Simulations of Strongly-Correlated Electrons

D. J. Scalapino

University of California, Santa Barbara

       The discovery of high temperature superconductivity in 1986 and the subsequent proliferation of model systems and theoretical explanations made it clear that it would be useful to understand the actual properties of some many-electron Hamiltonians. In the case of one-dimensional systems, analytic solutions for a few particular Hamiltonians have provided important insights. However, in two or three dimensions we have few exact analytic results. One approach to this problem has been the development of numerical techniques aimed at obtaining controlled results for finite lattices. Some examples include Lancoz diagonalization, quantum Monte Carlo (QMC), stochastic series expansion (SSE) and density matrix renormalization group (DMRG) methods.

       At present DMRG techniques provide a powerful approach to one-dimensional chain and some n-leg ladder models. QMC and particularly recent SSE methods allow one to study 2D and 3D unfrustrated spin models on large lattices. QMC techniques are also applicable to higher dimensional fermion models which have special particle-hole or time-reversal symmetries. However, controlled numerical techniques for studying frustrated spin systems and general many-fermion systems in 2D and 3D remain to be developed.

Here, I’ll give a few examples of what has been learned and list some open questions which relate to why we need more powerful methods and where they might come from:

  • Do we really understand the cuprates and if so, what is so special about Hg(1223) under pressure?

  • What is required of a Hamiltonian so that it has a spin-liquid state in two or
    higher dimensions?

  • Can ideas from the field of quantum information help us design better DMRG algorithms?

  • Can cold neutral atoms in an optical lattice or arrays of ionized atoms confined in an rf trap provide quantum simulators for these problems?