# Systematic interpolatory ansatz for one-dimensional polaron systems

### Submission summary

 As Contributors: Erik Jonathan Lindgren Preprint link: scipost_201908_00010v1 Date submitted: 2019-08-28 Submitted by: Lindgren, Erik Jonathan Submitted to: SciPost Physics Discipline: Physics Subject area: Quantum Physics Approaches: Theoretical, Computational

### Abstract

We explore a new variational principle for studying one-dimensional quantum systems in a trapping potential. We focus on the Fermi polaron problem, where a single distinguishable impurity interacts through a contact potential with a background of identical fermions. We can accurately describe this system, for various particle numbers, different trapping potentials and arbitrary finite repulsion, by constructing a truncated basis containing states at both zero and infinite repulsion. The results agree well with matrix product states methods and with the analytical result for two particles in a harmonic well.

###### Current status:
Editor-in-charge assigned

### Submission & Refereeing History

Submission scipost_201908_00010v1 on 28 August 2019

## Reports on this Submission

### Report

This paper describes a (numerical) variational procedure to approximately solve the $1+N_{\uparrow}$ fermionic problem in a harmonic trap in one dimension. The basic idea is to consider a number of eigenstates of the non-interacting system and a number of eigenstates in the strongly interacting limit, and use these to construct a new basis via the Gram-Schmidt procedure in which the Hamiltonian is then diagonalized. Although there are methods available which can (in principle) give the answer to this problem free from any systematic error, the advantage of this method is that it is less time-consuming (one needs to construct the basis once to get results for all values of the coupling strength). The results for the 1+1 and 1+2 problem convincingly show that the method can provide very accurate results. As explained below, my main concern is the fact that the number of particles remains small.

I believe the paper can be considered for publication in Scipost after some revisions and after some questions have been addressed.

Here is a list of my concerns/questions:

- My main concern is the fact that the number of particles remains small: at the end some results (position space density in the ground state) are shown for the 1+6 problem, for which discrepancy with MPS is already larger. It is not clear how well the method can reproduce the energy spectrum (for 6 and for larger values of $N_{\uparrow}$). It is said: “we focus on the Fermi polaron problem”. The Fermi polaron problem is a single distinguishable impurity which interacts with a non-interacting Fermi sea. One or two or six particles is arguably a Fermi sea. The authors should address the question of larger values of $N_{\uparrow}$ (if not, I would not call it the Fermi polaron problem).
- I guess the title is inspired by the title of Ref. 47. Here, however, a Gram-Schmidt procedure provides the basis in which the Hamiltonian is then numerically diagonalized. I therefore don’t like the term “interpolatory ansatz”, which suggests a simple form of the wave function with some variational parameters like in Ref. 47. I would urge the authors to choose a more suitable title, which also makes it clear that the method is quite different from the one presented in Ref. 47.
- Many important references are missing and some references seem less relevant to me. For example: although Ref. [2] is indeed a seminal paper on the superfluid to Mott insulator transition in a 3D optical lattice, the discussion is about 1D (I could add many other breakthrough papers if we include 3D). No mention of other important experimental papers in 1D, such as e.g. Liao et al, Nature 467, 567 (2010) or Yang et al, Phys. Rev. Lett. 119, 165701 (2017). Many references about many-body techniques for the Fermi polaron problem are missing or to experiments (e.g. Nature 485, 615 (2002)). These are just a few examples. I would urge the authors to improve/extend their bibliography.
- Is there a particular reason the discussion is limited to the repulsive case? Does the method have a problem with handling the attractive case? I think this should be addressed in the paper.
- The first sentence of the abstract reads: “we explore a new variational principle for…”. This sounds very strange since the variational principle is what it is. You are not exploring a new one. Please rephrase this.
- In section 5.1.1 the authors write “it is important to exclude such linearly dependent states to avoid singular behavior in the Gram-Schmidt orthogonalization process”. For the 1+1 problem, this can easily be done. Does this cause any problems for the general case of N_{\uparrow} particles?
- Please include which units are being used in the plots (for energy, length, etc.)
- Figure 2 shows the energy of the lowest six states for the 1+1 system in a harmonic trap. I expected the energies for the even-parity states to be quasi-identical to the ones plotted in Figure 1 of Ref. 47. (It is stated in Ref. 47 that the error there is bounded by about 0.03). However, the energies do not seem to agree. Why? Even if it’s just a matter of units or an energy shift, the relative energies of the fifth and the sixth states seem very different. Can the authors explain where the difference with Ref. 47 comes from?
- In section 5.1.3 the authors write “we do not claim to have that high numerical precision in neither our method nor in the numerical integral used for the analytical formula.” Can the authors provide an estimate of their numerical/extrapolation error? This allows the reader to know up to which point to trust figure 5.
- I compared figure 7 to figures 2 and 3 of Ref. 47. I again see quantitative and qualitative differences. Can the authors explain these differences?
- In the conclusion it is stated that “computing densities is however a significantly time consuming step”. Can the authors be more precise about how time-consuming it is and why?

Finally, I found some problems with notations:
- Notations are not always consistent with having n states at zero interaction and m states at infinite interaction. (See, e.g., page 3: $0 \leq i \leq n$, which says there are $n+1$ non-interacting states. This happens a number of times throughout the paper.
- On page 3, $\mathcal{M}_l$ is the set of points where $x_0$ is SMALLER than exactly $l$ of the $x_1, \ldots, x_N$. On pages 5 and 13 it is LARGER and GREATER, while on page 8, it is again SMALLER.
- In a number of places the superscripts (i) or ($\mu$) are dropped for the quantum numbers $k$ or $q$, but this not done in a consistent way (for example, in eq. (9) and on page 5 right above equation (11)). I suggest to include these everywhere.
- Section 3.3 starts with a mere repetition of what was said right before, but this time with different notation: what was $a^{(\mu)}_j$ before is now $\alpha_j$. Even more confusingly, the notation $\alpha_j$ is reused at the end of 3.4 to indicate the final basis states. Please stick to a single consistent notation.
- At the end of section 3.5., n and m suddenly indicate some many-body states. I didn’t quite understand what these states exactly are because the notation was not clearly introduced.

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