We consider the space of $n\times n$ non-Hermitian Hamiltonians ($n=2$, 3, ...) that are equivalent to a single $n\times n$ Jordan block. We focus on adiabatic transport around a closed path (i.e., a loop) within this space, in the limit as the time scale $T=1/\varepsilon$ taken to traverse the loop tends to infinity. We show that, for a certain class of loops and a choice of initial state, the state returns to itself and acquires a complex phase that is ${\varepsilon }^{-1}$ times an expansion in powers of ${\varepsilon }^{1/n}$. The exponential of the term of $n\mathrm{th}$ order (which is equivalent to the “geometric” or Berry phase modulo $2\pi$) is thus independent of $\varepsilon$ as $\varepsilon \to 0$; it depends only on the homotopy class of the loop and is an integer power of $e^{2\pi i/n}$. One of the conditions under which these results hold is that the state being transported is, for all points on the loop, that of slowest decay.

中文翻译：

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特殊地点空间中的非赫米特绝热运输

我们考虑的空间 $ñ\times ñ$ 非埃尔米特哈密顿量（$ñ=2$，3，...）相当于一个 $ñ\times ñ$乔丹街区。我们专注于围绕该空间内的闭合路径（即环路）的绝热运输，其极限是时间尺度$Ť=\mathrm{1个}/\varepsilon$遍历循环趋于无穷大。我们表明，对于特定类型的循环和初始状态的选择，状态会返回自身并获得一个复杂的相位，即${\varepsilon }^{-\mathrm{1个}}$ 倍增权力 ${\varepsilon }^{\mathrm{1个}/ñ}$。项的指数$ñ日$ 阶（等效于“几何”或贝里相位模 $2\pi$）因此独立于 $\varepsilon$ 如 $\varepsilon \to 0$; 它仅取决于循环的同伦类，并且是${Ë}^{2\pi \mathrm{一世}/ñ}$。这些结果成立的条件之一是，对于环路上的所有点，传输的状态都是最慢的衰减状态。