Parity dependent phase diagrams in spin-cluster two-leg ladders

Motivated by the recent experiment on ${\mathrm{K}}_2{\mathrm{Cu}}_3\mathrm{O}{\left({\mathrm{SO}}_4\right)}_3$, an edge-shared tetrahedral spin-cluster compound [M. Fujihala, T. Sugimoto, T. Tohyama, S. Mitsuda, R. A. Mole, D. H. Yu, S. Yano, Y. Inagaki, H. Morodomi, T. Kawae, H. Sagayama, R. Kumai, Y. Murakami, K. Tomiyasu, A. Matsuo, and K. Kindo, Phys. Rev. Lett. 120, 077201 (2018)], we investigate two-leg spin-cluster ladders with the plaquette number $n_p$ in each cluster up to 6 by the density-matrix renormalization-group method. We find that the phase diagrams of such ladders strongly depend on the parity of $n_p$. For even $n_p$, the phase diagrams have two phases; one is the Haldane phase, and the other is the cluster rung-singlet phase. For odd $n_p$, there are four phases, which are a cluster-singlet phase, a cluster rung-singlet phase, a Haldane phase, and an even Haldane phase. Moreover, in the latter case the region of the Haldane phase increases while that of the cluster-singlet phase and the even Haldane phase shrinks as $n_p$ increases. We thus conjecture that in the large $n_p$ limit the phase diagrams will become independent of $n_p$. By analyzing the ground-state energy and entanglement entropy we obtain the order of the phase transitions. In particular, for $n_p=1$ there is no phase transition between the even Haldane phase and the cluster rung-singlet phase while for other odd $n_p$ there is a first-order phase transition. Our paper provides comprehensive phase diagrams for these cluster-based models and may be helpful to understand experiments on related materials.

Figure 1

The sketch of the cluster-based spin ladder. The interactions corresponding to the model are marked. We have assumed that $J_c=J_{\parallel }$ and therefore the same color represents the same interaction strength. $n_p$ is the number of plaquettes in a single cluster and $k$ is the index of the cluster. ${\mathbf{S}}_{i,j}^{\left(k\right)}$ represents the spin operators in cluster $k$ with the leg index $i$ and rung index $j$, thus $1\le j\le n_p+1$.

Figure 2

Phase diagrams of Hamiltonian (1) for $n_p=2,4,6$. There are an HP phase and a CRS phase in each phase diagram (see text for details). The transition between the two phases is of the first order. The red solid line is the eye guide of the phase boundary. It is obvious that the phase boundary depends almost linearly on $J_{\perp }$ and as $n_p$ increases the slope of the phase boundary decreases. Along the dashed cyan line ($J_{\parallel }=0$) the rungs within a cluster are decoupled.

Figure 3

A phase transition for $n_p=2$ is determined from $E_0$, $E_1$, $S$, and ${\xi }_i$. Here $J_{\parallel }=3$ is fixed in our calculation, and $S$ and ${\xi }_i$ are obtained with equal system and environment sizes. $\left(\mathrm{a}\right)E_0$ and $E_1$ for $L_c=16$ are shown as a function of $J_{\perp }$. Level crossing occurs at $J_{\perp }=3.1571\left(2\right)$, indicating a first-order phase transition. $\left(\mathrm{b}\right)S$ for the cluster number $L_c=16,24,32,40$ is shown as a function of $J_{\perp }$. The jump in $S$ indicates a first-order phase transition. Inset: Finite-size extrapolation to determine the transition point $J_{\perp }=3.156\left(1\right)$ in the thermodynamic limit. $\left(\mathrm{c}\right)$ Entanglement spectrum ${\xi }_i$ and corresponding degeneracy for $L_c=40$ in the HP phase and the CRS phase are shown. In the HP phase the entanglement spectrum is evenfold degenerate, which is a characteristic feature of the symmetry-protected topological phase. In the CRS phase the lowest entanglement spectrum is nondegenerate and it is nearly zero, suggesting that the ground state is a product state of the system and the environment.

Figure 4

Phase diagrams for $n_p=1,3,5$. The blue lines between the CS phase and the HP phase, and between the HP phase and the EHP phase, represent continuous phase transitions. The dashed cyan line in (a) represents a crossing between EHP phase and CRS phase instead of a phase transition. It is determined from the edge states under OBC. Along the dashed purple line ($J_{\parallel }=0$), the rungs are decoupled. Other lines denote first-order phase transitions.

Figure 5

$S$, $\mathrm{\Delta }$, and ${\xi }_i$ for $n_p=3$ are plotted as a function of $J_{\parallel }$. In our calculation, $J_{\perp }=-3$ is fixed, and $S$ and ${\xi }_i$ are obtained with equal system and environment sizes. $\left(\mathrm{a}\right)S$ for $L_c=20$, 30, and 40. The two peaks in $S$ suggest two phase transitions. Inset: The positions of the right peak are extrapolated to the thermodynamic limit. $\left(\mathrm{b}\right)$ The excitation gap for $L_c=20$, 30, and 40. Inset: In the thermodynamic limit, the excitation gaps at the two dips close. Red filled circles, $L_c$ from 20 to 40 with step 5; blue open circles, $L_c$ from 20 to 80 with step 10. $\left(\mathrm{c}\right)$ The entanglement spectrum and corresponding degeneracy for $L_c=40$ in three phases. In the HP phase, they are evenfold degenerate.

Figure 6

$E_0$ and $E_1$ for $L_c=8,n_p=3$ are plotted as a function of $J_{\perp }$. The level crossing suggests a first-order phase transition.

Figure 7

$J_{\perp }/J_{\parallel }$, the estimated phase boundary in the large $J_{\perp }$ limit, is plotted as a function of $1/n_p$. The blue solid line is a quadratic polynomial fitting of the data. We have $J_{\perp }/J_{\parallel }=1.40\left(3\right)$ as $n_p\to \infty$, which is well consistent with expected result [53] $J_{\perp }/J_{\parallel }=1.401\cdots$.

Figure 8

Entanglement entropy corresponding to four different system-environment cuts $A,B,C,D$ is shown for $J_{\parallel }=0.02,1.0$, and 3.0. $n_p=2$, $L_c=16$, and OBC are used to illustrate our analysis. The cut $A$ separates the ladder into two equal parts. The HP phase (left) and the CRS phase (right) are separated by a jump of the entanglement entropy, which signals a first-order phase transition. The black dashed line below $S=1$ is at $S=ln2$ and that above $S=1$ is at $2ln2$.