We present a systematic geometric framework to study closed quantum systems based on suitably chosen variational families. For the purpose of (A) real time evolution, (B) excitation spectra, (C) spectral functions and (D) imaginary time evolution, we show how the geometric approach highlights the necessity to distinguish between two classes of manifolds: K\"ahler and non-K\"ahler. Traditional variational methods typically require the variational family to be a K\"ahler manifold, where multiplication by the imaginary unit preserves the tangent spaces. This covers the vast majority of cases studied in the literature. However, recently proposed classes of generalized Gaussian states make it necessary to also include the non-K\"ahler case, which has already been encountered occasionally. We illustrate our approach in detail with a range of concrete examples where the geometric structures of the considered manifolds are particularly relevant. These go from Gaussian states and group theoretic coherent states to generalized Gaussian states.
Authors / Affiliations: mappings to Contributors and OrganizationsSee all Organizations.
- 1 2 3 Lucas Hackl,
- 1 2 Tommaso Guaita,
- 4 5 Tao Shi,
- 6 Jutho Haegeman,
- 7 Eugene Demler,
- 1 2 Ignacio Cirac
- 1 Munich Center for Quantum Science and Technology [MCQST]
- 2 Max-Planck-Institut für Quantenoptik / Max Planck Institute of Quantum Optics [MPQ]
- 3 Københavns Universitet / University of Copenhagen [UCPH]
- 4 中国科学院大学 / University of Chinese Academy of Sciences [UCAS]
- 5 中国科学院理论物理研究所 / Institute of Theoretical Physics, Chinese Academy of Sciences [ITP]
- 6 Universiteit Gent / Ghent University
- 7 Harvard University