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Physical review. B, Condensed matter

Exact solutions for two interacting particles in a one-dimensional ring with a magnetic flux

Qiming Li, J. Callaway

Published Feb 1, 1991• 6 citations
Publisher

Abstract

We solve the one-dimensional extended Hubbard model with a magnetic flux analytically for the case of two particles. We find that bound states are ground states only for attractive interactions. We have studied the variation of the ground-state energy as a function of the flux. For the bound state, the period of the oscillation is half of a flux quantum. The dependence of the magnitude of the energy barrier on the size of the ring is discussed in connection with the possible existence of the flux quantization in the thermodynamic limit.

DOI
10.1103/physrevb.43.3278

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Journal of the Physical Society of Japan•Nov 29, 2016•0 citations
We solve the one-dimensional extended Hubbard model with a magnetic flux analytically for the case of two particles. We find that bound states are ground states only for attractive interactions. We have studied the variation of the ground-state energy as a function of the flux. For the bound state, the period of the oscillation is half of a flux quantum. The dependence of the magnitude of the energy barrier on the size of the ring is discussed in connection with the possible existence of the flux quantization in the thermodynamic limit.
We study the ground state of two interacting bosonic particles confined in a ring-shaped lattice potential and subjected to a synthetic magnetic flux. The system is described by the Bose-Hubbard model and solved exactly through a plane-wave Ansatz of the wave function. We obtain energies and correlation functions of the system both for repulsive and attractive interactions. In contrast with the one-dimensional continuous theory described by the Lieb-Liniger model, in the lattice case we prove that the center of mass of the two particles is coupled with its relative coordinate. Distinctive features clearly emerge in the persistent current of the system. While for repulsive bosons the persistent current displays a periodicity given by the standard flux quantum for any interaction strength, in the attractive case the flux quantum becomes fractionalized in a manner that depends on the interaction. We also study the density after a long time expansion of the system. Our results can be used to benchmark approximate schemes for the many-body problem such as the density matrix renormalization group or other variational schemes.
By means of the Bethe-ansatz method, the one-dimensional Hubbard model with repulsive on-site coupling in an external magnetic field is solved exactly. It is found that the energy of the ground state depends on the magnetic flux \ensuremath{\varphi} through the ring formed by a periodic lattice chain, and the lowest-energy state corresponds to \ensuremath{\varphi}=n(hc/2e), with n an integer and -e the electron charge. The calculations illustrate that the strongly correlated Hubbard ring with periodic boundary conditions in its ground state is a superconducting ring. It can also be shown that the system possesses an internal SU(2)\ifmmode\times\else\texttimes\fi{}SU(2) symmetry.
In a flat Bloch band, the kinetic energy is quenched and single particles cannot propagate since they are localized due to destructive interference. Whether this remains true in the presence of interactions is a challenging question because a flat dispersion usually leads to highly correlated ground states. Here we compute numerically the ground-state energy of lattice models with completely flat band structure in a ring geometry in the presence of an attractive Hubbard interaction. We find that the energy as a function of the magnetic flux threading the ring has a half-flux quantum ${\mathrm{\ensuremath{\Phi}}}_{0}/2=hc/(2e)$ period, indicating that only bound pairs of particles with charge 2e2e2e are propagating, while single quasiparticles with charge eee remain localized. For some one-dimensional lattice models, we show analytically that in fact the whole many-body spectrum has the same periodicity. Our analytical arguments are valid for both bosons and fermions, for generic interactions respecting some symmetries of the lattice and at arbitrary temperatures. Moreover, for the same one-dimensional lattice models, we construct an extensive number of exact conserved quantities. These conserved quantities are associated to the occupation of localized single quasiparticle states and force the single-particle propagator to vanish beyond a finite range. Our results suggest that in lattice models with flat bands preformed pairs dominate transport even above the critical temperature of the transition to a superfluid state.
We investigate anomalous oscillations due to the Aharonov-Bohm(AB) and Aharonov-Casher(AC) effects of the one-dimensional Hubbard ring with flux in the strong coupling limit. By using the exact diagonalization method and the Shiba transformation, we examine the energies of the ground state and few excited states in the presence of the flux producing the AB or AC effect, where the transformation not only reverses the sign of the interaction U, but also exchanges the role between the AB and AC effects in the model Hamiltonian. We classify the number of oscillation of the ground state as a function of the flux systematically and clarify the close relationship between the AB and AC effects. For example, it is shown that the number of the periods are given by N_L - N_e(N_L - N_↑ + N_↓) for the AB(AC) effect in the very strong attraction, where N_L, N_e, N_↑, and N_↓ are the system size, the number of the total electrons, the number of electrons with up-spin, and the number of electrons with down-spin, respectively, in the condition of N_L > N_e > N_↓ > N_↑. More special case, such as the half-filled band and spin balanced case (N_L = N_e and N_↓ = N_↑ ), we find the periodicity of the system to be 0(2) for the AB(AC) effect in the very strong repulsion. These results show us the nature of interesting phenomena originated by the interplay between the strong correlation and the quantum interference effect in the mesoscopic ring.