We investigate how free probability allows us to approximate the density of states in tight-binding models of disordered electronic systems. Extending our previous studies of the Anderson model in one dimension with nearest-neighbor interactions [Chen et al., Phys. Rev. Lett. 109, 036403 (2012)], we find that free probability continues to provide accurate approximations for systems with constant interactions on two- and three-dimensional lattices or with next-nearest-neighbor interactions, with the results being visually indistinguishable from the numerically exact solution. For systems with disordered interactions, we observe a small but visible degradation of the approximation. To explain this behavior of the free approximation, we develop and apply an asymptotic error analysis scheme to show that the approximation is accurate to the eighth moment in the density of states for systems with constant interactions, but is only accurate to sixth order for systems with disordered interactions. The error analysis also allows us to calculate asymptotic corrections to the density of states, allowing for systematically improvable approximations as well as insight into the sources of error without requiring a direct comparison to an exact solution.
Theoretical studies of localization, anomalous diffusion and ergodicity breaking require solving the electronic structure of disordered systems. We use free probability to approximate the ensemble-averaged density of states without exact diagonalization. We present an error analysis that quantifies the accuracy using a generalized moment expansion, allowing us to distinguish between different approximations. We identify an approximation that is accurate to the eighth moment across all noise strengths, and contrast this with perturbation theory and isotropic entanglement theory.
We provide a brief overview of approaches for calculating the density of states of quantum systems and random matrix Hamiltonians using the tools of free probability theory. For a given Hamiltonian of a quantum system or a generic random matrix Hamiltonian, which can be written as a sum of two non-commutating operators, one can obtain an expression for the density of states of the Hamiltonian from the known density of states of the two component operators by assuming that these operators are mutually free and by using the free additive convolution. In many examples of interacting quantum systems and random matrix models, this procedure is known to provide a reasonably accurate approximation to the exact numerical density of states. We review some of the examples that are known in the literature where this procedure works very well, and also discuss some of the limitations of this method in situations where the free probability approximation fails to provide a sufficiently accurate description of the exact density of states. Subsequently, we describe a perturbation scheme that can be developed from the subordination formulas for the Cauchy transform of the density of states and use it to obtain approximate analytical expressions for the density of states in various models, such as the Rosenzweig-Porter random matrix ensemble and the Anderson model with on-site disorder.
A simple and commonly employed approximate technique with which one can examine spatially disordered systems when strong electronic correlations are present is based on the use of real-space unrestricted self-consistent Hartree-Fock wave functions. In such an approach the disorder is treated exactly while the correlations are treated approximately. In this report we critique the success of this approximation by making comparisons between such solutions and the exact wave functions for the Anderson-Hubbard model. Due to the sizes of the complete Hilbert spaces for these problems, the comparisons are restricted to small one-dimensional chains, up to ten sites, and a 4x4 two-dimensional cluster, and at 1/2 filling these Hilbert spaces contain about 63,500 and 166 million states, respectively. We have completed these calculations both at and away from 1/2 filling. This approximation is based on a variational approach which minimizes the Hartree-Fock energy, and we have completed comparisons of the exact and Hartree-Fock energies. However, in order to assess the success of this approximation in reproducing ground-state correlations we have completed comparisons of the local charge and spin correlations, including the calculation of the overlap of the Hartree-Fock wave functions with those of the exact solutions. We find that this approximation reproduces the local charge densities to quite a high accuracy, but that the local spin correlations, as represented by < S_i . S_j >, are not as well represented. In addition to these comparisons, we discuss the properties of the spin degrees of freedom in the HF approximation, and where in the disorder-interaction phase diagram such physics may be important.
In a recent contribution [P. Corboz, R. Or\'us, B. Bauer, and G. Vidal, Phys. Rev. B 81, 165104 (2010)] fermionic projected entangled-pair states (PEPSs) were used to approximate the ground state of free and interacting spinless fermion models, as well as the $t\text{\ensuremath{-}}J$ model. This paper revisits these three models in the presence of an additional next-nearest hopping amplitude in the Hamiltonian. First we explain how to account for next-nearest neighbor Hamiltonian terms in the context of fermionic PEPS algorithms based on simulating time evolution. Then we present benchmark calculations for the three models of fermions and compare our results against analytical, mean-field, and variational Monte Carlo results, respectively. Consistent with previous computations restricted to nearest neighbor Hamiltonians, we systematically obtain more accurate (or better converged) results for gapped phases than for gapless ones.
Asymptotically accurate results are obtained for the average Green function and density of states of a disordered system for a renormalizable class of models (as opposed to the lattice model examined previously [I. M. Suslov, Zh. Éksp. Teor. Fiz. 106 , 560 (1994)]. For N ∼1 (where N is the order of perturbation theory), only the parquet terms corresponding to the higher powers of large logarithms are taken into account. For large N , this approximation is inadequate because of the higher rate of increase with respect to N of the coefficients for the lower powers of the logarithms. The latter coefficients are determined from the renormalization condition for the theory expressed in the form of a Callan-Symanzik equation using the Lipatov asymptote as boundary conditions. For calculating the self-energy at finite momentum, a modification of the parquet approximation, is used that allows the calculations to be done in an arbitrary finite logarithmic approximation, including the principal asymptote in N of the expansion coefficients. It is shown that the phase transition point moves in the complex plane, thereby ensuring regularity of the density of states for all energies and avoiding the “false” pole in such a way that the effective interaction remains logarithmically weak.
This work is devoted to the development of a systematic method for studying electron localization. The developed method is Typical Medium Dynamical Cluster Approximation (TMDCA) using the Anderson-Hubbard model. The TMDCA incorporates non-local correlations beyond the local typical environment in a self-consistent way utilizing the momentum resolved typical-density-of-states and the non-local hybridization function to characterize the localization transition. For the (non-interacting) Anderson model, I show that the TMDCA provides a proper description of the Anderson localization transition in one, two, and three dimensions. In three-dimensions, as a function of cluster size, the TMDCA systematically recovers the re-entrance behavior of the mobility edge and obtains the correct critical disorder strength for the various disorder configurations and the associated \textit{universal order-parameter-critical-exponent}
β and in lower-dimensions, the well-knowing scaling relations are reproduced in agreement with numerical exact results. The TMDCA is also extended to treat diagonal and off-diagonal disorder by generalizing the local Blackman-Esterling-Berk and the importance of finite cluster is demonstrated. It was further generalized for multiband systems. Applying the TMDCA to weakly interaction electronic systems, I show that incorporating Coulomb interactions into disordered electron system result in two competing tendencies: the suppression of the current due to correlations and the screening of the disorder leading to the homogenizing of the system. It is shown that the critical disorder strength (
WcU), required to localize all states, increases with increasing interactions (
U); implying that the metallic phase is stabilized by interactions. Using the results, a soft pseudogap at the intermediate
W close to
WcU is predicted independent of filling and dimension, and I demonstrate in three-dimensions that the mobility edge is preserved as long as the chemical potential,
μ, is at or beyond the mobility edge energy (
ωϵ). A two-particle formalism of electron localization is also developed within the TMDCA and used to calculate the direct-current conductivity, enabling direct comparison with experiments. Note significantly, the TMDCA benchmarks well with numerical exact results with a dramatic reduction in computational cost, enabling the incorporation of material's specific details as such provide an avenue for the possibility of studying electron localization in real materials.