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.
. Polaron, molecule and pairing in one-dimensional spin-1/2 Fermi gas with an attractive Delta-function interaction[J]. Frontiers of Physics, 2012, 7(1): 8-15.
Xi-Wen Guan. Polaron, molecule and pairing in one-dimensional spin-1/2 Fermi gas with an attractive Delta-function interaction. Front. Phys. , 2012, 7(1): 8-15.
P. Fulde and R. A. Ferrell, Phys. Rev. , 1964, 135(3A): A550
4
A. I. Larkin and Yu. N. Ovchinnikov, Sov. Phys. JETP , 1965, 20: 762
5
K. Yang, Phys. Rev. B , 2001, 63(14): 140511(R)
6
A. E. Feiguin and F. Heidrich-Meisner, Phys. Rev. B , 2007, 76(22): 220508(R)
7
M. Tezuka and M. Ueda, Phys. Rev. Lett. , 2008, 100(11): 110403
8
E. Zhao and W. V. Liu, Phys. Rev. A , 2008, 78(6): 063605
9
E. Zhao, X.-W. Guan, W. V. Liu, M. T. Batchelor, and M. Oshikawa, Phys. Rev. Lett. , 2009, 103(14): 140404
10
X.-W. Guan and T.-L. Ho, Phys. Rev. A , 2011, 84(2): 023616
11
X.-G. Yin, X.-W. Guan, S. Chen, and M. T. Bachelor, Phys. Rev. A , 2011, 84(1): 011602(R)
12
G. Orso, Phys. Rev. Lett. , 2007, 98(7): 070402
13
H. Hu, X.-J. Liu, and P. D. Drummond, Phys. Rev. Lett. , 2007, 98(7): 070403
14
X. W. Guan, M. T. Batchelor, C. Lee, and M. Bortz, Phys. Rev. B , 2007, 76(8): 085120
15
M. Casula, D. M. Ceperley, and E. J. Mueller, Phys. Rev. A , 2008, 78(3): 033607
16
T. Iida and M. Wadati, J. Phys. Soc. Jpn. , 2008, 77(2): 024006
17
Yean-an Liao, A. S. C. Rittner, T. Paprotta, Wenhui Li, G. B. Partridge, R. G. Hulet, S. K. Baur, and E. J. Mueller, Nature , 2010, 467(7315): 567
18
A. Schirotzek, Cheng-Hsun Wu, A. Sommer, and M. W. Zwierlein, Phys. Rev. Lett. , 2009, 102(23): 230402
19
S. Nascimbène, N. Navon, K. J. Jiang, L. Tarruell, M. Teichmann, J. McKeever, F. Chevy, and C. Salomon, Phys. Rev. Lett. , 2009, 103(17): 170402
20
R. Combescot and S. Giraud, Phys. Rev. Lett. , 2008, 101(5): 050404
21
G. M. Bruum and P. Massignan, Phys. Rev. Lett. , 2010, 105(2): 020403
22
C. J. M. Mathy, M. M. Parish, and D. A. Huse, Phys. Rev. Lett. , 2011, 106(16): 166404
23
M. M. Parish, Phys. Rev. A , 2011, 83(5): 051603(R)
24
M. J. Leskin, O. H. T. Nummi, F. Massel, and P?rm? , New. J. Phys. , 2010, 12: 073044
25
X.-W. Guan, in preparation
26
M. Olshanii, Phys. Rev. Lett. , 1998, 81(5): 938
27
Liming Guan, Shu Chen, Yupeng Wang, and Zhong-Qi Ma, Phys. Rev. Lett. , 2009, 102(16): 160402
28
C. N. Yang, Lectures given at the Karpacz Winter School of Physics, February 1970, see: Selected papers 1945-1980 with commentary, W. H. Freeman and Company, 1983: 430
29
M. Takahashi, Prog. Theor. Phys. , 1970, 44(4): 899
30
J. B. McGuire, J. Math. Phys. , 1965, 6(3): 432
31
J. B. McGuire, J. Math. Phys. , 1966, 7(1): 123
32
S. Giraud and R. Combescot, Phys. Rev. A , 2009, 79(4): 043615
33
M. T. Batchelor, M. Bortz, X. W. Guan, and N. Oelkers, J. Phys.: Conf. Ser. , 2006, 42: 5
34
J.-Y. Lee and X.-W. Guan, Nucl. Phys. B , 2011, 853(1): 125
35
C. N. Yang, Int. J. Mod. Phys. B , 1988, 2(6): 1325
36
M. Takahashi, Prog. Theor. Phys. , 1970, 44: 11
37
X.-W. Guan and Z.-Q. Ma, arXiv:1110.2821 , 2011
38
J. Catani, G. Lamporesi, D. Naik, M. Gring, M. Inguscio, and F. Minardi, arXiv:1106.0828, 2011
