Computing Masses from Effective Transfer Matrices

The density matrix renormalization group (DMRG) has been extended to study quantum phase transitions on random graphs of fixed connectivity. As a relevant example, we have analysed the random Ising model in a transverse field. If the couplings are random, the number of retained states remains reasonably low even for large sizes. The resulting quantum spin-glass transition has been traced down for a few disorder realizations, through the careful measurement of selected observa...

This thesis gives an extension for the Density Matrix Renormalisation Group (DMRG) to two dimensions and described a newly developed combination of the DMRG and a Green Function Monte Carlo simulation (GFMC). The first two chapters focus on the DMRG method. The properties are reviewed and it is shown that good quality results can be obtained for two-dimensional systems through finite-size scaling. In the third and fourth chapter the properties of the two-dimensional frust...

The overlap length of a three-dimensional Ising spin glass on a cubic lattice with Gaussian interactions has been estimated numerically by transfer matrix methods and within a Migdal-Kadanoff renormalization group scheme. We find that the overlap length is large, explaining why it has been difficult to observe spin glass chaos in numerical simulations and experiment.

We propose a new method for the calculation of thermodynamic properties of one-dimensional quantum systems by combining the TMRG approach with the corner transfer-matrix method. The corner transfer-matrix DMRG method brings reasonable advantage over TMRG for classical systems. We have modified the concept for the calculation of thermal properties of one-dimensional quantum systems. The novel QCTMRG algorithm is implemented and used to study two simple test cases, the classica...

A generalization of matrix product states (MPS) is introduced which is suitable for describing interacting quantum systems in two and three dimensions. These scale-renormalized matrix-product states (SR-MPS) are based on a course-graining of the lattice in which the blocks at each level are associated with matrix products that are further transformed (scale renormalized) with other matrices before they are assembled to form blocks at the next level. Using variational Monte Ca...

The Ising spin glass model in a transverse field has a zero temperature phase transition driven solely by quantum fluctuations. This quantum phase transition occuring at a critical transverse field strength has attracted much attention recently. We report the progress that has been made via Monte Carlo simulations of the finite dimensional, short range model.

We present Tethered Monte Carlo, a simple, general purpose method of computing the effective potential of the order parameter (Helmholtz free energy). This formalism is based on a new statistical ensemble, closely related to the micromagnetic one, but with an extended configuration space (through Creutz-like demons). Canonical averages for arbitrary values of the external magnetic field are computed without additional simulations. The method is put to work in the two dimensio...

We propose a numerical method of estimating various physical quantities in lattice (supersymmetric) quantum mechanics. The method consists only of deterministic processes such as computing a product of transfer matrix, and has no statistical uncertainties. We use the numerical quadrature to define the transfer matrix as a finite dimensional matrix, and find that it effectively works by rescaling variable for sufficiently small lattice spacings. For a lattice supersymmetric qu...

Different phenomenological RG transformations based on scaling relations for the derivatives of the inverse correlation length and singular part of the free-energy density are considered. These transformations are tested on the 2D square Ising and Potts models as well as on the 3D simple-cubic Ising model. Variants of RG equations yielding more accurate results than Nightingale's RG scheme are obtained. In the 2D case the finite-size equations which give the {\it exact} value...

We study the spectrum of two dimensional coupled arrays of continuum one-dimensional systems by wedding a density matrix renormalization group procedure to a renormalization group improved truncated spectrum approach. To illustrate the approach we study the spectrum of large arrays of coupled quantum Ising chains. We demonstrate explicitly that the method can treat the various regimes of chains, in particular the three dimensional Ising ordering transition the chains undergo ...