Abstract
Many-body physics of identical particles is commonly believed to be a sovereign territory of Quantum Mechanics. The aim of this contribution is to show that it is actually not the case and one gets useful insights into a quantum many-body system by using the theory of classical dynamical systems. In the contribution we focus on one paradigmatic model of many-body quantum physics - the Bose–Hubbard model which, in particular, describes interacting ultracold Bose atoms in an optical lattice . We show how one can find/deduce the energy spectrum of the Bose–Hubbard model by using a kind of the semiclassical approach.
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Notes
- 1.
This should be opposed to the Fermi–Hubbard model, where the spectrum can be found analytically by using the Betha ansatz.
- 2.
- 3.
The effective Planck constant \(\hbar _{eff}=1/N\) should not be mismatched with the fundamental Planck constant \(\hbar \) which we set to unity from now on. We also mention that within the discussed formalism \(a_l\) and \(a_l^*\) are the canonical variables, i.e., one does not interpret them as order parameters.
- 4.
- 5.
Slight asymmetry of \(\rho (E)\) with respect to \(E=0\) is related to the fact that L is odd. For even L (for example \(L=6\)) the distribution is perfectly symmetric, i.e., \(\rho (E)\) is an even function of E.
- 6.
Regular localized solutions of DNLSE are known as discrete solitons or breathers [4].
- 7.
For the periodic boundary conditions (which are used throughout the paper) the quantum BH model possesses additional, pure quantum integral of motion – the total quasimomentum \(\kappa =2\pi k/L\). Thus the whole spectrum can be decomposed into L independent spectra labeled by \(\kappa \). In Fig. 3 we choose \(\kappa =2\pi /L\) subspace. The results for other \(\kappa \) look similar, except the case \(\kappa =0\) where one should take into account the odd-even symmetry of the eigenstates.
- 8.
Berry–Robnik’s statistics gives level-spacing distribution for a system with mixed phase space and interpolates between Poisson and Wigner–Dyson statistics.
- 9.
Bloch oscillations are dynamical response of the system to an external static field. For non-interacting atoms these would be periodic oscillation of the mean atomic momentum with the Bloch frequency which is proportional to the field strength.
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Kolovsky, A.R. (2017). Treating Many-Body Quantum Systems by Means of Classical Mechanics. In: Mantica, G., Stoop, R., Stramaglia, S. (eds) Emergent Complexity from Nonlinearity, in Physics, Engineering and the Life Sciences. Springer Proceedings in Physics, vol 191. Springer, Cham. https://doi.org/10.1007/978-3-319-47810-4_4
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