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arXiv:cond-mat/9902007 · Journal of the Physical Society of Japan

Integrable Magnetic Model of Two Chains Coupled by Four-Body Interactions

Norihiro Muramoto, Minoru Takahashi

Published Jun 15, 1999• 44 citations
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Abstract

An exact solution for an XXZ chain with four-body interactions is obtained and its phase diagram is determined. The model can be reduced to two chains coupled by four-body interactions, and it is shown that the ground state of the two-chain model is magnetized in part. Furthermore, a twisted four-body correlation function of the anti-ferromagnetic Heisenberg chain is obtained.

DOI
10.1143/jpsj.68.2098

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An exact solution for an XXZ chain with four-body interactions is obtained and its phase diagram is determined. The model can be reduced to two chains coupled by four-body interactions, and it is shown that the ground state of the two-chain model is magnetized in part. Furthermore, a twisted four-body correlation function of the anti-ferromagnetic Heisenberg chain is obtained.
The phase diagram of a coupled tetrahedral Heisenberg model is obtained. The quantum chain has a local gauge symmetry and its eigenspectrum is obtained by the composition of the eigenspectra of $\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\ensuremath{-}\frac{1}{2}$ XXZ chains with arbitrary distribution of $\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\ensuremath{-}\frac{3}{2}$ impurities. The phase diagram is quite rich with an infinite number of phases with ferromagnetic, antiferromagnetic, or ferrimagnetic order. In some cases the ground state and the low-lying eigenlevels of the model can be exactly calculated since they coincide with the eigenlevels of the exactly integrable XXZ chain. The thermodynamical properties of the model at low temperatures are also studied through finite-size analysis.
A mixed Ising-Heisenberg spin system consisting of triangular XXZXXZXXZ-Heisenberg spin clusters assembled into a chain by alternating with Ising spins interacting to all three spins in the triangle is considered. The exact solution of the model is given in terms of the generalized decoration-iteration map and within the transfer-matrix technique. Exact expressions for thermodynamic functions are derived. Ground-state phase diagrams and thermodynamic and magnetic properties of the system are examined.
Introduction to Integrable Many-Body Systems II This is the second part of a three-volume introductory course about integrable systems of interacting bodies. The models of interest are quantum spin chains with nearest-neighbor interactions between spin operators, in particular Heisenberg spin-1/2 models. The Ising model in a transverse field, expressible as a quadratic fermion form by using the Jordan-Wigner transformation, is the subject of Sect. 12. The derivation of the coordinate Bethe ansatz for the XXZ Heisenberg chain and the determination of its absolute ground state in various regions of the anisotropy parameter are presented in Sect. 13. The magnetic properties of the ground state are explained in Sect. 14. Sect. 15 concerns excited states and the zero-temperature thermodynamics of the XXZ model. The thermodynamics of the XXZ Heisenberg chain is derived on the basis of the string hypothesis in Sect. 16; the thermodynamic Bethe ansatz equations are analyzed in high-temperature and low-temperature limits. An alternative derivation of the thermodynamics without using strings, leading to a non-linear integral equation determining the free energy, is the subject of Sect. 17. A nontrivial application of the Quantum Inverse Scattering method to the fully anisotropic XYZ Heisenberg chain is described in Sect. 18. Sect. 19 deals with integrable cases of isotropic spin chains with an arbitrary spin.