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Bulletin of the American Physical Society

Magnetic Susceptibilities of Finite Ising Chains in the Presence of Defect Sites

S. V. Goupalov, Daniel Charles Mattis

Published Mar 12, 2008• 0 citations
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Abstract

Any antiferromagnet with zero net magnetic moment exhibits limited response to an external homogeneous magnetic field. This changes dramatically in the presence of defect sites, even those that carry no spin. We examine the excess susceptibilities, longitudinal and transverse, due to one or more defects at arbitrary separations in a finite Ising chain with nearest-neighbor couplings. Adapting matrix methods to finite chains we derive exact formulas valid at all T 0.

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Any antiferromagnet with zero net magnetic moment exhibits limited response to an external homogeneous magnetic field. This changes dramatically in the presence of defect sites, even those that carry no spin. We examine the excess susceptibilities, longitudinal and transverse, due to one or more defects at arbitrary separations in a finite Ising chain with nearest-neighbor couplings. Adapting matrix methods to finite chains we derive exact formulas valid at all T 0.
Any antiferromagnet with zero net magnetic moment exhibits limited response to an external homogeneous magnetic field. This changes dramatically in the presence of defect sites, even those that carry no spin. We examine the excess susceptibilities, longitudinal and transverse, due to one or more defects at arbitrary separations in a finite Ising chain with nearest-neighbor couplings. Adapting matrix methods to finite chains we derive exact formulas valid at all $T\ensuremath{\geqslant}0$.
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.
Motivated by frustrated magnets and quasi-one-dimensional magnetic materials, we study the magnetic properties of one-dimensional (1D) Ising chains with nearest-neighbor (NN) and weaker next-to-nearest-neighbor (NNN) interactions in the presence of vacancy defects. The effect of a vacancy on the magnetic susceptibility of a spin chain is twofold: it reduces the length of the chain by an effective ``vacancy size'' and may also act as a free spin, a ``quasispin,'' with a Curie-type ${\ensuremath{\chi}}_{\text{quasi}}=\ensuremath{\langle}{S}^{2}\ensuremath{\rangle}/T$ contribution to the susceptibility. In chains with antiferromagnetic short-range order, the susceptibility of vacancy-free chains is exponentially suppressed at low temperatures, and quasispins dominate the effect of impurities on the chains' magnetic properties. For chains with antiferromagnetic NN interactions, the quasispin matches the value $\ensuremath{\langle}{S}^{2}\ensuremath{\rangle}=1$ of the Ising spins in the chain for ferromagnetic NNN interactions and vanishes for antiferromagnetic NNN interactions. For chains with ferromagnetic short-range order, quasispin effects are insignificant due to exponentially large low-temperature susceptibilities, and the dominant effect of a vacancy is effectively changing the length of the chain.
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.
We study the three-dimensional antiferromagnetic Ising model in both uniform longitudinal [Formula: see text] and transverse [Formula: see text] magnetic fields by using the effective-field theory (EFT) with finite cluster [Formula: see text] spin (EFT-1). We analyzed the behavior of the magnetic susceptibility to investigate the reentrant phenomena that we have seen in the same phase diagram previously obtained in other papers. Our results shows the presence of two divergences in the susceptibility that indicates the existence of a reentrant behavior.