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Science in China Series A-Mathematics Physics Astronomy & Technological Science (in Chinese)

有外磁场的二维Ising模型在T>Tc的能量-自旋关联函数

沐霖 阎, 宏康 赵

Published Jun 1, 1993• 0 citations
Publisher

Abstract

在标度极限下,即T→Tc+,H→0,R→∞,同时h=const·H|T—Tc|~(-15/8)固定,且很小,而r=R|T-Tc|固定,且很大时,我们研究了外磁场很小时 Ising模型的能量-自旋关联函数<σM1,N1σM1,N2εM3N3>.用集团展开方法得到其h2级与K0(nr)及K1(nr)有关的领头和次领头项.
DOI
10.1360/za1993-23-6-589

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[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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Quantum sequential machines (QSMs) are a quantum version of stochastic sequential machines (SSMs). Recently, we showed that two QSMs M_1 and M_2 with n_1 and n_2 states, respectively, are equivalent iff they are (n_1+n_2)^2--equivalent (Theoretical Computer Science 358 (2006) 65-74). However, using this result to check the equivalence likely needs exponential expected time. In this paper, we consider the time complexity of deciding the equivalence between QSMs and related problems. The main results are as follows: (1) We present a polynomial-time algorithm for deciding the equivalence between QSMs, and, if two QSMs are not equivalent, this algorithm will produce an input-output pair with length not more than (n_1+n_2)^2. (2) We improve the bound for the equivalence between QSMs from (n_1+n_2)^2 to n_1^2+n_2^2-1, by employing Moore and Crutchfield's method (Theoretical Computer Science 237 (2000) 275-306). (3) We give that two MO-1QFAs with n_1 and n_2 states, respectively, are equivalent iff they are (n_1+n_2)^2--equivalent, and further obtain a polynomial-time algorithm for deciding the equivalence between two MO-1QFAs. (4) We provide a counterexample showing that Koshiba's method to solve the problem of deciding the equivalence between MM-1QFAs may be not valid, and thus the problem is left open again.
[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)}] .