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arXiv:1905.06910 · physica status solidi (b)

The One‐Dimensional Bose–Fermi–Hubbard Model in the Limit of Fast Fermions

Alexander Mering, Michael Fleischhauer

Published Jul 18, 2019• 0 citations
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Abstract

The authors discuss the ground‐state phase diagram of the one‐dimensional Bose–Fermi–Hubbard model (BFHM) in the limit of fast fermions based on an effective boson model. The authors give a detailed derivation of the effective model with long‐range Rudermann–Kittel–Kasuya–Yosida (RKKY)‐type interactions, discuss its range of validity, and provide a deeper insight into its implications. In particular, they show that integrating out the fast fermion degrees of freedom in a naive way results in an ill‐behaved effective Hamiltonian and a proper renormalization is required. Based on the effective Hamiltonian, the phase diagram in the thermodynamic limit is constructed by analytic means and is compared to numerical results obtained by density matrix renormalization group (DMRG) techniques for the full BFHM. The most prominent feature of the phase diagram, the existence of a phase separation between Mott insulator (MI) and charge density wave (CDW) is discussed in depth including boundary effects.

DOI
10.1002/pssb.201900256

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The authors discuss the ground‐state phase diagram of the one‐dimensional Bose–Fermi–Hubbard model (BFHM) in the limit of fast fermions based on an effective boson model. The authors give a detailed derivation of the effective model with long‐range Rudermann–Kittel–Kasuya–Yosida (RKKY)‐type interactions, discuss its range of validity, and provide a deeper insight into its implications. In particular, they show that integrating out the fast fermion degrees of freedom in a naive way results in an ill‐behaved effective Hamiltonian and a proper renormalization is required. Based on the effective Hamiltonian, the phase diagram in the thermodynamic limit is constructed by analytic means and is compared to numerical results obtained by density matrix renormalization group (DMRG) techniques for the full BFHM. The most prominent feature of the phase diagram, the existence of a phase separation between Mott insulator (MI) and charge density wave (CDW) is discussed in depth including boundary effects.
In a recent work [1] we presented results for the Bose-Fermi-Hubbard model (BFHM) in the limit of ultrafast fermions. The present work gives an overview over the used methods and an deeper insight into the implications arising from the treated limit. Starting from the discussion of the phase diagram obtained by numerical means, we develop an analytic theory and derive an effective bosonic Hamiltonian. Arising issues in the Hamiltonian are overcome by inclusion of a back-action, renormalizing the solution of our system. Based on a detailed analysis of the effective Hamiltonian, the phase diagram in the thermodynamic limit is constructed by analytic means and comparison to numerical results obtained by density matrix renormalization group (DMRG) techniques for the full BFHM shows a very reasonable agreement. The most prominent feature of the phase diagram, the existence of a phase separation between Mott insulator (MI) and charge density wave (CDW) is discussed in depth with inclusion of important effects due to the boundary condition.
The ground-state phase diagram of mixtures of spin polarized fermions and bosons in a 1D periodic lattice is discussed in the limit of large fermion hopping and half filling of the fermions. Numerical simulations performed with the density matrix renormalization group (DMRG) show in addition to bosonic Mott insulating (MI), superfluid (SF), and charge density-wave phases (CDW) a yet unreported phase with spatial separation of MI and CDW regions. We derive an effective bosonic theory which allows for a complete understanding and quantitative prediction of the bosonic phase diagram. In particular the origin of CDW phase and the MI-CDW phase separation is revealed as an effective fermion-mediated long-range interaction between bosons with alternating sign.
We study the phase diagram of the zero-temperature, one-dimensional Bose-Fermi-Hubbard model for fixed fermion density in the limit of small fermionic hopping. This model can be regarded as an instance of a disordered Bose-Hubbard model with dichotomic values of the stochastic variables. Phase boundaries between compressible, incompressible (Mott-insulating), and partially compressible phases are derived analytically within a generalized strong-coupling expansion and numerically using density matrix renormalization group (DMRG) methods. We show that first-order correlations in the partially compressible phases decay exponentially, indicating a glass-type behavior. Fluctuations within the respective incompressible phases are determined using perturbation theory and are compared to DMRG results.
By means of the density matrix renormalization group technique, we accurately determine the zero-temperature phase diagram of the one-dimensional extended Bose–Hubbard model with on-site and nearest-neighbor interactions. We analyze the scaling of the charge and of the neutral ground-state energy gaps, as well as of various order parameters. In this way we come to an accurate location of the boundaries between the superfluid and the insulating phases. In this last region, we are able to distinguish between the conventional Mott insulating and density-wave phases and the Haldane insulator phase displaying long-range string ordering, as originally predicted by Dalla Torre et al (2006 Phys. Rev. Lett. 97 260401).
We carry out an extensive study of the phase diagrams of the extended Bose Hubbard model, with a mean filling of one boson per site, in one dimension by using the density matrix renormalization group and show that it contains Superfluid (SF), Mott-insulator (MI), density-wave (DW) and Haldane-insulator (HI) phases. We show that the critical exponents and central charges for the HI-DW, MI-HI and SF-MI transitions are consistent with those for models in the two-dimensional Ising, Gaussian, and Berezinskii-Kosterlitz-Thouless (BKT) universality classes, respectively; and we suggest that the SF-HI transition may be more exotic than a simple BKT transition. We show explicitly that different boundary conditions lead to different phase diagrams.
The zero-temperature phase diagram of the one-dimensional Bose-Hubbard model with nearest-neighbor interaction is investigated using the density-matrix renormalization group. Recently normal phases without long-range order have been conjectured between the charge-density wave phase and the superfluid phase in one-dimensional bosonic systems without disorder. Our calculations demonstrate that there is no intermediate phase in the one-dimensional Bose-Hubbard model but a simultaneous vanishing of crystalline order and appearance of superfluid order. The complete phase diagrams with and without nearest-neighbor interaction are obtained. Both phase diagrams show reentrance from the superfluid phase to the insulator phase.