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Peer-reviewedCC BY 4.0SciPost

Hilbert space diffusion in systems with approximate symmetries

  • Rahel L. Baumgartner1
  • Luca V. Delacrétaz2
  • Pranjal Nayak3
  • Julian Sonner1
  1. 1University of Geneva
  2. 2University of Chicago
  3. 3European Organization for Nuclear Research
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Abstract

Random matrix theory (RMT) universality is the defining property of quantum mechanical chaotic systems, and can be probed by observables like the spectral form factor (SFF). In this paper, we describe systematic deviations from RMT behaviour at intermediate time scales in systems with approximate symmetries. At early times, the symmetries allow us to organize the Hilbert space into approximately decoupled sectors, each of which contributes independently to the SFF. At late times, the SFF transitions into the final ramp of the fully mixed chaotic Hamiltonian. For approximate continuous symmetries, the transitional behaviour is governed by a universal process that we call Hilbert space diffusion. The diffusion constant corresponding to this process is related to the relaxation rate of the associated nearly conserved charge. By implementing a chaotic sigma model for Hilbert-space diffusion, we formulate an analytic theory of this process which agrees quantitatively with our numerical results for different examples.

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Cite

BibTeXRIS
BibTeX
@article{baumgartner2026hilbert,
  title = {{Hilbert space diffusion in systems with approximate symmetries}},
  author = {Rahel L. Baumgartner and Luca V. Delacrétaz and Pranjal Nayak and Julian Sonner},
  journal = {SciPost},
  year = {2026},
  month = oct,
  doi = {10.21468/scipostphys.21.4.101},
  url = {http://scipost.org/SciPostPhys.21.4.101},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Hilbert space diffusion in systems with approximate symmetries
AU  - Rahel L. Baumgartner
AU  - Luca V. Delacrétaz
AU  - Pranjal Nayak
AU  - Julian Sonner
PY  - 2026
DA  - 2026/10/22
JO  - SciPost
DO  - 10.21468/scipostphys.21.4.101
UR  - http://scipost.org/SciPostPhys.21.4.101
AB  - Random matrix theory (RMT) universality is the defining property of quantum mechanical chaotic systems, and can be probed by observables like the spectral form factor (SFF). In this paper, we describe systematic deviations from RMT behaviour at intermediate time scales in systems with approximate symmetries. At early times, the symmetries allow us to organize the Hilbert space into approximately decoupled sectors, each of which contributes independently to the SFF. At late times, the SFF transitions into the final ramp of the fully mixed chaotic Hamiltonian. For approximate continuous symmetries, the transitional behaviour is governed by a universal process that we call Hilbert space diffusion. The diffusion constant corresponding to this process is related to the relaxation rate of the associated nearly conserved charge. By implementing a chaotic sigma model for Hilbert-space diffusion, we formulate an analytic theory of this process which agrees quantitatively with our numerical results for different examples.
SN  - 2542-4653
ER  -