Theoretical modeling for quantum liquids from 1d to 2d dimensional crossover using quantum groups

Recent experimental and theoretical work in strongly correlated electron system necessitates a formulation to deal with 1d to 2d dimensional crossover and raises several interesting questions. A particularly interesting question is what happens to the 1d Luttinger liquid, as we go from 1d to 2d? The issue of dimensional crossover in lower dimensional systems is of great general interest, in particular the relationship between quantum liquids and dimensional crossover from 1d to 2d. Thus one may ask given a 1d Luttinger liquid, what happens if we go from 1d to 2d, does the Luttinger liquid end up as a Fermi liquid or a non-Fermi liquid [NFL] in 2d. More importantly and relevantly what are the conditions that determine which type of quantum liquid we will end up with. Looking ahead to future applications we need to know what parameters to tune in order to engineer 2d quantum liquids from their 1d counterparts. In other words what are the parameters that determine quantum phase transitions. Can the Luttinger liquid lead to stripe formation in high temperature superconducting [HTSC] materials in a natural way? Can we classify the quantum liquids existing in 1d, 2d and 3d under symmetry classes of quantum groups in practically meaningful way? What is the nature of dimensional phase transitions and how are these [i.e. dimensional phase transitions] made possible by interactions of quantum liquids. The main point of emphasis of the present note is that the Permanent address: Department of Physics, University of Peshawar, Peshawar, NWFP, Pakistan. Photon Factory, KEK, Tsukuba, Ibaraki 305, Japan