{(\ensuremath{-}1)}^{\ensuremath{\eta}}{s}_{i}$, so that the Gibbs potential is also invariant, $G(T, H, {H}^{\ensuremath{'}}, \mathcal{R})=G(T, {H}^{\ensuremath{'}}, H, \ensuremath{-}\mathcal{R})$. Using this symmetry, we make a scaling hypothesis about the special point $\ensuremath{\tau}[\ensuremath{\equiv}T\ensuremath{-}{T}_{c} (R=0)]=H={H}^{\ensuremath{'}}=R=0$, namely, that the Gibbs potential obeys the functional equation $G({\ensuremath{\lambda}}^{{a}_{\ensuremath{\tau}}}\ensuremath{\tau}, {\ensuremath{\lambda}}^{\mathrm{aH}}H, {\ensuremath{\lambda}}^{a{H}^{\ensuremath{'}}}{H}^{\ensuremath{'}}, {\ensuremath{\lambda}}^{a\mathcal{R}}\mathcal{R})=\ensuremath{\lambda}G(\ensuremath{\tau}, H, {H}^{\ensuremath{'}}, \mathcal{R})$; the four scaling powers are found to be ${a}_{\ensuremath{\tau}}=\frac{1}{2}$, ${a}_{H}={a}_{H}=\frac{15}{16}$, ${a}_{\mathcal{R}}=\frac{7}{8}$.

DOI
10.1103/physrevb.8.2273

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The first phase diagram explicitly exhibiting intersecting lines of tricritical points is presented. Their point of intersection is a critical point of order 4. The system is a simple assembly of ferromagnetic Ising planes coupled with an arbitrary interplanar interaction, with Hamiltonian H=\ensuremath−J(\ensuremathΣ〈ij〉xysisj+R\ensuremathΣ〈ij〉zsisj)\ensuremath−\ensuremathμH\ensuremathΣisi\ensuremath−\ensuremathμH\ensuremath′(\ensuremath−1)\ensuremathη\ensuremathΣisi,\mathcal{H}=\ensuremath{-}J\left(\ensuremath{\Sigma}\stackrel{\mathrm{xy}}{〈\mathrm{ij}〉}{s}_{i}{s}_{j}+\mathcal{R}\ensuremath{\Sigma}\stackrel{z}{〈\mathrm{ij}〉}{s}_{i}{s}_{j}\right)\ensuremath{-}\ensuremath{\mu}H\ensuremath{\Sigma}\stackrel{}{i}{s}_{i}\ensuremath{-}\ensuremath{\mu}{H}^{\ensuremath{'}}{(\ensuremath{-}1)}^{\ensuremath{\eta}}\ensuremath{\Sigma}\stackrel{}{i}{s}_{i},H=\ensuremath−J(\ensuremathΣ〈ij〉xy​si​sj​+R\ensuremathΣ〈ij〉z​si​sj​)\ensuremath−\ensuremathμH\ensuremathΣi​si​\ensuremath−\ensuremathμH\ensuremath′(\ensuremath−1)\ensuremathη\ensuremathΣi​si​, where J>0J>0J>0, si=\ifmmode±\else\textpm\fi1{s}_{i}=\ifmmode\pm\else\textpm\fi{}1si​=\ifmmode±\else\textpm\fi1, and \ensuremathμ\ensuremath{\mu}\ensuremathμ is the magnetic moment per spin. The first sum is over nearestneighbor (nn) spins in an x\ensuremath−yx\ensuremath{-}yx\ensuremath−y plane, the second sum over nn spins coupled along the zzz direction, HHH is the magnitude of a uniform magnetic field, and H\ensuremath′{H}^{\ensuremath{'}}H\ensuremath′ is a staggered magnetic field which acts oppositely on adjacent planes of constant zzz (\ensuremathη=0\ensuremath{\eta}=0\ensuremathη=0 on even planes, +1 on odd planes). The Hamiltonian is invariant under R\ensuremath→\ensuremath−R\mathcal{R}\ensuremath{\rightarrow}\ensuremath{-}\mathcal{R}R\ensuremath→\ensuremath−R, H\ensuremath→H\ensuremath′H\ensuremath{\rightarrow}{H}^{\ensuremath{'}}H\ensuremath→H\ensuremath′, and si\ensuremath→(\ensuremath−1)\ensuremathηsi{s}_{i}\ensuremath{\rightarrow}{(\ensuremath{-}1)}^{\ensuremath{\eta}}{s}_{i}si​\ensuremath→(\ensuremath−1)\ensuremathηsi​, so that the Gibbs potential is also invariant, G(T,H,H\ensuremath′,R)=G(T,H\ensuremath′,H,\ensuremath−R)G(T, H, {H}^{\ensuremath{'}}, \mathcal{R})=G(T, {H}^{\ensuremath{'}}, H, \ensuremath{-}\mathcal{R})G(T,H,H\ensuremath′,R)=G(T,H\ensuremath′,H,\ensuremath−R). Using this symmetry, we make a scaling hypothesis about the special point \ensuremathτ[\ensuremath≡T\ensuremath−Tc(R=0)]=H=H\ensuremath′=R=0\ensuremath{\tau}[\ensuremath{\equiv}T\ensuremath{-}{T}_{c} (R=0)]=H={H}^{\ensuremath{'}}=R=0\ensuremathτ[\ensuremath≡T\ensuremath−Tc​(R=0)]=H=H\ensuremath′=R=0, namely, that the Gibbs potential obeys the functional equation G(\ensuremathλa\ensuremathτ\ensuremathτ,\ensuremathλaHH,\ensuremathλaH\ensuremath′H\ensuremath′,\ensuremathλaRR)=\ensuremathλG(\ensuremathτ,H,H\ensuremath′,R)G({\ensuremath{\lambda}}^{{a}_{\ensuremath{\tau}}}\ensuremath{\tau}, {\ensuremath{\lambda}}^{\mathrm{aH}}H, {\ensuremath{\lambda}}^{a{H}^{\ensuremath{'}}}{H}^{\ensuremath{'}}, {\ensuremath{\lambda}}^{a\mathcal{R}}\mathcal{R})=\ensuremath{\lambda}G(\ensuremath{\tau}, H, {H}^{\ensuremath{'}}, \mathcal{R})G(\ensuremathλa\ensuremathτ​\ensuremathτ,\ensuremathλaHH,\ensuremathλaH\ensuremath′H\ensuremath′,\ensuremathλaRR)=\ensuremathλG(\ensuremathτ,H,H\ensuremath′,R); the four scaling powers are found to be a\ensuremathτ=12{a}_{\ensuremath{\tau}}=\frac{1}{2}a\ensuremathτ​=21​, aH=aH=1516{a}_{H}={a}_{H}=\frac{15}{16}aH​=aH​=1615​, aR=78{a}_{\mathcal{R}}=\frac{7}{8}aR​=87​.
The one-dimensional spin-½ Ising model with very long-range ferromagnetic interaction and first and second neighbor antiferromagnetic interactions is solved exactly in a direct magnetic field. A study of the finite temperature behavior confirms the existence of up to four first-order phase transitions ending in classical critical points. We find a confluence of three critical lines at a ``tricritical'' point with exponents β=14 , δ=5, γ=γ′=1,α=0, and α′=12. We find also a confluence of four critical lines at a ``tetracritical'' point with exponents β=14, δ=7, γ=γ′=1, α=0, and α′=23. We show the existence of triple lines and ``quadruple'' lines, along which three or four phases are simultaneously in contact with each other, respectively. For some special values of the interaction strengths, five phases are in contact with each other at a ``quintuple'' point. We remark that our ``multiple'' lines and ``multicritical'' points are not ``true'' multiple lines and multicritical points but that they would be so in a somewhat extended model. This model satisfies a generalized form of the ordinary phase rule of Gibbs. In distinction to various models considered recently, our model does not exhibit a singular phase-boundary diameter.
We explore the phase diagram of Ising spins on one-dimensional chains that criss-cross in two perpendicular directions and that are connected by interchain couplings. This system is of interest as a simpler, classical analog of a quantum Hamiltonian that has been proposed as a model of magnetic behavior in Nb_{12}O_{29} and also, conceptually, as a geometry that is intermediate between one and two dimensions. Using mean-field theory as well as Metropolis Monte Carlo and Wang-Landau simulations, we locate quantitatively the boundaries of four ordered phases. Each becomes an effective Ising model with unique effective couplings at large interchain coupling. Away from this limit, we demonstrate nontrivial critical behavior, including tricritical points that separate first- and second-order phase transitions. Finally, we present evidence that this model belongs to the two-dimensional Ising universality class.
The goal of this work is to provide an analysis of spaces of critical points for multicomponent systems. First, we propose the geometric concept of order O\mathcal{O}O for critical points; we distinguish it from a previous definition of a "multicritical" point. Specifically, we may define the intersection of spaces of critical points of order O\mathcal{O}O to be a space of critical points of order (O+1\mathcal{O}+1O+1). Ordinary critical points are defined to be of order O=2\mathcal{O}=2O=2, so that the tricritical points introduced by Griffiths are of order O=3\mathcal{O}=3O=3. We discuss more general examples of critical spaces of order O=3\mathcal{O}=3O=3 which are known for a wide variety of systems; we also propose several examples of models of magnetic systems showing critical points of order O=4\mathcal{O}=4O=4---i.e., systems having intersecting lines of tricritical points. The analysis of critical and coexistence spaces also provides a new form of the Gibbs phase rule suitable for complex magnetic models. Next we define---for the critical points of order O\mathcal{O}O of which examples have been given---special directions in terms of which to make a scaling hypothesis. We give the hypothesis for simple systems and then for tricritical points, and then, in a subsequent paper, part II, the special directions are used to make a scaling hypothesis at spaces of critical points of any order. Certain predictions (e.g., scaling laws and "single-power" scaling functions) follow in a simple and straightforward fashion. We consider the scaling hypothesis at a critical space of order O\mathcal{O}O in terms of a group of transformations. We can define a set of invariants of the group. It is possible, for $\mathcal{O}\ensuremath{\ge}3$, to make a second scaling hypothesis for the space of order $\mathcal{O}\ensuremath{-}1$ using certain of these invariants as independent variables. This is advantageous because certain "double-power" scaling functions then follow directly; these predict that for O=3\mathcal{O}=3O=3, experimental data collapse from a volume onto a line. This prediction is to be contrasted with ordinary scaling functions, which predict that data collapse by only a single dimension (e.g., from a volume onto a surface or from a surface onto a line).
The nnn model is an Ising model with nearest-neighbor antiferromagnetic interactions (J1<0{J}_{1}<0J1​<0) but also next-nearest-neighbor ferromagnetic exchange (J2>0{J}_{2}>0J2​>0). This model is analyzed in external magnetic field using the same techniques as applied to the meta model of Paper I. Again, the staggered susceptibility ${\ensuremath{\chi}}_{\mathrm{st}}$ appears to diverge along the critical line in the $H\ensuremath{-}T$ plane with a constant exponent ${\ensuremath{\gamma}}_{\mathrm{st}}=\frac{5}{4}$, consistent with the universality hypothesis. However, in contrast to the meta model, it is found that the direct susceptibility $\ensuremath{\chi}$ diverges at the tricritical point with an exponent $\overline{\ensuremath{\gamma}}\ensuremath{\simeq}\frac{1}{4}$. Implications of the scaling hypothesis at the tricritical point are discussed and the results for both the meta and nnn models are utilized to obtain the scaling power a‾2{\overline{a}}_{2}a2​ corresponding to the "weak" direction (in the sense of Griffiths and Wheeler). Included in this discussion is the double-power-law form, predicted to hold within the crossover region by tricritical-point scaling.
We explore the phase diagram of Ising spins on one-dimensional chains which criss-cross in two perpendicular directions and are connected by interchain couplings. This system is of interest as a simpler, classical analog of a quantum Hamiltonian which has been proposed as a model of magnetic behavior in Nb12_{12}12​O29_{29}29​ and also, conceptually, as a geometry which is intermediate between one and two dimensions. Using mean field theory as well as Metropolis Monte Carlo and Wang-Landau simulations, we locate quantitatively the boundaries of four ordered phases. Each becomes an effective Ising model with unique effective couplings at large interchain coupling. Away from this limit we demonstrate non-trivial critical behavior, including tricritical points which separate first and second order phase transitions. Finally, we find that the crossover of the magnetization critical exponent from the Ising to tricritical Ising value shows an unusual non-monotonic behavior.