The thermodynamics of the one-dimensional Heisenberg-Ising model for |Δ| ≪ 1 as well as of the X-Y-Z model is reduced to a set of non-linear integral equations under some plausible assumptions. It is remarkable that the number of unknown functions involved in them becomes finite when π/ cos -1Δ is a rational number for the Heisenberg-Ising model and when Kl /ζ is a rational number for the X-Y-Z model (where coupling constants Jx, Jy and Jz are parametrized by ζ, l, and Jz as Jx = Jz cn(2ζ, l) and Jy = Jz dn(2ζ, l); 1 ≥l ≥0, Kl ≥2ζ≥0 and Kl is the complete elliptic integral of the first kind of modulus l). The validity of our theory has been confirmed by the high-temperature expansion of the free energy through the second term for a general value of Δ and through the fourth them for Δ= ½.
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.
Using a modified Lanczos method we have studied the two-dimensional Heisenberg antiferromagnetic model (at zero temperature) with lattices up to 24 sites. The ground-state and first-excited-state energies are evaluated. We show that the model is gapless in the thermodynamic limit in agreement with recent Monte Carlo simulations. Accurate results for the ``square ladder'' are also presented showing that this model is massive. The finite-size dependence of the results is discussed. In addition, we calculate the energy of a resonating-valence-bond state on finite lattices and its overlap with the exact ground state.
Since the advent of high-
Tc cuprate superconductors in 1986, strongly correlated electron systems have attracted much attention. Since the cuprates are essentially two-dimensional, low-dimensional systems have moved into the focus of condensed-matter theory. From a theoretical point of view, one-dimensional systems are of particular interest because there are exact numerical and analytical methods which permit detailed studies and deep insights into the many-body problem. In the first part of this thesis, using the numerically exact methods Exact Diagonalization and the Density-Matrix Renormalization Group (DMRG), we investigate the properties of the Tomonaga–Luttinger liquid which is the generic metallic state of matter in one dimension. In particular, we concentrate on the investigation of the so-called Tomonaga–Luttinger liquid parameter which determines the critical exponent
α for the density of states near the Fermi energy. Experimental results for some quasi one-dimensional materials report
α≳1, which would imply
Kρ≲0.17, a value which cannot be reconciled with the bare Hubbard model where
KρH≥0.5, i.e.,
αH≲1/8. We develop new accurate numerical methods to obtain
Kρ and investigate how to obtain such small values for
Kρ for slightly doped charge-density-wave insulators. In the second part of this thesis, using the Thermodynamic Bethe Ansatz (TBA) as exact analytical method, we investigate the one-dimensional Hubbard model in the spindisordered regime, which is characterized by the temperature being much larger than the magnetic energy scale but small compared to the Mott–Hubbard gap. Our study is motivated by the controversy about the Mott–Hubbard insulator in infinite dimensions whose ground state is also spin-disordered. In this system the determination of the precise value of the critical interaction strength Uc where the Mott–Hubbard gap closes is still unsolved. Therefore, we provide an example of a Hubbard-type model with a disordered spin background which can be solved exactly. The thermodynamics of our model can be understood in terms of gapped charged excitations with an effective dispersion which are decoupled form the spin degrees of freedom; the latter contribute only entropically. An interpretation of this regime in terms of a putative interacting-electron system at zero temperature leads to a metal-insulator transition at a finite interaction strength above which the gap opens linearly. Our exact results indicate that the strong-coupling expansion of the ground-state energy cannot be used to locate
Uc. However, the strong-coupling expansion of the gap permits a reliable extrapolation of the critical interaction strength.