Contractor Renormalization Group Method Research Articles

Application of Contractor Renormalization Group to the Heisenberg Zig-Zag and the Hubbard Chain

The contractor renormalization group method was devised in 1994 by Morningstar and Weinstein. It was primarily aimed at extracting the physics of lattice quantum field theories (like lattice quantum chromodynamics). However, it is a general method of analyzing Hamiltonian lattice systems, e.g. Ising, Heisenberg or Hubbard models. The aim of this work is to show the application of contractor renormalization group method to one-dimensional quantum systems — the Heisenberg zig-zag model and the Hubbard chain. As a test of the method, the ground state energy of these systems will be calculated.

Quarks, gluons, and frustrated antiferromagnets

The contractor renormalization group method is used to establish the equivalence of various Hamiltonian free fermion theories and a class of generalized frustrated antiferromagnets. In particular, after a detailed discussion of a simple example, it is argued that a generalized frustrated $\mathrm{SU}(3)$ antiferromagnet whose single-site states have the quantum numbers of mesons and baryons is equivalent to a theory of free massless quarks. Furthermore, it is argued that for slight modification of the couplings which define the frustrated antiferromagnet Hamiltonian, the theory becomes a theory of quarks interacting with color gauge fields.

Probing the CORE of the Haldane conjecture

The COntractor REnormalization group method (CORE), originally developed for application to lattice gauge theories, is very well adapted the study of spin systems and systems with fermions. As an warmup exercise for studying Hubbard models this method is applied to spin- 1 2 and spin-1 anti-ferromagnets in one space dimension in order to see if it is able to explain the physics of the Haldane conjecture. The method not only provides support for Haldane's conjecture but provides insight into the physics of a more general class of spin-1 systems with Hamiltonians of the form H = Σ j →s(j) · →s(j + 1) − β( →s(j) · →s(j + 1)) 2 about which, until now, little was known.