The one-dimensional attractive lattice fermion gas equivalent to the Heisenberg-Ising spin-1/2 chain is studied for a ring geometry threaded by magnetic flux. We find that for charged fermions having interaction strength \ensuremath{\Delta}=cos(\ensuremath{\pi}/p) with p noninteger, the adiabatic ground state is periodic in the magnetic flux threading the chain, with period two flux quanta, as found by Shastry and Sutherland for the repulsive case. We find that, at particular values of the threading field, a sequence of initially zero-energy bound states form at the Fermi surface during the adiabatic process, the largest containing [p] (the integer part of p) fermions. This largest bound state moves rapidly around to the other Fermi point where it sequentially unbinds. We find Berry's phase for the whole process to be [p]\ensuremath{\pi}. For p integer, as \ensuremath{\Phi} increases, eventually all the particles in the system go into bound states of size [p]. The period in this case is of order the size of the system. The charge-carrying mass of the fermions is calculated by finding the energy of the adiabatic ground state with twisted boundary conditions.
We study the Heisenberg spin chain with twisted boundary conditions, focusing on the adiabatic flow of the energy spectrum as a function of the twist angle. In terms of effective field theory for the nearest-neighbor model, we show that the period 2 (in unit
2π) obtained by Sutherland and Shastry arises from irrelevant perturbations around the massless fixed point, and that this period may be rather general for one-dimensional interacting lattice models at half filling. In contrast, the period for the Haldane-Shastry spin model with
1/r2 interaction has a different and unique origin for the period, namely, it reflects fractional statistics in Haldane's sense.
We study spinless fermions in a flux threaded AB
2 chain taking into account nearest-neighbor Coulomb interactions. The exact diagonalization of the spinless AB
2 chain is presented in the limiting cases of infinite or zero nearest-neighbor Coulomb repulsion for any filling. Without interactions, the AB
2 chain has a flat band even in the presence of magnetic flux. We show that the respective localized states can be written in the most compact form as standing waves in one or two consecutive plaquettes. We show that this result is easily generalized to other frustrated lattices such as the Lieb lattice. A restricted Hartree-Fock study of the
V/t versus filling phase diagram of the AB
2 chain has also been carried out. The validity of the mean-field approach is discussed taking into account the exact results in the case of infinite repulsion. The ground-state energy as a function of filling and interaction
V is determined using the mean-field approach and exactly for infinite or zero
V. In the strong coupling limit, two kinds of localized states occur: one-particle localized states due to geometry and two-particle localized states due to interaction and geometry. These localized fermions create open-boundary regions for itinerant carriers. At filling $\ensuremath{\rho}=2/9$ and in order to avoid the existence of itinerant fermions with positive kinetic energy, phase separation occurs between a high-density phase ($\ensuremath{\rho}=2/3$) and a low-density phase ($\ensuremath{\rho}=2/9$) leading to a metal-insulator transition. The ground-state energy reflects such phase separation by becoming linear on filling above
2/9. We argue that for filling near or larger than 2/9, the spectrum of the t-V AB
2 chain can be viewed as a mix of the spectra of Luttinger liquids (LL) with different fillings, boundary conditions, and LL velocities.
We consider a system of charged spinless fermions in a ring governed by the antiferromagnetic Heisenberg-Ising Hamiltonian and in the presence of a magnetic flux. We find that the two broken-symmetry ground states evolve adiabatically with increasing flux with a period corresponding to two flux quanta, while the period of the total spectrum is one flux quantum. This behavior, already observed for this system in the gapless regime [B. Sutherland and B. S. Shastry, Phys. Rev. Lett. 65, 1833 (1990)], is shown to be a natural consequence of the relevant degrees of freedom involved in the low-energy physics of this system: antiferromagnetic domain walls or solitons, with half the charge of the original particles. A space-time approach is introduced to describe the dynamics of these objects, affording a complete topological classification of space-time histories of the system. This allows a physically complete understanding of the ground-state-subspace evolution with increasing flux in the antiferromagnetic broken-symmetry regime. In this case, the flux period doubling can be explained in terms of the Berry's phase gained by the two degenerate broken-symmetry ground states upon adiabatic switching of the flux.