arXiv subjects

N. S. Witte

Publications and source records attributed to N. S. Witte.

43 records · Page 3Linked to original sources

arXiv1999

Moment Formalisms applied to a solvable Model with a Quantum Phase Transition. I. Exponential Moment Methods

We examine the Ising chain in a transverse field at zero temperature from the point of view of a family of moment formalisms based upon the cumulant generating function, where we find exact solutions for the generating functions and cumulants at arbitrary couplings and hence for both the ordered and disordered phases of the model. In a t-expansion analysis, the exact Horn-Weinstein function E(t) has cuts along an infinite set of curves in the complex Jt-plane which are confined to the left-hand half-plane Im Jt < -1/4 for the phase containing the trial state (disordered), but are not so for the other phase (ordered). For finite couplings the expansion has a finite radius of convergence. Asymptotic forms for this function exhibit a crossover at the critical point, giving the excited state gap in the ground state sector for the disordered phase, and the first excited state gap in the ordered phase. Convergence of the t-expansion with respect to truncation order is found in the disordered phase right up to the critical point, for both the ground state energy and the excited state gap. However convergence is found in only one of the Connected Moments Expansions (CMX), the CMX-LT, and the ground state energy shows convergence right to the critical point again, although to a limited accuracy.

cond-mat.stat-mech↗

arXiv1999

The Lanczos Algorithm for extensive Many-Body Systems in the Thermodynamic Limit

We establish rigourously the scaling properties of the Lanczos process applied to an arbitrary extensive Many-Body System which is carried to convergence n to infinity and the thermodynamic limit N to infinity taken. In this limit the solution for the limiting Lanczos coefficients are found exactly and generally through two equivalent sets of equations, given initial knowledge of the exact cumulant generating function. The measure and the Orthogonal Polynomial System associated with the Lanczos process in this regime are also given explicitly. Some important representations of these Lanczos functions are given, including Taylor series expansions, and theorems controlling their general properties are proven.

math-ph↗

arXiv1997

The Analytic Lanczos Method

The classical formalism of the Moment Problem has been combined with a cumulant approach and applied to the extensive many-body problem. This has yielded many new exact results for many-body systems in the thermodynamic limit - for the ground state energy, for excited state gaps, and for arbitrary ground state averages. The method applies to any extensive Hamiltonian system, for any phase or symmetry arising in the model, whether on a lattice or in the continuum, and for any dimensionality. The theorems are of a nonperturbative nature with respect to any couplings occuring in the model.

cond-mat.stat-mech↗

arXiv1997

The exact realisation of the Lanczos Method for a quantum Many-Body System

The Lanczos process has been analytically and exactly carried out for the spin 1/2 isotropic XY chain in the thermodynamic limit, yielding a form for the Lanczos coefficient $β^2(s)$. This coefficient has a monotonic variation for real positive $s$ and confirms a general theorem on the ground state properties of extensive Many-body Systems. The Taylor expansion of the coefficient about s=0 has a finite radius of convergence, and ground state estimates based on a finite truncation of this are shown to be asymptotic.

cond-mat.stat-mech↗

arXiv1997

2D XXZ Model ground state Properties using an analytic Lanczos Expansion

We develop the formalism for calculating arbitrary expectation values for any extensive lattice Hamiltonian system using a new analytic Lanczos expansion, or plaquette expansion, and a recently proved exact theorem for ground state energies. The ground state energy, staggered magnetisation and the excited state gap of the 2D anisotropic antiferromagnetic Heisenberg Model are then calculated using this expansion for a range of anisotropy parameters and compared to other moment based techniques, such as the "t"-expansion, and spin-wave theory and series expansion methods. We find that far from the isotropic point all moment methods give essentially very similar results, but near the isotropic point the plaquette expansion is generally better than the others.

cond-mat.stat-mech↗

arXiv1997

Exact Solution to the Moment Problem for the XY Chain

We present the exact solution to the moment problem for the spin-1/2 isotropic antiferromagnetic XY chain with explicit forms for the moments with respect to the Neel state, the cumulant generating function, and the Resolvent Operator. We verify the correctness of the Horn-Weinstein Theorems, but the analytic structure of the generating function in the complex t-plane is quite different from that assumed by the "t"-Expansion and the Connected Moments Expansion due to the vanishing gap. This function has a finite radius of convergence about t=0, and for large t has a leading descending algebraic series E(t)-E_0 ~ At^{-2}. The Resolvent has a branch cut and essential singularity near the ground state energy of the form G(s)/s ~ B|s+1|^{-3/4} exp(C|s+1|^{1/2}). Consequently extrapolation strategies based on these assumptions are flawed and in practise we find that the CMX methods are pathological and cannot be applied, while numerical evidence for two of the "t"-expansion methods indicates a clear asymptotic convergence behaviour with truncation order.

cond-mat.stat-mech↗

arXiv1996

Analytic Solution for the Ground State Energy of the Extensive Many-Body Problem

A closed form expression for the ground state energy density of the general extensive many-body problem is given in terms of the Lanczos tri-diagonal form of the Hamiltonian. Given the general expressions of the diagonal and off-diagonal elements of the Hamiltonian Lanczos matrix, $α_n(N)$ and $β_n(N)$, asymptotic forms $α(z)$ and $β(z)$ can be defined in terms of a new parameter $z\equiv n/N$ ($n$ is the Lanczos iteration and $N$ is the size of the system). By application of theorems on the zeros of orthogonal polynomials we find the ground-state energy density in the bulk limit to be given in general by ${\cal E}_0 = {\rm inf}\,\left[α(z) - 2\,β(z)\right]$.

cond-mat↗