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Physical review. B, Condensed matter

Solution of the time-dependent Schrödinger equation for two-dimensional spin-1/2 Heisenberg systems

Pedro de Vries, Hans A. De Raedt

Published Apr 1, 1993• 48 citations
Publisher

Abstract

Numerical calculations on two-dimensional spin-1/2 XXZ Heisenberg models are presented. The computational technique entails solving the time-dependent Schr\"odinger equation. The time propagation of wave functions is performed employing Trotter-Suzuki product formulas. The algorithms are numerically stable and are used to study models involving \ensuremath{\approxeq}${10}^{7}$ states. With this approach both the thermodynamic properties as well as the many-particle dynamics can be obtained. Densities of states and zero-temperature dynamic magnetic structure factors are presented. For the isotropic Heisenberg antiferromagnet thermodynamic properties are computed for lattices up to 26 sites.

DOI
10.1103/physrevb.47.7929

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Numerical calculations on two-dimensional spin-1/2 XXZ Heisenberg models are presented. The computational technique entails solving the time-dependent Schr"odinger equation. The time propagation of wave functions is performed employing Trotter-Suzuki product formulas. The algorithms are numerically stable and are used to study models involving \ensuremath{\approxeq}107{10}^{7}107 states. With this approach both the thermodynamic properties as well as the many-particle dynamics can be obtained. Densities of states and zero-temperature dynamic magnetic structure factors are presented. For the isotropic Heisenberg antiferromagnet thermodynamic properties are computed for lattices up to 26 sites.
Numerical calculations on two-dimensional spin-½ XXZ Heisenberg models are presented. The computational technique entails solving the time-dependent Schrödinger equation. The time propagation of wave functions is performed employing Trotter-Suzuki product formulas. The algorithms are numerically stable and are used to study models involving ≈10^7 states. With this approach both the thermodynamic properties as well as the many-particle dynamics can be obtained. Densities of states and zero-temperature dynamic magnetic structure factors are presented. For the isotropic Heisenberg antiferromagnet thermodynamic properties are computed for lattices up to 26 sites.
The one-dimensional spin-1 Heisenberg antiferromagnetic chains are studied by a generalized two-time Green's-function method at finite temperature. 〈Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 is set to zero at each site to guarantee the lack of the long-range order in the one-dimensional Heisenberg system. The Green's functions are decoupled in terms of the correlation functions of spin operators. A set of self-consistent equations of the correlation functions are derived and solved numerically. Correlation functions of spin operators, thermodynamic properties such as the internal energy, and specific heat in the entire temperature region are obtained. The Haldane gap in the excitation spectrum appears naturally in the analytical result. When k is zero, there is a gap 2\ensuremath{\Delta}, which is 1.0J and close to the theoretical result of White and Huse. And there is a broad maximum in the temperature dependence of the specific heat. The properties of the thermodynamic quantities are consistent with the numerical results on the finite chains. \textcopyright{} 1996 The American Physical Society.
Two-dimensional spin 1/2 antiferromagnetic Heisenberg models are numerically studied using Entanglement Perturbation Theory, where the ground state wave function is described by a product of local matrices defined at every site, and each matrix is optimized variationally to minimize the energy. We first apply this method for the spin 1/2 antiferromagnetic Heisenberg model on the square lattices to reproduce correctly the known ground state energy and the spin structure factors. Then, we study the spin 1/2 antiferromagnetic Heisenberg model on the triangular lattice with spatially anisotropic nearest neighbor couplings, J and J ', to establish the ground state phase diagram as a function of J'/ J.
We study algorithm to calculate the time dependent Schrödinger equation numerically and to visualize the results. We extend the product formula by the Trotter-Suzuki decomposition to cases of Hamiltonians with vector potentials. This method is unconditionally stable and it can be used for a time dependent Hamiltonian so that one can apply it to various problems without any difficulty. After examining numerical errors for time evolutions of one dimensional wave packets, we present the errors under a constant magnetic field. Also we discuss a cyclotron motion in quantum mechanics, where an advantage for the unconditional stability is demonstrated. In order to visualize the results we make animations of the time evolutions, where the phase of the wave function is represented by colors and its absolute value is shown by brightness. Here we suggest a new way to represent a complex number by colors.
We compute all dynamical spin–spin correlation functions for the spin-1/2 XXZ anisotropic Heisenberg model in the gapless antiferromagnetic regime at zero temperature, using numerical sums of exact determinant representations for form factors of spin operators on the lattice. Contributions from intermediate states containing many particles and string (bound) states are included. We present modified determinant representations for the form factors valid in the general case with string solutions to the Bethe equations. Our results are such that the available sum rules are saturated to high precision. We Fourier transform our results back to real space, allowing us in particular to make a comparison with known exact formulae for equal-time correlation functions for small separations in zero field, and with predictions for the zero-field asymptotics from conformal field theory.
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.