The S =1 XXZ spin chain is studied by the finite-size scaling and phenomenological renormalization-group technique up to N =16. It is found that a two-dimensional Ising-model-type transition exists at λ=1.188±0.007, and a Kosterlitz-Thouless-type transition exists at λ=-0.01±0.03. It agrees with Haldane's conjecture completely.
Finite-size scaling (phenomenological renormalization) techniques are trusted and widely applied in low-dimensional magnetism and, particularly, in lattice gauge field theory. Recently, investigations have begun which subject the theoretical basis to systematic and intensive scrutiny to determine the validity of finite-size scaling in a variety of situations. The 2D ANNNI model is an example of a situation where finite-size scaling methods encounter difficulty, related to the occurrence of a disorder line (one-dimensional line). A second example concerns the behavior of the spin-1/2 antiferromagnetic XXZ model where the T=0 critical behavior is exactly known and features an essential singularity at the isotropic Heisenberg point. Standard finite-size scaling techniques do not convincingly reproduce the exact phase behavior and this is attributable to the essential singularity. The point is relevant in connection with a finite-size scaling analysis of a spin-one antiferromagnetic XXZ model, which claims to support a conjecture by Haldane that the T=0 phase behavior of integer-spin Heisenberg chains is significantly different from that of half-integer-spin Heisenberg chains.
The ground state of the S=1 antiferromagnetic Heisenberg chain with an axial-symmetry-breaking anisotropy Etsumj [(Sjx)2-(Sjy)2] is studied by the finite-size scaling and phenomenological renormalization-group technique based on numerical diagonalizations up to N=14. It is found that two phase transitions exist at Ec1=0.20\ifmmode\pm\else\textpm\fi{}0.01 and Ec2=1.71\ifmmode\pm\else\textpm\fi{}0.01, and both transitions belong to the same universality class as the two-dimensional Ising model.
One-dimensional S=1XXZ chains with uniaxial single-ion-type anisotropy are studied by numerical exact diagonalization of finite-size systems. The numerical data are analyzed using conformal field theory, the level spectroscopy, phenomenological renormalization-group, and finite-size scaling method. We thus present a quantitatively reliable ground-state phase diagram of this model. The ground states of this model contain the Haldane phase, large-D phase, N\'eel phase, two XY phases, and the ferromagnetic phase. There are four different types of transitions between these phases: the Brezinskii-Kosterlitz-Thouless-type transitions, the Gaussian-type transitions, the Ising-type transitions, and the first-order transitions. The locations of these critical lines are accurately determined.
The one dimensional S=1 XXZ model with next-nearest-neighbor interaction α and Ising-type anisotropy Δ is studied by using a numerical diagonalization technique. We discuss the ground state phase diagram of this model numerically by the twisted-boundary-condition level spectroscopy method and the phenomenological renormalization group method, and analytically by the spin wave theory. We determine the phase boundaries among the XY phase, the Haldane phase, the ferromagnetic phase and the Néel phase, and then we confirm the universality class. Moreover, we map this model onto the non-linear σ model and analyze the phase diagram in the α≪ -1 and Δ∼ 1 region by using the renormalization group method.
We investigate the one-dimensional quantum XXZ model in the presence of diagonal disorder. Recently the model has been analyzed with the help of field-theoretical renormalization group methods, and a phase diagram has been predicted. We study the model with exact diagonalization techniques up to chain lengths of 16 sites. Using finite-size scaling methods we estimate critical exponents and the phase diagram and find reasonably good agreement with the field-theoretical results, namely, that any amount of disorder destroys the superfluidity for XXZ anisotropy Δ between $ -1/2 $ and 1, while the superfluidity persists to finite disorder strength for $ -1 < Δ< -1/2$ and then undergoes a Kosterlitz-Thouless type transition.
Research in recent years has shown that combining finite-size scaling theory with the transfer matrix technique yields a powerful tool for the investigation of critical behavior. In particular, the method has been used to study two-dimensional statistical mechanical and one-dimensional quantum mechanical systems. We review finite-size scaling theory from the general point of view of renormalization group theory for both continuous and first-order transitions (both for systems with discrete and continuous symmetries). We review applications where a comparison with exact results can be made. These include the Ising, Baxter, and q-state Potts models and the Ising model with a defect line. Various other applications such as quantum systems, the self-avoiding random walk, percolation, and Kosterlitz-Thouless transitions are briefly reviewed. The Kosterlitz-Thouless transitions and the critical fan in the antiferromagnetic 3-state Potts model are discussed at somewhat greater length.