Coupled Luttinger Liquids

Many one--dimensional quantum systems, in particular interacting electron and spin systems, can be described a Luttinger liquids. Here, some basic ideas of this picture of one--dimensional systems are briefly reviewed. I then discuss the effect of interchain coupling for a finite number of parallel chains. In the case of spin chains coupled by exchange interactions, the low--energy properties are radically different according to whether the number of coupled chains is even or odd: even number of chains have a gap in the spin excitations, whereas odd numbers of chains are gapless. The effect of interchain tunneling is analyzed for two and three coupled chains of itinerant fermions: for repulsive interactions, the two--chain system is ``universally'' found to be a d--wave superconductor, with a gap in the spin excitation spectrum. On the other hand, for three chains the ground state depends both on the boundary conditions in the transverse direction and on the strength of the interactions. Weak repulsive interactions in all cases lead to dominant superconducting pairing of d--type. An example of a three--leg spin ladder with a spin gap is proposed. A general scheme to keep track of fermion anticommutation in the bosonization technique is developed.Comment: 14 pages, Revtex with epsf, 3 figures included as Postscript files, Rencontres de Moriond, Les Arcs, France, Jan. 199

Quantum fluctuations in the spiral phase of the Hubbard model

We study the magnetic excitations in the spiral phase of the two--dimensional Hubbard model using a functional integral method. Spin waves are strongly renormalized and a line of near--zeros is observed in the spectrum around the spiral pitch $\pm{\bf Q}$. The possibility of disordered spiral states is examined by studying the one--loop corrections to the spiral order parameter. We also show that the spiral phase presents an intrinsic instability towards an inhomogeneous state (phase separation, CDW, ...) at weak doping. Though phase separation is suppressed by weak long--range Coulomb interactions, the CDW instability only disappears for sufficiently strong Coulomb interaction.Comment: Figures are NOW appended via uuencoded postscript fil