The two-chain model of correlated fermions, appropriate e.g. for doped spin ladders or carbon nanotubes, is investigated in the weak-interaction limit. It is shown that the model scales to situations with high internal symmetries, generically SO(6), whereas on critical lines there is SO(5) or SO(3) symmetry. Consequences for the level spectroscopy of the model are discussed. In particular, the ``pi-mode'' is shown to be sixfold degenerate and contains both spin and charge excitations.
The two-chain model of correlated fermions, appropriate e.g. for doped spin ladders or carbon nanotubes, is investigated in the weak-interaction limit. It is shown that the model scales to situations with high internal symmetries, generically SO(6), whereas on critical lines there is SO(5) or SO(3) symmetry. Consequences for the level spectroscopy of the model are discussed. In particular, the ``pi-mode'' is shown to be sixfold degenerate and contains both spin and charge excitations.
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 a system of two Tomonaga-Luttinger models coupled by a small transverse hopping (a two-chain ladder). We use Abelian and non-Abelian bosonisation to show that the strong coupling regime at low energies can be described by an SO(5)1 WZW model (or equivalently 5 massless Majorana fermions) deformed by symmetry breaking terms that nonetheless leave the theory critical at T=0. The SO(5) currents of the theory comprise the charge and spin currents and linear combinations of the so-called pi operators (S.C. Zhang, Science 275, 1089 (1997)) which are local in terms both of the original fermions and those of the effective theory. Using bosonisation we obtain the asymptotic behaviour of all correlation functions. We find that the 5 component ``superspin'' vector has power law correlations at T=0; other fermion bilinears have exponentially decaying correlations and the corresponding tendencies are suppressed. Conformal field theory also allows us to obtain the energies, quantum numbers, and degeneracies of the low-lying states and fit them into deformed SO(5) multiplets.
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-Tc superconductivity.
We discuss properties of a recently proposed exactly SO(5)-symmetric ladder model. In the strong coupling limit we demonstrate how the SO(3)-symmetric description of spin ladders in terms of bond bosons can be upgraded to an SO(5)-symmetric bond-boson model, which provides a particularly simple example for the concept of SO(5) symmetry. Based on this representation we show that antiferromagnetism and superconductivity can be understood as condensation of either magnetic or charged bosons into a resonating-valence-bond vacuum. We identify exact eigenstates of a finite cluster with general multiplets of the SO(5) group, and present numerical results for the single particle, spin and charge spectra of the SO(5)-symmetric model and identify ``fingerprints'' of SO(5) symmetry in these. In particular we show that SO(5) symmetry implies a ``generalized rigid band behavior'' of the photoemission spectrum, i.e., spectra for the doped case are rigorously identical to spectra for spin-polarized states at half filling.
We present a low-energy effective field theory to describe the SO(n) bilinear-biquadratic spin chain. We start with n=6 and construct the effective theory by using six Majorana fermions. After determining various correlation functions, we characterize the phases and establish the relation between the effective theories for SO(6) and SO(5). Together with the known results for n=3 and 4, a reduction mechanism is proposed to understand the ground state for arbitrary SO(n). Also, we provide a generalization of the Lieb-Schultz-Mattis theorem for SO(n). The implications of our results for entanglement and correlation functions are discussed.
We study a system of two Tomonaga-Luttinger models coupled by a small transverse hopping (a two-chain ladder). We use Abelian and non-Abelian bosonization to show that the strong coupling regime at low energies can be described by an SO(5)1 Wess-Zumino-Witten model (or equivalently five massless Majorana fermions) deformed by symmetry-breaking terms that nonetheless leave the theory critical at T=0. The SO(5) currents of the theory comprise the charge and spin currents and linear combinations of the so-called pi operators [S.C. Zhang, Science 275, 1089 (1997)], which are local in terms both of the original fermions and those of the effective theory. Using bosonization we obtain the asymptotic behavior of all correlation functions. We find that the five-component ``superspin'' vector has power-law correlations at T=0; other fermion bilinears have exponentially decaying correlations and the corresponding tendencies are suppressed. Conformal field theory also allows us to obtain the energies, quantum numbers, and degeneracies of the low-lying states and fit them into deformed SO(5) multiplets.