This is part of a virtual seminar series in theoretical condensed matter and atomic physics, with the aim to connect researchers working at different Nordic universities.
Similarly to the ergodicity hypothesis in classical chaotic systems, in the quantum setting there is a similar concept, related to quantum thermalization and equipartition over degrees of freedom and dubbed as the eigenstate thermalization hypothesis. This concept is very useful as it provides a link between classical and quantum chaos.
The concept of multifractality of quantum wave-functions is a way to break the above ergodicity in terms of chaotization and equipartitioning over degrees of freedom in quantum systems. On the other hand, in quantum information theory it is the entanglement entropy which represents the main measure of ergodicity and thermalization. On the third side, in the eigenstate thermalization hypothesis the fluctuations of local observables and their scaling with the system size play the central role.
In this talk I will represent an exact relation between the above three measures, namely multifractal dimensions, scaling of fluctuations of local observables and the (Renyi) entanglement entropy. I will show that the fractal dimension of the non-ergodic wave function puts an upper bound on its entanglement entropy [1]. I will also provide a couple of explicit examples demonstrating that the entanglement entropy may reach its ergodic (Page) value when the wave function is still highly non-ergodic and occupies a zero fraction of the total Hilbert space. If time permits I will briefly discuss some other possible deviations from ergodicity relevant for the chaotic many-body systems [2-4].
[1] G. De Tomasi, I. M. K., “Multifractality meets entanglement: relation for non-ergodic extended states”, Phys. Rev. Lett. 124, 200602 (2020) [arXiv:2001.03173] [2] I. M. K., M. Haque, and P. McClarty, “Eigenstate Thermalization, Random Matrix Theory and Behemoths”, Phys. Rev. Lett. 122, 070601 (2019) [arXiv:1806.09631]. [3] M. Haque, P. A. McClarty, I. M. K. , “Entanglement of mid-spectrum eigenstates of chaotic many-body systems—deviation from random ensembles.” [arXiv:2008.12782]. [4] A. Bäcker, I. M. K., M. Haque,, “Multifractal dimensions for chaotic quantum maps and many-body systems”, Phys. Rev. E 100, 032117 (2019) [arxiv:1905.03099].
Zoom-link: https://nordita.org/zoom/nvcms
Condensed Matter Physics Seminar