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Efficient and Flexible Approach to Simulate Low-Dimensional Quantum Lattice Models with Large Local Hilbert Spaces

by Thomas Köhler, Jan Stolpp, Sebastian Paeckel

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Submission summary

Authors (as registered SciPost users): Thomas Köhler · Sebastian Paeckel
Submission information
Preprint Link: scipost_202009_00002v2  (pdf)
Date accepted: 2021-02-15
Date submitted: 2021-01-14 15:17
Submitted by: Paeckel, Sebastian
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
  • Quantum Physics
Approach: Theoretical


Quantum lattice models with large local Hilbert spaces emerge across various fields in quantum many-body physics. Problems such as the interplay between fermions and phonons, the BCS-BEC crossover of interacting bosons, or decoherence in quantum simulators have been extensively studied both theoretically and experimentally. In recent years, tensor network methods have become one of the most successful tools to treat such lattice systems numerically. Nevertheless, systems with large local Hilbert spaces remain challenging. Here, we introduce a mapping that allows to construct artificial $U(1)$ symmetries for any type of lattice model. Exploiting the generated symmetries, numerical expenses that are related to the local degrees of freedom decrease significantly. This allows for an efficient treatment of systems with large local dimensions. Further exploring this mapping, we reveal an intimate connection between the Schmidt values of the corresponding matrix\hyp product\hyp state representation and the single\hyp site reduced density matrix. Our findings motivate an intuitive physical picture of the truncations occurring in typical algorithms and we give bounds on the numerical complexity in comparison to standard methods that do not exploit such artificial symmetries. We demonstrate this new mapping, provide an implementation recipe for an existing code, and perform example calculations for the Holstein model at half filling. We studied systems with a very large number of lattice sites up to $L=501$ while accounting for $N_{\rm ph}=63$ phonons per site with high precision in the CDW phase.

List of changes

1. We moved Sec. 5.2 to the appendix and replaced it with a physically motivated reasoning of the observed numerical properties of our mapping.
2. We moved Sec. 6.2 into the appendix.
3. We reformulated Sec. 5.1 and simplified the overall notation, e.g., removing the dots indicating the orientation of the tensor legs.
4. We extended Sec. 2 presenting the general concept.
5. We introduced latin letters (e.g., $a_j$) for bonds labeling combined indices of irreducible representations and the corresponding block-dimensions, the first being labeled by the corresponding greek letter (e.g., $a_j \equiv (\alpha_j, m_{j; \alpha_j})$)

Published as SciPost Phys. 10, 058 (2021)

Reports on this Submission

Anonymous Report 1 on 2021-2-9 (Invited Report)


The new version of the manuscript is considerably improved. The authors now provide a very intuitive and pedagogical introduction with most of the cumbersome technical aspects moved to appendices, and have modified the notation to improve readability.


The authors have addressed all previous comments and suggestions. I recommend it for publications without changes.

  • validity: top
  • significance: high
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: excellent

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