Quantum phase transitions to topological Haldane phases in spin-one chains studied by linked-cluster expansions

We use linked-cluster expansions to analyze the quantum phase transitions between symmetry-unbroken trivial and topological Haldane phases in two different spin-one chains. The first model is the spin-one Heisenberg chain in the presence of a single-ion anisotropy, while the second one is the dimerized spin-one Heisenberg chain. For both models, we determine the ground-state energy and the one-particle gap inside the nontopological phase as a high-order series using perturbative continuous unitary transformations. Extrapolations of the gap series are applied to locate the quantum critical point and to extract the associated critical exponent. We find that this approach works unsatisfactorily for the anisotropic chain, since the quality of the extrapolation appears insufficient due to the large correlation length exponent. In contrast, extrapolation schemes display very good convergence for the gap closing in the case of the dimerized spin-one Heisenberg chain.

Figure 1

(i) The left part illustrates the Heisenberg chain in the presence of a single-ion anisotropy. Circles embody spin-one degrees of freedom, and black lines represent Heisenberg interactions with strength ${\lambda }_{\mathrm{ac}}$. The right part shows the energy spectrum of an isolated spin subject to a single-ion anisotropy. The three eigenstates are denoted by $|\alpha \rangle$, with $\alpha \in \{0,\pm 1\}$ labeling the quantum number of $S^z$. (ii) The left part illustrates the dimerized Heisenberg chain. The black box shows the dimer $d$ consisting of a left (red circles) and a right (white circles) spin one. Black lines represent interdimer Heisenberg interactions with strength ${\lambda }_{\mathrm{dc}}$. On the right, the energy spectrum of an isolated spin-one Heisenberg dimer is sketched. The lowest-energy eigenstate is the singlet $|s\rangle$. Excitations are triplet states $|t^{\alpha }\rangle$ with $\alpha \in \{0,\pm 1\}$ and quintuplet states $|q^{\beta }\rangle$ with $\beta \in \{0,\pm 1,\pm 2\}$.

Figure 2

Critical point ${\lambda }_{\mathrm{ac}}^{\mathrm{c}}$ as a function of the order $r$ extracted from different families of DlogPadé extrapolants $d\text{P}[L,M]$ with $L-M=\mathrm{const}$. The dashed horizontal line indicates ${\lambda }_{\mathrm{ac}}^{\mathrm{c}}=1.03258$ from Ref. [34]. Inset: Critical exponent $z\nu$ extracted from biased DlogPadé extrapolation using ${\lambda }_{\mathrm{ac}}^{\mathrm{c}}=1.03258$ in Eq. (11). The dashed horizontal line indicates $z\nu =1.472$ (taken also from Ref. [34]).

Figure 3

The critical point ${\lambda }_{\mathrm{dc}}^{\mathrm{c}}$ displayed as a function of the order $r$ in the upper (lower) panel using DlogPadé (Padé) extrapolation. Different symbols correspond to different families of extrapolants with $L-M=\mathrm{const}$. Inset: Critical exponent $z\nu$ extracted from DlogPadé extrapolation.

Figure 4

Linear scaling of the critical point ${\lambda }_{\mathrm{dc}}^{\mathrm{c}}$ as a function of $1/r$. Data points are the average of Padé extrapolants of the same order shown in Fig. 3. Error margins correspond to their variance. Solid and dashed lines are linear scalings through the data points and their variances, respectively.