Recently Lin, Balents, and Fisher have demonstrated that two-leg Hubbard ladders and armchair carbon nanotubes renormalize towards the integrable SO(8) Gross-Neveu model. Here, we exploit this integrability to extract non-trivial effects of interactions in these systems in their doped phase. We so obtain results that could not be expected a priori. Using thermodynamic Bethe ansatz, we compute exactly both the spin and single-particle gaps and the Luttinger parameter describing low-energy excitations. We show rigourously both the spin gap and the electron gap do not vanish at finite doping, while the Luttinger parameter remains close to its free Fermionic value of 1, even for larger values of doping. A similar set of conclusions is drawn for the undoped systems' behavior in a finite magnetic field. We also comment on the existence in these systems of the $\ensuremath{\pi}$ resonance, a hallmark of Zhang's SO(5) theory of high-${T}_{c}$ superconductivity.
Recently Lin, Balents, and Fisher have demonstrated that two-leg Hubbard ladders and armchair carbon nanotubes renormalize towards the integrable SO(8) Gross-Neveu model. Here, we exploit this integrability to extract non-trivial effects of interactions in these systems in their doped phase. We so obtain results that could not be expected a priori. Using thermodynamic Bethe ansatz, we compute exactly both the spin and single-particle gaps and the Luttinger parameter describing low-energy excitations. We show rigourously both the spin gap and the electron gap do not vanish at finite doping, while the Luttinger parameter remains close to its free Fermionic value of 1, even for larger values of doping. A similar set of conclusions is drawn for the undoped systems' behavior in a finite magnetic field. We also comment on the existence in these systems of the \ensuremathπ resonance, a hallmark of Zhang's SO(5) theory of high-Tc superconductivity.
Motivated by recent work of Lin, Balents, and Fisher [Phys. Rev. B 58, 1794 (1998)], we compute correlation functions at zero temperature for weakly coupled two-leg Hubbard ladders and (N,N) armchair carbon nanotubes. In this paper it was argued that such systems renormalize towards the SO(8) Gross-Neveu model, an integrable theory. We exploit this integrability to perform the computation at the SO(8) invariant point. Any terms breaking the SO(8) symmetry can be treated systematically in perturbation theory, leading to a model with the same qualitative features as the integrable theory. Using said correlators, we determine the optical conductivity, the single-particle spectral function, and the $I\ensuremath{-}V$ curve for tunneling into the system from an external metallic lead. The frequency, \ensuremath{\omega}, dependent optical conductivity is determined exactly for $\ensuremath{\omega}<3m$ (m being the fermion particle mass in the SO(8) Gross-Neveu model). It is characterized by a sharp ``exciton'' peak at $\ensuremath{\omega}=\sqrt{3}m,$ followed by the onset of the particle-hole continuum beginning at $\ensuremath{\omega}=2m.$ Interactions modify this onset to $\ensuremath{\sigma}(\ensuremath{\omega}+2m)\ensuremath{\sim}{\ensuremath{\omega}}^{1/2}$ and not the ${\ensuremath{\omega}}^{\ensuremath{-}1/2}$ one would expect from the van Hove singularity in the density of states. Similarly, we obtain the exact single-particle spectral function for energies less than 3m. The latter possesses a $\ensuremath{\delta}$ function peak arising from single-particle excitations, together with a two-particle continuum for $\ensuremath{\omega}>~2m.$ The final quantity we compute is the tunneling $I\ensuremath{-}V$ curve to lowest nonvanishing order in the tunneling matrix elements. For this quantity, we present exact results for voltages, V<(1+3)m. The resulting differential conductance is marked by a finite jump at $\ensuremath{\omega}=2m,$ the energy of the onset of tunneling into the continuum of two-particle states. Through integrability, we are able to characterize this jump exactly. All calculations are done through form-factor expansions of correlation functions. These give exact closed form expressions for spectral functions because the SO(8) Gross-Neveu model is massive: each term in the expansion has an energy threshold below which it does not contribute. Thus, we obtain exact results below certain thresholds by computing a finite number of terms in this series. Previous to this paper, the only computed form factor of SO(8) Gross-Neveu was the two-particle form factor of an SO(8) current with two fundamental fermions. In this paper we compute the set of all one- and two-particle form factors for all relevant fields, the currents as well as the kinks and fermions.
We revisit the problem of interacting electrons hopping on a two-leg ladder. A perturbative renormalization-group analysis reveals that at half-filling the model scales onto an exactly soluble Gross-Neveu model for arbitrary finite-ranged interactions, provided they are sufficiently weak. The Gross-Neveu model has an enormous global SO(8) symmetry, manifest in terms of eight real Fermion fields that, however, are highly nonlocal in terms of the electron operators. For generic repulsive interactions, the two-leg ladder exhibits a Mott insulating phase at half-filling with d-wave pairing correlations. Integrability of the Gross-Neveu model is employed to extract the exact energies, degeneracies, and quantum numbers of all the low-energy excited states, which fall into degenerate SO(8) multiplets. One SO(8) vector includes two charged Cooper pair excitations, a neutral s=1 triplet of magnons, and three other neutral s=0 particle-hole excitations. A triality symmetry relates these eight two-particle excitations to two other degenerate octets, which are comprised of single-electron-like excitations. In addition to these 24 degenerate ``particle'' states costing an energy (mass) m to create, there is a 28-dimensional antisymmetric tensor multiplet of ``bound'' states with energy 3m. Doping away from half-filling liberates the Cooper pairs, leading to quasi-long-range d-wave pair field correlations, but maintaining a gap to spin and single-electron excitations. For very low doping levels, integrability allows one to extract exact values for these energy gaps. Enlarging the space of interactions to include attractive interactions reveals that there are four robust phases possible for the weak coupling two-leg ladder. While each of the four phases has a (different) SO(8) symmetry, they are shown to all share a common SO(5) symmetry---the one recently proposed by Zhang as a unifying feature of magnetism and superconductivity in the cuprates.
We study the dynamical spin response of doped two-leg Hubbard-like ladders in the framework of a low-energy effective field theory description given by the SO(6) Gross-Neveu model. Using the integrability of the SO(6) Gross-Neveu model, we derive the low energy dynamical magnetic susceptibility. The susceptibility is characterized by an incommensurate coherent mode near $(\ensuremath{\pi},\ensuremath{\pi})$ and by broad two excitation scattering continua at other k-points. In our computation we are able to estimate the relative weights of these contributions. All calculations are performed using form factor expansions which yield exact low energy results in the context of the SO(6) Gross-Neveu model. To employ this expansion, a number of hitherto undetermined form factors were computed. To do so, we developed a general approach for the computation of matrix elements of semilocal SO(6) Gross-Neveu operators. While our computation takes place in the context of SO(6) Gross-Neveu, we also consider the effects of perturbations away from an SO(6) symmetric model, showing that small perturbations at best quantitatively change the physics.
We study the effect of doping away from half-filling in weakly (but finitely) interacting N-leg Hubbard ladders using renormalization-group and bosonization techniques. For a small on-site repulsion U, the N-leg Hubbard ladders are equivalent to an N-band model, where at half-filling the Fermi velocities are ${v}_{1}{=v}_{N}<{v}_{2}{=v}_{N\ensuremath{-}1}<\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}.$ We then obtain a hierarchy of energy scales, where the band pairs $(j,N+1\ensuremath{-}j)$ are successively frozen out. The low-energy Hamiltonian is then the sum of N/2 [or $(N\ensuremath{-}1)/2$ for N odd] two-leg ladder Hamiltonians without gapless excitations (plus a single chain for N odd with one gapless spin mode)---similar to the N-leg Heisenberg spin ladders. The energy scales lead to a hierarchy of gaps. Upon doping away from half-filling, the holes first enter the band(s) with the smallest gap: For odd N, the holes enter first the nonbonding band (N+1)/2 and the phase is a Luttinger liquid, while for even N, the holes enter first the band pair (N/2,N/2+1) and the phase is a Luther-Emery liquid, similar to numerical treatments of the $t\ensuremath{-}J$ model, i.e., at and close to half-filling, the phases of the Hubbard ladders for small and large U are the same. For increasing doping, hole pairs subsequently enter at critical dopings the other band pairs $(j,N+1\ensuremath{-}j)$ (accompanied by a diverging compressibility): The Fermi surface is successively opened by doping, starting near the wave vector $(\ensuremath{\pi}/2,\ensuremath{\pi}/2).$ Explicit calculations are given for the cases N=3 and 4.
A proper theoretical description of doped antiferromagnets is until now an unresolved issue. In this work we study chain and ladder systems, and in particularly we focus on the physically most interesting two leg ladder. Our starting point is the Spin Fermion model in which an antiferromagnetic background of localized spins interacts with mobile holes via a rotation invariant Kondo-like term. The interesting region of the phase diagram is that close to the Mott-insulator transition where the doping delta is zero. At zero doping in fact the system is an insulator and the spins organize themselves in a spin liquid, a rotational invariant state characterized by a finite correlation length and an energy gap (a spin gap) above the ground state. Such a state is also the ground state of a field theory, the non linear sigma model (NLsigmaM) which is recognized as the low energy effective theory for antiferromagnetic spin ladder. The first question we answer is how such a spin state evolves as one moves off from the Mott phase by increasing the doping. Integrating out the fermions in our model we obtain an effective theory for the spins which we are able to evaluate in the continuum limit. The effective theory is again a NLsigmaM with coupling constants which depend on the concentration of dopant holes. In contrast to existing mean field calculation our theory predicts a lowering of the spin gap with doping and a consequent increase in the correlation length. Indeed a lowering of the spin gap due to doping is also observed in numerical simulation and on NMR experiments on $Sr14-xCaxCu24O41 with which we obtain very good agreement. Secondly we concentrated on the behavior of the fermions. The general paradigm of interacting fermions in one dimension is that of the Luttinger Liquid characterized by bosonic excitations and spin charge separation. By generalizing our approach we are able to access the one particle fermion propagator as a quantity averaged over the NLsigmaM which controls the spin background. We see that in the limit of zero doping the quasiparticle weight Z is non-zero in a neighborhood of the Fermi energy. This in turn implies that the Luttinger liquid parameter Krho goes to one as the doping delta goes to zero as was first argued by Schulz. Our stronger result allows us to assert that in the very low doping regime the fermions constitute a Fermi liquid.
We investigate a metallic zigzag carbon nanotube by means of a Hubbard model, which includes both on-site and nearest-neighbor interactions. Assuming weak interactions, a renormalization group analysis of the equivalent two-leg ladder followed by bosonization and refermionization results in a Gross-Neveu model with an enlarged symmetry relative to the original Hamiltonian. For the undoped case the symmetry of the Gross-Neveu model is SO(8), but for the doped case the particle-hole symmetry is broken and the symmetry reduces to SO(6). Four ground-state phases are found in the undoped carbon nanotube with repulsive interactions, a d-wave Mott insulator, an s-wave Mott insulator, a p-density wave, and a charge density wave. The doped case has two ground-state phases, a d-wave superconductor and a phase where a p density wave and a charge density wave coexist. We also explore the global phase diagram with a general interaction profile and find several additional states, including a chiral current phase where current flows around the nanotube along the zigzag bonds.