Thijssen Ch. 11 discusses "Transfer Matrix methods"; Need Lanczos (sec. 11.3) or QMC methods (12.6) for solution of complex cases

Other cases discussed in notes and in the references given

1. Examples of many-body problems
   • Hubbard Model
     • Review by Dagotto for problems in HiTc related systems
     • Setting up Hamiltonian in convenient basis (see Lin and Gubernatis)
     • Example from Class project, 1996, by Sengupta
   • Spin systems, Heisenberg Model - Derived in limit of Hubbard Model
     • Magnetic order?
     • Finite size scaling
   • Spin systems, the Ising model - See review by Kogut
     • Real model for classical, quantum spins
     • Isomorphic to Lattice gauge theories
       See review by Kogut; Thijssen, Ch. 13
     • Transfer matrix method
       • Large eigenvalue gives free energy
       • Simple 1d spin 1/2 case (Thijssen Ch. 11)
       • Approach to 2d - very complex (Thijssen Ch. 11)
       • Solution by Lanczos (Thijssen Ch. 11) or QMC (Thijssen Ch. 12)
     • The 1d Ising model in a transverse field
       • Maps onto model for quantum double-well tunneling model for ferroelectrics
       • Hamiltonian for intracting quantum Ising spins
       • Perturbation theory analysis - shows the key points of ground state energy and correlation functons
       • Phase transition at critical coupling constant
       • Duality of Ising model shows analytically where transition is expected
       • A gap vanishes at this point (gap between ground state and lowest excited state.)
   • Solution by Lanczos methods
     • Consider finite systems with a finite (but large basis)
     • The hamiltonian operator as a sparse matrix
     • Generation of the basis in which H is tridiagonal
       (This is the tricky part for a many body system, because the basis is so large one must order the computations carefully)
     • Examples of solutions
       • Ising model from Roomany
       • Hubbard Model from Sengupta
     • Finite size scaling and the phase transition at infinite size
   • References
     • E. Dagotto, Rev. Mod. Phys. 66, 763 (1994).
     • J. Kogut, Rev. Mod. Phys. 51, 651 (1979).
     • H. Roomany, H. W. Wyld, and L. Holloway, Phys. Rev. D21, 1557 (1980), and thesis, U of I, 1980.
     • H. Q. Lin and J. E. Gubernatis, Computers in Physics 7, 400 (1993).
     • E. R. Gagliano, et. al., Phys. Rev. B34, 1677 (1986).
     • B. Bernu, C. Lhuillier, and L. Pierre, Phys. Rev. Letters 69, 2590 (1992).