Comparative density-matrix renormalization group study of symmetry-protected topological phases in spin-1 chain and Bose-Hubbard models

We reexamine the one-dimensional spin-1 $XXZ$ model with on-site uniaxial single-ion anisotropy as to the appearance and characterization of the symmetry-protected topological Haldane phase. By means of large-scale density-matrix renormalization group (DMRG) calculations the central charge can be determined numerically via the von Neumann entropy, from which the ground-state phase diagram of the model can be derived with high precision. The nontrivial gapped Haldane phase shows up in between the trivial gapped even Haldane and Néel phases, appearing at large single-ion and spin-exchange interaction anisotropies, respectively. We furthermore carve out a characteristic degeneracy of the lowest entanglement level in the topological Haldane phase, which is determined using a conventional finite-system DMRG technique with both periodic and open boundary conditions. Defining the spin and neutral gaps in analogy to the single-particle and neutral gaps in the intimately connected extended Bose-Hubbard model, we show that the excitation gaps in the spin model qualitatively behave just as for the bosonic system. We finally compute the dynamical spin structure factor in the three different gapped phases and find significant differences in the intensity maximum which might be used to distinguish these phases experimentally.

Figure 1

(a) Valence bond picture of the AKLT state in a spin-1 $XXZ$ chain. Each of the two $S=1/2$ spins connected by an ellipse form a singlet ($1/\sqrt{2}$)($\uparrow \downarrow -\downarrow \uparrow$). The two free edge spins cause the fourfold degeneracy of the ground-state energy. (b) To simulate the AKLT state within OBC DMRG, the free edge spins have to be excluded from the system (dashed circles). (c) With PBC the AKLT state can be simulated without any changes, so that the lowest entanglement level exhibits a fourfold degeneracy according to the two edge spins.

Figure 2

Central charge $c^{*}\left(L\right)$ as obtained by DMRG for the spin-1 $XXZ$ model with $D=0$ and PBC. The OH-AFM transition can be assigned to $J_z/J=1.186\pm 0.001$ with $c=1/2$, where a pronounced peak appears [see also the magnifying inset which shows $c^{*}\left(L\right)$ close to the transition point].

Figure 3

$c^{*}\left(L\right)$ for $D=0.5$ near the OH-AFM transition. (b) shows that the Ising transition point $J_{z,c2}/J\left(L\right)$ obtained from $c^{*}\left(L\right)$ can be linearly extrapolated to the thermodynamic limit.

Figure 4

(a) $J_z$ dependence of the two lowest energy eigenvalues at $D/J=1.5$, using TBC and $L=32$. Obviously the energies of the OH state (squares) and the EH state (circles) cross at the EH-OH transition point (dashed line). (b) Critical points $J_{z,\mathrm{c}1}/J$ obtained by the level spectroscopy technique (stars) [via the central charge (pluses)] as obtained in (a) [(c)] vs the inverse of the system size squared at $D/J=1.5$ for $L$ up to 128. (c) Central charge $c^{*}$ of the 1D spin-1 $XXZ$ model (1) with $D/J=1.5$, indicating the EH-OH (OH-AFM) transition point with $c=1$ ($c=1/2$). The solid line denotes the EH-OH transition extracted from (a) and (b), which is in accordance with the position of the maximum in $c^{*}\left(L\right)$. Turning on a perturbation $\delta \widehat{\mathcal{H}}$ that breaks the lattice-inversion symmetry, the central charge $c^{*}\left(L\right)$ (solid symbols) becomes zero for large enough system sizes ($L\ge 64$).

Figure 5

Extrapolated data for the spin gap ${\Delta }_{\mathrm{s}}$ (squares) and neutral gap ${\Delta }_{\mathrm{n}}$ (open circles) as a function of $J_z/J$ for (a) $D/J=0$ and (b) $D/J=1.5$. The solid squares in (b) give ${\Delta }_{\mathrm{s}}$ with a finite inversion-symmetry-breaking perturbation $g/J=0.1$ [see Eq. (2)].

Figure 6

Entanglement spectrum ${\xi }_{\alpha }$ of the 1D spin-1 $XXZ$ model (1) with $D=1.5$. The fourfold degeneracy of the lowest entanglement level can be taken as an indication of a nontrivial Haldane state in the case of DMRG simulations with PBC. As the system size increases, the degeneracy appears in the whole HI phase; compare data for $L=64$ [(a)] with those for 128 [(b)]. Using OBC and taking spins ($S=1/2$) at the edges into account, an almost perfect double degeneracy is obtained for the OH phase even for small systems with $L=128$, as demonstrated in (c).

Figure 7

Entanglement spectrum ${\xi }_{\alpha }$ if an inversion-symmetry-breaking term is added to the spin-1 chain (1) with $D=1.5$, where $g/J=0.1$ (left panels) and $0.2$ (right panels). Data obtained by DMRG with PBC (upper panels) and OBC with half-spin edges (lower panels).

Figure 8

Entanglement spectrum ${\xi }_{\alpha }$ for (a) $D/J=2.88$, (b) $2.90$, and (c) $2.92$ close to the EH-OH-AFM multicritical point with $L=1024$ and OBC, showing the disappearance of the double degeneracy of the lowest entanglement levels.

Figure 9

Central charge $c^{*}$ of the 1D spin-1 $XXZ$ model (1), (a) near the multicritical point $D_{\mathrm{mc}}/J$ and (b) across the first-order EH-AFM transition for $D/J=3.7$. The dashed line in (b) denotes the EH-AFM transition point $D/J\simeq 3.9446$ according to Ref. [33].

Figure 10

DMRG ground-state phase diagram of the 1D spin-1 $XXZ$ model with single-ion anisotropy (1). Shown are the even Haldane (EH), odd Haldane (OH), and antiferromagnetic (AFM) phases. The EH-OH (squares) and OH-AFM (circles) transition points are determined from the central charge $c=1$ and $c=1/2$, respectively, which was extracted from the von Neumann entropy via Eq. (7). The EH-OH transition line was confirmed by a careful finite-size scaling of the two low-lying energy levels with TBC. The solid diamond gives the EH-OH-AFM multicritical point determined from the entanglement analysis. Error bars are smaller than symbols. The dashed (dotted) line denotes the MI-HI (HI-DW) transition in the EBHM with $n_b=2$ bosons per site (taken from Ref. [18]).

Figure 11

Intensity plots of the dynamical structure factor $S^{zz}(k,\omega )$ for (a) $J_z/J=1$, (b) $J_z/J=1.694\simeq J_{z,c1}/J$, (c) $J_z/J=1.9$, (d) $J_z/J=2.138\simeq J_{z,c2}/J$, and (e) $J_z/J=3$. Data are obtained by the DDMRG technique for $L=64$, using PBC and a Lorenzian broadening $\eta =0.1t$. Crosses give the maximum value of $S^{zz}(k,\omega )$ at fixed momenta $k=2\pi j/L$ with $j=1,...,L/2$.

Figure 12

Finite-size scaling of the excitation gaps at the Heisenberg point ($D=0$ and $J_z/J=1$).