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.
An exactly soluble
n-component continuous spin model with tricritical points is shown to be equivalent, in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$, to a one-component model with anharmonic long-range interactions.
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 ${a}_{\ensuremath{\tau}}=\frac{1}{2}$,
aH=aH=1615,
aR=87.
The one-dimensional Ising model in an external magnetic field with uniform long-range interactions and random short-range interactions satisfying bimodal annealed distributions is studied. This generalizes the random model discussed by Paladin et al. (J. Phys. I France 4, 1994, p. 1597). Exact results are obtained for the thermodynamic functions at arbitrary temperatures, and special attention is given to the induced and spontaneous magnetization. At low temperatures the system can exist in a “ferrimagnetic” phase with magnetization 0 < σ < 1, in addition to the usual paramagnetic, ferromagnetic and antiferromagnetic phases. For a fixed distribution of the random variables the system presents up to three tricritical points for different intensities of the long-range interactions. Field-temperature diagrams can present up to four critical points. 1
Models with competing nearest-neighbor and very long-range interactions are solved exactly for several one-dimensional cases, including the usual Ising chain (ICCI), the XY model (XYCI), the transverse Ising model (TICCI), and the spherical model (SCCI). For certain ratios of the competing interaction strengths, ICCI and XYCI display triple points and two critical points in a field. In addition TICCI has an apparently enclosed phase in an H-T phase diagram; however, this phase is really paramagnetic as can be seen when an extended phase diagram is used. The extended phase diagrams for ICCI, XYCI, and TICCI display tricritical points and the tricritical exponents have different values from the usual classical values. In contrast to the rich phase behavior of the preceding models, SCCI shows very simple phase behavior, which is directly related to the ground state. Finally, the introduction of a staggered field to ICCI and a simple transformation allows reinterpretation as a metamagnetic model. Using ICCI as a guide the observability of the tricritical exponents is discussed.