We study the spectrum of fermions hopping on a chain with a weak incommensuration close to dimerization; both
q, the deviation of the wave number from $\ensuremath{\pi},$ and $\ensuremath{\delta},$ the strength of the incommensuration, are small. For free fermions, we use a continuum Dirac theory to show that there are an infinite number of bands which meet at zero energy as q approaches zero. In the limit that the ratio $q/\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\delta}}0,$ the number of states lying inside the
q=0 gap is nonzero and equal to $2\ensuremath{\delta}/{\ensuremath{\pi}}^{2}.$ Thus the limit $\stackrel{\ensuremath{\rightarrow}}{q}0$ differs from
q=0; this can be seen clearly in the behavior of the specific heat at low temperature. For interacting fermions or the
XXZ spin-
1/2 chain close to dimerization, we use bosonization and a renormalization group analysis to argue that similar results hold; as $\stackrel{\ensuremath{\rightarrow}}{q}0,$ there is a nontrivial density of states near zero energy. However, the limit $\stackrel{\ensuremath{\rightarrow}}{q}0$ and
q=0 give the same results near commensurate wave numbers which are different from $\ensuremath{\pi};$ for both free and interacting fermions, we find that a nonzero value of q is necessary to close the gap. Our results for free fermions are applied to the Azbel-Hofstadter problem of electrons hopping on a two-dimensional lattice in the presence of a magnetic field. Finally, we discuss the complete energy spectrum of free fermions with incommensurate hopping by going up to higher orders in $\ensuremath{\delta}.$
It is well established that at zero magnetic field
h=0, the spin diffusion constant of the spin-
21 XXZ chain is for anisotropy $\mathrm{\ensuremath{\Delta}}>1$ finite for both low and high temperatures, implying that the type of spin transport is normal diffusive. Although it is expected that this holds for all finite temperatures
T>0, the calculation of the spin diffusion constant for all
T is typically intractable. Here we consider a class of energy eigenstates that exist both for anisotropies $\mathrm{\ensuremath{\Delta}}=1$ and $\mathrm{\ensuremath{\Delta}}>1$. We show that at the isotropic point their contributions are behind the diffusion constant being infinite, spin transport being anomalous superdiffusive for
T>0. That for $\mathrm{\ensuremath{\Delta}}>1$ such states do not contribute to the diffusion constant is shown to imply it is finite, spin transport being normal diffusive for
T>0. By combining the connection through a Jordan-Wigner transformation of the spin-
21 XXZ chain to the one-dimensional (1D) lattice spinless fermion model at zero chemical potential $\ensuremath{\mu}=0$, for $V/J\ensuremath{\ge}1$ with its Bethe-ansatz solution, where
V is the nearest-neighbor Coulomb repulsion and
J=2t is twice the hopping integral
t, in this paper we also address the issue of the
T>0 charge transport of that model at $\ensuremath{\mu}=0$. It is found to be anomalous superdiffusive for
V/J=1 and normal diffusive for
V/J>1. Our results thus open the door to a key advance in the understanding for all finite temperatures
T>0 of the spin transport in the spin-
21 XXZ chain at
h=0 for anisotropy $\mathrm{\ensuremath{\Delta}}\ensuremath{\ge}1$ and of the charge transport in the 1D lattice spinless fermion model at $\ensuremath{\mu}=0$ for $V/J\ensuremath{\ge}1$.
A Green-function formalism is presented to study a Peierls-Hubbard Hamiltonian in two dimensions. The lattice consists of parallel dimerized chains with alternating nearest-neighbor hoppings t and ${\mathit{t}}_{\mathrm{\ensuremath{\parallel}}}$ and another hopping ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$ between different chains. The method treats the interdimer hopping as a perturbation and yields exact results in the uncorrelated case and for isolated dimers. The calculated spectral functions exhibit a number of narrow subbands with typical low-dimensional singularities. The dependence of the gap at the Fermi level on the electron-electron interaction U agrees qualitatively with the exact result in the known one-dimensional nondimerized limit. The paramagnetic susceptibility shows a maximum structure at low temperatures that is enhanced by U and by dimerization and a Curie-Weiss behavior at high temperatures.
The behavior in an external magnetic field is studied exactly for a wide class of multichain quantum spin models. It is shown that the magnetic field together with the interchain couplings cause commensurate–incommensurate phase transitions between the gapless phases in the ground state. The conformal limit of these models is studied and it is shown that the low-lying excitations for the incommensurate phases are not independent, because they are governed by the same magnetic field (chemical potential for excitations). A scenario for the transition from one to two space dimensions for the exactly integrable multichain quantum spin models is proposed, and it is shown that the incommensurate phases in an external magnetic field disappear in the limit of an infinite number of coupled spin chains. The similarities in the external field behavior for the quantum multichain spin models and a wide class of quantum field theories are discussed. The scaling exponents for the appearance of the gap in the spectrum of low-lying excitations of the quantum multichain models due to the relevant perturbations of the integrable theories are calculated.
The low energy properties of different one-dimensional fermionic lattice models are investigated using the bosonization technique. We attach much importance to a proper consideration of the Klein factors which are neglected or inaccurately treated in the majority of the literature. In order to deal with the nonlinear terms, arising from the dimerization or similar scattering processes, we use the self-consistent harmonic approximation (SCHA), a variational method where the nonlinearities are replaced by a quadratic potential. We construct an extension of the SCHA in order to account for the existence of Klein factors in bosonized Hamiltonians. As prototypical models we consider one-dimensional dimerized spinless fermions and Hubbard models with periodic modulation of the hopping and of the chemical potential, respectively. In an attempt to describe more realistic systems we also consider a model where one-dimensional electrons are coupled to a three-dimensional conduction band via a local interaction. In order to see how the one-dimensional electron system is modified in the presence of the three-dimensional environment, we calculate the charge and the spin susceptibility, and analyze the collective excitations within RPA. We find that the spin excitations remain unaffected while in the charge channel, the collective modes of the one-dimensional and three-dimensional subsystems are coupled.
We consider the interacting Aubry-Andr\'e model describing fermions on a one-dimensional lattice with an incommensurate potential and a short-range many-body interaction. The single-particle spectrum has infinitely many gaps in the extended phase and at zero temperature is an insulator for almost all the chemical potentials. The many-body interaction has the effect that the gaps are strongly decreased or increased depending on the attractive or repulsive nature of the interaction, but even the smallest gaps remain open. The system is a band insulator for generic chemical potentials even in the presence of interaction, and a quantum phase transition is excluded at weak coupling.