Extension theory for lattice Green functions

A method is presented of using the known Green functions or densities of states (DOS) for a given Hamiltonian H to find Green functions or DOS for any lattice Hamiltonian Hₑ in the algebra of H, or Hₑ=f(H). The procedure is further developed to permit using the known Green functions and DOS of two Hamiltonians H and K to obtain those of any Hamiltonian Hₑ in the algebra generated by the direct products H{}I and I{}K. The method does not depend on translation symmetry, and rather general extension formulas are derived. Several analytical and numerical results are presented as examples. Green functions well known for the nearest-neighbor lattice Hamiltonian in one dimension are used to solve a one-dimensional (1D) second-neighbor model, a 1D model with infinite range, a 2D model on a square lattice with first-, second-, and third-neighbor interactions, a 2D square-lattice model with edges and corners, and a simple-cubic lattice Hamiltonian with surfaces, edges, and corners in three dimensions. Green functions for the threefold-coordinated Sierpi {n} \'{n}{}ski lattice Hamiltonian treated by O'Shaughnessy and Procaccia are extended to the treatment of a Sierpi {n} \'{n}{}ski model with first and second neighbors, and then are further extended to treat a new fractal Hamiltonian on the Cartesian product of two Sierpi {n} \'{n}{}ski lattices. Green functions of the 1D Fibonacci lattice treated by Kohmoto are extended to the treatment of quasiperiodic Fibonacci plaid lattices in two and three dimensions. These lattices are quasiperiodic and admit inflation-deflation transformations, but do not have interesting or forbidden rotation symmetry.