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

TL;DR

This study explores how the phase diagrams of spin-cluster two-leg ladders depend on the parity of the number of plaquettes, revealing distinct phases and transitions for even and odd plaquette counts, with implications for experimental materials.

Contribution

It provides the first comprehensive analysis of parity-dependent phase diagrams in spin-cluster ladders using DMRG, revealing new phases and transition behaviors.

Findings

Even $n_p$ yields two phases: Haldane and cluster rung-singlet.

Odd $n_p$ yields four phases: cluster-singlet, cluster rung-singlet, Haldane, and even Haldane.

Phase regions evolve with increasing $n_p$, approaching parity-independent behavior.

Abstract

Motivated by the recent experiment on $\rm{K_2Cu_3O\left(SO_4\right)_3}$, an edge-shared tetrahedral spin-cluster compound [M. Fujihala \textit{et al.}, Phys. Rev. Lett. \textbf{120}, 077201 (2018)], we investigate two-leg spin-cluster ladders with the plaquette number $n_p$ in each cluster up to six by the density-matrix renormalization group method. We find that the phase diagram of such ladders strongly depends on the parity of $n_p$. For even $n_p$, the phase diagram has 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 the cluster-singlet phase and the even Haldane phase shrink as $n_p$ increases. We thus conjecture that in the large $n_p$ limit, the phase diagram will become independent of $n_p$. By analysing the ground-state energy and entanglement entropy we obtain the order of the phase transtions. In particular, for $n_p=1$ there is no phase transition between the even Haldane phase and the cluster-singlet phase while for other odd $n_p$ there is a first-order phase transition. Our work provides comprehensive phase diagrams for these cluster-based models and may be helpful to understand experiments on related materials.

Equations

$$\mathcal{H}=\sum _{k=1}^{L_c}{\mathcal{H}}_{\mathrm{intra}}^{(k)}+\sum _{k=1}^{L_c-1}{\mathcal{H}}_{\mathrm{inter}}^{(k:k+1)}$$

$${\mathcal{H}}_{\mathrm{intra}}^{(k)}$$

$$+J_c\sum _{j=1}^{n_p}\sum _{a=0,1}{\mathbf{S}}_{1,j+a}^{(k)}\cdot {\mathbf{S}}_{2,j+1-a}^{(k)}$$

$${\mathcal{H}}_{\mathrm{inter}}^{(k:k+1)}=J_{\mathrm{inter}}\sum _{i=1,2}{\mathbf{S}}_{i,n_p+1}^{(k)}\cdot {\mathbf{S}}_{i,1}^{(k+1)}$$

Full text

I Introduction

An integer is either even or odd, which is known as the parity. The properties of some physical systems associated with different parity may be fundamentally differentPink (1975); Dzero et al. (2003); Machens et al. (2013); Wei et al. (2015); Weichselbaum et al. (2018). A well-known example stems from Haldane’s conjectureHaldane (1983a, b) that the lowest excitation of spin chains with integer spin is gapful while those with half integer spin is gapless. Correspondingly, the lowest excitation of spin-$\frac{1}{2}$ ladders with even legs are gapful but those with odd legs are gaplessDagotto and Rice (1996); Oitmaa et al. (1996); Greven et al. (1996). This conjecture was soon confirmed by various numerical and experimental worksNightingale and Blöte (1986); Buyers et al. (1986); Renard et al. (1988); Golinelli et al. (1992); White (1996) and therefore the gapful phase is called Haldane phase. Recently, further theoretical works show that the Haldane phase in spin chains with odd integer spins and even integer spins are actually differentPollmann et al. (2012). The former one is protected by some symmetries, such as time-reversal, spacial inversion and dihedral symmetry but the latter is not although edge states may exist in both of them. Hereafter, following literature, we will just call the former one as Haldane $\left(\rm HP\right)$ phase but the latter one as even Haldane $\left(\rm EHP\right)$ phase. Now we know that the HP phase is actually a symmetry-protected topological phaseGu and Wen (2009); Pollmann et al. (2010); Schuch et al. (2011); Chen et al. (2013). These theoretical progresses have stimulated extensive efforts to search for such topologically nontrivial phase in quasi one-dimensional materials as well as artificial structures and the HP phase has been reported in a variety of experimentsPopović et al. (2004); Senko et al. (2015); Zhou et al. (2017); Nakagawa and Kawakami (2017); Xu et al. (2018).

Very recently, evidences for the HP phase in spin-cluster materials were first reported in the compound $\rm{K_{2}Cu_{3}O\left(SO_{4}\right)_{3}}$ by M. Fujihala et al.Fujihala et al. (2018). This compound consists of edge-shared tetrahedral spin clusters. Spin-$\frac{1}{2}$ $\rm{Cu^{2+}}$ sits at the corners of the tetrahedra. These clusters are connected via $\rm{SO_{4}^{2+}}$ along the $b$ axis. In other directions, they are connected via nonmagnetic ions or no exchange path is allowed and thus interactions can be neglected. These identify $\rm{K_{2}Cu_{3}O\left(SO_{4}\right)_{3}}$ as a quasi one-dimensional compound. Various experimental measurements in combination with theoretical analysisFujihala et al. (2018); Furrer et al. (2018) reveal that its ground state is an HP phase. In addition to $\rm{K_{2}Cu_{3}O\left(SO_{4}\right)_{3}}$, some other cluster-type one-dimensional materials have also been reportedJohnsson et al. (2000); Choi et al. (2009); Baek et al. (2012); Fujihala et al. (2013); Choi et al. (2014); Hase et al. (2017), such as $\rm{Cu_{2}Te_{2}O_{5}X_{2}}$ with $\rm{X=Cl,Br}$. These experiments call for a systematical investigation on the low-energy properties of spin-cluster ladders.

For this purpose, we study a Hamiltonian written as follows

[TABLE]

where $L_{c}$ is the number of clusters. The Hamiltonian has two parts, the first is the interaction within one cluster, and the other is the interaction between two nearest-neighbor clusters. Such intra-cluster and inter-cluster Hamiltonians are given byFujihala et al. (2018):

[TABLE]

[TABLE]

where $n_{p}$ is the number of the plaquettes within one cluster, which corresponds to the number of the tetrahedra within one cluster in compounds. ${\bf{S}}_{i,j}^{(k)}$ is the spin operator in the $k$th cluster with the leg index $i$ and rung index $j$. A schematic representation of the model and the couplings $J_{\perp},J_{\parallel},J_{c}$ and $J_{\rm{inter}}$ are plotted in Fig. 1. This model was proposedFujihala et al. (2018) for $\rm{K_{2}Cu_{3}O\left(SO_{4}\right)_{3}}$, where $n_{p}$ takes 2 and $J_{c}=J_{\parallel}$ due to the symmetry of a tetrahedron. Although in known compounds, $n_{p}$ is limited to 1 or 2, in our theoretical work we will consider general $n_{p}$ and extrapolate our conclusions to the large $n_{p}$ limit. Moreover, for simplicity, we assume that $J_{c}=J_{\parallel}$ is satisfied for all $n_{p}$ and set $J_{\rm{inter}}=1$ as the energy unit.

To study the low-energy properties of Hamiltonian $\left(\ref{HAM}\right)$, we resort to the state-of-art numerical algorithm, density-matrix renormalization group (DMRG)White (1992, 1993); Schollwöck (2005, 2011). To avoid edge effect, periodic boundary condition (PBC) is employed unless stated explicitly otherwise. We keep up to 3000 optimal bases thus the largest truncation error is smaller than $10^{-10}$. $U(1)$ symmetry is used to accelerate the computation therefore the Hamiltonian is diagonalized in the sector with fixed $z$-component of the total spin. All target states in the given sector are used with equal weight to construct the reduced density matrix. Several relevant quantities such as the ground state, the first excited state and corresponding energy $E_{0}$, $E_{1}$, the entanglement entropy and entanglement spectrum of the ground state are calculated. All the data in our figures are obtained in the sector with the $z$-component of the total spin zero. As we will show, this is enough for our model since all phases are gapful. And then the excitation gap $\Delta\equiv E_{1}-E_{0}$. To calculate the entanglement entropy and entanglement spectrum, we arrange the ladder into one chain in the rung-major order, and split the chain into two halves. One is the system and the other is the environment. After tracing out the freedom of the environment, we obtain the reduced density matrix $\rho$. The entanglement entropyEisert et al. (2010); Nishioka (2018) $S$ is then calculated by its definition $S=-\sum_{i}\rho_{i}\ln\rho_{i}$ with $\rho_{i}$ the eigenvalues of $\rho$. The entanglement spectrum $\xi_{i}=-\ln\rho_{i}$ is thus readily available.

We find that the phase diagrams of this model depend strongly on the parity of $n_{p}$. For even $n_{p}$, we have two phases, and for odd $n_{p}$, we have four phases. The particular features for $n_{p}=1$ and large $n_{p}$ are also discussed. The rest of the paper is organized as follows. In Sec. II, we present the phase diagrams for even $n_{p}$. In Sec. III, we present the phase diagrams for odd $n_{p}$. In Sec. IV, we show the results in the large $n_{p}$ limit and in Sec. V we conclude our work.

II Phase Diagram for Even $n_{p}$

In this section, we will discuss the phase diagrams for even $n_{p}$. In Fig. 2, we show our results for $n_{p}=2,4$ and $6$. We find two phases, which are an HP phase and a cluster rung-singlet (CRS) phase. The former is common in spin-1 chains and spin-$\frac{1}{2}$ two-leg laddersGlarum et al. (1991); Buyers et al. (1986); Chaboussant et al. (1998). The latter is a trivial product state of singlets. In this phase, $J_{\perp}$ dominates over $J_{\parallel}$, and each rung within a cluster (rung index $1<j\leq n_{p}$) forms a singlet. The four spins in the plaquette connecting the two nearest-neighbor clusters also form a singlet. In Appendix A, we provide some numerical evidences for the CRS phase.

The phase diagram and the phase boundary can be determined by the ground state and first excited state as well as by the entanglement entropy and entanglement spectrum. In the following we will illustrate our procedure that determines the phase boundary for $n_{p}=2$ and for other even $n_{p}$ it is similar. In Fig. 3 (a), we plot the energy of the ground state and the first excited state for $L_{c}=16$ as a function of $J_{\perp}$. An energy-level crossing occurs at $J_{\perp}=3.1571(2)$, and this signals a first-order phase transition between the HP phase and the CRS phase. Moreover, we calculate the excitation gap $\Delta$ and find that both phases are gapful. In Fig. 3(b), we show that the entanglement entropy $S$ for $L_{c}=16,24,32$ and $40$. In this case, the length of the system and the environment is equal. We observe a jump in $S$. This is interpreted as a first-order phase transition. After a finite-size extrapolation, the transition point is determined at $J_{\perp}=3.156(1)$. Such phase transition is also reflected in the entanglement spectrum. As we show in Fig.3 (c), the degeneracy of the entanglement spectrum is different in the two phases. In the HP phase, all the entanglement spectrum is even-fold degenerate, which is a characteristic feature of the symmetry-protected topological phaseLevin and Wen (2006); Kitaev and Preskill (2006); Li and Haldane (2008); Pollmann et al. (2010); Fidkowski (2010); Turner et al. (2010); Rao et al. (2014). However, in the CRS phase, some of the entanglement spectrum is even-fold degenerate and other is odd-fold degenerate.

III Phase Diagram for Odd $n_{p}$

When $n_{p}$ is odd the phase diagrams in Fig. 4 are different from those for even $n_{p}$. In addition to the HP phase and the CRS phase, we find two more phases. One is the cluster singlet (CS) phase and the other is the EHP phase. Similar to the method extracting the information of the CRS phase for $n_{p}=2$ in the appendix A, the properties of the CS phase can be determined as well from the entanglement entropy with various cuts. In the CS phase, the intra-cluster interaction $J_{\parallel}$ is dominant and each cluster is a singlet. The dominant term in the ground state is the product state of these singlets. Moreover, as $J_{\parallel}$ increases, quantum fluctuation around such product state becomes smaller. Therefore, $S$ of the cuts separating one cluster converges adiabatically to a nonzero constant while that of the cut separating two nearest-neighbor clusters converges to zero in the large $J_{\parallel}$ limit. As we explain in Sec. I, the EHP phase is different from the HP phase. Particularly, it is not protected by symmetries. Therefore, it can evolve into a product state accompanied with a phase transition or without a phase transitionPollmann et al. (2010). $n_{p}=3,5$ belong to the former case. There is a first-order phase transition between the EHP phase and the CRS phase. But $n_{p}=1$ belongs to the latter case, i.e., there is no phase transition between the EHP phase and the CRS phase, agreeing with previous worksBrenig and Becker (2001); Totsuka and Mikeska (2002). Even so, these two phases can be distinguished by their edge states. To show this, we calculate low-energy states under both PBC and open boundary condition (OBC). We find that under both PBC and OBC the ground state of the CRS phase is unique. The ground state of the EHP phase is unique under PBC but it is nine-fold degenerate under OBC, demonstrating the presence of a nearly free $S=1$ effective spin at each end of the ladder.

Similar to those with even $n_{p}$, the phase diagrams with odd $n_{p}$ are determined as well by $E_{0}$, $E_{1}$, $S$ and $\xi_{i}$. We demonstrate this for $n_{p}=3$ in Fig. 5 and Fig. 6. Let us first see Fig. 5. For simplicity, we fix $J_{\perp}=-3$. In panel (a), we show the entanglement entropy $S$ for $L_{c}=20,30$ and $40$ as a function of $J_{\parallel}$. Two sharp peaks are clearly observed, suggesting a phase transition near each of them. The accurate positions of the two critical points can be obtained by finite-size extrapolation, as we show in the inset for the right one. In the thermodynamic limit, they are $J_{\parallel}=0.216(2)$ and $1.406(2)$. Moreover, contrary to that in Fig. 3, there is no discontinuity in $S$, and this suggests a continuous phase transition. In panel (b), we show the excitation gap for various $L_{c}$ and two minimums are found. In the inset, we show the extrapolation of the excitation gaps at $J_{\parallel}=0.216$ and $J_{\parallel}=1.406$. In the thermodynamic limit they become zero. We also confirm that the phases are gapful. These results are in good agreement with those obtained from the entanglement entropy. Since a characteristic feature of the HP phase is the even-fold-degenerate entanglement spectrum, we calculate them with equal system length and environment length. As we show in panel (c), the spectrum is indeed even-fold degenerate in the HP phase but in the EHP phase and CS phase they are not. Moreover, we calculate the degeneracy of the ground state in the HP phase under both OBC and PBC. We find that the ground state is unique under PBC but it is 4-fold degenerate under OBC. These provide further information supporting our phase diagrams.

In Fig. 6, the two lowest energies are shown near the phase-transition points. $J_{\parallel}=0.4,1.0$ and $3.0$ correspond to the EHP to CRS, HP to CRS and CS to CRS transitions, respectively. In all three cases, a level-crossing occurs, suggesting a first-order phase transition.

IV Large $n_{p}$ Limit

Now, we have the phase diagrams for various $n_{p}$. From Fig. 2 and 4, it is obvious that these phase diagrams depend on the parity of $n_{p}$. For even $n_{p}$, the phase diagram has an HP phase and a CRS phase. But for odd $n_{p}$, there are four phases, an HP phase , a CRS phase , a CS phase and an EHP phase. Moreover, our results show that the region of the CS and EHP phases becomes smaller as $n_{p}$ increases. Therefore, we expect that in the large $n_{p}$ limit the phase diagrams will include only an HP phase and a CRS phaseE (11). For small $n_{p}$, Hamiltonian (3) can not be neglected even if $J_{\rm{inter}}$ is much smaller in comparison with $J_{\perp}$ or $J_{\parallel}$ because it connects the two nearest-neighbor clusters. However, in the large $n_{p}$ limit, the bulk properties are determined solely by Hamiltonian (2). Hamiltonian (3) only have some edge effect and thus can be neglected.

First, we consider the properties of Hamiltonian (2) and try to gain some insight from them. Actually, Hamiltonian (2) in the large $n_{p}$ limit has been extensively studiedGelfand (1991); Bose and Gayen (1993); Xian (1995); Weihong et al. (1998); Hakobyan et al. (2001); Liu et al. (2008); Li and Lin (2012); Li et al. (2012); Barcza et al. (2012) and its phase diagram is already known. It includes an HP phase and a rung-singlet(RS) phase. A first-order phase transition is exactly knownXian (1995) to occur at $J_{\perp}/J_{\parallel}=1.401\cdots$.

To verify our analysis, let us examine the phase boundary in Fig. 2 and 4. We notice that when $J_{\perp}$ is large enough the phase boundary is almost linear to $J_{\perp}$. This means that $J_{\perp}/J_{\parallel}$ is nearly a constant. In the large $n_{p}$ limit, we also expect that the phase transition depends only on $J_{\perp}/J_{\parallel}$. In Fig. 7, we extrapolate $J_{\perp}/J_{\parallel}$ as a quadratic polynomial of $1/n_{p}$ to the large $n_{p}$ limit and obtain $J_{\perp}/J_{\parallel}=1.40(3)$, and this agrees well with the expected transition point $J_{\perp}/J_{\parallel}=1.401\cdots$. Moreover, in the large $n_{p}$ limit, the edge configuration can be neglected and CRS phase becomes RS phase. These provide strong numerical evidence supporting our analysis.

V Conclusions

In conclusion, we study cluster-based two-leg ladders with the plaquette number $n_{p}$ up to six. These models are direct extensions of the model proposed for an edge-shared tetrahedral spin-cluster compound $\rm{K_{2}Cu_{3}O\left(SO_{4}\right)_{3}}$. The numerically exact ground-phase phase diagrams are mapped out by using density-matrix renormalization group method. We find that they are closely associated with the parity of $n_{p}$. For even $n_{p}$, there are two phases in the phase diagram, which includes an HP phase and a CRS phase. For odd $n_{p}$, in addition to the HP phase and the CRS phase, there are two more phases, which are a CS phase and an EHP phase. Moreover, the region of such two phases shrinks as $n_{p}$ increases, which leads to our conjecture that in the large $n_{p}$ limit they may disappearE (11). By extrapolating the phase transition points to the large $n_{p}$ limit, we can reproduce the phase transition point of the Hamiltonian without the intercluster coupling (i.e. $J_{\rm{inter}}=0$), which verifies our conjecture. By analysing the energy and entanglement entropy, we determine the order of the phase transitions. The transition from HP phase to CS phase or to EHP phases in odd $n_{p}$ are continuous. There is no phase transition from the EHP phase to the CRS phase for $n_{p}=1$, and all other phase transitions are first order.

It was argued in Ref. Fujihala et al., 2018 that there may be an HP phase for all even $n_{p}$. Our work show that an HP phase is present in the phase diagram for all $n_{p}$. This can be understood from the Hamiltonian (2) in the large $n_{p}$ limit. In such case, there is an HP phase in the Hamiltonian (2), no matter $n_{p}$ is even or odd. This phase may persist after turning on the $J_{\rm{inter}}$ because it does not break the time-reversal symmetry.

Acknowledgments

We acknowledge useful discussion with Zheng-Xin Liu, Gao-Yang Li, Chong Chen and Chen Cheng. This work was supported by NSFC(Grants No. 11874188, 11834005, 11674139, 11704166, 11474029), PCSIRT (Grant No. IRT-16R35) and the Fundamental Research Funds for the Central Universities.

Appendix A Characteristics of the CRS phase from the entanglement entropy

In this Appendix, we demonstrate some properties of CRS phase in Fig. 2 from the aspect of the entanglement entropy of four different cuts. These cuts are readily available in the DMRG sweeps. For simplicity, we use $n_{p}=2$ to illustrate our analysis and for other $n_{p}$(both even and odd), it is similar. Moreover, the same analysis is applicable to the CS phase. In the following, we focus on the CRS phase only. As we show in Fig. 8, the entanglement entropy $S$ of the cut $B,C$ and $D$ in CRS phase is almost independent of $J_{\parallel}$. In particular, $S$ of the cut $C$ is nearly zero. This suggests that the ground state is a product state of the system and the environment. Moreover, $S$ of the cut $D$ is nearly $\ln 2$, which is just that of a singlet formed by two spin-$\frac{1}{2}$’s. After considering the symmetry, we may conclude that the two spin-$\frac{1}{2}$’s connected by the rung cutted by $D$ forms a singlet. We notice that $S$ of the cut $B$ is nearly $\ln 2$. This suggests that the rightmost two spins of the left cluster and the leftmost two spins of the right cluster form a singlet. However, it is not a product state of two rung singlets, which becomes clear when we study the $S$ of the cut $A$. To reflect such difference from the RS phase, we use CRS to mark this phase.

The nearly zero $S$ of the cut $C$ suggests that the largest $\rho_{i}$ is almost $1$ and all others are nearly zero. In the DMRG simulations, it is thus difficult to select optimal bases according to $\rho_{i}$. To obtain accurate results, more bases should be kept, in particular, for excited states.