content_image/370437.jpg content_image/370436.jpg content_image/370434.jpg
at small energies where a half filled 2-leg ladder can be well described by the O(8) Gross- Neveu model. 25 The Gross-Neveu model is exactly solvable for all semi-simple symmetry groups and a great deal is known about its thermodynamics and correlation functions. In the SO(8) case the correlation functions were studied in Refs. (26,27). Since the model itself has Lorentz symmetry, all excitation branches have relativistic dispersion laws: 

<img src='content_image/370437.jpg'>

The spectrum consists of three octets of particles of mass Δ and a multiplet of 29 excitons with mass √$_{3Δ.}$ Two octets consist of quasi-particles of different chirality transforming according to the two irreducible spinor representations of SO(8), while the third octet con- sists of vector particles. The latter include magnetic excitations as well as the Cooperon (a particle with charge ± 2 e ). The 16 kink fields, carrying charge, spin, orbit, and parity indices, are direct descendants of the original electron lattice operators on the ladders. 

The SO(8) GN model describes several different phases related to one another by particle- hole transformations. Which phase is realized depends on the bare interaction. In this paper we assume that it is in the so-called D-Mott phase (in the terminology of Ref. 25). On the two two-leg ladders ( A and B ), the superconducting (SC) order parameters are given by 

<img src='content_image/370436.jpg'>

The distinct feature of the half filled ladder is that this order parameter is purely real and has a Z$_{2}$ symmetry. However, the symmetry is restored to U(1) and the phase stiffness becomes non-zero as soon as doping is introduced. It is an interesting feature of the SO(8) GN model that the only mode which becomes gapless at finite doping is the Cooperon. Neither magnetic excitations, nor quasi-particles become gapless. 29 When the doping increases the SO(8) GN model gradually crosses over to the SO(6) GN one plus the U(1) Gaussian model. The latter model describes the fluctuations of the superconducting phase. The effective low energy bosonized Lagrangian density for the Cooperon field, Φ, is 

<img src='content_image/370434.jpg'>

where φ is the field dual to θ . (Here – according to Ref. (28) – the Luttinger parameter K depends weakly on doping and is always in the range 1 > K > 0 . 9. On the other hand, the phase velocity is strongly doping dependent.) 

