Non-Hermitian adiabatic transport in spaces of exceptional points

Abstract

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 $T=1/\epsilon$ 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 ${\epsilon }^{-1}$ times an expansion in powers of ${\epsilon }^{1/n}$. The exponential of the term of $n$th order (which is equivalent to the "geometric" or Berry phase modulo $2\pi$), is thus independent of $\epsilon$ as $\epsilon \to 0$; 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.

arXiv comment4+3 pages. v2: slight title change; 9 pages, now in regular article format; as published

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• Symmetry-protected quantization of complex Berry phases in non-Hermitian many-body systems
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• Exploring Nonreciprocal Noise Transfer under Onsager-Casimir Symmetry in Synthetic-Field Optomechanics
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