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Hierarchical Anderson modelKritchevski, Evgenij. January 2008 (has links)
In this thesis, we study the spectral properties of the hierarchical Anderson model. This model is an approximation of the Anderson tight-binding model on Zd , with the usual discrete Laplacian replaced by a hierarchical long-range interaction operator. In the hierarchical Anderson model, we are given a countable set X endowed with a hierarchical structure. The free hierarchical Laplacian is a self-adjoint operator Delta acting on the Hilbert space l 2( X ). The spectrum of Delta consists of isolated infinitely degenerate eigenvalues. We look at small random perturbations of the operator Delta. The disorder is modeled by a random potential Vo, (Vopsi)(x) = o( x)psi(x) for psi ∈ l 2( X ). The numbers o(x) are identically distributed independent random variables with a bounded density. The hierarchical Anderson model is the random self-adjoint operator Ho = Delta + Vo. We prove the following two results. If the model has a spectral dimension dsp ≤ 4 then, almost surely, the spectrum of Ho is dense pure-point. The second result is on eigenvalue statistics. For dsp < 1, the energy levels for Ho are asymptotically a Poisson point process in the thermodynamic limit, after a proper rescaling.
Hierarchical Anderson modelKritchevski, Evgenij. January 2008 (has links)
No description available.
Algorithms for discrete and continuous quantum systemsMukherjee, Soumyodipto January 2018 (has links)
This thesis is divided into three chapters. In the first chapter we outline a simple and numerically inexpensive approach to describe the spectral features of the single-impurity Anderson model. The method combines aspects of the density matrix embedding theory (DMET) approach with a spectral broadening approach inspired by those used in numerical renormalization group (NRG) methods. At zero temperature for a wide range of U , the spectral function produced by this approach is found to be in good agreement with general expectations as well as more advanced and complex numerical methods such as DMRG-based schemes. The theory developed here is simply transferable to more complex impurity problems The second chapter outlines the density matrix embedding methodology in the context of electronic structure applications. We formulate analytical gradients for energies obtained from DMET focusing on two scenarios: RHF-in-RHF embedding and FCI-in-RHF embedding. The former involves solving the small embedded system at Restricted Hartree-Fock (RHF) level of theory. This serves to check the validity of the formulas by reproducing the RHF results on the full system for energies and gradients. The latter scenario employs full configuration interaction (FCI) as a high level solver for the small embedded system. Our results show that only Hellmann-Feynman terms, which involve derivatives of one- and two-electron terms in the atomic orbital basis, are required to calculate energy gradients in both cases. We applied our methodology to the problem of H 10 ring dissociation where the analytical gradients matched those obtained numerically. The gradient formulation is applicable to geometry optimization of strongly correlated molecules and solids. It can also be used in ab initio molecular dynamics where forces on nuclei are obtained from DMET energy gradients. In the final chapter we focus on the study of finite-temperature equilibrium properties of quantum systems in continuous space. We formulate an expansion of the partition function in continuous-time and use Monte Carlo to sample terms in the resulting infinite series. Such a strategy has been highly successful in quantum lattice models but has found scant application in off-lattice systems. The Monte Carlo estimate of the average energy of quantum particles in continuous space subject to simple model potentials is found to converge with low statistical error to the exact solutions even when very high perturbation order is required. We outline two ways in which the algorithm can be applied to more complex problems. First, by drawing an analogy between the formulation in continuous-time with the discretized, Trotter factorized version of standard path integral Monte Carlo (PIMC). This allows one to use the suite of standard PIMC moves to carry out the position sampling required to obtain the weight of each time configuration. Finally, we propose an alternate route by fitting the many-particle potential with multidimensional Gaussians which provides an analytical form for the position integrals.
Anderson localization in two-channel wires with correlated disorder DNA as an application /Baǧcı, V. M. Kemal. Krokhin, Arkadii, January 2007 (has links)
Thesis (Ph. D.)--University of North Texas, Dec., 2007. / Title from title page display. Includes bibliographical references.
Densidade espectral para o modelo de Anderson de duas impurezas sem correlação eletrônica / Spectral density for the two-impurity Anderson model without electronic correlationSilva, Marcelo Ferreira da 27 March 1998 (has links)
Este trabalho calcula analítica e numericamente a densidade espectral para o modelo de Anderson de duas impurezas sem correlação eletrônica (U=0). Nossos resultados servem como passo inicial para se entender o modelo com a correlação eletrônica. O modelo estudado descreve a interação entre elétrons de um metal e impurezas magnéticas localizadas, e a simplificação, U = 0, torna o Hamiltoniano quadrático permitindo assim que se divida o mesmo em dois termos: um envolvendo apenas operadores pares (canal par) e outro envolvendo apenas operadores ímpares (canal ímpar). Cada termo encontrado difere pouco do Hamiltoniano de Nível Ressonante. Nossos resultados abrangem tanto a diagonalização analítica como a numérica pelo método do Grupo de Renormalização, adaptado para o caso de duas impurezas. A simplicidade do Hamiltoniano permite que (1) se identifique características do modelo que afetam adversamente a precisão do cálculo numeríco e (2) se encontre uma maneira de circundar tais dificuldades. Os resultados aqui encontrados ajudaram o desenvolvimento do cálculo da densidade espectral do modelo correlacionado, desenvolvido paralelamente em nosso grupo de pesquisa. / This work calculates analytically and numerically the spectral density for the two impurity uncorrelated Anderson model (U = O). Our results serve as an initial step towards understanding models with electronic correlation. The studied model describes the interaction between conduction-band electrons of a metal and localized magnetic impurities. The simplification U = O turns the Hamiltonian quadratic, allowing us to split it into two parts: one involving only even operators (even channel), the other involving odd operators (odd channel). Each term has a form differing a little from that for the Resonant Level Hamiltonian. Our results include analytic diagonalization as well as numerical calculations using the method of the Renormalization Group, adapted for the two impurity case. The traditional tridiagonalization method imposes particle-hole symmetry, while our treatment preserves the energy dependence of the coupling, between the impurities and the conduction-band, and consequently, the natural asymmetry of the model. The simplicity of the Hamiltonian allowed us to (1) identify characteristics of the model that affect adversely the acuracy of the numeric calculation and (2) find a way to surround such difficulties. The results here found helped the development of the calculation of the spectral density of the correlated model, developed simultaneously in our research group.
Stochastic processes in random environmentOrtgiese, Marcel January 2009 (has links)
We are interested in two probabilistic models of a process interacting with a random environment. Firstly, we consider the model of directed polymers in random environment. In this case, a polymer, represented as the path of a simple random walk on a lattice, interacts with an environment given by a collection of time-dependent random variables associated to the vertices. Under certain conditions, the system undergoes a phase transition from an entropy-dominated regime at high temperatures, to a localised regime at low temperatures. Our main result shows that at high temperatures, even though a central limit theorem holds, we can identify a set of paths constituting a vanishing fraction of all paths that supports the free energy. We compare the situation to a mean-field model defined on a regular tree, where we can also describe the situation at the critical temperature. Secondly, we consider the parabolic Anderson model, which is the Cauchy problem for the heat equation with a random potential. Our setting is continuous in time and discrete in space, and we focus on time-constant, independent and identically distributed potentials with polynomial tails at infinity. We are concerned with the long-term temporal dynamics of this system. Our main result is that the periods, in which the profile of the solutions remains nearly constant, are increasing linearly over time, a phenomenon known as ageing. We describe this phenomenon in the weak sense, by looking at the asymptotic probability of a change in a given time window, and in the strong sense, by identifying the almost sure upper envelope for the process of the time remaining until the next change of profile. We also prove functional scaling limit theorems for profile and growth rate of the solution of the parabolic Anderson model.
Numerical studies on quantum phase transition of Anderson models. / Numerical studies on quantum phase transition of Anderson models.January 2007 (has links)
Li, Ying Wai = 安德森模型下量子相變的數值研究 / 李盈慧. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (leaves 69-72). / Text in English; abstracts in English and Chinese. / Li, Ying Wai = Andesen mo xing xia liang zi xiang bian de shu zhi yan jiu / Li, Yinghui. / Chapter 1 --- Review on Anderson Models and Quantum Phase Transitions --- p.1 / Chapter 1.1 --- The Anderson Impurity Model --- p.1 / Chapter 1.2 --- The Periodic Anderson Model --- p.2 / Chapter 1.3 --- Quantum Phase Transitions (QPTs) --- p.3 / Chapter 1.4 --- Motivation of this project --- p.4 / Chapter 2 --- Studies on the Ground State Energy of Periodic Anderson Model --- p.7 / Chapter 2.1 --- Background --- p.7 / Chapter 2.2 --- Hamiltonian and Physical Meanings of Lattice Anderson Model --- p.8 / Chapter 2.2.1 --- The first term: -t ´iσ （c+̐ưσci+lσ + h.c.) --- p.8 / Chapter 2.2.2 --- The second term: Ef´iσ̐ưfiσ --- p.9 / Chapter 2.2.3 --- The third term: V ´ ̐ưσ (c+iσ̐ư̐ưσ + h.c.) --- p.9 / Chapter 2.2.4 --- The fourth term: U ̐ưσ´ nfitnfi↓ --- p.9 / Chapter 2.2.5 --- The whole Hamiltonian --- p.10 / Chapter 2.3 --- Non-Interacting Case of Lattice Anderson Model --- p.10 / Chapter 2.3.1 --- The Hamiltonian in momentum space --- p.11 / Chapter 2.3.2 --- The conduction band eK --- p.12 / Chapter 2.3.3 --- The band energies ±K --- p.12 / Chapter 2.3.4 --- The energy band gap Δ --- p.14 / Chapter 2.3.5 --- Green's functions at finite temperature --- p.14 / Chapter 2.4 --- Perturbation in U for symmetric model --- p.16 / Chapter 2.4.1 --- Previous Results --- p.16 / Chapter 2.4.2 --- Ground state energy at finite temperature by time-dependent perturbation theory --- p.18 / Chapter 3 --- Numerical Integration using Wang-Landau Sampling --- p.22 / Chapter 3.1 --- Background --- p.22 / Chapter 3.2 --- Wang-Landau integration --- p.25 / Chapter 3.2.1 --- Description of the method --- p.25 / Chapter 3.2.2 --- Correspondence between Wang-Landau sampling for physical systems and Wang-Landau integration --- p.27 / Chapter 3.3 --- Results --- p.28 / Chapter 3.3.1 --- Application to one- and two-dimensional test integrals . --- p.28 / Chapter 3.3.2 --- An example of a potential application: Perturbative calculation of the lattice Anderson model --- p.31 / Chapter 3.3.3 --- Discussion and summary --- p.35 / Chapter 4 --- Studies on QPT of Anderson Impurity Model by Quantum Entanglement --- p.38 / Chapter 4.1 --- Background --- p.38 / Chapter 4.2 --- Formalism --- p.39 / Chapter 4.2.1 --- Hamiltonian --- p.39 / Chapter 4.2.2 --- Conditions Used in Our Study --- p.40 / Chapter 4.2.3 --- Quantifying Quantum Entanglement: Entropy and Concurrence --- p.41 / Chapter 4.3 --- Numerical Results --- p.45 / Chapter 4.3.1 --- Method --- p.45 / Chapter 4.3.2 --- Finite Size Effects of the Ground State Energy --- p.46 / Chapter 4.3.3 --- Finite Size Effects of the Von Neumann Entropy --- p.49 / Chapter 4.3.4 --- Finite Size Effects of the Fermionic Concurrence --- p.53 / Chapter 4.4 --- Summary --- p.58 / Chapter 5 --- Fidelity in Critical Phenomena --- p.59 / Chapter 5.1 --- Background --- p.59 / Chapter 5.2 --- Ground State Fidelity and Dynamic Structure Factor --- p.60 / Chapter 5.3 --- Mixed-state fidelity and thermal phase transitions --- p.63 / Chapter 5.4 --- Summary --- p.64 / Chapter 6 --- Conclusion --- p.66 / Bibliography --- p.69
Smoothed universal correlations in the two-dimensional Anderson modelUski, V., Mehlig, B., Romer, R. A., Schreiber, M. 30 October 1998 (has links) (PDF)
We report on calculations of smoothed spectral correlations in the twodimensional Anderson model for weak disorder. As pointed out in (M. Wilkinson, J. Phys. A: Math. Gen. 21, 1173 (1988)), an analysis of the smoothing dependence of the correlation functions provides a sensitive means of establishing consistency with random matrix theory. We use a semiclassical approach to describe these fluctuations and offer a detailed comparison between numerical and analytical calculations for an exhaustive set of two-point correlation functions. We consider parametric correlation functions with an external Aharonov-Bohm flux as a parameter and discuss two cases, namely broken time-reversal invariance and partial breaking of time-reversal invariance. Three types of correlation functions are considered: density-of-states, velocity and matrix element correlation functions. For the values of smoothing parameter close to the mean level spacing the semiclassical expressions and the numerical results agree quite well in the whole range of the magnetic flux.
Theory of electron localization in disordered systems /Arnold, Wolfram Till, January 2000 (has links)
Thesis (Ph. D.)--University of Oregon, 2000. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 199-204). Also available for download via the World Wide Web; free to UO users.
Localization properties for the unitary Anderson modelHamza, Eman F. January 2007 (has links) (PDF)
Thesis (Ph. D.)--University of Alabama at Birmingham, 2007. / Description based on contents viewed Feb. 12, 2009; title from PDF t.p. Includes bibliographical references (p. 75-77).
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