〈ノート〉多体問題とグリーン関数との関係の研究--グリーン関数と多体問題(24) 量子統計力学(16)

邦夫橋爪

Published Dec 1, 2012• 0 citations

Abstract

[synopsis] In this paper, the Bethe-Peierls approximation of the Ising model is interpreted. It is an improvement over the Bragg-Williams approximation of the assumpion N _<++>/2/1γN＝(N_+/N)^2 of a relation between N_<++> and N_+. In the Bethe-Peierls approximation we focus our attention on a sublattice composed of any lattice site plus its γ nearest neighbors. And we find a more accurate relation between N_<++> and N_+. P(S,n) is the probability that n of the nearest neighbors are in a spin-up state while the canter site has the spin state S. From the meaning given to P(S,n) it follows that {(1+L)/2}=N_+/N=Σ^^γ__ P(+1,n) and {(1+σ)/2}=N_<++>/(_γN/2)=(1/γ)Σ^^γ__ nP(+1,n). In our approximation we require that Σ^^γ__ P(+1,n)=(1/γ)x Σ^^γ__ n{P(+1,n)+P(-1,n)}. We assume that P(±1,n)=(1/q)_γC_ne^<1/(k_BT)(±2n±γ)>z^n. z={(1+ze^<2ε/k_BT>)/(z+e^<2ε/k_BT>)}^<γ-1>. L^^-={(z^x-1)/(z^x)+1}, σ^^-[2z^2/{(1+ze)^^<2ε/k_BT>)(1+z^x)}]-1, x≡γ/(γ-1). The internal energy of the Ising lattice in H=0 is U(0,T)=-(1/2)εγ(2σ^^--2L^^-+1). The Curie tempererature T_c is T_c=[(2ε/k_B)/log_e{γ/(γ-2)}] .

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〈ノート〉多体問題とグリーン関数との関係の研究--グリーン関数と多体問題(23)量子統計力学(15)

邦夫橋爪

[Synopsis] In this paper, the Bragg-Williamsa pproximation of the Ising model is interpreted. The number N_＋/N is a measure of the "long-range order" in the lattice, and it is a fraction of up spins of all spins. The number N_<＋＋>/(_γN/2) is a measure of the "short-range order" in the lattice, and if it is definite that a given spin is up, the number is the fraction of its nearest neighbor with spin up. the parameter of long-range oeder L and that of short-range order σ are defined as N_+/N≡(1/2)(L+1) (-1L≦L≦＋1) and as N_<＋＋>/(γN/2)≡(1/2)(σ＋1)(-1≦σ≦＋1) respectively. The Bragg-Williams approximation consists of putting N_<＋＋>/(γN/2)≒N_<＋＋>/N. It states that there is no short-range order. The energy per spin is (1/N)E(L)＝-(1/2)εγL^2-βHL. The partition function in the Bragg-Williams approximation is Z(H, T)＝Σ__{S_i}e^<1/ N(1/2εγL^2＋βHL)>. We calculated the thermodynamic functions id' est' Helmholtz's free energy F, the spontaneous magnetization per spin , the internal energy U, the specific heat at constant volume C_V. The equation of state for the lattice gas is as follows : P_G＝βH-(1/8)ε_0γ(1＋ ^^^-)-(1/2)k_BTlog_e((1- ^^^-)/4) and (1/ν)＝(1/2)(1＋(h/2)L^^-) in the Bragg-Williams approximation.

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〈ノート〉多体問題とグリーン関数との関係の研究--グリーン関数と多体問題(22)量子統計力学(14)

邦夫橋爪

[Synopsis] The energy Hamiltonian, given by Heisenberg in 1928, to discuss the physical properties and the atomics tructure of ferromagnetic substance is as follows: H＝—2Σ__ J_ S_i・S_j.-gβΣ__iS_i・H. A spinv ariable S_i is associated with each lattice site. In the Ising model of J_ ＝J_ ＝0 and J_ ≠0, proposed by E. Ising in Zeit. Phys. 31 253(1925), the above Hamiltonian is simplified as follows : H＝-2Σ__ J_ S_ S_ -gβHΣ__iS_ ・S_ ＝±1/2. The system is an array of N fixed points calld lattice sites that form an n-dimensional periodic lattice. The energy of the system in the configuration specified by {S_i} is as follows : E{S_i}＝εΣ__ S_iS_j-βHΣ__iS. S_i＝＋1 denotes that the ith site has spin ip and S_i＝-1 detotes that the ith site has spin down. The partition function is Z(H, T)＝Σ__ Σ__ ･･･Σ__ e^<- / >＝e^<<1/k_BT>N(1/2γε-βH)>Σ^^N_ e^<-2・1/ (εγ-βH)N_＋>Σ^1__ g(N_＋,N_＋＋)e^<4・1/ εN_＋＋>. Next the Ising model can be made to simulate the system of a lattice gas and a binary alloy of other than a ferromagnetic substance by a change of names.

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