Generalization of the Affleck-Kennedy-Lieb-Tasaki Model for Quantum Ferromagnetism

Abstract

We study a spin-$S$ ferromagnetic model with exactly-written ground states, known as the partially-magnetized valence bond solid (VBS) states with magnetization $m=(S-1)/S$, which is a ferromagnetic generalization of the Affleck-Kennedy-Lieb-Tasaki model. We find that the VBS state and an antiferromagnetic ground state with magnetization $m=0$ are degenerate for $S=3/2$ and $S=2$ by using the Lanczos method and the density matrix renormalization group method (DMRG). However, increasing $S$, the magnetization of the ground states is uniquely determined as the fraction $m=(S-1)/S$. This is not just a ferromagnet, but a quantum ferromagnet due to quantum entanglement inherent in VBS states. In the low-energy excitation spectrum, we find the coexistence of the Haldane gap and Goldstone-like ferromagnetic magnon excitation. This ``magnetic chimera'' clearly appears under a finite magnetic field. Finally, we discuss an application to the measurement-based quantum computation and an extension of the Haldane's conjecture.

Figures (9)

• Figure 1: (Color online)Schematic picture of the ground states (a) in traditional spin-$S$ chain models and (b) in "quantum" ferromagnetism of spin-$S$ chain models. Properties about total spin $S_{\rm tot}$ of the ground states and low energy excitation $\Delta E_\pm$ are also summarized. For $\Delta E_+$, the spin-1/2 antiferromagnet (AF) has $\Delta E_+\propto |q|$, i.e., des Cloizeaux-Pearson modePR.128.2131 while the spin-1 AF has the Haldane gap $\Delta E_+>0$PRL.50.1153. On the other hand, $\Delta E_-\propto q^2$ is Goldstone-type gapless one-magnon modeISBN.0521551439.
• Figure 2: (Color online)Ferromagnetic AKLT state ${\left|\Phi\right\rangle_{\!}}$ in Eq. (\ref{['VBS']}) written with spin-singlets ${\left|\phi_s\right\rangle_{\!}}_{i,i+1}$ and background ferromagnetic Ising states ${\left|S-1\right\rangle_{\!}}_{i,F}$. Spin-$S$ operator $\hbox{\boldmath $\hat{S}$}_i$ at the $i$-th site is decomposed into spin-$(S-1)$ operator $\hbox{\boldmath $\hat{S}$}_{i,F}$ and two spin-1/2 operators $\hbox{\boldmath $\hat{s}$}_{i,L}$, $\hbox{\boldmath $\hat{s}$}_{i,R}$. This is identical to Oshikawa's stateJPCM.4.7469 and is depicted in the bottom panel of Fig. \ref{['figV']} (b).
• Figure 3: (Color online)$E_{S_{\rm tot}=0}$ and $S_{\rm tot}$ as a function of the number of sweeps in the finite size method for $S=2$, and $N=24$ obtained by DMRG with $\chi=100$. All data points are positive: $E_{S_{\rm tot}=0} > 0$ and $S_{\rm tot}>0$. The unit of energy $E_{+,N=4}$ will be given by Eq. (\ref{['def:scale']}) in the next section § \ref{['sec:Gp']}.
• Figure 4: (Color online)System-size $N$ dependence of the Haldane gap $\Delta E_+=E_+$ of ${\hat{H}}^{(S)}(\beta_S)$ defined in Eq. (\ref{['def:DeltaE+']}) with the unit of energy $E_{+,N=4}$ in Eq. (\ref{['def:scale']}). Using ${\hat{H}}^{(S=1)}(\beta_1)\propto \hat{H}_{0}^{(S=1)}$, the $S=1$ data (solid line) was obtained from a table in a previous studyPRB.88.245118 and the $S=1$ data points without a solid line were calculated by DMRG additionally.
• Figure 5: (Color online)Low energy excitation $\Delta E_{-,q}=E_{-,q}$ as a function $q$ with $\Delta S=-1$ ($S_{{\rm tot}}=N(S-1)-1$), obtained using the Lanczos method. For each $S$, larger point indicates a larger system size $N$. The solid lines are fitted curves using the dispersion function in Eq. (\ref{['def:DeltaE-']}). $E_{+,N=4}$ is the unit of energy in Eq. (\ref{['def:scale']}).