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arXiv:0712.4306 · Progress of Theoretical Physics

An Exact Calculation of the Transverse Susceptibility for an Antiferromagnetic Ising Chain

N. Kunisada, Y. Fukumoto

Published Jun 1, 2008• 1 citation

Abstract

We study the transverse susceptibility of the fully frustrated antiferromagnetic Ising Δ-chain, extending Minami's transfer-matrix method to the transverse susceptibility of general-type Ising linear-chains [J. Phys. Soc. Jpn. 67 (1998), 2255]. For the transverse fields Γ1 in tip spin sites and Γ2 in bottom spin sites, we calculate zero-field transverse susceptibilities χxtip = lim Γ1 →0Mxtip/Γ1 and χxbottom = lim Γ2 →0Mxbottom/Γ2 while keeping the ratio Γ1/Γ2 fixed, where Mxtip (bottom) denotes the magnetization for tip (bottom) spin sites. Both the transverse susceptibilities follow Curie's law at low temperatures. We also calculate χxbottom(Γ1 > 0), the transverse susceptibility of the bottom spin chain under a finite tip-spin transverse field, to understand the Curie-type behavior in zero-field susceptibility. Using the second-order perturbation theory, we also discuss the Γ1 dependence of χxbottom (Γ1) at zero temperature.

DOI: 10.1143/ptp.119.913

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We study the transverse susceptibility of the fully frustrated antiferromagnetic Ising Δ-chain, extending Minami's transfer-matrix method to the transverse susceptibility of general-type Ising linear-chains [J. Phys. Soc. Jpn. 67 (1998), 2255]. For the transverse fields Γ1 in tip spin sites and Γ2 in bottom spin sites, we calculate zero-field transverse susceptibilities χxtip = lim Γ1 →0Mxtip/Γ1 and χxbottom = lim Γ2 →0Mxbottom/Γ2 while keeping the ratio Γ1/Γ2 fixed, where Mxtip (bottom) denotes the magnetization for tip (bottom) spin sites. Both the transverse susceptibilities follow Curie's law at low temperatures. We also calculate χxbottom(Γ1 > 0), the transverse susceptibility of the bottom spin chain under a finite tip-spin transverse field, to understand the Curie-type behavior in zero-field susceptibility. Using the second-order perturbation theory, we also discuss the Γ1 dependence of χxbottom (Γ1) at zero temperature.

Series expansions are presented for the magnetisation, susceptibility, magnetic field derivatives of the susceptibility and 'specific heat' of the zero-temperature transverse Ising model. Coefficients in these series have been calculated to tenth order in lambda 2 (where lambda is the transverse field) for the linear chain, to eighth order for the square lattice and to seventh order for the triangular lattice. These series yield estimates of the low-temperature critical exponents alpha ', beta ', gamma ' and Delta ' of the two- and three-dimensional Ising models. They provide good evidence for the symmetry of exponents above and below the critical point, e.g. gamma '= gamma .

Exact solution of the magnetic susceptibility for an S = 1/2 defective finite Ising chain is obtained by employing the transfer matrix method. The thermal dependence of susceptibility multiplied by temperature for the chain coupled to different magnetic impurities is investigated at all T ≥ 0. The results show that in the zero temperature limit the product of these two quantities equals the square of the net spin, demonstrating that Curie’s constants are truly invariant at low temperatures. The intrinsic properties of net spin are also reflected through their different responses to the parity of chain length at low temperature. In addition, the impacts of various host–impurity couplings as well as single-ion anisotropies on the susceptibility are discussed as well.

An antiferromagnetic Ising model on the Δ chain, with a macroscopic ground state degeneracy and exponentially decaying spin correlation functions, is studied. By introducing transverse magnetic fields Γ on spins at tips of the triangles and λΓ on spins on the bottom chain, we study the ground state as a function of λ and Γ. The λ=0 problem is exactly solved and the ground state has a staggered magnetization on the bottom chain for any Γ>0. Because the transverse field on the bottom chain with finite λ destroys the staggered order, the order–disorder transition occurs as λ is increased. We calculate the transition point λ c (Γ/ J ) as a function of Γ/ J , where J denotes the magnitude of the exchange interaction, by using the series expansion method around the λ=0 limit and conclude that λ c (Γ/ J =+0)≃0.9459. It is shown that states in the degenerate ground state manifold at Γ=0 are crucial in getting a finite value of λ c (+0).