A Hamiltonian H, with locally smeared Ising-type s-d exchange between s-electrons and magnetic impurities, in a dilute magnetic alloy, is investigated. The Feynman-Kac theorem, Laplace expansion and Bogolyubov inequality are applied to obtain a lower and upper bound (lb and ub) on the system’s free energy per conducting electron f(H,β). The two bounds differ, in the infinite-volume limit by a term lt > 0, linear in impurity concentration: lb = f(h,β) −lt, ub = f(h,β), h denoting the Hamiltonian of the approximating mean-field s-d system. h represents randomly positioned impurities interacting with a mean field implemented by the gas of conduction s-electrons, the latter interacting with the field of barriers and wells (according to the s-electron’s spin orientation) localized at the impurity sites. The inequality f(h,β) −lt ≤ f(H,β) ≤ f(h,β) demonstrates increasing accuracy of the mean-field h-theory, with decreasing impurity concentration.

中文翻译：

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具有单组分杂质和 s 电子自旋的随机 sd 系统的无限体积极限

哈密​​顿量H，研究了稀磁性合金中 s 电子和磁性杂质之间的局部涂抹 Ising 型 sd 交换。应用 Feynman-Kac 定理、拉普拉斯展开和 Bogolyubov 不等式来获得系统每个导电电子的自由能的下限和上限（lb 和 ub）F(H,β). 这两个界限不同，在无限体积限制中，一个术语lt > 0，杂质浓度线性：lb = F(H,β) -lt, ub = F(H,β),H表示近似平均场 sd 系统的哈密顿量。H表示随机定位的杂质与由传导 s 电子气体实现的平均场相互作用，后者与位于杂质位点的势垒和阱（根据 s 电子的自旋方向）的场相互作用。不平等F(H,β) -lt ≤ F(H,β) ≤ F(H,β)表明平均场的准确性越来越高H- 理论，杂质浓度降低。