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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Mahan, G. D. Many Particle Physics (Plenum Press, 1981).
Bouganne, R., Aguilera, M. B., Ghermaoui, A., Beugnon, J. & Gerbier, F. Anomalous decay of coherence in a dissipative many-body system. Nat. Phys. 16, 21–25 (2020).
Syassen, N. et al. Strong dissipation inhibits losses and induces correlations in cold molecular gases. Science 320, 1329–1331 (2008).
Barontini, G. et al. Controlling the dynamics of an open many-body quantum system with localized dissipation. Phys. Rev. Lett. 110, 035302 (2013).
Tomita, T., Nakajima, S., Danshita, I., Takasu, Y. & Takahashi, Y. Observation of the Mott insulator to superfluid crossover of a driven-dissipative Bose–Hubbard system. Sci. Adv. 99, e1701513 (2017).
Sponselee, K. et al. Dynamics of ultracold quantum gases in the dissipative Fermi–Hubbard model. Quantum Sci. Technol. 4, 014002 (2018).
Li, J. et al. Observation of parity–time symmetry breaking transitions in a dissipative Floquet system of ultracold atoms. Nat. Commun. 10, 855 (2019).
Tomita, T., Nakajima, S., Takasu, Y. & Takahashi, Y. Dissipative Bose–Hubbard system with intrinsic two-body loss. Phys. Rev. A 99, 031601 (2019).
Lee, T. E. Anomalous edge state in a non-Hermitian lattice. Phys. Rev. Lett. 116, 133903 (2016).
Leykam, D., Bliokh, K. Y., Huang, C., Chong, Y. D. & Nori, F. Edge modes, degeneracies and topological numbers in non-Hermitian systems. Phys. Rev. Lett. 118, 040401 (2017).
Shen, H., Zhen, B. & Fu, L. Topological band theory for non-Hermitian Hamiltonians. Phys. Rev. Lett. 120, 146402 (2018).
Yao, S. & Wang, Z. Edge states and topological invariants of non-Hermitian systems. Phys. Rev. Lett. 121, 086803 (2018).
Kunst, F. K., Edvardsson, E., Budich, J. C. & Bergholtz, E. J. Biorthogonal bulk-boundary correspondence in non-Hermitian systems. Phys. Rev. Lett. 121, 026808 (2018).
Gong, Z. et al. Topological phases of non-Hermitian systems. Phys. Rev. X 8, 031079 (2018).
Chen, Y. & Zhai, H. Hall conductance of a non-Hermitian Chern insulator. Phys. Rev. B 98, 245130 (2018).
Kawabata, K., Shiozaki, K., Ueda, M. & Sato, M. Symmetry and topology in non-Hermitian physics. Phys. Rev. X 9, 041015 (2019).
Lee, C. H. & Thomale, R. Anatomy of skin modes and topology in non-Hermitian systems. Phys. Rev. B 99, 201103 (2019).
Lee, J. Y., Ahn, J., Zhou, H. & Vishwanath, A. Topological correspondence between Hermitian and non-Hermitian systems: anomalous dynamics. Phys. Rev. Lett. 123, 206404 (2019).
Zeuner, J. M. et al. Observation of a topological transition in the bulk of a non-Hermitian system. Phys. Rev. Lett. 115, 040402 (2015).
Zhou, H. et al. Observation of bulk Fermi arc and polarization half charge from paired exceptional points. Science 359, 1009–1012 (2018).
Harari, G. et al. Topological insulator laser: theory. Science 359, eaar4003 (2018).
Bandres, M. A. et al. Topological insulator laser: experiments. Science 359, eaar4005 (2018).
Helbig, T. et al. Observation of bulk boundary correspondence breakdown in topolectrical circuits. Preprint at https://arxiv.org/abs/1907.11562 (2019).
Xiao, L. Non-Hermitian bulk-boundary correspondence in quantum dynamics. Nat. Phys. (2020); https://doi.org/10.1038/s41567-020-0836-6
Sachdev, S. Quantum Phase Transitions (Cambridge Univ. Press, 1999).
Kamenev, A. Field Theory for Non-Equilibrium Systems (Cambridge Univ. Press, 2011).
Pourfath, M.The Non-Equilibrium Green’s Function Method for Nanoscale Device Simulation (ed. Selberherr, S.) (Springer, 2014).
Zinn-Justin, J. Phase Transitions and Renormalisation Group (Oxford Univ. Press, 2007).
Witczak-Krempa, W., Sørensen, E. S. & Sachdev, S. The dynamics of quantum criticality revealed by quantum Monte Carlo and holography. Nat. Phys. 10, 361–366 (2014).
Giamarchi, T. Quantum Physics in One Dimension (Oxford Science, 2004).
Lieb, E. H. & Liniger, W. Exact analysis of an interacting Bose gas. I. The general solution and the ground state. Phys. Rev. 130, 1605–1616 (1963).
Imambekov, A. & Glazman, L. I. Exact exponents of edge singularities in dynamic correlation functions of 1D Bose gas. Phys. Rev. Lett. 100, 206805 (2008).
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.).
The authors declare no competing interests.
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Pan, L., Chen, X., Chen, Y. et al. Non-Hermitian linear response theory. Nat. Phys. (2020). https://doi.org/10.1038/s41567-020-0889-6