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arXiv:1611.00626 · Physical Review A

Trap effects and the continuum limit of the Hubbard model in the presence of a harmonic potential

Davide Nigro

Published Sep 7, 2017• 4 citations
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Abstract

We discuss how to perform the continuum limit of the $d$-dimensional Hubbard model in the presence of a harmonic confining potential at zero temperature and fixed particle number. While for $d\ensuremath{\ge}3$ the system can be mapped into a noninteracting two-component Fermi gas in a harmonic trap, for $d=1$ and 2 the two-body interaction is described by a Dirac $\ensuremath{\delta}$. We show that the properties of this continuum limit can be put in one-to-one correspondence with those obtained by applying the trap-size scaling formalism to the confined Hubbard model in the so called dilute regime (fixed number of particles and weak confinement). The correspondence for $d<3$ has been tested, in the case of two particles with opposite spin, by comparing numerical exact-diagonalization results of the lattice system with those obtained in the continuum limit.

DOI
10.1103/physreva.96.033608

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We conduct a theoretical study of SU(N\mathrm{SU}(NSU(N) fermions confined by a one-dimensional harmonic potential. First, we introduce a numerical approach for solving the trapped interacting few-body problem, by which one may obtain accurate energy spectra across the full range of interaction strengths. In the strong-coupling limit, we map the SU(N\mathrm{SU}(NSU(N) Hamiltonian to a spin-chain model. We then show that an existing, extremely accurate ansatz---derived for a Heisenberg SU(2) spin chain---is extendable to these NNN-component systems. Lastly, we consider balanced SU(N\mathrm{SU}(NSU(N) Fermi gases that have an equal number of particles in each spin state for N=2,3,4N=2,\phantom{\rule{0.16em}{0ex}}3,\phantom{\rule{0.16em}{0ex}}4N=2,3,4. In the weak- and strong-coupling regimes, we find that the ground-state energies rapidly converge to their expected values in the thermodynamic limit with increasing atom number. This suggests that the many-body energetics of NNN-component fermions may be accurately inferred from the corresponding few-body systems of NNN distinguishable particles.
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Analytical results on the correlation functions of strongly correlated many-body systems are rare in the literature and their importance cannot be overstated. We present determinant representations for the space-, time-, and temperature-dependent correlation functions of the strongly interacting one-dimensional Hubbard model in the presence of an external trapping potential. These representations are exact and valid in both equilibrium and nonequilibrium scenarios like the ones initiated by a sudden change of the confinement potential. In addition, they can be implemented numerically very easily, significantly outperforming other numerical approaches. As applications of our results we investigate the single-particle spectral functions of systems with harmonic trapping and show that dynamical quasicondensation occurs for both fermionic and bosonic spin-1/21/21/2 systems released from a Mott insulator state.
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