This dissertation deals with the dynamics of interacting quantum and classical spin models
and the question of whether and to which degree the dynamics of these models agree with
each other.
For this purpose, XXZ models are studied on different lattice geometries of finite size,
ranging from one-dimensional chains and quasi-one-dimensional ladders to two-dimensional
square lattices. Particular attention is paid to the high-temperature analysis of the temporal
behavior of autocorrelation functions for both the local density of magnetization (spin)
and energy, which are closely related to transport properties of the considered models. Due
to the conservation of total energy and total magnetization, the dynamics of such densities
are expected to exhibit hydrodynamic behavior for long times, which manifests itself in
a power-law tail of the autocorrelation function in time. From a quantum mechanical
point of view, the calculation of these autocorrelation functions requires solving the linear
Schrödinger equation, while classically Hamilton’s equations of motion need to be solved.
An efficient numerical pure-state approach based on the concept of typicality enables
circumventing the costly numerical method of exact diagonalization and to treat quantum
autocorrelation functions with up to N = 36 lattice sites in total.
While, in full generality, a quantitative agreement between quantum and classical dy-
namics can not be expected, contrarily, based on large-scale numerical results, it is
demonstrated that the dynamics of the quantum S = 1/2 and classical spins coincide, not
only qualitatively, but even quantitatively, to a remarkably high level of accuracy for all
considered lattice geometries. The agreement particularly is found to be best in the case
of nonintegrable quantum models (quasi-one-dimensional and two-dimensional lattice),
but still satisfactory in the case of integrable chains, at least if transport properties are
not dominated by the extensive number of conservation laws.
Additionally, in the context of disordered spin chains, such an agreement of the dynamics
is found to hold even in the presence of small values of disorder, while at strong disorder
the agreement is pronounced most for larger spin quantum numbers.
Finally, it is shown that a putative many-body localization transition within the one-
dimensional spin chain is shifted to stronger values of disorder with increasing spin
quantum number. It is concluded that classical or semiclassical simulations might provide
a meaningful strategy to investigate the quantum dynamics of strongly interacting quantum
spin models, even if the spin quantum number is small and far from the classical limit.
| Identifer | oai:union.ndltd.org:uni-osnabrueck.de/oai:osnadocs.ub.uni-osnabrueck.de:ds-202206137059 |
| Date | 13 June 2022 |
| Creators | Schubert, Dennis |
| Contributors | Prof. Dr. Robin Steinigeweg, Prof. Dr. Jochen Gemmer |
| Source Sets | Universität Osnabrück |
| Language | English |
| Detected Language | English |
| Type | doc-type:doctoralThesis |
| Format | application/zip, application/pdf |
| Rights | Attribution 3.0 Germany, http://creativecommons.org/licenses/by/3.0/de/ |
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