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Physical review. B, Condensed matter

Frustrated nearest-neighbord=2Ising model: Exact results

R. Shankar, Ganpathy N. Murthy

Published Mar 1, 1987• 13 citations
Publisher

Abstract

We study a d=2 Ising model with vertical coupling K>0 and random horizontal coupling ${K}_{H}$(${K}_{H}$:${K}_{1}$\ensuremath{\le}${K}_{H}$,\ensuremath{\le}${K}_{2}$), which is the same within a row. This generalizes the McCoy-Wu model since it allows for frustration. By mapping it into a random-field model in d=1, we show that the free energy F has infinitely differentiable singularities at ${K}^{\mathrm{*}}$=${K}_{2}$ (Griffith's singularity in temperature); ${K}^{\mathrm{*}}$=〈K〉, where 〈K〉 is the average horizontal bond (infinite correlation length and \ensuremath{\nu}=1), and ${K}^{\mathrm{*}}$=${K}_{1}$ (Griffith's singularity, if ${K}_{1}$0, its dual version if not). We show for the last case that F(t)\ensuremath{\approxeq}${t}^{1/t}$.

DOI
10.1103/physrevb.35.3671

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We study a d=2 Ising model with vertical coupling K>0 and random horizontal coupling KH{K}_{H}KH​(KH{K}_{H}KH​:K1{K}_{1}K1​\ensuremath{\le}KH{K}_{H}KH​,\ensuremath{\le}K2{K}_{2}K2​), which is the same within a row. This generalizes the McCoy-Wu model since it allows for frustration. By mapping it into a random-field model in d=1, we show that the free energy F has infinitely differentiable singularities at K∗{K}^{\mathrm{*}}K∗=K2{K}_{2}K2​ (Griffith's singularity in temperature); K∗{K}^{\mathrm{*}}K∗=〈K〉, where 〈K〉 is the average horizontal bond (infinite correlation length and \ensuremath{\nu}=1), and K∗{K}^{\mathrm{*}}K∗=K1{K}_{1}K1​ (Griffith's singularity, if K1{K}_{1}K1​0, its dual version if not). We show for the last case that F(t)\ensuremath{\approxeq}t1/t{t}^{1/t}t1/t.
We study a d=2 Ising model where the veritcal bonds are fixed and ferromagnetic and the horizontal bonds can vary randomly in sign and in magnitude (within some limits) but are same within each now. The model therefore generalizes that of McCoy and Wu since it allows for the interesting case of frustration. We use the transfer matrix to map our problem to a collection of random field d=1 problems about which a lot is known. We find generally three transitions: a Griffiths transition, its dual version, and one with infinite correlation length and index \ensuremath{\nu}=1. In all cases the free energy has infinitely differentiable singularities. In addition there are some zero-temperature transitions.
The Landau-Ginzburg-Wilson Hamiltonian for the McCoy-Wu strip-random Ising model on a two-dimensional lattice is studied. In the Landau-Ginzburg-Wilson Hamiltonian, the random temperature only depends on one of the coordinates (xxx), the horizontal coordinate. The difference (rather than differential) saddle point equation is solved numerically. In the critical regime, there exist locally ordered regions, where the saddle point solutions are not zero. Due to the translational invariance in the vertical direction, the locally ordered regions are striplike in the vertical direction. In the excited state solutions, there exist horizontal domain walls across the locally ordered regions. It is found that these horizontal domain walls are stabilized by free-energy barriers due to the discreteness. The stability conditions for such domain walls are studied. Usually, the domain wall across multiple locally ordered regions can be stable. These multiple locally ordered regions form an elementary cluster. The criterion for multiple locally ordered regions forming an elementary cluster is obtained. At the saddle point level, we get an effective Hamiltonian in which the elementary clusters behave as Ising spins and are coupled with their neighbors. For the random temperatures with narrow uniform probability distributions, the distribution of elementary cluster size, the horizontal and vertical bonds between elementary clusters are calculated numerically. The vertical bonds are extremely strong and the horizontal bonds are extremely weak. The distribution of vertical bonds is exponential. There is an exponentially small probability of elementary clusters with extremely large size. These elementary clusters with extremely large size will cause the susceptibility divergent in a range of temperature. The random temperature with binary probability distributions is also studied. The qualitative properties are the same. The Griffiths-McCoy singularity in the saddle point solutions does not depend on the distribution of random temperature.
We continue our investigation of an Ising model with immobile random impurities by studying the spin-spin correlation functions. These correlations are not probability-1 objects and have a probability distribution. When the random bonds have the particular distribution function studied in the first paper of this series, we demonstrate that the average value and the second moment of the temperature derivatives of these correlations are infinitely differentiable but fail to be analytic at Tc{T}_{c}Tc​, the temperature at which the observable specific heat fails to be analytic. When T<TcT<{T}_{c}T<Tc​, we consider ${S}_{\ensuremath{\infty}}(l)=limit of\text{}〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{l,m}〉\text{as}m\ensuremath{\rightarrow}\ensuremath{\infty}$. This limit is not independent of lll. In the special case that the random bonds are symmetrically distributed about the lllth row, the geometric mean of ${S}_{\ensuremath{\infty}}(l)$ is computed and shown to vanish exponentially rapidly when $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$. We contrast this with a lower bound that shows that the spontaneous magnetization can vanish no more rapidly than ${T}_{c}\ensuremath{-}T$, and present a description of how the local magnetization ${S}_{\ensuremath{\infty}}{(l)}^{\frac{1}{2}}$ behaves as $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$.
A soluble, but nontrivial, model of a dilute Ising ferromagnet is studied, with infinite-range interactions but finite average connectivity c. The density of (Yang-Lee) zeros of the partition function in the complex z=exp(-2H) plane (where H is the external magnetic field in units of the temperature) is calculated explicitly in the high-temperature phase for large but finite c and small \ensuremath{\Vert}H\ensuremath{\Vert}. The density of zeros on the unit circle H=i\ensuremath{\theta} has the form \ensuremath{\rho}(\ensuremath{\theta})\ensuremath{\sim}exp{-[cf(K)/\ensuremath{\Vert}\ensuremath{\theta}\ensuremath{\Vert}]ln/B (1/\ensuremath{\Vert}\ensuremath{\theta}\ensuremath{\Vert})} for \ensuremath{\Vert}\ensuremath{\theta}\ensuremath{\Vert}\ensuremath{\rightarrow}0. The function f(K) (K=J/T) vanishes at the critical coupling KC{K}_{C}KC​(c). Heuristic arguments are given for the form of \ensuremath{\rho}(\ensuremath{\theta}) expected for systems with short-range interactions.