The classical limit of quantum spin systems

Élliott H. Lieb

Élliott H. Lieb

Nick G. Duffield

P Lslash ugiewicz, Robert Olkiewicz

Environmentally induced classical properties in quantum spin systems in the thermodynamic limit are discussed. In particular, it is shown that such a system, subjected to a specific interaction with another quantum system, may be described for all practical purposes as a classical dynamical system. This provides an alternative approach to the classical ℏ → 0 limit for the emergence of reversible classical evolution from dissipative quantum dynamics.

Heinz–Jürgen Schmidt

We study the relation between quantum mechanical and classical Gibbs states of spin systems with spin quantum number $s$. It is known that quantum states and observables can be represented by functions defined on the phase space ${\mathcal S}$, which in our case is the $N$-fold product of unit spheres. Therefore, the classical limit $s\to\infty$ of (suitably scaled) quantum Gibbs states can be described as the limit of functions defined on ${\mathcal S}$. We choose to approximate the exponential function of the Hamiltonian by a polynomial of degree $n$ and thus have to deal with the problem of the limit of double sequences (depending on $n$ and $s$) treated in the theorem of Moore-Osgood. The convergence of quantum Gibbs states to classical ones is illustrated by the example of the Heisenberg dimer. We apply our method to the explicit calculation of the phase space function describing spin monomials, and finally add some general remarks on the theory of spin coherent states.

H.-J. Mikeska

This chapter presents an investigation of the dynamics of one anisotropic spin in an external, periodically time-dependent field. The simple quantum system displays chaotic behavior in the classical limit. Chaotic behavior in classical deterministic systems is characterized by an extremely sensitive dependence on initial conditions. The statistics of quasienergy levels are discussed for our driven spin system with a twofold purpose. The statistics of quasienergy levels is equivalent to the energy level statistics in a time-independent system and the dependence of the quasienergy level statistics on the quantum parameter is investigated. The statistical properties of the spectrum of a quantum problem in its classical limit are associated with the regularity properties of the corresponding classical system. Using the complementary approaches of phase space analysis and level statistics we have shown in both schemes that for large spins, i.e. in the semiclassical regime, stochasticity can be discussed using concepts developed for classical systems.