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Density Matrix Renormalization Group calculations for the Ising Model with a Transverse Field

Arnold H. Kole

Published Jan 1, 2018• 1 citation
OpenAlex

Abstract

In this thesis, the Density Matrix Renormalization Group algorithm is introduced. Some key concepts from quantum mechanics are discussed to aid in understanding the method and it is put into the context of the Renormalization Group. The algorithm is discussed from a mostly practical perspective. As a model to test the algorithm, the Ising Model with a Transverse Field was used. This model is interesting because it is one of the simplest one-dimensional models that displays a phase transition at a finite external field. It can also be solved analytically, which makes it possible to easily check the results of DMRG calculations. The DMRG algorithm was implemented from scratch in C++ for the Transverse Field Ising Model. Using this implementation, the errors in DMRG calculations were explored, as well as the phase transition in the Transverse Field Ising Chain.

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•Aug 26, 2013•0 citations
In this thesis, the Density Matrix Renormalization Group algorithm is introduced. Some key concepts from quantum mechanics are discussed to aid in understanding the method and it is put into the context of the Renormalization Group. The algorithm is discussed from a mostly practical perspective. As a model to test the algorithm, the Ising Model with a Transverse Field was used. This model is interesting because it is one of the simplest one-dimensional models that displays a phase transition at a finite external field. It can also be solved analytically, which makes it possible to easily check the results of DMRG calculations. The DMRG algorithm was implemented from scratch in C++ for the Transverse Field Ising Model. Using this implementation, the errors in DMRG calculations were explored, as well as the phase transition in the Transverse Field Ising Chain.
White's density-matrix renormalization-group (DMRG) method has been applied to the one-dimensional Ising model in a transverse field (ITF), in order to study the accuracy of the numerical algorithm. Due to the exact solubility of the ITF for any finite chain length, the errors introduced by the basis truncation procedure could have been directly analyzed. By computing different properties, like the energies of the low-lying levels or the ground-state one- and two-point correlation functions, we obtained a detailed picture of how these errors behave as functions of the various model and algorithm parameters. Our experience with the ITF contributes to a better understanding of the DMRG method, and may facilitate its optimization in other applications.
The density matrix renormalization group (DMRG) is a numerical method for studying low dimensional strongly correlated models. The study of these models is one of the most active areas of condensed matter physics today and has been spurred not only by the discovery of high temperature superconductivity but also by experiments on new compounds. In this thesis we study fundamentals of the density matrix renormalization group method as well as applications of the method to Heisenberg spin chains. The introductory text is meant as a brief overview of the DMRG as well as Heisenberg spin systems and provides an introduction to the papers. In papers I and II we investigate the density matrix renormalization group with the aim to better understand the underlying structure of the method. We find that if the renormalization converges to a fixed point in the thermodynamic limit, the resulting basis states can be written as a matrix product form. We also show an explicit relation of the DMRG to variational calculations using the fact that the matrix product states can be rederived through a variational ansatz without any reference to the DMRG calculation. It is then shown how the matrix product states can be used to define a set of Bloch waves for describing elementary excitations of the Heisenberg spin-1 chain. The low lying energy spectrum is calculated. In papers III and IV we investigate the thermodynamics of impurities in a spin-1/2 chain. For this purpose we have shown that a recently developed transfer matrix DMRG method can be used to calculate impurity properties. Both impurity corrections to thermodynamic properties and local properties at finite temperature can be obtained. The method gives quicker and more accurate results than quantum Monte Carlo. The numerical results show good agreement with field theory calculations for a number of impurity configurations and the cross-over function for the magnetic susceptibility from high- to low-temperature behavior shows the expected data collapse. Our results also give quantitative predictions for experiments on doped spin-1/2 chains.
We present two new analytic formulations of the density matrix renormalization group (DMRG) method. In these formulations we combine the block renormalization group (BRG) procedure with the variational and Fokker-Planck methods. The BRG method is used to reduce the lattice size while the latter are used to construct approximate target states to compute the block density matrix. We apply our DMRG methods to the Ising model in a transverse field (ITF model) and compute several of its critical properties, which are then compared with the old BRG results.
The spin-1/21/21/2 quantum Ising chain in a transverse random magnetic field is studied by means of the density-matrix renormalization group. The system evolves from an ordered to a paramagnetic state as the amplitude of the random field is increased. The dependence of the magnetization on a uniform magnetic field in the zzz direction and the spontaneous magnetization as a function of the amplitude of the transverse random magnetic field are determined. The behavior of the spin-spin correlation function both above and at criticality is studied. The scaling laws for magnetization and correlation functions are tested against previous numerical and renormalization-group results.
We have adjusted the density-matrix renormalization method to handle two-dimensional systems of limited width. The key ingredient for this extension is the incorporation of symmetries in the method. The advantage of our approach is that we can force certain symmetry properties to the resulting ground-state wave function. Combining the results obtained for system sizes up to $30\ifmmode\times\else\texttimes\fi{}6$ and finite-size scaling, we derive the phase-transition point and the critical exponent for the gap in the Ising model in a transverse field on a two-dimensional square lattice.
There has been much recent interest in the study of out of equilibrium quantum systems due to experimental advances in trapped cold atomic gases that allow systematic studies of these effects.In this work we study a quantum Ising model in the presence of transverse and longitudinal magnetic fields.The model lacks an analytical solution in the most general case and has to be treated numerically.We first investigate the static properties of the model, including the ground state energy and the zero temperature phase diagram, using the Density Matrix Renormalization Group (DMRG) method.We identify ordered and disordered phases separated by a quantum phase transition.We then consider the time evolution of the system following an abrupt quench of the fields.We study the response of the order parameter to the quenches within the ordered and disordered phases as well as quenches between the phases using the time-dependent DMRG method.The results are in qualitative agreement with experiments and the implemented numerical method in this work can be extended to allow for more general interactions.iii I would like to thank my supervisor, Dr. Malcolm Kennett, whose constant support and hope in the most disappointing situations, his helpful guidance, and endless energy were all new and very invaluable to me.I like to thank my