Exact results for polaron and molecule in one-dimensional spin-1/2 Fermi gas

Using exact Bethe ansatz solutions, we show that a spin-down fermion immersed into a fully polarized spin-up Fermi sea with a weak attraction is dressed by the surrounding spin-up fermions to form the one-dimensional analog of a polaron. As the attraction becomes strong, the spin-down fermion binds with one spin-up fermion to form a tightly bound molecule. Throughout the whole interaction regime, a crossover from the polaron to a molecule state is fully demonstrated through exact results of the excitation spectrum, the effective mass, binding energy and kinetic energy. Furthermore, a clear distinction between the polaron and molecule is conceived by the probability distribution, single-particle reduced density matrix, and density-density correlations, which are calculated directly from the Bethe ansatz wave function. Such a polaron-molecule crossover presents a universal nature of an impurity immersed into a fermionic medium with an attraction in one dimension.

Figure 1

Schematic configuration of the polaron-molecule crossover of a single attractive impurity in the 1D Fermi gas. In the weak attractive limit, the single impurity (red), a spin-down fermion dressed by the surrounding scattered spin-up fermions (blue) from the medium, behaves like a polaron (dashed oval) with an effective mass $m^{*}\approx m$. For strong attraction, it binds with one spin-up fermion from the Fermi sea to a tightly bound molecule (black circle) of two atoms with an effective mass $m^{*}\approx 2m$.

Figure 2

Binding energy $E_b$. The solid line is the numerical result and the circles represent the result from the analytical expressions (17) and (19) for strong and weak attractions, respectively. The unit of energy is $\frac{{\hslash }^2{n}^2}{2m}$ and $\gamma$ is the dimensionless interaction strength. Here $N=10,L=1$.

Figure 3

(a) The kinetic energy as a function of the square of the momentum $q$ for different values of interaction strength. The unit of energy is ${\hslash }^2/\left(2m{L}^2\right)$ and the unit of $q^2$ is $L^{-2}$. (b) Effective mass $m^{*}/m$ as a function of interaction strength $c$. The line is the numerical result and the circles denote the analytical result obtained from (16) for the strong-coupling regime. For the weak-coupling regime, the leading-order contribution to the effective mass is identified as an order of $c^2$. The mass ratio $m^{*}/m$ has no unit and the unit of $c$ is $L^{-1}$. Here $N=10$ and $L=1$.

Figure 4

Probability distribution of the ground state in the case of three fermions for $c=0,-1,-4,-40$. When $c=0$, the ground state corresponds to $\{k_1,k_2,k_3\}=\{0,0,2\pi /L\}$. Here the distribution unit is $L^{-3}$.

Figure 5

One-body correlations of the ground state at $c=0,-0.04,-40,-400$. When $c=0$, the momenta are $\{0,0,\pm 2\pi ,\pm 4\pi ,\pm 6\pi \}$. The black lines are the numerical results of the real part of the correlation function; the open circles denote the analytical results of $c=0$. The insets show the absolute values of the one-body correlations for $N=8$. Here the unit of correlations is $L^{-1}$.

Figure 6

Density-density correlations between up-spins in the ground sate for $c=0,-0.04,-40,-400$. When $c=0$, the momenta are $\{0,0,\pm 2\pi ,\pm 4\pi ,\pm 6\pi \}$. The black lines are the numerical results; the red line is the analytical result of $c=0$. $N=8$. The unit of length is $L$ and the unit of correlations is $L^{-2}$.

Figure 7

(a) Density-density correlations between up-spin and down-spin in the ground sate for $c=0,-0.04,-4,-40,-100$. For clarity, the curves for $c\ne 0$ have shifted upwards by $0.1,0.2,0.3,0.4$, respectively. When $c=0$, the momenta are $\{0,0,\pm 2\pi ,\pm 4\pi ,\pm 6\pi \}$. The different symbol lines are the numerical results for different interaction; the red circles are the analytical result for $c=0$. (b) The full width at half maximum (FWHM) as a function of the interaction strength $c$. $N=8$. The unit of length is $L$, the unit of correlations is $L^{-2}$, and the unit of $c$ is $L^{-1}$.