Density functional theory in one-dimension for contact-interacting fermions

A density functional theory is developed for fermions in one dimension, interacting via a delta-function. Such systems provide a natural testing ground for questions of principle, as the local density approximation should work well for short-ranged interactions. The exact-exchange contribution to the total energy is a local functional of the density. A local density approximation for correlation is obtained using perturbation theory and Bethe-Ansatz results for the one-dimensional contact-interacting uniform Fermi gas. The ground-state energies are calculated for two finite systems, the analogs of Helium and of Hooke's atom. The local approximation is shown to be excellent, as expected.Comment: 10 pages, 7 Figure

Modified conjugated gradient method for diagonalising large matrices

We present an iterative method to diagonalise large matrices. The basic idea is the same as the conjugated gradient (CG) method, i.e, minimizing the Rayleigh quotient via its gradient and avoiding reintroduce errors to the directions of previous gradients. Each iteration step is to find lowest eigenvector of the matrix in a subspace spanned by the current trial vector and the corresponding gradient of the Rayleigh quotient, as well as some previous trial vectors. The gradient, together with the previous trail vectors, play a similar role of the conjugated gradient of the original CG algorithm. Our numeric tests indicate that this method converges significantly faster than the original CG method. And the computational cost of one iteration step is about the same as the original CG method. It is suitably for first principle calculations.Comment: 6 Pages, 2EPS figures. (To appear in Phys. Rev. E

Operator-Based Truncation Scheme Based on the Many-Body Fermion Density Matrix

In [S. A. Cheong and C. L. Henley, cond-mat/0206196 (2002)], we found that the many-particle eigenvalues and eigenstates of the many-body density matrix $\rho_B$ of a block of $B$ sites cut out from an infinite chain of noninteracting spinless fermions can all be constructed out of the one-particle eigenvalues and one-particle eigenstates respectively. In this paper we developed a statistical-mechanical analogy between the density matrix eigenstates and the many-body states of a system of noninteracting fermions. Each density matrix eigenstate corresponds to a particular set of occupation of single-particle pseudo-energy levels, and the density matrix eigenstate with the largest weight, having the structure of a Fermi sea ground state, unambiguously defines a pseudo-Fermi level. We then outlined the main ideas behind an operator-based truncation of the density matrix eigenstates, where single-particle pseudo-energy levels far away from the pseudo-Fermi level are removed as degrees of freedom. We report numerical evidence for scaling behaviours in the single-particle pseudo-energy spectrum for different block sizes $B$ and different filling fractions \nbar. With the aid of these scaling relations, which tells us that the block size $B$ plays the role of an inverse temperature in the statistical-mechanical description of the density matrix eigenstates and eigenvalues, we looked into the performance of our operator-based truncation scheme in minimizing the discarded density matrix weight and the error in calculating the dispersion relation for elementary excitations. This performance was compared against that of the traditional density matrix-based truncation scheme, as well as against a operator-based plane wave truncation scheme, and found to be very satisfactory.Comment: 22 pages in RevTeX4 format, 22 figures. Uses amsmath, amssymb, graphicx and mathrsfs package

Decoupling of the S=1/2 antiferromagnetic zig-zag ladder with anisotropy

The spin-1/2 antiferromagnetic zig-zag ladder is studied by exact diagonalization of small systems in the regime of weak inter-chain coupling. A gapless phase with quasi long-range spiral correlations has been predicted to occur in this regime if easy-plane (XY) anisotropy is present. We find in general that the finite zig-zag ladder shows three phases: a gapless collinear phase, a dimer phase and a spiral phase. We study the level crossings of the spectrum,the dimer correlation function, the structure factor and the spin stiffness within these phases, as well as at the transition points. As the inter-chain coupling decreases we observe a transition in the anisotropic XY case from a phase with a gap to a gapless phase that is best described by two decoupled antiferromagnetic chains. The isotropic and the anisotropic XY cases are found to be qualitatively the same, however, in the regime of weak inter-chain coupling for the small systems studied here. We attribute this to a finite-size effect in the isotropic zig-zag case that results from exponentially diverging antiferromagnetic correlations in the weak-coupling limit.Comment: to appear in Physical Review

Computational Nuclear Physics and Post Hartree-Fock Methods

We present a computational approach to infinite nuclear matter employing Hartree-Fock theory, many-body perturbation theory and coupled cluster theory. These lectures are closely linked with those of chapters 9, 10 and 11 and serve as input for the correlation functions employed in Monte Carlo calculations in chapter 9, the in-medium similarity renormalization group theory of dense fermionic systems of chapter 10 and the Green's function approach in chapter 11. We provide extensive code examples and benchmark calculations, allowing thereby an eventual reader to start writing her/his own codes. We start with an object-oriented serial code and end with discussions on strategies for porting the code to present and planned high-performance computing facilities.Comment: 82 pages, to appear in Lecture Notes in Physics (Springer), "An advanced course in computational nuclear physics: Bridging the scales from quarks to neutron stars", M. Hjorth-Jensen, M. P. Lombardo, U. van Kolck, Editor

Symmetry of the Atomic Electron Density in Hartree, Hartree-Fock, and Density Functional Theory

The density of an atom in a state of well-defined angular momentum has a specific finite spherical harmonic content, without and with interactions. Approximate single-particle schemes, such as the Hartree, Hartree-Fock, and Local Density Approximations, generally violate this feature. We analyze, by means of perturbation theory, the degree of this violation and show that it is small. The correct symmetry of the density can be assured by a constrained-search formulation without significantly altering the calculated energies. We compare our procedure to the (different) common practice of spherically averaging the self-consistent potential. Kohn-Sham density functional theory with the exact exchange-correlation potential has the correct finite spherical harmonic content in its density; but the corresponding exact single particle potential and wavefunctions contain an infinite number of spherical harmonics.Comment: 11 pages, 6 figures. Expanded discussion of spherical harmonic expansion of Hartree density. Some typos corrected, references adde

Exchange Interaction in Binuclear Complexes with Rare Earth and Copper Ions: A Many-Body Model Study

We have used a many-body model Hamiltonian to study the nature of the magnetic ground state of hetero-binuclear complexes involving rare-earth and copper ions. We have taken into account all diagonal repulsions involving the rare-earth 4f and 5d orbitals and the copper 3d orbital. Besides, we have included direct exchange interaction, crystal field splitting of the rare-earth atomic levels and spin-orbit interaction in the 4f orbitals. We have identified the inter-orbital $4f$ repulsion, U$_{ff}$ and crystal field parameter, $\Delta_f$ as the key parameters involved in controlling the type of exchange interaction between the rare earth $4f$ and copper 3d spins. We have explored the nature of the ground state in the parameter space of U$_{ff}$, $\Delta_f$, spin-orbit interaction strength $\lambda$ and the $4f$ filling n$_f$. We find that these systems show low-spin or high-spin ground state depending on the filling of the $4f$ levels of the rare-earth ion and ground state spin is critically dependent on U$_{ff}$ and $\Delta_f$. In case of half-filling (Gd(III)) we find a reentrant low-spin state as U$_{ff}$ is increased, for small values of $\Delta_f$, which explains the recently reported apparent anomalous anti-ferromagnetic behaviour of Gd(III)-radical complexes. By varying U$_{ff}$ we also observe a switch over in the ground state spin for other fillings . We have introduced a spin-orbit coupling scheme which goes beyond L-S or j-j coupling scheme and we find that spin-orbit coupling does not significantly alter the basic picture.Comment: 22 pages, 11 ps figure

Correlation effects in ionic crystals: I. The cohesive energy of MgO

High-level quantum-chemical calculations, using the coupled-cluster approach and extended one-particle basis sets, have been performed for (Mg2+)n (O2-)m clusters embedded in a Madelung potential. The results of these calculations are used for setting up an incremental expansion for the correlation energy of bulk MgO. This way, 96% of the experimental cohesive energy of the MgO crystal is recovered. It is shown that only 60% of the correlation contribution to the cohesive energy is of intra-ionic origin, the remaining part being caused by van der Waals-like inter-ionic excitations.Comment: LaTeX, 20 pages, no figure

From Coherent Modes to Turbulence and Granulation of Trapped Gases

The process of exciting the gas of trapped bosons from an equilibrium initial state to strongly nonequilibrium states is described as a procedure of symmetry restoration caused by external perturbations. Initially, the trapped gas is cooled down to such low temperatures, when practically all atoms are in Bose-Einstein condensed state, which implies the broken global gauge symmetry. Excitations are realized either by imposing external alternating fields, modulating the trapping potential and shaking the cloud of trapped atoms, or it can be done by varying atomic interactions by means of Feshbach resonance techniques. Gradually increasing the amount of energy pumped into the system, which is realized either by strengthening the modulation amplitude or by increasing the excitation time, produces a series of nonequilibrium states, with the growing fraction of atoms for which the gauge symmetry is restored. In this way, the initial equilibrium system, with the broken gauge symmetry and all atoms condensed, can be excited to the state, where all atoms are in the normal state, with completely restored gauge symmetry. In this process, the system, starting from the regular superfluid state, passes through the states of vortex superfluid, turbulent superfluid, heterophase granular fluid, to the state of normal chaotic fluid in turbulent regime. Both theoretical and experimental studies are presented.Comment: Latex file, 25 pages, 4 figure

Atomic X-ray Spectroscopy of Accreting Black Holes

Current astrophysical research suggests that the most persistently luminous objects in the Universe are powered by the flow of matter through accretion disks onto black holes. Accretion disk systems are observed to emit copious radiation across the electromagnetic spectrum, each energy band providing access to rather distinct regimes of physical conditions and geometric scale. X-ray emission probes the innermost regions of the accretion disk, where relativistic effects prevail. While this has been known for decades, it also has been acknowledged that inferring physical conditions in the relativistic regime from the behavior of the X-ray continuum is problematic and not satisfactorily constraining. With the discovery in the 1990s of iron X-ray lines bearing signatures of relativistic distortion came the hope that such emission would more firmly constrain models of disk accretion near black holes, as well as provide observational criteria by which to test general relativity in the strong field limit. Here we provide an introduction to this phenomenon. While the presentation is intended to be primarily tutorial in nature, we aim also to acquaint the reader with trends in current research. To achieve these ends, we present the basic applications of general relativity that pertain to X-ray spectroscopic observations of black hole accretion disk systems, focusing on the Schwarzschild and Kerr solutions to the Einstein field equations. To this we add treatments of the fundamental concepts associated with the theoretical and modeling aspects of accretion disks, as well as relevant topics from observational and theoretical X-ray spectroscopy.Comment: 63 pages, 21 figures, Einstein Centennial Review Article, Canadian Journal of Physics, in pres