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arXiv:1611.01073 · Physical Review A

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

Runxin Mao, Xin Guan, Biao Wu

Published Oct 27, 2016• 13 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.94.043645

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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.
Using solutions of the discrete Bethe ansatz equations, we study in detail the quantum impurity problem of a spin-down fermion immersed into a fully ploarized spin-up Fermi sea with weak attraction. We prove that this impurity fermion in the one-dimensional (1D) fermionic medium behaves like a polaron for weak attraction. However, as the attraction grows, the spin-down fermion binds with one spin-up fermion from the fully-polarized medium to form a tightly bound molecule. Thus it is seen that the system undergos a crossover from a mean field polaron-like nature into a mixture of excess fermions and a bosonic molecule as the attraction changes from weak attraction into strong attraction. This polaron-molecule crossover is universal in 1D many-body systems of interacting fermions. In a thermodynamic limit, we further study the relationship between the Fredholm equations for the 1D spin-1/2 Fermi gas with weakly repulsive and attractive delta-function interactions.
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.
Abstract Using the variational method, we study the properties of a spin-down impurity immersed in a one-dimensional (1D) spin-up Fermi sea. With repulsive interactions between them, the impurity is dressed up by surrounding particles in Fermi sea and forms a polaron. We clearly calculate the binding energy, effective mass, momentum distribution, Tan contact, and pair correlation. Even in strong repulsive regimes, the results can agree with the exact Bethe Ansatz results. The repulsive polaron energy E + is below Fermi energy E F and no negative effective masses are found in whole interaction regimes, unequal masses polarons are also calculated. We show a clear momentum distribution and calculate the Tan contact from three different aspects. Furthermore, we explore the particle-hole excitation and find that the hole terms in Fermi sea have a great influence on the polaron energy and contact in repulsive regime. These results show that the variational method can still be used effectively in 1D repulsive polaron system .
We address the problem of a single ``spin-down'' impurity atom interacting attractively with a spin-up Fermi gas in two dimensions (2D). We consider the case where the mass of the impurity is greater than or equal to the mass of a spin-up fermion. Using a variational approach, we resolve the questions raised by previous studies and show that there is, in fact, a transition between polaron and molecule (dimer) ground states in 2D. For the molecule state, we use a variational wave function with a single particle-hole excitation on the Fermi sea and we find that its energy matches that of the exact solution in the limit of infinite impurity mass. Thus, we expect the variational approach to provide a reliable tool for investigating 2D systems.
Using both the exact Bethe ansatz method and the variational method, we study properties of the one-dimensional Fermi polaron. We focus on the binding energy, effective mass, momentum distributions, Tan contact and correlation functions. As the attraction increases, the impurity is more tightly bound and correlated with the surrounding particles, and the size of formed polaron decreases. In addition, compared with the Bethe ansatz method, the variational method is totally qualified to study the one-dimensional Fermi polaron. The intrinsic reason is that the number of particle-hole excitations in a Fermi sea, caused by a single impurity, is always rather small. The variational method can be well extended to other impurity systems .
Based on the recently developed interaction renormalization for the one-dimensional ppp-wave interaction, we study the problem of a single impurity immersed in a highly polarized Fermi sea. They interact through a narrow ppp-wave Feshbach resonance, so the effective range r0{r}_{0}r0​ naturally appears in the system. We use the variational approach limited to single-particle--hole excitations on top of the unperturbed Fermi sea. The polaron exhibits two branches of solutions, namely, the attractive and repulsive branches, with varying scattering length across the resonance. We calculate the energy spectrum, residue, and effective mass for each of the branches. We compare the polaronic energy with the energy of the dressed molecule and find that the molecular state is energetically favored when increasing the interaction strength. The critical interaction strength for the polaron-to-molecule transition will shift to the BCS side of the ppp-wave resonance as the effective range increases.