Models with range-free frustrated Ising spin interaction and additional Hubbard interaction are treated exactly by means of the discrete time slicing method of Grassmann-field theory. Critical and tricritical points, spin and charge correlations, and the fermion propagator are derived as a function of temperature and chemical potential $\ensuremath{\mu},$ of the Hubbard coupling U and the spin-glass energy J. U is allowed to be either repulsive
(U>0) or attractive
(U<0). Cuts through the multidimensional phase diagram are obtained. Analytical and numerical evaluations take important replica symmetry-breaking (RSB) effects into account. Results for the ordered phase are given at least in a one-step approximation (1RSB), and for
T=0 we report the first two-, three-, and four-step calculations (4RSB) for fermionic spin glasses. The use of exact relations and invariances under RSB together with 2RSB calculations for all fillings and 4RSB solutions for half filling allow to model exact solutions by interpolation. For
T=0, our numerical results provide strong evidence that the exact spin-glass pseudogap obeys $\ensuremath{\rho}{(E)=c}_{1}|E\ensuremath{-}{E}_{F}|$ for energies close to the Fermi level with ${c}_{1}\ensuremath{\approx}0.13.$ Rapid convergence of ${\ensuremath{\rho}}^{\ensuremath{'}}{(E}_{F})$ under increasing orders of RSB is observed and ${\ensuremath{\rho}}^{\ensuremath{''}}(E)$ is evaluated to estimate subleading powers. Over a wide range of the pseudogap and after a small transient regime $\ensuremath{\rho}(E)$ regains a linear shape with a larger slope and a small S-like perturbation. The leading term resembles the Efros-Shklovskii Coulomb pseudogap of two-dimensional localized disordered fermionic systems. Beyond half filling we obtain a $\ensuremath{\nu}\ensuremath{-}1\ensuremath{\sim}(\ensuremath{\mu}\ensuremath{-}{U)}^{2},$ $\ensuremath{\mu}>~U$ dependence of the fermion filling factor $\ensuremath{\nu}.$ We find a half filling transition between a phase for $U>\ensuremath{\mu},$ where the Fermi level lies inside the Hubbard gap, into a phase where $\ensuremath{\mu}(>U)$ is located at the center of the upper spin-glass (SG) pseudogap (SG gap). For $\ensuremath{\mu}>U$ the Hubbard gap combines with the lower one of two SG gaps (phase I), while for $\ensuremath{\mu}<U$ it joins the sole SG gap which exists in this half-filling regime (phase II). Shoulders of the combined gaps are shaped by RSB due to spin-glass order. We predict scaling behavior at the half-filling transition which becomes continuous due to $\ensuremath{\infty}\mathrm{RSB}.$ Implications of the half-filling transition between the deeper insulating phase II and phase I for the eventual delocalization by additional hopping processes in itinerant model extensions are discussed. Possible metal-insulator transition scenarios are described. Generalizations to random Hubbard coupling and alloy models as well as frustrated magnetic interactions with ferro- or antiferromagnetic components are discussed.
This thesis is concerned with the mean field theory of spin glasses within the replica method approach. We have studied the Sherrington-Kirkpatrick model [33,4δ] both with Ising and m-component spins. It is shown that the replica symmetric solution gives an instability in the condensed phase which requires a breaking of replica symmetry in the spin glass phase. The various broken replica symmetry solutions proposed thus far (Bray and Moore [52], Blandin et al [53], Parisi [54]) are reviewed and particular attention is given to the one proposed by Parisi. The order parameter susceptibility and the stability of the solution are studied within this breaking of replica symmetry scheme which give results in agreement with the predictions of TAP [47] and Bray and Moore [52]. The breakdown of linear response theory in the spin glass phase is also discussed.
A class of infinite-ranged random model Hamiltonians is defined as a limiting case in which the appropriate form of mean-field theory, order parameters and phase diagram to describe spin-glasses may be established. It is believed that these Hamiltonians may be exactly soluble, although a complete solution is not yet available. Thermodynamic properties of the model for Ising and
XY spins are evaluated using a many-replica procedure. Results of the replica theory reproduce properties at and above the ordering temperature which are also predicted by high-temperature expansions, but are in error at low temperatures. Extensive computer simulations of infinite-ranged Ising spin-glasses are presented. They confirm the general details of the predicted phase diagram. The errors in the replica solution are found to be small, and confined to low temperatures. For this model, the extended mean-field theory of Thouless, Anderson, and Palmer gives physically sensible low-temperature predictions. These are in quantitative agreement with the Monte Carlo statics. The dynamics of the infinite-ranged Ising spin-glass are studied in a linearized mean-field theory. Critical slowing down is predicted and found, with correlations decaying as ${e}^{{\ensuremath{-}[\frac{(T\ensuremath{-}{T}_{c})}{T}]}^{2}t}$ for
T greater than
Tc, the spin-glass transition temperature. At and below
Tc, spin-spin correlations are observed to decay to their long-time limit as ${t}^{\ensuremath{-}\frac{1}{2}}$.