The linear σ-ω model in the thermo field dynamics

Kouji Nakamura, Y. Yamanaka, Hiroomi Umezawa

Mizuhiko Saeki

Y. Georgelin, Y. Leblanc

H. Matusumoto

Rivani Adistia Dewi, Imam Nugraha Albania, Rizky Rosjanuardi +1 more

Spin-1/2 system is one of the most important phenomenon in quantum mechanics.Such a system is known to be expressed as Pauli matrices, that isand 𝜎 3 = ( 1 0 0 -1 ) .By inserting the identity matrix 𝐼, we can construct a C*-algebra 𝐶 * ({𝐼, 𝜎 1 , 𝜎 2 , 𝜎 3 }) with |{𝐼, 𝜎 1 , 𝜎 2 , 𝜎 3 }⨂{𝐼, 𝜎 1 , 𝜎 2 , 𝜎 3 }| = 16 = |{𝐸 𝑖𝑗 : 𝑖, 𝑗 = 1,2,3,4} |, where 𝐸 𝑖𝑗 is the canonical basis of C*-algebra 𝑀 4 (ℂ).In this study, by using the linear independence concept, we got 𝐶 * ({𝐼, 𝜎 1 , 𝜎 2 , 𝜎 3 })⨂𝐶 * ({𝐼, 𝜎 1 , 𝜎 2 , 𝜎 3 }) ≅ 𝑀 4 (ℂ).This means, that for any observable in the composite of two spin-1/2 systems, by self-adjointness of Pauli matrices, it can be expressed by the field reduction from ℂ to ℝ in 𝑠𝑝𝑎𝑛{𝐼, 𝜎 1 , 𝜎 2 , 𝜎 3 }⨂{𝐼, 𝜎 1 , 𝜎 2 , 𝜎 3 } where the structure itself is nonassociative algebra.Furthermore, given a Hamiltonian 𝐻(𝑡) = (𝜎 1 ⨂𝜎 1 ) + 𝒥 𝑠𝑖𝑛(𝜔𝑡) (𝜎 3 ⨂𝐼) + 𝒥 𝑠𝑖𝑛 (𝜔𝑡 + 𝜋 2 ) (𝐼⨂𝜎 3 ) and the initial state which is a linear combination of Bell basis, by using the integration factor method and the commutation fact of the exponential operator, we got the following timedependent state |𝜓(𝑡)⟩ = 1 √2 (𝑒 ( 𝑖𝑡 ℏ )( 𝑖𝒥 ℏ𝜔 𝑐𝑜𝑠(𝜔𝑡))( 𝑖𝒥 ℏ𝜔 𝑠𝑖𝑛(𝜔𝑡))𝜈 𝑖(5-𝑖) ) 𝑖=1 4 with 𝜈 𝑖𝑗 = 𝛿 (𝑖+𝑗)5 (-1) 𝑚𝑎𝑘𝑠{𝑖,𝑗} , where 𝛿 𝑖𝑗 is Kronecker delta.

G. A. T. Allan, Donald D. Betts

Fisher has found exactly for the square and honeycomb lattice the static initial perpendicular susceptibility of the quantum-mechanical version of the Ising model defined by the Hamiltonian H=−J ∑ nnσizσjz−mHx(t) ∑ iσix,where all symbols have their usual meanings. As the magnetization, Mx = mΣiσix, does not commute with the Hamiltonian, the system can relax and Kubo's linear response theory can be applied. We have been able to evaluate the response function exactly and hence obtain simply the exact frequency-dependent initial perpendicular susceptibility for the honeycomb lattice Jχ⊥(ω,T)/Nm2=¾[〈σ0zσ1z〉−〈σ0zσ1zσ2zσ3z〉]ω02/(ω02−ω2)+3[¾〈σ0zσ1z〉+¼〈σ0zσ1zσ2zσ3z〉]ω02/(9ω02−ω2),where ω0 = 2J/ℏ and 〈σ0zσ1z〉 is the nearest-neighbor correlation evaluated by Onsager. A similar expression is obtained for the square lattice. For the four-particle correlation 〈σ0zσ1zσ2zσ3z〉 the other three spins are nearest neighbors to spin zero, and we have obtained its value from Fisher's transformation theory. The behavior of χ⊥ as a function of T and ω is described and its physical significance is discussed.