Journal of the Physical Society of Japan

Phase Diagram ofS=1/2 AntiferromagneticXXZChainwith Next-Nearest-Neighbor Interactions

K. Nomura, Kiyomi Okamoto

Published Apr 15, 1993• 73 citations

Abstract

We numerically investigate the critical lines of the fluid-dimer and the Néel-dimer transitions of the S =1/2 antiferromagnetic XXZ chain with next-nearest interactions, assuming the asymptotic behavior to be determined by the quantum sine-Gordon model. Our result is different from that of Tonegawa, Harada and Kaburagi, who used the phenomenological renormalization group (PRG). We show that the PRG may lead to a false critical point when it is applied to the Kosterlitz-Thouless-type transition.

DOI: 10.1143/jpsj.62.1123

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Journal of the Physical Society of Japan

Phase Diagram ofS=1/2 AntiferromagneticXXZChainwith Next-Nearest-Neighbor Interactions

K. Nomura, Kiyomi Okamoto

Published Apr 15, 1993• 73 citations

Publisher

Abstract

DOI: 10.1143/jpsj.62.1123

Related papers

Critical properties of S= 1/2 antiferromagnetic XXZ chain with next-nearest-neighbour interactions

K. Nomura, Kazuki Okamoto

We investigate numerically the critical lines and the critical properties of the fluid-dimer and the Neel-dimer transitions of the S= 1/2 antiferromagnetic XXZ chain with next-nearest interactions, and we confirm that the universality class of this model belongs to the quantum sine-Gordon model, as is expected from the bosonization. The method which we use in this paper to calculate the critical lines is free from the logarithmic corrections on the Kosterlitz-Thouless (K-T)-type transition line, which have made the K-T critical point difficult to obtain. By the use of this method, it is possible to determine the K-T critical line with high precision from small size data, and to identify the universality class.We investigate numerically the critical lines and the critical properties of the fluid-dimer and the Neel-dimer transitions of the S= 1/2 antiferromagnetic XXZ chain with next-nearest interactions, and we confirm that the universality class of this model belongs to the quantum sine-Gordon model, as is expected from the bosonization. The method which we use in this paper to calculate the critical lines is free from the logarithmic corrections on the Kosterlitz-Thouless (K-T)-type transition line, which have made the K-T critical point difficult to obtain. By the use of this method, it is possible to determine the K-T critical line with high precision from small size data, and to identify the universality class.

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Shunsuke Furukawa, Masahiro Sato, Akira Furusaki

We study the ground-state phase diagram of a one-dimensional spin- $\frac{1}{2}$ easy-plane XXZ model with a ferromagnetic nearest-neighbor (NN) coupling ${J}_{1}$ and a competing next-nearest-neighbor (NNN) antiferromagnetic coupling ${J}_{2}$ in the parameter range $0<{J}_{2}/|{J}_{1}|<0.4$ . When ${J}_{2}/|{J}_{1}|\ensuremath{\lesssim}1/4$, the model is in a Tomonaga-Luttinger liquid phase which is adiabatically connected to the critical phase of the XXZ model of ${J}_{2}=0$ . On the basis of the effective (sine-Gordon) theory and numerical analyses of low-lying energy levels of finite-size systems, we show that the NNN coupling induces phase transitions from the Tomonaga-Luttinger liquid to gapped phases with either N\'eel or dimer order. Interestingly, these two types of ordered phases appear alternately as the easy-plane anisotropy is changed toward the isotropic limit. The appearance of the antiferromagnetic (N\'eel) order in this model is remarkable, as it is strongly unfavored by both the easy-plane ferromagnetic NN coupling and the antiferromagnetic NNN coupling in the classical-spin picture. We argue that emergent trimer degrees of freedom play a crucial role in the formation of the N\'eel order.We study the ground-state phase diagram of a one-dimensional spin-$\frac{1}{2}$ easy-plane XXZ model with a ferromagnetic nearest-neighbor (NN) coupling ${J}_{1}$ and a competing next-nearest-neighbor (NNN) antiferromagnetic coupling ${J}_{2}$ in the parameter range $0<{J}_{2}/|{J}_{1}|<0.4$. When ${J}_{2}/|{J}_{1}|\ensuremath{\lesssim}1/4$, the model is in a Tomonaga-Luttinger liquid phase which is adiabatically connected to the critical phase of the XXZ model of ${J}_{2}=0$. On the basis of the effective (sine-Gordon) theory and numerical analyses of low-lying energy levels of finite-size systems, we show that the NNN coupling induces phase transitions from the Tomonaga-Luttinger liquid to gapped phases with either N\'eel or dimer order. Interestingly, these two types of ordered phases appear alternately as the easy-plane anisotropy is changed toward the isotropic limit. The appearance of the antiferromagnetic (N\'eel) order in this model is remarkable, as it is strongly unfavored by both the easy-plane ferromagnetic NN coupling and the antiferromagnetic NNN coupling in the classical-spin picture. We argue that emergent trimer degrees of freedom play a crucial role in the formation of the N\'eel order.

arXiv:1003.4517Physical Review BMar 26, 201032 citations

We have studied the phase diagram of the one-dimensional

S=\frac{1}{2}

XXZ

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.

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.

We calculate the quantum phase diagram of an XXZ chain with nearest-neighbor (NN)

{J}_{1}

and next-NN exchange

{J}_{2}

with anisotropies ${\ensuremath{\Delta}}_{1}$ and ${\ensuremath{\Delta}}_{2},$ respectively. In particular we consider the case ${\ensuremath{\Delta}}_{1}=\ensuremath{-}{\ensuremath{\Delta}}_{2},$ to interpolate between the XX chain $({\ensuremath{\Delta}}_{i}=0)$ and the isotropic model with ferromagnetic phase

{J}_{2}.

For ${\ensuremath{\Delta}}_{1}<\ensuremath{-}1,$ a ferromagnetic phase and two antiferromagnetic phases exist. For $|{\ensuremath{\Delta}}_{i}|<1,$ the boundary between the dimer and spin fluid phases is determined by the method of crossing of excitation spectra. For large

{J}_{2}{/J}_{1},

this method seems to indicate the existence of a second spin fluid critical phase. However, an analysis of the spin stiffness and magnetic susceptibility for ${\ensuremath{\Delta}}_{1}={\ensuremath{\Delta}}_{2}=1$ suggests that a small gap is present.

We study the ground-state phase diagram of a one-dimensional spin-

\frac{1}{2}

easy-plane XXZ model with a ferromagnetic nearest-neighbor (NN) coupling

{J}_{1}

and a competing next-nearest-neighbor (NNN) antiferromagnetic coupling

{J}_{2}

in the parameter range

0<{J}_{2}/|{J}_{1}|<0.4

. When ${J}_{2}/|{J}_{1}|\ensuremath{\lesssim}1/4$, the model is in a Tomonaga-Luttinger liquid phase which is adiabatically connected to the critical phase of the XXZ model of

{J}_{2}=0

. On the basis of the effective (sine-Gordon) theory and numerical analyses of low-lying energy levels of finite-size systems, we show that the NNN coupling induces phase transitions from the Tomonaga-Luttinger liquid to gapped phases with either N\'eel or dimer order. Interestingly, these two types of ordered phases appear alternately as the easy-plane anisotropy is changed toward the isotropic limit. The appearance of the antiferromagnetic (N\'eel) order in this model is remarkable, as it is strongly unfavored by both the easy-plane ferromagnetic NN coupling and the antiferromagnetic NNN coupling in the classical-spin picture. We argue that emergent trimer degrees of freedom play a crucial role in the formation of the N\'eel order.