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Physical review. B, Solid state

Exact solution in an external magnetic field of Ising models with three-spin interactions

Luke Lokia Liu, Eugene Stanley

Published Oct 1, 1974• 17 citations
Publisher

Abstract

The three-spin interaction Hamiltonian ${\mathcal{H}}^{(3)}\ensuremath{\equiv}\ensuremath{-}{J}_{3}\ensuremath{\Sigma}{〈\mathrm{ijk}〉}^{}{s}_{i}{\ensuremath{\sigma}}_{j}{\ensuremath{\sigma}}_{k}\ensuremath{-}H\ensuremath{\Sigma}{i}^{}{s}_{i}\ensuremath{-}{H}^{\ensuremath{'}}\ensuremath{\Sigma}{j}^{}{\ensuremath{\sigma}}_{j}$ (the $s$ and $\ensuremath{\sigma}$ spins belong to different sublattices) is solved for some two-dimensional lattices by a generalized star-triangle transformation. The ${H}^{\ensuremath{'}}=0$ internal energy, specific heat, and magnetization are explicitly calculated, and the singularity structures of other functions (e.g.$x$ are studied. Although the actual critical-point exponents remain the same as for systems with two-spin interaction, the three-spin interaction gives rise to different amplitude and background terms. In particular, the ${H}^{\ensuremath{'}}=0$ magnetization is nonzero for all finite $T$, because of the lack of spin-reversal symmetry; the diameter is seen explicitly to have an energy singlularity, with critical exponent $1\ensuremath{-}\ensuremath{\alpha}$.

DOI
10.1103/physrevb.10.2958

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The three-spin interaction Hamiltonian H(3)\ensuremath≡\ensuremath−J3\ensuremathΣ〈ijk〉si\ensuremathσj\ensuremathσk\ensuremath−H\ensuremathΣisi\ensuremath−H\ensuremath′\ensuremathΣj\ensuremathσj{\mathcal{H}}^{(3)}\ensuremath{\equiv}\ensuremath{-}{J}_{3}\ensuremath{\Sigma}{〈\mathrm{ijk}〉}^{}{s}_{i}{\ensuremath{\sigma}}_{j}{\ensuremath{\sigma}}_{k}\ensuremath{-}H\ensuremath{\Sigma}{i}^{}{s}_{i}\ensuremath{-}{H}^{\ensuremath{'}}\ensuremath{\Sigma}{j}^{}{\ensuremath{\sigma}}_{j}H(3)\ensuremath≡\ensuremath−J3​\ensuremathΣ〈ijk〉si​\ensuremathσj​\ensuremathσk​\ensuremath−H\ensuremathΣisi​\ensuremath−H\ensuremath′\ensuremathΣj\ensuremathσj​ (the sss and \ensuremathσ\ensuremath{\sigma}\ensuremathσ spins belong to different sublattices) is solved for some two-dimensional lattices by a generalized star-triangle transformation. The H\ensuremath′=0{H}^{\ensuremath{'}}=0H\ensuremath′=0 internal energy, specific heat, and magnetization are explicitly calculated, and the singularity structures of other functions (e.g.xxx are studied. Although the actual critical-point exponents remain the same as for systems with two-spin interaction, the three-spin interaction gives rise to different amplitude and background terms. In particular, the H\ensuremath′=0{H}^{\ensuremath{'}}=0H\ensuremath′=0 magnetization is nonzero for all finite TTT, because of the lack of spin-reversal symmetry; the diameter is seen explicitly to have an energy singlularity, with critical exponent 1\ensuremath−\ensuremathα1\ensuremath{-}\ensuremath{\alpha}1\ensuremath−\ensuremathα.
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We calculate the scaled constant-magnetization and constant magnetic field specific heats $\frac{[{C}_{M}(\ensuremath{\epsilon},M)\ensuremath{-}{C}_{M}(\ensuremath{\epsilon},0)]}{{M}^{\ensuremath{-}\frac{\ensuremath{\alpha}}{\ensuremath{\beta}}}}$ and $\frac{[{C}_{H}(\ensuremath{\epsilon},H)\ensuremath{-}{C}_{H}(\ensuremath{\epsilon},0)]}{{H}^{\ensuremath{-}\frac{\ensuremath{\alpha}}{\ensuremath{\beta}\ensuremath{\delta}}}}$ as functions of the scaled variable $x\ensuremath{\equiv}\frac{\ensuremath{\epsilon}}{{M}^{\frac{1}{\ensuremath{\beta}}}}$ and $y\ensuremath{\equiv}\frac{\ensuremath{\epsilon}}{{H}^{\frac{1}{\ensuremath{\beta}\ensuremath{\delta}}}}$, respectively [where $\ensuremath{\epsilon}\ensuremath{\equiv}\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$ and Tc{T}_{c}Tc​ is the critical temperature],for the spin-\textonehalf{} Ising model (bcc lattice) and the spin-\textonehalf{}, $\ensuremath{\infty}$ Heisenberg models (fcc lattice). Our calculations are based on previously calculated scaling functions for the M(H,T)M(H,T)M(H,T) equation of state. We also calculate the amplitudes of the zero-field specific heat for T>TcT>{T}_{c}T>Tc​ and T<TcT<{T}_{c}T<Tc​. Thus, we obtain the functions ${C}_{M}(\ensuremath{\epsilon},M)$ and ${C}_{H}(\ensuremath{\epsilon},H)$, though for the Heisenberg models we cannot obtain the finite nonzero constant CM(0,0)=CH(0,0){C}_{M}(0,0)={C}_{H}(0,0)CM​(0,0)=CH​(0,0). We compare our calculated functions with the data of Connelly, Loomis, and Mapother on nickel.