Dimerization and effective decoupling in two spin-1 generalizations of the spin-$\frac{1}{2}$ Majumdar-Ghosh chain

We perform a systematic DMRG investigation of the two natural spin-1 generalizations of the spin-1/2 Majumdar-Ghosh chain, the spin-1 $J_1\text{−}{J}_2$ Heisenberg chain, where $J_2$ is a next-nearest-neighbor Heisenberg coupling, and the spin-1 $J_1\text{−}{J}_3$ model, where $J_3$ refers to a three-site interaction defined by $J_3\left[({\mathbf{S}}_{i-1}·{\mathbf{S}}_i)({\mathbf{S}}_i·{\mathbf{S}}_{i+1})+\mathrm{H}.\mathrm{c}.\right]$. Although both models are rigorously equivalent to the Majumdar-Ghosh chain for spin 1/2, their physics appears to be quite different for spin 1. Indeed, when all couplings are antiferromagnetic, the spin-1 $J_1\text{−}{J}_2$ model undergoes an effective decoupling into two next-nearest-neighbor (NNN) Haldane chains upon increasing $J_2$, while the $J_1\text{−}{J}_3$ chain undergoes a spontaneous dimerization similar to the spin-1/2 Majumdar-Ghosh chain upon increasing $J_3$. By extending the phase diagram to all signs of the couplings, we show that both the dimerized and the NNN-Haldane phase are actually present in the $J_1\text{−}{J}_3$ model, the former one adjacent to the Haldane one, the latter to the ferromagneric one, with an Ising transition between them. By contrast, the $J_1\text{−}{J}_2$ chain only has a NNN-Haldane phase between the Haldane phase and the ferromagnetic phase for positive $J_2$. In both cases, our DMRG data are consistent with a continuous Kosterlitz-Thouless transition between the NNN-Haldane and the ferromagnetic phases.

Figure 1

Phase diagram of the $J_1\text{−}{J}_3$ model defined by the Hamiltonian of Eq. (2). The transition between the Haldane and dimerized phases at ${\theta }_1\approx 0.03519\pi$ is continuous in the WZW SU(2)${}_2$ universality class [3]. The transition between the dimerized and NNN-Haldane phases is Ising and takes place at ${\theta }_2\approx 0.8913\pi$. Kosterlitz-Thouless transition to the ferromagnetic phase takes place at ${\theta }_3\approx 0.9474\pi$. The transition between the ferromagnetic and Haldane phases at ${\theta }_4\approx 1.33\pi$ is first order.

Figure 2

Ground-state energy per bond calculated in the middle of finite-size chains as a function of $\theta$. In the ferromagnetic phase, the DMRG data coincide exactly with the analytic prediction ${\epsilon }_{FM}=cos\theta +2sin\theta$.

Figure 3

Scaling of the dimerization at the critical Ising point. (a) Log-log plot of the dimerization. The linear curve corresponds to the Ising critical point, and the slope to the critical exponent. This leads to ${\theta }_c=0.8913\pi$, and to a slope 0.141, that agrees within $15\%$ with the prediction $1/8$ for Ising critical point. (b) Site dependence of $D(j,N)$ at the critical point fitted to $1/{[Nsin(\pi j/N)]}^d$ for $N=300$ and $N=400$. This leads to an exponent $d=0.140$, that also agrees with the Ising prediction $1/8$ within $15\%$. For clarity the data for $N=300$ are shifted by 50 sites.

Figure 4

Entanglement spectrum for an open chain with $N=150$ sites as a function of $\theta$ when cut across (a) an odd and (b) an even bond. Only the lower part of the spectrum is shown. The dots show the multiplicity of the Schmidt values. Insets: VBS sketches of the various boundaries created by the cut of the chain inside the dimerized and NNN-Haldane phases. The two spins $1/2$ created at each edge in the NNN-Haldane phase couple with each other and form a triplet, which is represented as an orange ellipse. This leads to the threefold degeneracy of the lowest state of the entanglement spectrum.

Figure 5

Energy gap while interpolating between the $J_1\text{−}{J}_3$ model ($\alpha =0$) with $J_3/J_1=tan\left(0.9\pi \right)$ and the $J_1\text{−}{J}_2$ model ($\alpha =1$) with $J_1=-J_2$, which is known to be in the NNN-Haldane phase. Since the gap remains open for all the values of the interpolation parameter $\alpha$, the nondimerized phase in the $J_1\text{−}{J}_3$ model is identified as a NNN-Haldane phase.

Figure 6

Scaling of the entanglement entropy of open chains after removing the Friedel oscillations with conformal distance $d\left(n\right)$ for $\theta =0.8913\pi$ and different system sizes. The extracted values of the central charge agree within $8\%$ with the field-theory prediction $c=1/2$ for the Ising critical theory. Data are shifted vertically for clarity.

Figure 7

Left panels: Linear scaling of the ground-state energy per site in open chains with $1/N^2$ after subtracting ${\varepsilon }_0$ and ${\varepsilon }_1$ terms for (a) even and (b) odd number of sites. The extracted values of the velocities agree with each other within $15\%$. Right panel: Energy gaps in the singlet (blue) and triplet (green) sectors for open boundary conditions as a function of $1/N$ for even (circles) and odd (diamonds) number of sites.

Figure 8

Sketch of the nonmagnetic domain wall between the NNN-Haldane phase and the dimerized phase.

Figure 9

Local magnetization along a chain with $N=200$ sites and as a function of the parameter $\theta$ that changes linearly along the chain interpolating between the dimerized phase on the left and NNN-Haldane phase on the right. (a) A spin 1 is localized at the dimerized edge of the chain. (b) A spin 1 is localized at each edge. (c) A spin 1 is localized at each edge and, as follows from (d), there is also a spin 1 delocalized over the dimerized domain. Panel (d) shows the difference between (c) and (b).

Figure 10

Example of fit of the spin-spin correlations to the Ornstein-Zernicke form given by Eq. (16) for $N=600$ and $\theta =0.913\pi$. In the first step (a), we extract the correlation length discarding the oscillations. In the second step (b), we fit the reduced correlation function to extract the wave vector $q$.

Figure 11

(a) Inverse of the correlation length and (b) wave vector $q$ as a function of $\theta$ between the dimerized and the ferromagnetic phases. Light-gray symbols stand for the points where the extracted correlation length is comparable to the total length of the chain. Inset: divergence of the correlation length with the distance to ${\theta }_3=0.9474\pi$ in a log-log scale. In panel (b), the black line corresponds to the $q$ vector given by Eq. (15) that minimizes the energy of the one-magnon instability while the red line is the result of a fit $q\propto |\theta -{\theta }_c{|}^{\nu }$ with $\nu \approx 0.40$. The inset shows the scaling towards the critical point in log-log scale.

Figure 12

Phase diagram of $J_1\text{−}{J}_2$ model given by the Hailtonian of Eq. (1). The transition between the Haldane and the NNN-Haldane phase is first order and takes place at ${\theta }_1\approx 0.204\pi$. The system undergoes a continuous transition to the ferromagnetic phase at ${\theta }_2\approx 0.9220\pi$. The transition between the ferromagnetic and the Haldane phase is first order and is at $\theta =3\pi /2$.

Figure 13

(a) Inverse of the correlation length and (b) wave vector $q$ as a function of $\theta$. In panel (b), the black line corresponds to the $q$ vector given by Eq. (20) that minimizes the energy of the one-magnon instability while the red line is the result of a fit $q\propto |\theta -{\theta }_c{|}^{\nu }$ with $\nu \approx 0.41$. The inset shows the scaling towards the critical point in log-log scale.

Figure 14

Phase diagram of $J_1\text{−}{J}_3$ model as a function of nearest-neighbor coupling $J_1$ with fixed coupling constant $J_3=1$.

Figure 15

Phase diagram of $J_1\text{−}{J}_2$ model as a function of nearest-neighbor coupling $J_1$ with fixed coupling constant $J_2=1$.