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Physical Review A

Ising Chain with Competing Interactions

John F. Nagle

Published Nov 1, 1970• 107 citations
Publisher

Abstract

The one-dimensional spin-\textonehalf{} Ising model is solved exactly with very long-range, equivalent-neighbor ferromagnetic interactions of strength ${J}_{\mathrm{LR}}$ superimposed on nearest-neighbor anti-ferromagnetic interactions of strength $\ensuremath{-}{J}_{\mathrm{sr}}$. For $\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}} >R=3.1532$... there is a critical point of classical type in zero field. For $2\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}}=R$ the critical exponents take the values $\ensuremath{\beta}=\frac{1}{4}$, $\ensuremath{\gamma}=1={\ensuremath{\gamma}}^{\ensuremath{'}}$, $\ensuremath{\delta}=5$, and $0=\ensuremath{\alpha}\ensuremath{\ne}{\ensuremath{\alpha}}^{\ensuremath{'}}=\frac{1}{2}$. For $2<\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}}<R$ the transition in zero field is of first order, and the critical points which occur in finite field are of classical type. For $0<\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}}<2$ there is no phase transition in zero field, although there are classical critical points in a field.

DOI
10.1103/physreva.2.2124

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The one-dimensional spin-\textonehalf{} Ising model is solved exactly with very long-range, equivalent-neighbor ferromagnetic interactions of strength JLR{J}_{\mathrm{LR}}JLR​ superimposed on nearest-neighbor anti-ferromagnetic interactions of strength \ensuremath−Jsr\ensuremath{-}{J}_{\mathrm{sr}}\ensuremath−Jsr​. For \ensuremath−JlrJsr>R=3.1532\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}} >R=3.1532\ensuremath−Jsr​Jlr​​>R=3.1532... there is a critical point of classical type in zero field. For 2\ensuremath−JlrJsr=R2\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}}=R2\ensuremath−Jsr​Jlr​​=R the critical exponents take the values \ensuremathβ=14\ensuremath{\beta}=\frac{1}{4}\ensuremathβ=41​, \ensuremathγ=1=\ensuremathγ\ensuremath′\ensuremath{\gamma}=1={\ensuremath{\gamma}}^{\ensuremath{'}}\ensuremathγ=1=\ensuremathγ\ensuremath′, \ensuremathδ=5\ensuremath{\delta}=5\ensuremathδ=5, and 0=\ensuremathα\ensuremath≠\ensuremathα\ensuremath′=120=\ensuremath{\alpha}\ensuremath{\ne}{\ensuremath{\alpha}}^{\ensuremath{'}}=\frac{1}{2}0=\ensuremathα\ensuremath=\ensuremathα\ensuremath′=21​. For 2<\ensuremath−JlrJsr<R2<\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}}<R2<\ensuremath−Jsr​Jlr​​<R the transition in zero field is of first order, and the critical points which occur in finite field are of classical type. For 0<\ensuremath−JlrJsr<20<\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}}<20<\ensuremath−Jsr​Jlr​​<2 there is no phase transition in zero field, although there are classical critical points in a field.
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.
The Ising linear chain with nearest-neighbor interactions of strength J s and infinite long-range ferromagnetic exponential interactions of strength J l was solved exactly in the presence of the external field. The spontaneous magnetization and the susceptibility are obtained. The transition temperature depends on J s and J l . For J s >0, the model shows the second order phase transion. The specific heat shows both a broad peak and a sharp peak. The value of sharp peak of the specific heat depends on J s and J l . For J s <0, the properties of the transition agree with those of Nagle model with long-range equivalent-neighbor ferromagnetic interactions. The exact free energy and the long-range order of our model are identical with those of the variational theory of a quasi-one-dimensional model.
A long-range Ising chain, which has ferromagnetic 1/r 3 type interactions, has been studied by exact numerical calculation of thermodynamic properties of the sequence of systems of N spins (N<or=30). By expressing the sequence in terms of a series and using extrapolation methods such as the Pade approximant, the authors have found a consistent way to obtain critical temperatures which is accurate to 0.4% for the very long-range equivalent neighbour model. The authors have estimated the high-temperature susceptibility and the low- temperature long-range correlations of the infinite chain and thence have made estimates for the critical exponents gamma and beta . Although the nature of the phase transition appears to be classical for 1<s<or approximately=1.3, it is certainly not classical for 1.6<s<2; in particular, as s approaches 2 from below, gamma increases regularly to about 2 or greater and beta decreases regularly to near 0.
As in many problems, one-dimensional models can give some insight into the physics of spin glasses because they are easier to solve than those in higher dimensionality, but this advantage can be limited because the physics of the one-dimensional problem may be essentially different. Typically, for example, the higher-dimensional problem we really want to study has a broken symmetry phase at low temperature, while one-dimensional systems (with short-range interactions) do not exhibit broken symmetry at finite temperature. Nevertheless, short-range one-dimensional systems can have broken symmetry at zero temperature, so one can study this relatively simple kind of order and the approach to it as the temperature is lowered to zero. This zero-temperature phase transition can be compared with finite-temperature transitions in higher dimensionality. This is the motivation for Section 12.1, where we will see that the critical point at T = h = 0 in a one-dimensional Ising spin glass with nearest-neighbour interactions has interesting features in common with finite-temperature spin glass transitions, and that they can be calculated analytically, sometimes rather simply. Further insight can be gained by studying one-dimensional systems with long-range (typically power law) interactions. Then if the exponent describing the falloff of the interaction strength with distance is small enough, finite-temperature phase transitions are possible. Though such problems are harder to solve than nearest-neighbour ones, they can still be easier than the full higher-dimensional ones.
A real-space, zero-temperature renormalization-group method is used to study a Heisenberg antiferromagnetic spin chain with long-range interactions varying as ${(\mathrm{distance})}^{\ensuremath{-}p}$. Simple renormalization-group equations are obtained and studied analytically, and the results are checked against a more accurate numerical calculation. The calculations provide evidence for a phase transition at $p\ensuremath{\approx}1.85$ characterized by a change in the behavior of zero-temperature correlation functions. For $p\ensuremath{\gtrsim}1.85$ the large-distance physics is that of the nearest-neighbor antiferromagnet, while there is a line of fixed points for $p\ensuremath{\lesssim}1.85$. I conjecture that $p\ensuremath{\approx}1.85$ also marks an order-disorder transition at finite temperature such as occurs in the Ising model with power-law interactions.
We consider an Ising chain with Hamiltonian $H=\ensuremath{\Sigma}{i>j}^{}\frac{({\ensuremath{\epsilon}}_{\mathrm{ij}}{S}_{i}{S}_{j})}{{(a|i\ensuremath{-}j|)}^{\ensuremath{\sigma}}}$, where the ${\ensuremath{\epsilon}}_{\mathrm{ij}}$ are independent random variables. We find a phase transition for $\frac{1}{2}<\ensuremath{\sigma}<1$. For $\frac{1}{2}<\ensuremath{\sigma}<\frac{2}{3}$ the critical exponents exhibit mean-field classical behavior. Near $\ensuremath{\sigma}=1$ we find a smoothly varying specific heat. We investigate the critical behavior near the upper and lower critical range by means of an $\ensuremath{\epsilon}$ expansion around $\ensuremath{\sigma}=1 \mathrm{and} \frac{2}{3}$.