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CRC Press eBooks

One-Dimensional Spin Chain Models in Condensed Matter Theory

M. S. Ramkarthik, Payal D. Solanki

Published Jun 29, 2021• 0 citations
Publisher

Abstract

Hamiltonians of one dimensional spin half models in condensed matter physics are expounded in this chapter. Hamiltonian of spins interacting with an external magnetic field and separate Hamiltonians for the direct exchange spin-spin interactions in X, Y, Z directions, Heisenberg Hamiltonian and the Dzyaloshinskii-Moriya interaction are dealt with in detail. It is emphasized here that the Hamiltonians are not only for the nearest-neighbour interaction but it is for the rth nearest-neighbour interaction, this increases the scope and usability of the recipies. Other important variants such as the Ising model, transverse Ising model, Majumdar-Ghosh model and a plethora of other spin chain models can be easily constructed using our recipes given and it has been highlighted via examples.

DOI
10.1201/9781003164678-6

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Hamiltonians of one dimensional spin half models in condensed matter physics are expounded in this chapter. Hamiltonian of spins interacting with an external magnetic field and separate Hamiltonians for the direct exchange spin-spin interactions in X, Y, Z directions, Heisenberg Hamiltonian and the Dzyaloshinskii-Moriya interaction are dealt with in detail. It is emphasized here that the Hamiltonians are not only for the nearest-neighbour interaction but it is for the rth nearest-neighbour interaction, this increases the scope and usability of the recipies. Other important variants such as the Ising model, transverse Ising model, Majumdar-Ghosh model and a plethora of other spin chain models can be easily constructed using our recipes given and it has been highlighted via examples.
Starting from a Calogero-Sutherland model with the hyperbolic interaction confined by an external field with Morse potential, we construct a Heisenberg spin chain with exchange interaction varies as 1/sinh 2 x on a lattice given in terms of the zeros of Laguerre polynomials. Varying the strength of the Morse potential, the Haldane-Shastry and harmonic spin chains are reproduced. The spectrum of the models in this class is found to be that of a classical one-dimensional Ising chain with non-uniform nearest-neighbour coupling in a non-uniform magnetic field which allows us to study the thermodynamics in the limit of infinite chains.
Starting from a Calogero--Sutherland model with hyperbolic interaction confined by an external field with Morse potential we construct a Heisenberg spin chain with exchange interaction ∝1/sinh⁡2x\propto 1/\sinh^2 x∝1/sinh2x on a lattice given in terms of the zeroes of Laguerre polynomials. Varying the strength of the Morse potential the Haldane--Shastry and harmonic spin chains are reproduced. The spectrum of the models in this class is found to be that of a classical one-dimensional Ising chain with nonuniform nearest neighbour coupling in a nonuniform magnetic field which allows to study the thermodynamics in the limit of infinite chains.
A one-dimensional Ising chain in which nearest neighbour (n.n.) and next nearest neighbour (n.n.n.) exchange interactions depend linearly on the distance between the spins is considered. The magnetic field and temperature dependence of the chain length is exactly calculated by use of a transfer matrix. It is shown that the chain length changes variously because of the competition among the n.n. and n.n.n. exchange interactions and the magnetic field. The zero-temperature phase diagram is also determined, which is composed of four spin states ↑, ↑↓, ↑↑↓ and ↑↑↓↓ (the smallest units of repetitions are shown). Dimerization and trimerization of the lattice are found in the ↑↑↓↓ and ↑↑↓ states respectively.