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DMRG och dess tillämpning på den endimensionella (1D) Heisenberg modellen

Jimmy Johansson

Published Jan 1, 2026• 0 citations
OpenAlex

Abstract

In this thesis, we analyze how the DMRG method works and use it on the one-dimensional (1D) antiferromagnetic Heisenberg Chain with open boundaries. We explicitly demonstrate how DMRG works for both the spin 1/2 and the spin 1 model and write programs for them in Python. A detailed process of the 1D spin 1/2 Heisenberg chain for four sites is shown to get exact values for its ground state energy values and to show how each step of the DMRG process works. We then compare the DMRG method to the analytically exact values and get very precise results. The DMRG method uses what is called the density matrix, formed by the eigenvector of the system we want to approximate. The exact ground state eigenvalue is obtained for N=14 sites and comparing that value with the DMRG method with only the 40 (D=40) most important states kept of the density matrix we get a difference from the exact result by about 3.5*10^(-13) after using four sweeps with the finite DMRG method, calculated in a small fraction of the time. We calculate the ground state eigenvalues for bigger systems up to N=50 and see the ground state energy per site E_0/N approach the Bethe ansatz result E_0/N=1/4-ln(2) in the thermodynamic limit (when N goes to infinity). The same approach is used for the spin 1 chain and we write a program, computing ground state eigenvalues ranges of N up to 50 and here in the thermodynamic limit E_0/N approaches approximately -1.401485(2). Finally we use DMRG to target the lowest five eigenstates of both the spin 1/2 and spin 1 systems to visualize the energy gap from the ground state to the first excited energy level in the thermodynamic limit. We get the energy gap approaching 0 for the spin 1/2 model and approaching the Haldane gap delta E =0.4105(1) for the spin 1 chain with open boundaries with a fourfold degeneracy of the ground state.

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In this thesis, we analyze how the DMRG method works and use it on the one-dimensional (1D) antiferromagnetic Heisenberg Chain with open boundaries. We explicitly demonstrate how DMRG works for both the spin 1/2 and the spin 1 model and write programs for them in Python. A detailed process of the 1D spin 1/2 Heisenberg chain for four sites is shown to get exact values for its ground state energy values and to show how each step of the DMRG process works. We then compare the DMRG method to the analytically exact values and get very precise results. The DMRG method uses what is called the density matrix, formed by the eigenvector of the system we want to approximate. The exact ground state eigenvalue is obtained for N=14 sites and comparing that value with the DMRG method with only the 40 (D=40) most important states kept of the density matrix we get a difference from the exact result by about 3.5*10^(-13) after using four sweeps with the finite DMRG method, calculated in a small fraction of the time. We calculate the ground state eigenvalues for bigger systems up to N=50 and see the ground state energy per site E_0/N approach the Bethe ansatz result E_0/N=1/4-ln(2) in the thermodynamic limit (when N goes to infinity). The same approach is used for the spin 1 chain and we write a program, computing ground state eigenvalues ranges of N up to 50 and here in the thermodynamic limit E_0/N approaches approximately -1.401485(2). Finally we use DMRG to target the lowest five eigenstates of both the spin 1/2 and spin 1 systems to visualize the energy gap from the ground state to the first excited energy level in the thermodynamic limit. We get the energy gap approaching 0 for the spin 1/2 model and approaching the Haldane gap delta E =0.4105(1) for the spin 1 chain with open boundaries with a fourfold degeneracy of the ground state.
We discuss the ground state of the Heisenberg spin-1/2 chain in one dimension, considering nearest neighbor interactions. In the thermodynamic limit the energy is calculated analytically in several ways. We start by applying the Bethe ansatz to calculate the exact ground state energy of the XXX Heisenberg model. Next we study small perturbations of the spin anisotropy, the parameter for the interaction in the z-direction for the XXZ model, by applying time-independent perturbation theory. In addition we expand an exact integral expression of the energy of the XXZ model around the spin anisotropy in the thermodynamic limit. The results are compared to numerical solutions, which are obtained by means of exact diagonalization up to twelve lattice sites. We find that for ten or more lattice sites the numerics agree well with the expectation from the analytical calculations. Finally the concept of integrability is discussed in the context of the Heisenberg model.
Linear Heisenberg antiferromagnets (HAFs) are chains of spin-SSS sites with isotropic exchange JJJ between neighbors. Open and periodic boundary conditions return the same ground-state energy per site in the thermodynamic limit, but not the same spin SG{S}_{G}SG​ when $S\ensuremath{\ge}1$. The ground state of open chains of NNN spins has SG=0{S}_{G}=0SG​=0 or SSS, respectively, for even or odd NNN. Density-matrix renormalization-group calculations with different algorithms for even and odd NNN are presented up to N=500N=500N=500 for the energy and spin densities $\ensuremath{\rho}(r,N)$ of edge states in HAFs with S=1S=1S=1, 3/2, and 2. The edge states are boundary-induced spin density waves (BI-SDWs) with $\ensuremath{\rho}(r,N)\ensuremath{\propto}{(\ensuremath{-}1)}^{r\ensuremath{-}1}$ for r=1,2,...,Nr=1,2,...,Nr=1,2,...,N. The SDWs are in phase when NNN is odd, are out of phase when NNN is even, and have finite excitation energy $\mathrm{\ensuremath{\Gamma}}(N)$ that decreases exponentially with NNN for integer SSS and faster than 1/N1/N1/N for half integer SSS. The spin densities and excitation energy are quantitatively modeled for integer SSS chains longer than $5\ensuremath{\xi}$ spins by two parameters, the correlation length $\ensuremath{\xi}$ and the SDW amplitude, with $\ensuremath{\xi}=6.048$ for S=1S=1S=1 and 49.0 for S=2S=2S=2. The BI-SDWs of S=3/2S=3/2S=3/2 chains are not localized and are qualitatively different for even and odd NNN. Exchange between the ends for odd NNN is mediated by a delocalized effective spin in the middle that increases $|\mathrm{\ensuremath{\Gamma}}(N)|$ and weakens the size dependence. The nonlinear sigma model ($\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$) has been applied to the HAFs, primarily to S=1S=1S=1 with even NNN, to discuss spin densities and exchange between localized states at the ends as $\mathrm{\ensuremath{\Gamma}}(N)\ensuremath{\propto}{(\ensuremath{-}1)}^{N}exp(\ensuremath{-}N/\ensuremath{\xi})$. S=1S=1S=1 chains with odd NNN are fully consistent with the $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$; S=2S=2S=2 chains have two gaps $\mathrm{\ensuremath{\Gamma}}(N)$ with the same $\ensuremath{\xi}$ as predicted whose ratio is 3.45 rather than 3; the $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$ is more approximate for S=3/2S=3/2S=3/2 chains with even NNN and is modified for exchange between ends for odd NNN.
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.
The one-dimensional Heisenberg antiferromagnet of S = 1 spins is studied by numericaldiagonalization method based on the Lanczos algorithm. The Haldane gap of this system isnumerically estimated from finite-size clusters under the twisted boundary condition up tothe 30-site system. The finite-size results are analyzed by a convergence-acceleration technique; the multistep procedure of the convergence acceleration is successfully carried out.The present estimate for the gap magnitude is more precisely than the previously reportedvalue from a Lanczos-diagonalization study. It also agrees well with the result reported ina study based on the density matrix renormalization group calculation. The present studyclarifies that Lanczos-diagonalization calculations can sufficiently contribute for preciselyestimating an energy gap although this method can treat only small clusters.
We develop an efficient method to perform density matrix renormalization group simulations of the SU(NNN) Heisenberg chain with open boundary conditions taking full advantage of the SU(NNN) symmetry of the problem. This method is an extension of the method previously developed for exact diagonalizations and relies on a systematic use of the basis of standard Young tableaux. Concentrating on the model with the fundamental representation at each site (i.e., one particle per site in the fermionic formulation), we have benchmarked our results for the ground-state energy up to N=8N=8N=8 and up to 420 sites by comparing them with Bethe ansatz results on open chains, for which we have derived and solved the Bethe ansatz equations. The agreement for the ground-state energy is excellent for SU(3) (12 digits). It decreases with NNN, but it is still satisfactory for N=8N=8N=8 (six digits). Central charges ccc are also extracted from the entanglement entropy using the Calabrese-Cardy formula and agree with the theoretical values expected from the SU(N)1\mathrm{SU}{(N)}_{1}SU(N)1​ Wess-Zumino-Witten conformal field theories.
In this thesis, we deal with 1-dimensional anisotropic spin models more specifically with the chain and ladder geometries. These models are fundamental to the study of condensed matter theory and quantum magnetism because of their simplicity. Research into the chain model dates back to 1930s with Hans Bethe. One method to study these systems is through numerical analysis including renormalization group techniques. These allow for the diagonalization of the Hamiltonian without sacrificing computation to a large virtual Hilbert space. In this thesis, we present results for both the spin chain and spin ladder geometries in open boundary conditions. This work uses the density matrix renormalization group technique to calculate system energy. Initially we will present a study into the spin energy gap of the spin-\sfrac{1}{2} anisotropic (XXZ) Heisenberg chain in open boundary conditions (OBCs). The energy gap is shown to be reduced to half in the ground sector when compared to periodic boundary conditions (PBCs) due to edge effects. Secondly, a full phase diagram for the spin-\sfrac{1}{2} anisotropic Heisenberg ladder is presented which shows the emergence of a rich schematic with a variety of phases. Lastly, weak coupling limit maps are compared to quantum field theory phase transition predictions. This research was motivated by the lack of comprehensive phase diagram results for ladders and by real materials being investigated at the University of Kent.