Classical analysis of quantum phase transitions in a bilayer model

In this Brief Report we extend the classical analysis performed on the schematic model proposed in [T. Moreira, G. Q. Pellegrino, J. G. Peixoto de Faria, M. C. Nemes, F. Camargo, and A. F. R. Toledo Piza, Phys. Rev. E 77, 051102 (2008)] concerning quantum phase transitions in a bilayer system. We show that appropriate integrations along the classical periodic orbits reproduce with excellent agreement both the quantum spectrum and the expected mean value for the number of excitons in the system, quantities which are directly related to the observed boson-fermion quantum phase transition.

Figure 1

Classical phase spaces for values (a) ${\delta }^{\prime }/g^{\prime }=1.0$ and (b) ${\delta }^{\prime }/g^{\prime }=2.5$. The separatrix orbit is shown as the thick curve corresponding to energy zero.

Figure 2

(a) Quantum spectra $E_k\times k$ for different values of $\delta /g$, and semiclassical spectra $\mathcal{E}\times k$ for the corresponding values of ${\delta }^{\prime }/g^{\prime }$ with $k$ calculated following relation (11). From bottom to top, curves are drawn for values $\delta /g=0,20,20\sqrt{8},80$. (b) Magnification of part of the curves for $\delta /g=20$. Triangles refer to the classical estimate.

Figure 3

Quantum expectation value for the number of bosons $⟨n_b\left(k\right)⟩$ (solid curves) and semiclassical estimate for its classical analog $n_b\left(\mathcal{E}\right)\equiv \frac{N}{4}\left(1+{\overline{j}}_z\right)$ (points), as a function of the coupling parameter $\delta /g$. In both cases, states $E_k$ are chosen for (a) $k=1$, (b) $k=200$, (c) $k=300$, and (d) $k=500$.