ON EXACT FREQUENCY-DEPENDENT SUSCEPTIBILITY OF 2-DIMENSIONAL ISING MODEL

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=-Jnnσi zσjz-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 J χ⊥(ω,T)/Nm2= 3/4[〈σ0zσ1z〉- 〈σ0zσ1zσ 2zσ3z〉] ω02/(ω02- ω2)+3[3/4〈σ0zσ 1z〉+1/4〈σ0zσ 1zσ2zσ3 z〉]ω02/(9ω0 2-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 〈σ0zσ1zσ 2zσ3z〉 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. © 1968 The American Institute of Physics.