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Permanent variational wave functions for bosons

by J. M. Zhang, H. F. Song, Y. Liu

Submission summary

As Contributors: Yu Liu · Jiang-min Zhang
Arxiv Link: https://arxiv.org/abs/2106.14679v3 (pdf)
Date submitted: 2021-11-19 07:18
Submitted by: Zhang, Jiang-min
Submitted to: SciPost Physics
Academic field: Physics
Specialties:
  • Atomic, Molecular and Optical Physics - Theory
  • Condensed Matter Physics - Computational
  • Quantum Physics
Approaches: Theoretical, Computational

Abstract

We study the performance of permanent states (the bosonic counterpart of the Slater determinant state) as approximating functions for bosons, with the intention to develop variational methods based upon them. For a system of $N$ identical bosons, a permanent state is constructed by taking a set of $N$ arbitrary (not necessarily orthonormal) single-particle orbitals, forming their product and then symmetrizing it. It is found that for the one-dimensional Bose-Hubbard model with the periodic boundary condition and at unit filling, the exact ground state can be very well approximated by a permanent state, in that the permanent state has high overlap (at least 0.96 even for 12 particles and 12 sites) with the exact ground state and can reproduce both the ground state energy and the single-particle correlators to high precision. For a generic model, we have devised a greedy algorithm to find the optimal set of single-particle orbitals to minimize the variational energy or maximize the overlap with a target state. It turns out that quite often the ground state of a bosonic system can be well approximated by a permanent state by all the criterions of energy, overlap, and correlation functions. And even if the error is apparent, it can often be remedied by including more configurations, i.e., by allowing the variational wave function to be a combination of multiple permanent states. The algorithm is used to study the stability of a two-particle system, with great success. All these suggest that permanent states are very effective as variational wave functions for bosonic systems, and hence deserve further studies.

Current status:
Editor-in-charge assigned


Author comments upon resubmission

We have added a subsection to illustrate the application of the algorithm to a two-particle stability problem.

It is a cubic lattice as large as 21*21*21.

The overlap is amazingly high.

List of changes

(1) The first and last sentences of the abstract are rewritten.

(2) The fourth paragraph of the Introduction is greatly expanded.

(3) A few sentences are added in the last paragraph of the Introduction. They are

'Note that this was not considered previously, but is necessary and effective for improving accuracy.'

'This enables us to use the algorithm to study the stability of a two-boson system, which is analogous to the negative ion of hydrogen.'

'We would like to mention that the whole paper is actually a by-product of studying these open problems.'

(4) The subsection containing equation 74 is new. It deals with a two-particle stability problem in a 3d lattice.

(5) The paragraph containing equations 56 and 57 is new.

(6) In the second paragraph of the Conclusion, we have added 'Note also that in practice many models of interest have only $N=2$ or $N=3$ particles. For such small values of $N$, the permanent computation is of course not an issue at all.'

(7) The 3rd paragraph of the Conclusion is new.

(8) The 4th paragraph of the Conclusion is expanded.

(9) In the 3rd last paragraph of the Conclusion, the following sentence is new: 'Given a primitive wave packet $\phi(\vec{x})$, we can imagine a many-body permanent state constructed with the wave packets $ \phi(\vec{x} - \vec{R}_m)$, where $\vec{R}_m $ runs through an $n$-dimensional lattice.'

(10) In the bibliography, refs. 41-44, 46, 55 are new.

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