The antiferromagnetic Heisenberg chain with axial anisotropy and applied magnetic field parallel to the axis, is probed through a new variational approach based on the Bethe–Ansatz solution at zero field. The variational ket is built by rotating the spins in the Bethe solution, with the angles and the value of the local magnetic moment as variational parameters. This procedure allows for a description of the various magnetic phases in the presence of a parallel field, such as ferromagnetic (F), antiferromagnetic (AF), and spin-flop (SF) phases. The approximation reduces to the exact Bethe solution for vanishing field. We compare with exact numerical results (Lanczos calculations for finite systems), and exact solutions when available.
The antiferromagnetic Heisenberg chain with axial anisotropy and applied magnetic field parallel to the axis, is probed through a new variational approach based on the Bethe–Ansatz solution at zero field. The variational ket is built by rotating the spins in the Bethe solution, with the angles and the value of the local magnetic moment as variational parameters. This procedure allows for a description of the various magnetic phases in the presence of a parallel field, such as ferromagnetic (F), antiferromagnetic (AF), and spin-flop (SF) phases. The approximation reduces to the exact Bethe solution for vanishing field. We compare with exact numerical results (Lanczos calculations for finite systems), and exact solutions when available.
Building on the fundamentals introduced in part I, we employ the Bethe ansatz to study some ground-state properties (energy, magnetization, susceptibility) of the one-dimensional s=1/2 Heisenberg antiferromagnet in zero and nonzero magnetic field. The 2-spinon triplet and singlet excitations from the zero-field ground state are discussed in detail, and their energies are calculated for finite and infinite chains. Procedures for the numerical calculation of real and complex solutions of the Bethe ansatz equations are discussed and applied. The paper is designed as a tutorial for beginning graduate students. It includes 10 problems for further study.
We present a variational approach to the Heisenberg antiferromagnetic model for spin 1/2 and anisotropic exchange, with an external longitudinal and transverse magnetic field. In the one-dimensional case, we calculate exactly with our trial function the expectation values of energy, sublattice magnetization, and magnetic susceptibility. On minimizing the energy, we obtain several phases, which depend on the strength of the field and anisotropy parameter. Subsequently we approach the problem numerically and solve it for a chain of 12 spins using the Lanczos method. The two approaches are in excellent concordance, particularly for the critical fields of the transitions, over a wide range of the parameters of the model. Beside its precision and mathematical simplicity, the method has the important advantage of accounting for the different magnetic phases and their transitions with a single trial function that has a compact mathematical expression. The formalism is expected to work better in higher dimensions, but the corresponding calculations are omitted because of the lack of numerical field-dependent data with which to compare.
A recently developed numerical method, entanglement perturbation theory (EPT), is used to study the antiferromagnetic Heisenberg spin chains with z-axis anisotropy λ and magnetic field B. To demonstrate the accuracy, we first apply EPT to the isotropic spin-1/2 antiferromagnetic Heisenberg model, and find that EPT successfully reproduces the exact Bethe Ansatz results for the ground state energy, the local magnetization, and the spin correlation functions (Bethe ansatz result is available for the first 7 lattice separations). In particular, EPT confirms for the first time the asymptotic behavior of the spin correlation functions predicted by the conformal field theory, which realizes only for lattice separations larger than 1000. Next, turning on the z-axis anisotropy and the magnetic field, the 2-spin and 4-spin correlation functions are calculated, and the results are compared with those obtained by Bosonization and density matrix renormalization group methods. Finally, for the spin-1 antiferromagnetic Heisenberg model, the ground state phase diagram in λ space is determined with help of the Roomany-Wyld RG finite-size-scaling. The results are in good agreement with those obtained by the level-spectroscopy method.
A recently developed numerical method, entanglement perturbation theory (EPT), is used to study the antiferromagnetic Heisenberg spin chains with z-axis anisotropy λ and magnetic field B. To demonstrate the accuracy, we first apply EPT to the isotropic spin1 2 antiferromagnetic Heisenberg model, and find that EPT successfully reproduces the exact Bethe Ansatz results for the ground state energy, the local magnetization, and the spin correlation functions (Bethe ansatz result is available for the first 7 lattice separations). In particular, EPT confirms for the first time the asymptotic behavior of the spin correlation functions predicted by the conformal field theory, which realizes only for lattice separations larger than 1000. Next, turning on the z-axis anisotropy and the magnetic field, the 2-spin and 4-spin correlation functions are calculated, and the results are compared with those obtained by Bosonization and density matrix renormalization group methods. Finally, for the spin-1 antiferromagnetic Heisenberg model, the ground state phase diagram in λ space is determined with help of the Roomany-Wyld RG finite-size-scaling. The results are in good agreement with those obtained by the level-spectroscopy method.
In this work we study the Heisenberg XXZ antiferromagnetic model for spin S and dimension d in the presence of an external longitudinal magnetic field. First, we essay a variational approach which uses as trial function a version of the analytic expression for the ground state given by the paired nonmagnetic excitation (PNME) theory [M. Lagos and G. G. Cabrera, Solid State Commun. 67, 221 (1988); Phys. Rev. B 38, 659 (1988)], generalized in order to incorporate the external field. The ground-state energy, sublattice magnetization, and magnetic susceptibility are obtained. Subsequently, we solve the problem numerically for a chain of 12 spins (S=1 and S=3/2) using the Lancz\"os method [E. Dagotto and A. Moreo, Phys. Rev. D 31, 865 (1985)]. The two approaches give excellent concordance over a wide range of the parameters of the model. We show that our analytic trial function represents accurately the ground state of the system for anisotropies ranging from the Ising limit to the almost isotropic Heisenberg model for all values of the field. Moreover, it accounts for the several antiferromagnetic and ferromagnetic phases occurring for different values of the magnetic field with the same precision. The critical fields of the transitions are predicted correctly. Also, results for various spin values S in two and three dimensions are presented.
The antiferromagnetic spin-1/2 Heisenberg model, with axial anisotropy and applied magnetic field parallel to the axis, is solved numerically by Lanczos diagonalization of finite systems. We devise an extremely accurate method that allows for a reliable extrapolation to the infinite size limit. In the above limit, our results are compared with exact analytical solutions obtained through the Bethe ansatz. As an important example, we probe the behavior of the Critical Magnetic Field in the whole anisotropic region. We find indications that suggest the presence of two different regimes for the scaling of size effects as a function of the anisotropy.