Uniform Existence of the Integrated Density of States for Randomly Weighted Hamiltonians on Long-Range Percolation Graphs

We consider random Hamiltonians defined on long-range percolation graphs over $\mathbb {Z}^{d}$ . The Hamiltonian consists of a randomly weighted Laplacian plus a random potential. We prove uniform existence of the integrated density of states and express the IDS using a Pastur-Shubin trace formula.     

Uniform Existence of the Integrated Density of States on Metric Cayley Graphs

Given an arbitrary, finitely generated, amenable group we consider ergodic Schrödinger operators on a metric Cayley graph with random potentials and random boundary conditions. We show that the normalised eigenvalue counting functions of finite volume parts converge uniformly. The integrated density of states (IDS) as the limit can be expressed by a Pastur-Shubin formula. The spectrum supports the corresponding measure and discontinuities correspond to the existence of compactly supported eigenfunctions. In this context, the present work generalises the hitherto known uniform IDS approximation results for operators on the d-dimensional metric lattice to a very large class of geometries.     

Lipschitz-Continuity of the Integrated Density of States for Gaussian Random Potentials

The integrated density of states of a Schrödinger operator with random potential given by a homogeneous Gaussian field whose covariance function is continuous, compactly supported and has positive mean, is locally uniformly Lipschitz-continuous. This is proven using a Wegner estimate.     

Transience, Recurrence and Critical Behavior¶for Long-Range Percolation

We study the behavior of the random walk on the infinite cluster of independent long-range percolation in dimensions d= 1,2, where x and y are connected with probability . We show that if d<s<2d, then the walk is transient, and if s≥ 2d, then the walk is recurrent. The proof of transience is based on a renormalization argument. As a corollary of this renormalization argument, we get that for every dimension d≥ 1, if d>s>2d, then there is no infinite cluster at criticality. This result is extended to the free random cluster model. A second corollary is that when d≥& 2 and d>s>2d we can erase all long enough bonds and still have an infinite cluster. The proof of recurrence in two dimensions is based on general stability results for recurrence in random electrical networks. In particular, we show that i.i.d. conductances on a recurrent graph of bounded degree yield a recurrent electrical network. Received: 27 October 2000 / Accepted: 29 November 2001     

The Integrated Density of States of the Random Graph Laplacian

We analyse the density of states of the random graph Laplacian in the percolating regime. A symmetry argument and knowledge of the density of states in the nonpercolating regime allows us to isolate the density of states of the percolating cluster (DSPC) alone, thereby eliminating trivially localised states due to finite subgraphs. We derive a nonlinear integral equation for the integrated DSPC and solve it with a population dynamics algorithm. We discuss the possible existence of a mobility edge and give strong evidence for the existence of discrete eigenvalues in the whole range of the spectrum.     

Discontinuities of the Integrated Density of States for Random Operators on Delone Sets

Despite all the analogies with usual random models, tight binding operators for quasicrystals exhibit a feature that clearly distinguishes them from the former: the integrated density of states may be discontinuous. This phenomenon is identified as a local effect, due to the occurrence of eigenfunctions with bounded support.Research partly supported by the DFG in the priority program Quasicrystals     

Groupoids,von Neumann Algebras and the Integrated Density of States

We study spectral properties of random operators in the general setting of groupoids and von Neumann algebras. In particular, we establish an explicit formula for the canonical trace of the von Neumann algebra of random operators and define an abstract density of states. While the treatment applies to a general framework we lay special emphasis on three particular examples: random Schrödinger operators on manifolds, quantum percolation and quasi–crystal Hamiltonians. For these examples we show that the distribution function of the abstract density of states coincides with the integrated density of states defined via an exhaustion procedure.     

Continuity of the Integrated Density of States on Random Length Metric Graphs

We establish several properties of the integrated density of states for random quantum graphs: Under appropriate ergodicity and amenability assumptions, the integrated density of states can be defined using an exhaustion procedure by compact subgraphs. A trace per unit volume formula holds, similarly as in the Euclidean case. Our setting includes periodic graphs. For a model where the edge lengths are random and vary independently in a smooth way we prove a Wegner estimate and related regularity results for the integrated density of states. These results are illustrated for an example based on the Kagome lattice. In the periodic case we characterise all compactly supported eigenfunctions and calculate the position and size of discontinuities of the integrated density of states.      

Ladder Invariants and Coherent States for Time-Dependent Non-Hermitian Hamiltonians

International Journal of Theoretical Physics - We extend the Dodonov–Malkin–Man’ko–Trifonov (DMMT) invariant method (Malkin et al. Phys. Rev. D 2, 1371 1, J. Math. Phys. 14,...     

The Diameter of a Long-Range Percolation Cluster on Generalized Pre-Sierpinski Carpet and Regular Tree

Journal of Statistical Physics - We consider some kinds of graphs obtained by generalizing the pre-Sierpinski carpet, which is one of well known fractal lattices. Here, we deal with not only some...     

Proof of the Absence of Long-Range Temporal Orders in Gibbs States

Journal of Statistical Physics - We address the question whether time translation symmetry can be spontaneously broken in a quantum many-body system. One way of detecting such a symmetry breaking...     

Density functionals and model Hamiltonians: Pillars of many-particle physics

Density-functional theory (DFT) and model Hamiltonians are conceptually distinct approaches to the many-particle problem, which can be developed and applied independently. In practice, however, there are multiple connections between the two. This review focuses on these connections. After some background and introductory material on DFT and on model Hamiltonians, we describe four distinct, but complementary, connections between the two approaches: (i) the use of DFT as input for model Hamiltonians, in order to calculate model parameters such as the Hubbard U and the Heisenberg J. (ii) The use of model Hamiltonians as input for DFT, as in the LDA + U functional. (iii) The use of model Hamiltonians as theoretical laboratories to study aspects of DFT. (iv) The use of special formulations of DFT as computational tools for studying spatially inhomogeneous model Hamiltonians. We mostly focus on this fourth combination, model DFT, and illustrate it for the Hubbard model and the Heisenberg model. Other models that have been treated with DFT, such as the PPP model, the Gaudin–Yang δ$\delta$-gas model, the XXZ chain, variations of the Anderson and Kondo models and Hooke’s atom are also briefly considered. Representative applications of model DFT to electrons in crystal lattices, atoms in optical lattices, entanglement measures, dynamics and transport are described.     

Band Edge Behavior of the Integrated Density of States of Random Jacobi Matrices in Dimension 1

Let H be a Jacobi matrix acting on and V a random potential of Anderson type. Let H = H+V . We give a general formula relating the decay of the integrated density of states of H at the edges of the almost sure spectrum of H to the decay of the integrated density of states of H at the edges of the spectrum of H.     

Asymptotic Behavior of the Integrated Density of States of Acoustic Operators with Random Long Range Perturbations

Journal of Statistical Physics - In this paper we study the behavior of the integrated density of states of random acoustic operators of the form Aω=—?1/?ω?. When...     

Density of States for Random Band Matrices

 By applying the supersymmetric approach we rigorously prove smoothness of the averaged density of states for a three dimensional random band matrix ensemble, in the limit of infinite volume and fixed band width. We also prove that the resulting expression for the density of states coincides with the Wigner semicircle with a precision 1/W ² , for W large but fixed. Received: 6 February 2002 / Accepted: 17 July 2002 Published online: 7 November 2002 RID="*" ID="*" Supported by NSF grant DMS 9729992     

Accuracy of the Time-Dependent Hartree–Fock Approximation for Uncorrelated Initial States

This article concerns the time-dependent Hartree–Fock (TDHF) approximation of single-particle dynamics in systems of interacting fermions. We find that the TDHF approximation is accurate when there are sufficiently many particles and the initial many-particle state is any Gibbs equilibrium state for noninteracting fermions (with Slater determinants as a special example). Assuming a bounded two-particle interaction, we obtain a bound on the error of the TDHF approximation, valid for short times. We further show that the error of the TDHF approximation vanishes at all times in the mean field limit.