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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

Published Dec 1, 2011• 0 citations

Abstract

[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.

Related papers

〈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.

•Dec 1, 2011•0 citations

〈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

[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)}] .

•Dec 1, 2012•0 citations

<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(11)量子統計力学(3)

邦夫橋爪

ln this paper, next subject is discussed, §32. The equation of state of an ideal gas. Here we consider the equation of state of a spinless ideal Fermi gas. ln high temperatures and low density, we obtain the form of a virial expansion of the equation of state. The corrections to the classical ideal gas law are due to quantum effects. The second virial coefficient is deduced. In low temperature and high densities, we obtain the expansion for the chemical potential in the expansion parameter /<ε_F>. Next we obtain the asynptotic expansion of the internal energyU in the expansion parameter /<ε_F>. We also obtain the specific heat at constant volume C_V.~

•Dec 1, 2009•0 citations

<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(15)量子統計力学(7)

邦夫橋爪

[Synopsis]In this paper, next subject is discussed. § 40. Magnetic properties of an imperfect electron gas. We discuss the effects of interparticle interaction affecting the magnetic properties of an electron gas. Two electrons prefer to be in an antisymmetric orbital wave function to minimize the repulsive energy. But in fermions, total wave function of two electrons with spin-1/2 must be antisymmetric. Therefore an antysimmetric orbital wave function requires a symmetric spin wave function. Thus the repulsive interaction would align the spins of the electrons. And the effect of a repulsive interaction enhances the paramagnetism. We here demonstrate this effect by considering an imperfect electron gas at very low temperature.

•Dec 1, 2010•0 citations

量子モンテカルロ法による1次元Heisenberg spin modelのシミュレーションおよ

裕宮崎

Kagoshima Kenritsu Tanki Daigaku Chiiki Kenkyūjo kenkyū nenpō•Apr 30, 1999•0 citations

1a-A7-4 Ising spin glassの計算機実験 : モード分解による解析(統計力学・物性基礎論)

一高山, 幸児根本

年会講演予稿集•Mar 31, 1985•0 citations

[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.

[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.

[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)}] .

ln this paper, next subject is discussed, §32. The equation of state of an ideal gas. Here we consider the equation of state of a spinless ideal Fermi gas. ln high temperatures and low density, we obtain the form of a virial expansion of the equation of state. The corrections to the classical ideal gas law are due to quantum effects. The second virial coefficient is deduced. In low temperature and high densities, we obtain the expansion for the chemical potential in the expansion parameter /<ε_F>. Next we obtain the asynptotic expansion of the internal energyU in the expansion parameter /<ε_F>. We also obtain the specific heat at constant volume C_V.~

[Synopsis]In this paper, next subject is discussed. § 40. Magnetic properties of an imperfect electron gas. We discuss the effects of interparticle interaction affecting the magnetic properties of an electron gas. Two electrons prefer to be in an antisymmetric orbital wave function to minimize the repulsive energy. But in fermions, total wave function of two electrons with spin-1/2 must be antisymmetric. Therefore an antysimmetric orbital wave function requires a symmetric spin wave function. Thus the repulsive interaction would align the spins of the electrons. And the effect of a repulsive interaction enhances the paramagnetism. We here demonstrate this effect by considering an imperfect electron gas at very low temperature.