Abstract

We analyze the ground-state phase diagram of an ultracold Bose-Fermi mixture placed in an optical lattice. The quantum phases involve pairing of fermions and one or several bosons. Depending on the physical parameters these composites can form a Fermi liquid, a density wave, a superfluid or a domain insulator. We determine by means of a mean-field formalism the phase boundaries for finite tunneling, and analyze the experimental feasibility of these sort of phases.

© 2004 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  4. F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, “Josephson Junction Arrays with Bose-Einstein Condensates,” Science 293, 843 (2001).
    [Crossref] [PubMed]
  5. S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D.M. Stamper-Kurn, and W. Ketterle, “Observation of Feshbach resonances in a Bose-Einstein condensate,” Nature (London) 392, 151 (1998).
    [Crossref]
  6. S. L. Cornish, N. R. Claussen, J. L. Roberts, E. A. Cornell, and C. E. Wieman, “Stable 85Rb Bose-Einstein Condensates with Widely Tunable Interactions,” Phys. Rev. Lett. 85, 1795 (2000).
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  8. M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature 415, 39 (2002).
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  13. A. G. Truscott, K. E. Strecker, W. I. McAlexander, G. B. Partridge, and R. G. Hulet, “Observation of Fermi Pressure in a Gas of Trapped Atoms,” Science 291, 2570 (2001).
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  14. F. Schreck, L. Khaykovich, K. L. Corwin, G. Ferrari, T. Bourdel, J. Cubizolles, and C. Salomon, “Quasipure Bose-Einstein Condensate Immersed in a Fermi Sea,” Phys. Rev. Lett. 87, 080403 (2001).
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    [Crossref]
  20. M. J. Bijlsma, B. A. Heringa, and H. T. C. Stoof, “Phonon exchange in dilute Fermi-Bose mixtures: Tailoring the Fermi-Fermi interaction,” Phys. Rev. A 61, 053601 (2000).
    [Crossref]
  21. H. Pu, W. Zhang, M. Wilkens, and P. Meystre, “Phonon Spectrum and Dynamical Stability of a Dilute Quantum Degenerate Bose-Fermi Mixture,” Phys. Rev. Lett. 88, 070408 (2002).
    [Crossref] [PubMed]
  22. P. Capuzzi and E. S. Hernández, “Zero-sound density oscillations in Fermi-Bose mixtures,” Phys. Rev. A 64, 043607 (2001).
    [Crossref]
  23. X.-J. Liu and H. Hu, “Collisionless and hydrodynamic excitations of trapped boson-fermion mixtures,” Phys. Rev. A 67, 023613 (2003).
    [Crossref]
  24. A. Albus, S. A. Gardiner, F. Illuminati, and M. Wilkens, “Quantum field theory of dilute homogeneous Bose-Fermi mixtures at zero temperature: General formalism and beyond mean-field corrections,” Phys. Rev. A 65, 053607 (2002).
    [Crossref]
  25. R. Roth, “Structure and stability of trapped atomic boson-fermion mixtures,” Phys. Rev. A 66, 013614 (2002).
    [Crossref]
  26. L. Viverit and S. Giorgini, “Ground-state properties of a dilute Bose-Fermi mixture,” Phys. Rev. A 66, 063604 (2002).
    [Crossref]
  27. K. K. Das, “Bose-Fermi Mixtures in One Dimension,” Phys. Rev. Lett. 90, 170403 (2003).
    [Crossref] [PubMed]
  28. M. A. Cazalilla and A. F. Ho, “Instabilities in Binary Mixtures of One-Dimensional Quantum Degenerate Gases,” Phys. Rev. Lett. 91, 150403 (2003).
    [Crossref] [PubMed]
  29. A. Albus, F. Illuminati, and J. Eisert, “Mixtures of bosonic and fermionic atoms in optical lattices,” Phys. Rev. A 68, 023606 (2003).
    [Crossref]
  30. H. P. Büchler and G. Blatter, “Supersolid versus Phase Separation in Atomic Bose-Fermi Mixtures,” Phys. Rev. Lett. 91, 130404 (2003).
    [Crossref] [PubMed]
  31. R. Roth and K. Burnett, “Quantum phases of atomic boson-fermion mixtures in optical lattices,” http://xxx.lanl.gov/abs/cond-mat/0310114.
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    [Crossref] [PubMed]
  33. M. Lewenstein, L. Santos, M. A. Baranov, and H. Fehrmann, “Atomic Bose-Fermi mixtures in an optical lattice,” http://xxx.lanl.gov/abs/cond-mat/0306180.
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    [Crossref]
  39. R. Shankar, “Renormalization-group approach to interacting fermions,” Rev. Mod. Phys,  66, 129 (1994).
    [Crossref]
  40. W. Krauth, M. Caffarel, and J.-P. Bouchard, “Gutzwiller wave function for a model of strongly interacting bosons,” Phys. Rev. B 45, 3137 (1992).
    [Crossref]
  41. K. Sheshadri, H. R. Krishnamurthy, R. Pandit, and T. V. Ramakrishnan, “Superfluid and insulating phases in an interacting-boson model: mean-field theory and the RPA,” Europhys. Lett. 22, 257 (1993).
    [Crossref]
  42. See e.g.P. G. de GennesSuperconductivity in metals and alloysW. A. Benjamin (1966).
  43. W. Hänsel, P. Hommelhoff, T. W. Hänsch, and J. Reichel, “Bose-Einstein condensation on a microelectronic chip,” Nature 413, 498 (2001).
    [Crossref] [PubMed]
  44. F. Schreck, L. Khaykovich, K. L. Corwin, G. Ferrari, T. Bourdel, J. Cubizolles, and C. Salomon, “Quasipure Bose-Einstein Condensate Immersed in a Fermi Sea,” Phys. Rev. Lett. 87, 080403 (2001);
    [Crossref] [PubMed]
  45. A. Görlitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J. R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W. Ketterle, “Realization of Bose-Einstein Condensates in Lower Dimensions,” Phys. Rev. Lett. 87, 130402 (2001).
    [Crossref] [PubMed]
  46. M. Greiner, I. Bloch, O. Mandel, T. W. Hänsch, and T. Esslinger, “Exploring Phase Coherence in a 2D Lattice of Bose-Einstein Condensates,” Phys. Rev. Lett. 87, 160405 (2001);
    [Crossref] [PubMed]
  47. S. Burger, F. S. Cataliotti, C. Fort, P. Maddaloni, F. Minardi, and M. Inguscio, “Quasi-2D Bose-Einstein condensation in an optical lattice,” Europhys. Lett. 57, 1 (2002).
    [Crossref]
  48. H. Bethe, “On the Theory of Metals, I. Eigenvalues and Eigenfunctions of a Linear Chain of Atoms,” Z. Phys. 74, 205 (1931).
  49. J. D. Johnson, S. Krinsky, and B. M. McCoy, “Vertical-Arrow Correlation Length in the Eight-Vertex Model and the Low-Lying Excitations of the X-Y-Z Hamiltonian,” Phys. Rev. A 8, 2526 (1973).
    [Crossref]
  50. M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, “Boson localization and the superfluid-insulator transition,” Phys. Rev. B 40, 546 (1989).
    [Crossref]
  51. D. van Oosten, P. van der Straten, and H. T. C. Stoof, “Quantum phases in an optical lattice,” Phys. Rev. A 63, 053601 (2001).
    [Crossref]
  52. D. Jaksch, V. Venturi, J. I. Cirac, C. J. Williams, and P. Zoller, “Creation of a Molecular Condensate by Dynamically Melting a Mott Insulator,” Phys. Rev. Lett. 89, 040402 (2002).
    [Crossref] [PubMed]
  53. G. G. Batrouni, V. Rousseau, R. T. Scalettar, M. Rigol, A. Muramatsu, P. J. H. Denteneer, and M. Troyer, “Mott Domains of Bosons Confined on Optical Lattices,” Phys. Rev. Lett. 89, 117203 (2002).
    [Crossref] [PubMed]

2003 (8)

A. Recati, P. O. Fedichev, W. Zwerger, and P. Zoller, “Spin-Charge Separation in Ultracold Quantum Gases,” Phys. Rev. Lett. 90, 020401 (2003).
[Crossref] [PubMed]

Z. Hadzibabic, S. Gupta, C. A. Stan, C. H. Schunck, M. W. Zwierlein, K. Dieckmann, and W. Ketterle, “Fiftyfold Improvement in the Number of Quantum Degenerate Fermionic Atoms,” Phys. Rev. Lett. 91, 160401 (2003).
[Crossref] [PubMed]

X.-J. Liu and H. Hu, “Collisionless and hydrodynamic excitations of trapped boson-fermion mixtures,” Phys. Rev. A 67, 023613 (2003).
[Crossref]

K. K. Das, “Bose-Fermi Mixtures in One Dimension,” Phys. Rev. Lett. 90, 170403 (2003).
[Crossref] [PubMed]

M. A. Cazalilla and A. F. Ho, “Instabilities in Binary Mixtures of One-Dimensional Quantum Degenerate Gases,” Phys. Rev. Lett. 91, 150403 (2003).
[Crossref] [PubMed]

A. Albus, F. Illuminati, and J. Eisert, “Mixtures of bosonic and fermionic atoms in optical lattices,” Phys. Rev. A 68, 023606 (2003).
[Crossref]

H. P. Büchler and G. Blatter, “Supersolid versus Phase Separation in Atomic Bose-Fermi Mixtures,” Phys. Rev. Lett. 91, 130404 (2003).
[Crossref] [PubMed]

A. B. Kuklov and B. V. Svistunov, “Counterflow Superfluidity of Two-Species Ultracold Atoms in a Commensurate Optical Lattice,” Phys. Rev. Lett. 90, 100401 (2003).
[Crossref] [PubMed]

2002 (10)

A. Albus, S. A. Gardiner, F. Illuminati, and M. Wilkens, “Quantum field theory of dilute homogeneous Bose-Fermi mixtures at zero temperature: General formalism and beyond mean-field corrections,” Phys. Rev. A 65, 053607 (2002).
[Crossref]

R. Roth, “Structure and stability of trapped atomic boson-fermion mixtures,” Phys. Rev. A 66, 013614 (2002).
[Crossref]

L. Viverit and S. Giorgini, “Ground-state properties of a dilute Bose-Fermi mixture,” Phys. Rev. A 66, 063604 (2002).
[Crossref]

H. Pu, W. Zhang, M. Wilkens, and P. Meystre, “Phonon Spectrum and Dynamical Stability of a Dilute Quantum Degenerate Bose-Fermi Mixture,” Phys. Rev. Lett. 88, 070408 (2002).
[Crossref] [PubMed]

Z. Hadzibabic, C. A. Stan, K. Dieckmann, S. Gupta, M. W. Zwierlein, A. Görlitz, and W. Ketterle, “Two-Species Mixture of Quantum Degenerate Bose and Fermi Gases,” Phys. Rev. Lett. 88, 160401 (2002).
[Crossref] [PubMed]

G. Modugno, G. Roati, F. Riboli, F. Ferlaino, R. J. Brecha, and M. Inguscio, “Collapse of a Degenerate Fermi Gas,” Science 297, 2240 (2002).
[Crossref] [PubMed]

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature 415, 39 (2002).
[Crossref] [PubMed]

S. Burger, F. S. Cataliotti, C. Fort, P. Maddaloni, F. Minardi, and M. Inguscio, “Quasi-2D Bose-Einstein condensation in an optical lattice,” Europhys. Lett. 57, 1 (2002).
[Crossref]

D. Jaksch, V. Venturi, J. I. Cirac, C. J. Williams, and P. Zoller, “Creation of a Molecular Condensate by Dynamically Melting a Mott Insulator,” Phys. Rev. Lett. 89, 040402 (2002).
[Crossref] [PubMed]

G. G. Batrouni, V. Rousseau, R. T. Scalettar, M. Rigol, A. Muramatsu, P. J. H. Denteneer, and M. Troyer, “Mott Domains of Bosons Confined on Optical Lattices,” Phys. Rev. Lett. 89, 117203 (2002).
[Crossref] [PubMed]

2001 (12)

D. van Oosten, P. van der Straten, and H. T. C. Stoof, “Quantum phases in an optical lattice,” Phys. Rev. A 63, 053601 (2001).
[Crossref]

O. Morsch, J. H. Müller, M. Cristiani, D. Ciampini, and E. Arimondo, “Bloch Oscillations and Mean-Field Effects of Bose-Einstein Condensates in 1D Optical Lattices,” Phys. Rev. Lett. 87, 140402 (2001).
[Crossref] [PubMed]

W. K. Hensinger, H. Häffner, A. Browaeys, N. R. Heckenberg, K. Helmerson, C. McKenzie, G. J. Milburn, W. D. Phillips, S. L. Rolston, H. Rubinsztein-Dunlop, and B. Upcroft, “Dynamical tunnelling of ultracold atoms,” Nature 412, 52 (2001);
[Crossref] [PubMed]

F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, “Josephson Junction Arrays with Bose-Einstein Condensates,” Science 293, 843 (2001).
[Crossref] [PubMed]

B. Paredes, P. Fedichev, J. I. Cirac, and P. Zoller, “1/2-Anyons in Small Atomic Bose-Einstein Condensates,” Phys. Rev. Lett. 87, 010402 (2001).
[Crossref] [PubMed]

A. G. Truscott, K. E. Strecker, W. I. McAlexander, G. B. Partridge, and R. G. Hulet, “Observation of Fermi Pressure in a Gas of Trapped Atoms,” Science 291, 2570 (2001).
[Crossref] [PubMed]

F. Schreck, L. Khaykovich, K. L. Corwin, G. Ferrari, T. Bourdel, J. Cubizolles, and C. Salomon, “Quasipure Bose-Einstein Condensate Immersed in a Fermi Sea,” Phys. Rev. Lett. 87, 080403 (2001).
[Crossref] [PubMed]

P. Capuzzi and E. S. Hernández, “Zero-sound density oscillations in Fermi-Bose mixtures,” Phys. Rev. A 64, 043607 (2001).
[Crossref]

W. Hänsel, P. Hommelhoff, T. W. Hänsch, and J. Reichel, “Bose-Einstein condensation on a microelectronic chip,” Nature 413, 498 (2001).
[Crossref] [PubMed]

F. Schreck, L. Khaykovich, K. L. Corwin, G. Ferrari, T. Bourdel, J. Cubizolles, and C. Salomon, “Quasipure Bose-Einstein Condensate Immersed in a Fermi Sea,” Phys. Rev. Lett. 87, 080403 (2001);
[Crossref] [PubMed]

A. Görlitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J. R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W. Ketterle, “Realization of Bose-Einstein Condensates in Lower Dimensions,” Phys. Rev. Lett. 87, 130402 (2001).
[Crossref] [PubMed]

M. Greiner, I. Bloch, O. Mandel, T. W. Hänsch, and T. Esslinger, “Exploring Phase Coherence in a 2D Lattice of Bose-Einstein Condensates,” Phys. Rev. Lett. 87, 160405 (2001);
[Crossref] [PubMed]

2000 (3)

M. J. Bijlsma, B. A. Heringa, and H. T. C. Stoof, “Phonon exchange in dilute Fermi-Bose mixtures: Tailoring the Fermi-Fermi interaction,” Phys. Rev. A 61, 053601 (2000).
[Crossref]

N. K. Wilkin and J. M. F. Gunn, “Condensation of “Composite Bosons” in a Rotating BEC,” Phys. Rev. Lett. 84, 6 (2000).
[Crossref] [PubMed]

S. L. Cornish, N. R. Claussen, J. L. Roberts, E. A. Cornell, and C. E. Wieman, “Stable 85Rb Bose-Einstein Condensates with Widely Tunable Interactions,” Phys. Rev. Lett. 85, 1795 (2000).
[Crossref] [PubMed]

1998 (4)

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices,” Phys. Rev. Lett. 81, 3108 (1998).
[Crossref]

S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D.M. Stamper-Kurn, and W. Ketterle, “Observation of Feshbach resonances in a Bose-Einstein condensate,” Nature (London) 392, 151 (1998).
[Crossref]

K. Mølmer, “Bose Condensates and Fermi Gases at Zero Temperature,” Phys. Rev. Lett. 80, 1804–1807 (1998).
[Crossref]

B. P. Anderson and M. Kasevich, “Macroscopic Quantum Interference from Atomic Tunnel Arrays”, Science 282, 1686 (1998).
[Crossref] [PubMed]

1994 (1)

R. Shankar, “Renormalization-group approach to interacting fermions,” Rev. Mod. Phys,  66, 129 (1994).
[Crossref]

1993 (1)

K. Sheshadri, H. R. Krishnamurthy, R. Pandit, and T. V. Ramakrishnan, “Superfluid and insulating phases in an interacting-boson model: mean-field theory and the RPA,” Europhys. Lett. 22, 257 (1993).
[Crossref]

1992 (1)

W. Krauth, M. Caffarel, and J.-P. Bouchard, “Gutzwiller wave function for a model of strongly interacting bosons,” Phys. Rev. B 45, 3137 (1992).
[Crossref]

1989 (1)

M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, “Boson localization and the superfluid-insulator transition,” Phys. Rev. B 40, 546 (1989).
[Crossref]

1973 (1)

J. D. Johnson, S. Krinsky, and B. M. McCoy, “Vertical-Arrow Correlation Length in the Eight-Vertex Model and the Low-Lying Excitations of the X-Y-Z Hamiltonian,” Phys. Rev. A 8, 2526 (1973).
[Crossref]

1960 (1)

M. Girardeau, “Relationship between systems of impenetrable bosons and fermions in one dimension,” J. Math. Phys. 1, 516 (1960).
[Crossref]

1931 (1)

H. Bethe, “On the Theory of Metals, I. Eigenvalues and Eigenfunctions of a Linear Chain of Atoms,” Z. Phys. 74, 205 (1931).

Abo-Shaeer, J. R.

A. Görlitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J. R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W. Ketterle, “Realization of Bose-Einstein Condensates in Lower Dimensions,” Phys. Rev. Lett. 87, 130402 (2001).
[Crossref] [PubMed]

Albus, A.

A. Albus, F. Illuminati, and J. Eisert, “Mixtures of bosonic and fermionic atoms in optical lattices,” Phys. Rev. A 68, 023606 (2003).
[Crossref]

A. Albus, S. A. Gardiner, F. Illuminati, and M. Wilkens, “Quantum field theory of dilute homogeneous Bose-Fermi mixtures at zero temperature: General formalism and beyond mean-field corrections,” Phys. Rev. A 65, 053607 (2002).
[Crossref]

Anderson, B. P.

B. P. Anderson and M. Kasevich, “Macroscopic Quantum Interference from Atomic Tunnel Arrays”, Science 282, 1686 (1998).
[Crossref] [PubMed]

Andrews, M. R.

S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D.M. Stamper-Kurn, and W. Ketterle, “Observation of Feshbach resonances in a Bose-Einstein condensate,” Nature (London) 392, 151 (1998).
[Crossref]

Arimondo, E.

O. Morsch, J. H. Müller, M. Cristiani, D. Ciampini, and E. Arimondo, “Bloch Oscillations and Mean-Field Effects of Bose-Einstein Condensates in 1D Optical Lattices,” Phys. Rev. Lett. 87, 140402 (2001).
[Crossref] [PubMed]

Auerbach, A.

A. Auerbach, Interacting Electrons and Quantum magnetism, (Springer, New York, 1994).
[Crossref]

Baranov, M. A.

M. Lewenstein, L. Santos, M. A. Baranov, and H. Fehrmann, “Atomic Bose-Fermi mixtures in an optical lattice,” http://xxx.lanl.gov/abs/cond-mat/0306180.

H. Fehrmann, M. A. Baranov, B. Damski, M. Lewenstein, and L. Santos, “Mean-field theory of Bose-Fermi mixtures in optical lattices,” http://xxx.lanl.gov/abs/cond-mat/0307635.

Batrouni, G. G.

G. G. Batrouni, V. Rousseau, R. T. Scalettar, M. Rigol, A. Muramatsu, P. J. H. Denteneer, and M. Troyer, “Mott Domains of Bosons Confined on Optical Lattices,” Phys. Rev. Lett. 89, 117203 (2002).
[Crossref] [PubMed]

Benjamin, W. A.

See e.g.P. G. de GennesSuperconductivity in metals and alloysW. A. Benjamin (1966).

Bethe, H.

H. Bethe, “On the Theory of Metals, I. Eigenvalues and Eigenfunctions of a Linear Chain of Atoms,” Z. Phys. 74, 205 (1931).

Bijlsma, M. J.

M. J. Bijlsma, B. A. Heringa, and H. T. C. Stoof, “Phonon exchange in dilute Fermi-Bose mixtures: Tailoring the Fermi-Fermi interaction,” Phys. Rev. A 61, 053601 (2000).
[Crossref]

Blatter, G.

H. P. Büchler and G. Blatter, “Supersolid versus Phase Separation in Atomic Bose-Fermi Mixtures,” Phys. Rev. Lett. 91, 130404 (2003).
[Crossref] [PubMed]

Bloch, I.

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature 415, 39 (2002).
[Crossref] [PubMed]

M. Greiner, I. Bloch, O. Mandel, T. W. Hänsch, and T. Esslinger, “Exploring Phase Coherence in a 2D Lattice of Bose-Einstein Condensates,” Phys. Rev. Lett. 87, 160405 (2001);
[Crossref] [PubMed]

Bouchard, J.-P.

W. Krauth, M. Caffarel, and J.-P. Bouchard, “Gutzwiller wave function for a model of strongly interacting bosons,” Phys. Rev. B 45, 3137 (1992).
[Crossref]

Bourdel, T.

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Figures (4)

Fig. 1.
Fig. 1.

(a) Formation of composite fermions between a fermion and a bosonic hole; (b) Composite fermions between a fermion and a boson.

Fig. 2.
Fig. 2.

(a) Phase space as a function of the boson chemical potential, µ̄, and the ratio α between the boson-fermion and the boson-boson interaction strengths. Roman numbers indicate the number of particles forming a composite fermion. The presence (absence) of a bar over the Roman number indicates that the composite fermions are formed by a fermion and bosonic holes (a fermion and bosons). The subindices A and R indicate whether the composite fermions attract or repel each other; (b) Full phase diagram for the region 0<µ̄<1, for the case ρF =0.4 and J/V=0.02. Different phases are present, including fermionic domains (FD), superfluid (SF), Fermi liquid (FL) and density-wave phase (DW).

Fig. 3.
Fig. 3.

Phase diagram as a function of the hopping JB /V, the fermion-boson interactions α=U/V, and the bosonic chemical potential µ̃=µ/V, for the case of JF =0. The lobes denote the analytical phase boundaries calculated using our mean-field approach. For ñ-1<µ̃<ñ, and µ̄-ñ+s<α<µ̄-ñ+s+1, the number of bosons, n, and the number of fermions, m, satisfy n+sm=ñ, and composites with one fermion and s bosonic holes (-s bosons) are formed.

Fig. 4.
Fig. 4.

(a) Phase diagram as a function of the hopping J/V, and the bosonic chemical potential µ̃=µ/V, for α=0.25 and ρf =0.25. (b) Same diagram but for α=0.75. Dotted lines indicate our analytical results for JF =0, solid lines our numerical results for JF =0, and dashed lines our numerical results for JF =JB . Phases A are formed by a Mott-Insulator phase for the bosons and a Fermi liquid for the fermions. Phases B are characterized by the formation of fermionic composites with one fermion and one bosonic hole.

Equations (33)

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H BFH = H 0 + H 1
H 0 = i [ 1 2 V n i ( n i 1 ) μ n i ] + U i n i m i
H 1 = i j ( J B b i b j + J F f i f j + h . c . ) ,
{ f ˜ i , f ˜ i } = ( n ˜ s ) ! n ˜ ! [ m i ( n i + s ) ! n i ! + ( 1 m i ) n i ! ( n i s ) ! ] .
E ϕ = H eff ϕ = { H 0 + 𝒫 H 1 [ E 𝒬 ( H 0 + H 1 ) ] 1 H 1 } ϕ
H eff = E 0 + 𝒫 H 1 1 E 0 H 0 H 1 + 𝒫 H 1 1 E 0 H 0 𝒬 H 1 1 E 0 H 0 H 1 + 𝒪 ( H 1 4 ) ,
H eff = J eff i j ( f ˜ i f ˜ j + h . c . ) + K eff i j m ˜ i m ˜ j
K eff = 4 J 2 V { n ˜ ( n ˜ + 1 s ) 1 + α s + ( n ˜ s ) ( n ˜ + 1 ) 1 α + s + 1 α s n ˜ ( n ˜ + 1 ) ( n ˜ s ) ( n ˜ + 1 s ) } ,
J eff = J ( in region I ) ,
J eff = 4 J 2 α V ( in region II ¯ ) ,
J eff = 4 J 2 α ( 2 α ) V ( in region II ) ,
| ϕ = i ( cos θ i 2 | 0 i + sin θ i 2 e ϕ i | 1 i ) .
k ( v k | 00 k , k + u k | 1 , 1 k , k ) ,
Z = Z 0 i D ψ i ( τ ) D ψ i * ( τ ) exp [ S ( ψ ) ]
S ( ψ ) = i , j d τ ( J B 1 ) i j ψ i * ( τ ) ψ j ( ψ ) i ln T τ exp [ d τ ψ i ( τ ) b i ( τ ) + h . c . ] 0 ,
i , j d τ 1 d τ 2 ψ i * ( τ 1 ) 𝒢 i j ( τ 1 τ 2 ) ψ j ( τ 2 ) ,
E 0 ( n , m ) = 1 2 V n ( n 1 ) + U n m μ n ,
S ( ψ ) = N ( 1 z J F 𝒢 ( ω = 0 ) ) ψ 2 N r ψ 2 ,
𝒢 ( ω = 0 ) = ( n ˜ + 1 ε ( n ˜ , 0 ) n ˜ ε ( n ˜ 1 , 0 ) ) ( 1 ρ F )
+ ( n ˜ s + 1 ε ( n ˜ s , 1 ) n ˜ s ε ( n ˜ s 1 , 1 ) ) ρ F ,
r = 1 2 d J F { ( n ˜ + 1 ε ( n ˜ , 0 ) n ˜ ε ( n ˜ 1 , 0 ) ) ( 1 ρ F ) + ( n ˜ s + 1 ε ( n ˜ s , 1 ) n ˜ s ε ( n ˜ s 1 , 1 ) ) ρ F }
ψ = i n = 0 N max m = 0 1 f n , m ( i ) n m ,
Ψ H Ψ = J B < k l > Φ ( k ) * Φ ( l ) J F < k l > n f n , 1 ( k ) * f n , 0 ( k ) + f n , 0 ( l ) * f n , 1 ( l )
k n ( + V 2 m = 0 1 f n , m ( k ) * f n , m ( k ) ( n ( n 1 ) ) + U f n , 1 ( k ) * f n , 1 ( k ) n
μ m = 0 1 f n , m ( k ) * f n , m ( k ) n μ F f n , 1 ( k ) * f m , 1 ( k ) ) ,
f 0 , 0 ( i ) = cos ( α 0 ( i ) )
f 0 , 1 ( i ) = sin ( α 0 ( i ) ) cos ( α 0 ( i ) )
f N max , 0 ( i ) = sin ( α 0 ( i ) ) sin ( α 2 N n max 2 ( i ) ) cos ( α N namx 1 ( i ) )
f N n max , 0 ( i ) = sin ( α 0 ( i ) ) sin ( α 2 N n max 2 ( i ) ) cos ( α N namx 1 ( i ) )
i h ¯ f nm ( i ) = ( V 2 n ( n 1 ) + U mn ) f n , m ( i )
J B ( ϕ ( i ) * n + 1 f n + 1 , m ( i ) + ϕ ( i ) n f n 1 , m ( i ) )
J F n ( f n , 1 ( i ) * f n , 0 ( i ) + f n , 0 ( i ) * f n , 1 ( i ) ) ,

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