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Communications in Theoretical Physics

Energy Spectrum of Ground State and Excitation Spectrum of Quasi-particle for Hard-Core Boson in Optical Lattices

Han Jiu-Rong, Haichao Zhang, Yuzhu Wang

Published Nov 1, 2005• 2 citations
Publisher

Abstract

We investigate the energy spectrum of ground state and quasi-particle excitation spectrum of hard-core bosons, which behave very much like spinless noninteracting fermions, in optical lattices by means of the perturbation expansion and Bogoliubov approach. The results show that the energy spectrum has a single band structure, and the energy is lower near zero momentum; the excitation spectrum gives corresponding energy gap, and the system is in Mott-insulating state at Tonks limit. The analytic result of energy spectrum is in good agreement with that calculated in terms of Green's function at strong correlation limit.

DOI
10.1088/6102/44/5/863

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We investigate the energy spectrum of ground state and quasi-particle excitation spectrum of hard-core bosons, which behave very much like spinless noninteracting fermions, in optical lattices by means of the perturbation expansion and Bogoliubov approach. The results show that the energy spectrum has a single band structure, and the energy is lower near zero momentum; the excitation spectrum gives corresponding energy gap, and the system is in Mott-insulating state at Tonks limit. The analytic result of energy spectrum is in good agreement with that calculated in terms of Green's function at strong correlation limit.
We investigate the energy spectrum of fermionized bosonic atoms, which behave very much like spinless noninteracting fermions, in optical lattices by means of the perturbation expansion and the retarded Green's function method. The results show that the energy spectrum splits into two energy bands with single-occupation; the fermionized bosonic atom occupies nonvanishing energy state and left hole has a vanishing energy at any given momentum, and the system is in Mott-insulating state with a energy gap. Using the characteristic of energy spectra we obtained a criterion with which one can judge whether the Tonks-Girardeau (TG) gas is achieved or not.
We theoretically study the excitation spectrum of confined macroscopic optical lattices in the Mott-insulating limit. For large systems, a fast numerical method is proposed to calculate the ground state filling and excitation energies. We introduce many-particle on-site energies capturing multi-band effects and discuss tunnelling on a perturbative level using an effectively restricted Hilbert space. Results for small one-dimensional lattices obtained by this method are in good agreement with the exact multi-band diagonalization of the Hamiltonian. Spectral properties associated with the formation of regions with constant filling, so-called Mott shells, are investigated and interfaces between the shells with strong particle fluctuations are characterized by gapless local excitations.
We study a mixture of NbN_{b}Nb​ bosons with point hard-core boson-boson interactions and NfN_{f}Nf​ noninteracting spinless fermions with point hard-core boson-fermion interactions in 1D optical lattice with external harmonic confine potential. Using an extended Jordan-Winger transformation (JWT) which maps the hard-core Bose-Fermi mixture into two component noninteracting spinless fermions with hard-core interactions between them, we get the ground states of the system. Then we determine in details the one particle density matrix, density profile, momentum distribution, the natural orbitals and their occupations based on the constructed ground state wavefunctions. We also discuss the ground state properties of the system with large but finite interactions which lead to the lift of ground degeneracy. Our results show that, although the total density profile is almost not affected, the distributions for bosons and fermions strongly depend on the relative strengthes of boson-boson interactions and boson-fermion interactions.
In this paper properties of a boson gas at zero temperature are investigated by means of field-theoretic methods. Difficulties arising from the depletion of the ground state are resolved in a simple way by the elimination of the zero-momentum state. The result of this procedure when applied to the calculation of the Green's functions of the system is identical to that of Beliaev. It is then shown generally that for a repulsive interaction the energy E(k)E(\mathrm{k})E(k) of a phonon of momentum k, which is found as the pole of a one-particle Green's function, approaches zero for zero momentum, which means that the phonon spectrum does not exhibit an energy gap.The Green's function method is applied to the calculation of the properties of a low-density boson gas. The next order term beyond that calculated by Lee and Yang, and Beliaev for the ground-state energy is obtained and the general form of the series expansion is found to be $(\frac{{E}_{0}}{\ensuremath{\Omega}})=\frac{1}{2}{n}^{2}{f}_{0}[1+a{(n{{f}_{0}}^{3})}^{\frac{1}{2}}+b(n{{f}_{0}}^{3})\mathrm{ln}n{{f}_{0}}^{3}+c(n{{f}_{0}}^{3})+d{(n{{f}_{0}}^{3})}^{\frac{3}{2}}\mathrm{ln}(n{{f}_{0}}^{3})+\ensuremath{\cdots}],$ where nnn is the density and f0{f}_{0}f0​ is the scattering length for the assumed two-body interaction between the bosons. The coefficients aaa and bbb are independent of the shape of the interaction, and are the only terms thus far calculated. The coefficient bbb is in agreement with the hard-sphere gas calculations of Wu and of Sawada.A discussion is given of the intermediate-density calculation of Brueckner and Sawada, and certain possible improvements in the method of summing a selected set of higher-order terms are proposed.
Abstract A strongly interacting Bose gas in an optical lattice is studied using a hard‐core interaction. Two different approaches are introduced, one is based on a spin‐1/2 Fermi gas with attractive interaction, the other one on a functional integral with an additional constraint (slave‐boson approach). The relation between fermions and hard‐core bosons is briefly discussed for the case of a one‐dimensional Bose gas. For a three‐dimensional gas we identify the order parameter of the Bose‐Einstein condensate through a Hubbard‐Stratonovich transformation and treat the corresponding theories within a mean‐field approximation and with Gaussian fluctuations. This allows us to evaluate the phase diagram, including the Bose‐Einstein condensate and the Mott insulator, the density‐density correlation function, the static structure factor, and the quasiparticle excitation spectrum. The role of quantum and thermal fluctuations are studied in detail for both approaches, where we find good agreement with the Gross‐Pitaevskii equation and with the Bogoliubov approach in the dilute regime. In the dense regime, which is characterized by the phase transition between the Bose‐Einstein condensate and the Mott insulator, we discuss a renormalized Gross‐Pitaevskii equation. This equation can describe the macroscopic wave function of the Bose‐Einstein condensate in the dilute regime as well as close to the transition to the Mott insulator. Finally, we compare the results of the attractive spin‐1/2 Fermi gas and those of the slave‐boson approach and find good agreement for all physical quantities.