Non-Hermitian linear response theory


Linear response theory lies at the heart of studying quantum matters, because it connects the dynamical response of a quantum system to an external probe to correlation functions of the unprobed equilibrium state. Thanks to linear response theory, various experimental probes can be used for determining equilibrium properties. However, so far, both the unprobed system and the probe operator are limited to Hermitian ones. Here, we develop a non-Hermitian linear response theory that considers the dynamical response of a Hermitian system to a non-Hermitian probe, and we can also relate such a dynamical response to the properties of an unprobed Hermitian system at equilibrium. As an application of our theory, we consider the real-time dynamics of momentum distribution induced by one-body and two-body dissipations. Remarkably, for a critical state with no well-defined quasi-particles, we find that the dynamics are slower than the normal state with well-defined quasi-particles, and our theory provides a model-independent way to extract the critical exponent in the real-time correlation function. We find surprisingly good agreement between our theory and a recent cold atom experiment on the dissipative Bose–Hubbard model. We also propose to further quantitatively verify our theory by performing experiments on dissipative one-dimensional Luttinger liquid.

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Fig. 1: Schematic of real-time dynamics of the momentum distribution induced by dissipations.
Fig. 2: Reanalysis, using our theory, of experimental data for the dissipative 2D Bose–Hubbard model reported in ref. 2.
Fig. 3: Experimental proposal of dissipative Luttinger liquids.

Data availability

All the data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.


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This work is supported by the Beijing Outstanding Young Scientist Program (H.Z.), NSFC grant no. 11734010 (H.Z. and Y.C.), NSFC grant no. 11604225 (Y.C.), MOST grant no. 2016YFA0301600 (H.Z.) and the Beijing Natural Science Foundation (Z180013; Y.C.).

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H.Z. initiated the project. H.Z. and Y.C. led the project. All authors contributed to performing calculations, discussing the results and writing the paper.

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Correspondence to Yu Chen or Hui Zhai.

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Pan, L., Chen, X., Chen, Y. et al. Non-Hermitian linear response theory. Nat. Phys. (2020).

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