Correlation functions and polaron-molecule crossover in one-dimensional attractive Fermi gases

Maolin Chang, Xiangguo Yin, Li Chen, and Yunbo Zhang

Phys. Rev. A 107, 053312 – Published 18 May 2023

Abstract

We study the correlation functions in a one-dimensional attractive Fermi polaron system of spin-1/2 Fermi gas with one flipped spin as an impurity immersed in the fully polarized spin-up media. By means of the exact Bethe ansatz wave function with string solutions for the momentum, we manage to explicitly deduce the asymptotic form of the correlation functions in the infinite attractive limit and show that the system undergoes a polaron-molecule crossover when the attraction grows. The correlation functions in the infinite attractive limit can be decomposed into the free fermion term with the number of particles reduced by one and the molecular term indicated by the emergence of a sharp peak or dip in the central area of the correlation. These analytical results are checked by the numerical Monte Carlo methods for multiple integrals. The calculation is further extended to the excited states, i.e., the super-Tonks-Girardeau gas, and there is no peak or dip found in the vicinity of the correlation center. Tan contact is found to be proportional to the cubic of the interaction strength and the total energy is verified to satisfy the Tan adiabatic theorem in the strong attraction regime.

Received 31 January 2023
Accepted 5 May 2023

DOI:https://doi.org/10.1103/PhysRevA.107.053312

Physics Subject Headings (PhySH)

Fermi gases Polarons Quantum correlations, foundations & formalism

Bethe ansatz

Atomic, Molecular & Optical

Authors & Affiliations

Maolin Chang, Xiangguo Yin, and Li Chen

Institute of Theoretical Physics and State Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University, Taiyuan 030006, China

Yunbo Zhang^*

Department of Physics and Key Laboratory of Optical Field Manipulation of Zhejiang Province, Zhejiang Sci-Tech University, Hangzhou 310018, China

^*ybzhang@zstu.edu.cn

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Issue

Vol. 107, Iss. 5 — May 2023

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Images

Figure 1
One-body correlation functions for the ground states of $N = 8$ fermions with the interaction strength $c^{'} = - 0.1, - 10, - 100, - 1000$ , as well as the exact results for the free fermion gas and molecular state in the strong attraction limit $c^{'} = - \infty$ . The dashed lines with markers are numerical results from Monte Carlo integration methods, while the solid lines are analytical results. The inset shows the enlargement of the central peak, where the red dashed line is numerical result for $c^{'} = - 10^{3}$ , the brown solid line and black solid line are analytical results for $c^{'} = - 10^{3}$ and $- 10^{10}$ , respectively. The unit of $c^{'}$ is $L^{- 1}$ .
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Figure 2
Up-up correlation functions for the ground states of $N = 8$ fermions with the interaction strength $c^{'} = - 0.1, - 10, - 100, - 1000$ , as well as the exact results for free fermion gas and in the strong attraction limit. The dashed lines with markers are the Monte Carlo numerical results, while the solid lines are analytical results. The inset shows the enlargement of the central dip, where the red dashed lines are numerical result for $c^{'} = - 10^{3}$ , the brown and black solid lines are analytical results for $c^{'} = - 10^{3}$ and $- 10^{10}$ , respectively. The unit of $c^{'}$ is $L^{- 1}$ .
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Figure 3
Up-down correlation for the ground state of $N = 8$ fermions with the interaction strength $c^{'} = - 0.1, - 10, - 100, - 1000$ , the exact results for free fermion gas and in the strong attraction limit. The dashed line with markers are Monte Carlo numerical results, while the solid lines are analytical results. The unit of $c^{'}$ is $L^{- 1}$ .
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Figure 4
The energy spectrum of $N = 4$ fermions with one impurity in the full interacting regime. The red lines are bound states while the blue ones refer to the gas states for attractive interaction. The black lines on the left ( $- 1 / c^{'} < 0$ ) are repulsive gas states. The STG gas refers to the many-body excited state for the particular case $c^{'} \to - \infty$ . The unit of $E$ is $ℏ^{2} / 2 m$ and the unit of $- 1 / c^{'}$ is $L$ .
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Figure 5
Exact results of one-body correlation functions of $N = 8$ fermions for the free fermion gas, the ground state of strong attraction, and the excited STG gas state of strong attraction with minimum energy. For the ground state (red dashed line), the interaction strength $c^{'}$ is taken as $- 10^{10}$ . The unit of $c^{'}$ is $L^{- 1}$ .
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Figure 6
Exact results of up-up correlation functions of $N = 8$ fermions for the free fermion gas, the ground state of strong attraction, and the excited STG gas state of strong attraction with minimum energy. For the ground state (red dashed line), the interaction strength $c^{'}$ is taken as $- 10^{10}$ . The unit of $c^{'}$ is $L^{- 1}$ .
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