Crossover from Fermi to Bose polarons in one- and two-dimensional interacting Fermi gases

Yue-Ran Shi, Ruijin Liu, and Wei Zhang

Phys. Rev. A 110, 013324 – Published 25 July 2024

Abstract

We investigate the Fermi-polaron system with one mobile impurity immersed atop a two-component interacting fermion bath in one- and two-dimensional square lattices, where the impurity atom interacts with only one species (the spin-up component) of the background. The ground state with a given total momentum can be approximated by an extended Gaussian state, which is constructed from the combination of Gaussian states and a non-Gaussian polaronic transformation (Lee-Low-Pines transformation). In the few-body limit, the variational energies of two- and three-body systems show good agreement with exact results, which indicates the validity of the non-Gaussian variational method. We then move on to the many-body limit, i.e., the polaron problem with an interacting background. We choose two representative fillings of the background fermions, $ρ = 1 / 4, 1 / 2$ , and obtain the corresponding ground states. Our results show that the system will undergo a smooth crossover from a Fermi-polaron to a Bose-polaron system as the interaction between background fermions increases. We further analyze the double occupancy and momentum distribution of the background fermions and find that the impurity will not significantly affect the background pairing.

Received 26 February 2024
Revised 10 June 2024
Accepted 8 July 2024

DOI:https://doi.org/10.1103/PhysRevA.110.013324

Physics Subject Headings (PhySH)

BCS-BEC crossover Polarons

Ultracold gases

Hubbard model Variational approach

Atomic, Molecular & Optical

Authors & Affiliations

Yue-Ran Shi ^1,2,3, Ruijin Liu^4,*, and Wei Zhang ^1,2,5,†

¹Department of Physics and Beijing Key Laboratory of Opto-electronic Functional Materials and Micro-nano Devices, Renmin University of China, Beijing 100872, China
²Key Laboratory of Quantum State Construction and Manipulation (Ministry of Education), Renmin University of China, Beijing 100872,China
³Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
⁴Institute of Theoretical Physics, University of Science and Technology Beijing, Beijing 100083, China
⁵Beijing Academy of Quantum Information Sciences, Beijing 100193, China

^*Contact author: rjliu@ustb.edu.cn
^†Contact author: wzhangl@ruc.edu.cn

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Issue

Vol. 110, Iss. 1 — July 2024

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Images

Figure 1
Schematic plot of the 1D lattice energy dispersions $E_{k} = - 2 cos k$ . The dotted line represents the Fermi energy determined by background particle fillings. There are three kinds of particles in our model, namely, the impurity particle (blue) and spin- $↑$ and spin- $↓$ particles (red). We consider the interaction between spin- $↑$ and spin- $↓$ particles, denoted $U_{ff}$ , and the interaction between impurity and spin- $↑$ component, denoted $U_{bf}$ .
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Figure 2
Two-body binding energy for (a) a 1D chain with size $L = 60$ and (b) a 2D square lattice with $L = 10$ . Exact solutions for both cases are obtained by solving Eq. (23) numerically in the thermodynamic limit. Here, we turn on only one interaction and fix the other one to be zero. The markers here represent $U = U_{ff}$ (red crosses) and $U = U_{bf}$ (blue circles). (c) The three-body ground-state energy versus $U_{bf}$ , with different $U_{ff}$ : $- 1$ (red), $- 4$ (green), $- 8$ (blue), and $- 12$ (black). The dotted lines are corresponding ground-state energies obtained using the MPS method. (d) The systematic errors for the three-body energy, defined as $| E_{MPS} - E | / | E_{MPS} |$ .
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Figure 3
The polaron energy [shifted by $E_{0} = E (U_{bf} = 0)$ ] for 1D and 2D systems, with particle fillings $ρ = 1 / 4$ and $ρ = 1 / 2$ . The system size is $L = 60$ for the 1D case and $L = 10$ for the 2D case. The black dashed and dotted lines represent the Fermi-polaron and Bose-polaron energies obtained using the Chevy-like ansatz, respectively. The behavior within the weak- $U_{bf}$ limit is closely shown in the insets. With increasing background interaction $U_{ff} = - 1, - 12, - 24$ , we observe the polaron energy of our model go through a smooth crossover between two limits for both 1D and 2D systems.
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Figure 4
Double occupancy $d$ for one- and two-dimensional square lattices with different particle fillings $ρ = 1 / 4$ and $ρ = 1 / 2$ , and impurity-background interactions, $U_{bf}$ = $- 1$ (red crosses) and $- 12$ (blue circles), varying with background interaction $U_{ff}$ .
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Figure 5
Momentum distribution $n_{σ} (k)$ for one- and two-dimensional square lattices with different particle fillings: $ρ = 1 / 4$ and $ρ = 1 / 2$ . The 1D and 2D results are plotted along $k = k_{x}$ and $k = k_{x} = k_{y}$ . We choose three different background interactions, $U_{ff} = - 1, - 6, - 12$ , represented by solid, dashed, and dotted lines, respectively, while the impurity-background interaction is fixed as $U_{bf} = - 12$ . The inset of (b) shows the 1D momentum distribution for the Bose-polaron limit $U_{bf} = - 1, U_{ff} = - 24$ .
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