Colla, Alessandra (2018) Dynamics of Quantum Spin Chains. [Magistrali biennali]
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In the present thesis we perform a step towards understanding the effec- tiveness of Hamiltonian perturbation theory in quantum dynamics, which is made possible by the Hamiltonian framing of quantum mechanics. To such a purpose, we focus our attention on quantum spin chains, namely on the isotropic Heisenberg ferromagnet, which is well known to be equivalent, via the Jordan-Wigner transformation, to a system of interacting fermions, and is also known to be exactly integrable through the Bethe ansatz. The interaction term of the Hamiltonian, of order four in fermionic creation and annihilation operators, is treated as a perturbation in the regime of small excitations: small number of reversed spins or, equivalently, small number of fermions per site. The calculation of the first order normal form Hamiltonian amounts to erasing all the nonresonant interaction terms, and allows to draw interesting conclusions on the dynamics of the system, e.g. to determine ap- proximately conserved quantities. Moreover, we get a good correspondence between our perturbative energy spectrum and the exact one computed via the Bethe ansatz, the approximation being of course the more accurate the smaller is the number of fermions per site. The present method, here tested in an exactly solvable case, can be applied to any nonintegrable system of weakly interacting fermions (and/or bosons).
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