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arXiv:cond-mat/0405385 · Journal of the Physical Society of Japan

Phase Diagram ofS=1 XXZ Chain with Next-Nearest-Neighbor Interaction

Takahiro Murashima, Keigo Hijii, K. Nomura, Takashi Tonegawa

Published May 15, 2005• 9 citations
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Abstract

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 $α$ $\ll$ -1 and $Δ$ $\sim$ 1 region by using the renormalization group method.

DOI
10.1143/jpsj.74.1544

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The one dimensional S=1 XXZ model with next-nearest-neighbor interaction ααα and Ising-type anisotropy ΔΔΔ 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 σσσ model and analyze the phase diagram in the ααα ≪\ll≪ -1 and ΔΔΔ ∼\sim∼ 1 region by using the renormalization group method.
One-dimensional S=1S=1S=1 XXZ\mathrm{XXZ}XXZ 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\mathrm{XY}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.
We have studied the phase diagram of the one-dimensional S=12S=\frac{1}{2}S=21​ XXZXXZXXZ model with ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor interactions. We have applied the quantum renormalization group (QRG) approach to obtain stable fixed points and the scaling of coupling constants. The QRG prescription has to be implemented to the second order of interblock interactions to obtain a self-similar Hamiltonian after each step of the QRG. This model shows a rich phase diagram which includes quantum spin-fluid and dimer phases in addition to the classical N\'eel and ferromagnetic ones. We have found the border between different phases by tracing the scaling of coupling constants.
By using mainly numerical methods, we investigate the ground-state phase diagram (GSPD) of an S=1S=1S=1 ferromagnetic-antiferromagnetic bond-alternating chain with the XXZXXZXXZ and the on-site anisotropies. This system can be mapped onto an anisotropic spin-2 chain when the ferromagnetic interaction is much stronger than the antiferromagnetic interaction. Since there are many quantum parameters in this system, we numerically obtained the GSPD on the plane of the magnitude of the antiferromagnetic coupling versus its XXZXXZXXZ anisotropy, by use of the exact diagonalization, the level spectroscopy as well as the phenomenological renormalization group. The obtained GSPD consists of six phases. They are the $XY1,thelarge−1, the large-1,thelarge−D$ (LD), the intermediate-DDD (ID), the Haldane (H), the spin-1 singlet dimer (SD), and the Néel phases. Among them, the LD, the H, and the SD phases are the trivial phases, while the ID phase is the symmetry-protected topological phase. The former three are smoothly connected without any quantum phase transitions. It is also emphasized that the ID phase appears in a wider region compared with the case of the GSPD of the anisotropic spin-2 chain with the XXZXXZXXZ and the on-site anisotropies. We also compare the obtained GSPD with the result of the perturbation theory.
By use of mainly numerical methods, we investigate the ground-state phase diagram of an S = 2 quantum spin chain with the XXZ and on-site anisotropies described by = Σ j ( S j x S j +1 x + S y j S y j +1 + Δ S z j S z j +1 + D Σ j ( S z j ) 2 , where Δ denotes the XXZ anisotropy parameter of the nearest-neighbor interactions and D the on-site anisotropy parameter. In the phase diagram with Δ > 0 and D > 0 there appear the XY phase, the Haldane/large- D (LD) phase, the intermediate- D (ID) phase, and the Néel phase. Among them the ID phase was proposed by Oshikawa in 1992 and has been considered to be absent since the DMRG studies in the latter half of 1990's. And also, the Haldane state and the LD state belong to the same phase, which should be named Haldane/LD phase.
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We investigate the inversion phenomenon between the XXZ anisotropies of the Hamiltonian and the wave function in quantum spin chains, focusing mainly on the S = 1/2 trimerized XXZ model with the next-nearest-neighbor interactions. We have obtained the ground-state phase diagram by use of the degenerate perturbation theory and the level spectroscopy analysis of the numerical data calculated by the Lanczös method. In some parameter regions, the spin-fluid phase is realized for the Ising-like anisotropy, and the Néel phase for the XY-like anisotropy, against the ordinary situation.