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A Density-Matrix Renormalization Group Study of Quantum Spin Models with Ring Exchange

Alexander H. L. Chan

Published Jul 4, 2012• 0 citations
Publisher

Abstract

In this thesis we discuss in detail the density-matrix renormalization group (DMRG) for simulating low-energy properties of quantum spin models. We implement an original DMRG routine on the S=1/2 antiferromagnetic Heisenberg chain and benchmark its efficiency against exact results (energies, correlation functions, etc.) as well as conformal field-theoretical calculations due to finite-size scaling (ground-state energy and spin gap logarithmic corrections). Moreover, we apply the DMRG to a two-leg square ladder system, where in addition to bilinear exchange terms, we also consider an additional cyclic four-spin ring-exchange. The transposition of four spins gives rise to biquadratic exchange terms which are non-trivial to implement in the DMRG. Intermediate results of the ring-exchange are presented along with the difficulties presently encountered.

DOI
10.71548/14411

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In this thesis we discuss in detail the density-matrix renormalization group (DMRG) for simulating low-energy properties of quantum spin models. We implement an original DMRG routine on the S=1/2 antiferromagnetic Heisenberg chain and benchmark its efficiency against exact results (energies, correlation functions, etc.) as well as conformal field-theoretical calculations due to finite-size scaling (ground-state energy and spin gap logarithmic corrections). Moreover, we apply the DMRG to a two-leg square ladder system, where in addition to bilinear exchange terms, we also consider an additional cyclic four-spin ring-exchange. The transposition of four spins gives rise to biquadratic exchange terms which are non-trivial to implement in the DMRG. Intermediate results of the ring-exchange are presented along with the difficulties presently encountered.
The effect of a ring exchange on doped two-leg ladders is investigated combining exact diagonalization (ED) and density matrix renormalization group (DMRG) computations. We focus on the nature and weights of the low-energy magnetic excitations and on superconducting pairing. The stability with respect to this cyclic term of a remarkable resonant mode originating from a hole pair-magnon bound state is examined. We also find that, near the zero-doping critical point separating rung-singlet and dimerized phases, doping reopens a spin gap.
A variant of White's density-matrix renormalization-group scheme, which is designed to compute low-lying energies of one-dimensional quantum lattice models with a large number of degrees of freedom per site is described. The method is tested on two exactly solvable models---the spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg chain and a dimerized XY\mathrm{XY}XY spin chain. To illustrate the potential of the method, it is applied to a model of spins interacting with quantum phonons. It is shown that the method accurately resolves a number of energy gaps on periodic rings that are sufficiently large to afford an accurate investigation of critical properties via the use of finite-size scaling theory.
We report a density-matrix renormalization-group (DMRG) study of S=1/2 antiferromagnetic Heisenberg spin chains. We show the Friedel oscillations in open spin chains within bosonization theory. The bosonization predictions are in excellent agreement with the DMRG calculation. The chain length dependence of the ground state and several low-energy excited state energies are compared with conformal field theory results. Based upon the DMRG calculations for various quantities, the interplay between the system size and the number of optimal states kept in the DMRG is carefully discussed. \textcopyright{} 1996 The American Physical Society.
The density matrix renormalization group (DMRG) method generates the low-energy states of linear systems of NNN sites with a few degrees of freedom at each site by starting with a small system and adding sites step by step while keeping constant the dimension of the truncated Hilbert space. DMRG algorithms are adapted to open chains with inversion symmetry at the central site, to cyclic chains and to weakly coupled chains. Physical properties rather than energy accuracy is the motivation. The algorithms are applied to the edge states of linear Heisenberg antiferromagnets with spin S≥1S \ge 1S≥1 and to the quantum phases of a frustrated spin-1/2 chain with exchange between first and second neighbors. The algorithms are found to be accurate for extended Hubbard and related 1D models with charge and spin degrees of freedom.
A density matrix renormalization group (DMRG) algorithm is presented for the Bethe lattice with connectivity Z=3Z=3Z=3 and antiferromagnetic exchange between nearest-neighbor spins s=1/2s=1/2s=1/2 or 1 sites in successive generations ggg. The algorithm is accurate for s=1s=1s=1 sites. The ground states are magnetic with spin S(g)=2gsS(g)={2}^{g}sS(g)=2gs, staggered magnetization that persists for large g>20g>20g>20, and short-range spin correlation functions that decrease exponentially. A finite energy gap to S>S(g)S>S(g)S>S(g) leads to a magnetization plateau in the extended lattice. Closely similar DMRG results for sss === 1/2 and 1 are interpreted in terms of an analytical three-site model.
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.