## PT phase transition in multidimensional quantum systems

Carl M. Bender, David J. Weir

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, $$H=p^2/2+x^2/2+q^2/2+y^2/2+igx^2y$$, $$H=p^2/2+x^2/2+q^2/2+y^2+igx^2y$$, $$H=p^2/2+x^2/2+q^2/2+y^2/2+r^2/2+z^2/2+igxyz$$, and $$H=p^2/2+x^2/2+q^2/2+y^2+r^2/2+3z^2/2+igxyz$$ 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.

http://arxiv.org/abs/1206.5100
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph)