Many-body spectral statistics of relativistic quantum billiard systems

Xianzhang Chen, Zhen-Qi Chen, Liang Huang* (Corresponding Author), Celso Grebogi

*Corresponding author for this work

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In the field of quantum chaos, spectral or energy-level spacing statistics are one of the most extensively investigated characteristics. In spite of a large body of existing literature, the effects of many-body interactions on the spectral statistics of relativistic quantum systems remained poorly understood. Treating electron-electron interactions through the one-orbital mean-field Hubbard
model, we address this fundamental issue using graphene billiards with the geometric shape of a circular sector as prototypical systems. Our approach is to consider the two characteristically different cases where the statistics are Poisson and GOE (Gaussian orthogonal ensemble) so the corresponding classical dynamics are typically integrable and chaotic, respectively, and to systematically investigate how the statistics change as the Hubbard interaction strength increases from zero. We find that, for energies near the Dirac point, the Hubbard interactions have a significant effect on the spectral statistics. In particular, regardless of the type of spectral statistics to begin with, increasing the Hubbard interaction strength up to a critical value causes the statistics to approach
GOE, rendering more applicable the random matrix theory. As the interaction strength increases beyond the critical value, the statistics evolve toward Poisson, due to the emergence of an energy gap rendering the quasiparticles massive. We also find that the energy levels above and below the Dirac point can exhibit different statistics, and the many-body interactions have little effect on the statistics for levels far from the Dirac point. These results reveal the intriguing interplay between many-body interactions and the spectral statistics, which we develop a physical picture to understand.
Original languageEnglish
Article number013050
Number of pages14
JournalPhysical Review Research
Early online date26 Jan 2023
Publication statusPublished - 26 Jan 2023

Bibliographical note

This work was supported by NSFC under Grant Nos. 12175090, 11775101, and 12047501, and by the 111 Project under Grant No. B20063. X.C. acknowledges
the financial support from the China Scholarship Council, and thanks B. Dietz for discussions and the suggestion of using Eq. (5) to interpolate between Poisson and
GOE statistics. The efforts at Arizona State University were supported by the Air Force of Scientific Research through Grant No. FA9550-21-1-0186.


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