PT phase transition in multidimensional quantum systems

Non-Hermitian PT-symmetric quantum-mechanical Hamiltonians generally exhibit a phase transition that separates two parametric regions, (i) a region of unbroken PT symmetry in which the eigenvalues are all real, and (ii) a region of broken PT symmetry in which some of the eigenvalues are complex. This transition has recently been observed experimentally in a variety of physical systems. Until now, theoretical studies of the PT phase transition have generally been limited to one-dimensional models. Here, four nontrivial coupled PT-symmetric Hamiltonians, $H=p^2\mathrm{/}2+x^2\mathrm{/}2+q^2\mathrm{/}2+y^2\mathrm{/}2+ig{x}^{2}y$, $H=p^2\mathrm{/}2+x^2\mathrm{/}2+q^2\mathrm{/}2+y^2+ig{x}^{2}y$, $H=p^2\mathrm{/}2+x^2\mathrm{/}2+q^2\mathrm{/}2+y^2\mathrm{/}2+r^2\mathrm{/}2+z^2\mathrm{/}2+igxyz$, and $H=p^2\mathrm{/}2+x^2\mathrm{/}2+q^2\mathrm{/}2+y^2+r^2\mathrm{/}2+3z^2\mathrm{/}2+igxyz$ are examined. Based on extensive numerical studies, this paper conjectures that all four models exhibit a phase transition. The transitions are found to occur at $g\approx 0.1$, $g\approx 0.04$,$g\approx 0.1$,and$g\approx 0.05$. These results suggest that the PT phase transition is a robust phenomenon not limited to systems having one degree of freedom.