SciPost Submission Page

Quantum Many-body Bootstrap

by Xizhi Han

Submission summary

As Contributors: Xizhi Han
Arxiv Link: (pdf)
Date submitted: 2020-09-16 03:15
Submitted by: Han, Xizhi
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Computational
  • Quantum Physics
Approaches: Theoretical, Computational


A numerical bootstrap method is proposed to provide rigorous and nontrivial bounds in general quantum many-body systems with locality. In particular, lower bounds on ground state energies of local lattice systems are obtained by imposing positivity constraints on certain operator expectation values. Complemented with variational upper bounds, ground state observables are constrained to be within a narrow range. The method is demonstrated with the Hubbard model in one and two dimensions, and bounds on ground state double occupancy and magnetization are discussed.

Current status:
Editor-in-charge assigned

Submission & Refereeing History

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Submission 2006.06002v2 on 16 September 2020

Reports on this Submission

Anonymous Report 1 on 2021-1-20 Invited Report


1. simple but powerful and general way of determining lower bounds for ground state energies


1. no comparison with existing literature that is very similar to this paper
2. no mentioning of computational cost
3. difficult to follow as there are almost no details worked out


This interesting paper revisits the powerful problem of finding lower bounds to strongly correlated quantum many-body systems through positivity constraints on the reduced density matrices. The set-up has been very popular in the quantum chemistry community, but has now been mostly abandoned because it is not extensive. The author has a slightly different take on the subject, which might make it more powerful.
In this particular case, the author imposes translational invariance, and solves a semi-definite problem very similar to the one performed in reference [8]. The results are remarkably accurate, although no details are given about the actual computational cost and/or scaling. The paper definitely needs to be expanded to be readable to a general audience.

The main issue, briefly addressed by the author in the introduction, is that it is not clear that this bootstrap method will do any better than exact diagonalization on a small sized system (the so-called Anderson bound). The author should definitely compare his method to such as Lancsoz method, and give arguments why his method is better and more wider applicable. The author should also compare his result (and the cost) with the results in the paper [8].

Requested changes

1. work out more details
2. prove that it does better than the Anderson bound
3. compare with RDM methods in the literature

  • validity: good
  • significance: good
  • originality: good
  • clarity: low
  • formatting: acceptable
  • grammar: good

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