arXiv 2008.040122020-08quant-phmath-phmath.FAmath.MP

非厄米谐振子中的量子相变

Quantum phase transitions in nonhermitian harmonic oscillator

Miloslav Znojil

备注：: 18 pp, 1 figure
发表信息：: Scientific Reports 10(1) (2020) 18523
DOI：: 10.1038/s41598-020-75468-w

摘要：

斯通定理要求在物理希尔伯特空间${\cal H}$中，一个稳定的量子系统的时间演化是么正的当且仅当相应的哈密顿量$H$是自伴的。有时候，演化的一个更简单的图像可以在一个明显非物理的希尔伯特空间${\cal K}$中构建，其中$H$是非厄米的，但${\cal PT}-$是对称的。不幸的是，在应用中，人们很少能成功地绕过物理希尔伯特空间${\cal H}$的必要重建中的关键技术障碍。对于一个${\cal PT}-$对称版本的尖峰谐振子，我们表明，在动力学制度的不可避免的水平交叉，这样的重建${\cal H}$成为可行的，而且，可通过非数值手段。这种对${\cal H}$的重构的一般形式使人们能够把每一个例外的不可避免的交叉点作为一个真正的、现象学上最吸引人的量子相变瞬间来处理。

英文摘要：

The Stone theorem requires that in a physical Hilbert space ${\cal H}$ the time-evolution of a stable quantum system is unitary if and only if the corresponding Hamiltonian $H$ is self-adjoint. Sometimes, a simpler picture of the evolution may be constructed in a manifestly unphysical Hilbert space ${\cal K}$ in which $H$ is nonhermitian but ${\cal PT}-$symmetric. In applications, unfortunately, one only rarely succeeds in circumventing the key technical obstacle which lies in the necessary reconstruction of the physical Hilbert space ${\cal H}$. For a ${\cal PT}-$symmetric version of the spiked harmonic oscillator we show that in the dynamical regime of the unavoided level crossings such a reconstruction of ${\cal H}$ becomes feasible and, moreover, obtainable by non-numerical means. The general form of such a reconstruction of ${\cal H}$ enables one to render every exceptional unavoided-crossing point tractable as a genuine, phenomenologically most appealing quantum-phase-transition instant.

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