Resonating dimer-monomer liquid state in a magnetization plateau of a spin-$\frac{1}{2}$ kagome-strip Heisenberg chain

Highly frustrated spin systems such as the kagome lattice (KL) are a treasure trove of new quantum states with large entanglements. We thus study the spin-$\frac{1}{2}$ Heisenberg model on a kagome-strip chain (KSC), which is one-dimensional KL, using the density-matrix renormalization group (DMRG) method. Calculating central charge and entanglement spectrum for the KSC, we find a novel gapless spin liquid state with doubly degenerate entanglement spectra in a 1/5 magnetization plateau. We also obtain a gapless low-lying continuum in the dynamic spin structure calculated by dynamical DMRG method. We propose a resonating dimer-monomer liquid state that would meet these features.

Highly frustrated spin systems such as the kagome lattice (KL) are a treasure trove of new quantum states with large entanglements. We thus study the spin- 1 2 Heisenberg model on a kagomestrip chain (KSC), which is one-dimensional KL, using the density-matrix renormalization group (DMRG) method. Calculating central charge and entanglement spectrum for the KSC, we find a novel gapless spin liquid state with doubly degenerate entanglement spectra in a 1/5 magnetization plateau. We also obtain a gapless low-lying continuum in the dynamic spin structure calculated by dynamical DMRG method. We propose a resonating dimer-monomer liquid state that would meet these features.
Quantum entanglement is a very important concept for research such as quantum information and quantum magnetism [1]. Low-dimensional quantum spin systems are attracting attention owing to emerging their ground states with strong quantum entanglement. Using quantum entanglement analysis for quantum spin models has recently attracted attention because it is able to characterize various phases [2][3][4][5][6][7][8][9][10], even in the states without the translational symmetry breaking and any longrange-dipole orders, such as the Haldane state in integer spin chains [11] and the Tomonaga-Luttinger liquid (TLL) state in half-integer spin chains. According to the conformal field theory, the central charge c charactering excitation properties can be obtained by calculating the entanglement entropy (EE) [2][3][4][5][6]. In the TLL state, there is a gapless excitation with a non-zero integer value of c, while in the Haldane state, there is a gapped excitation with c = 0. Moreover, the Haldane state can be characterized by the degeneracy of entanglement spectrum (ES) [7][8][9][10]. For example, the spin-1 chain exhibits a doubly degenerate ES [7].
In this Letter, we investigate the 1/5 plateaus of the KSC using the density-matrix renormalization group (DMRG) method. We determine a magnetic phase diagram of the 1/5 plateau and we find two new plateau phases that have not been identified in the previous study [35]. Our main result is that, even though one of the two new phases exhibits a gapless spin liquid behavior with c = 1, the ES of the phase is doubly degenerate. This means that this plateau phase has both properties of half-integer spin chain and spin-1 chain. Furthermore, we calculate the dynamical spin structure factor (DSSF) in this phase by mean of the dynamical DMRG (DDMRG) [38]. The DSSF of the S z (magnetic field) direction exhibits gapless and dispersionless low-energy excitations. In order to describe these novel properties in the new phase, we propose resonating dimer-monomer liquid (RDML) state that is a mixed state of singlet dimers and up-spin monomers.
The Hamiltonian for the spin-1 2 KSC in magnetic field is defined as where S i is the spin-1 2 operator, i, j runs over the nearest-neighbor spin pairs, J i,j corresponds to one of J X , J 1 , and J 2 in Fig. 1, and h is the magnitude of magnetic field. In the following we set J X = 1 as energy unit. We perform DMRG calculations at zero temperature are less than 5 × 10 −7 . For the calculation of the dynamical spin structure factors, we use DDMRG (for details, see Ref. [38]) for N = 60 and N = 80 under the periodic boundary condition (PBC). In our DDMRG calculations, m is set to be 1000. We first determine the phase diagram of the KSC at M/M sat = 1/5. We obtain six 1/5 plateau phases using N = 120 − 1000 clusters under the open boundary condition (OBC) as shown in Fig. 2(a). The regions of the phases I-VI are denoted by different colors. The white region has no magnetization plateau. The phases II and III are found in the present study, while other phases have already been found in the previous study [35]. Figure 2(b) shows nearest-neighbor spin-spin correlation S i · S j − S z i S z j and local magnetization S z i for the six phases. The lines connecting two nearest-neighbor sites denote the sign and magnitude of spin-spin correlation by color and thickness, respectively. The circle on each site represents S z i . The stability of the phases I, IV, V, and VI can be explained by energy gain due to local strong spin correlation as evidenced from thick lines in Fig. 2(b). On the other hand, the phases II and III do not have such a distinguished thick line. In the phase III, the spin-spin correlation is almost uniform and there is no symmetry breaking. In the phase II, periodic magnetic structure is not seen (see Supplemental Material [39]). Figure 3 shows magnetization curves at J 1 = 1.0 and J 2 = 0.9 (phase II), and J 1 = 0.9 and J 2 = 0.3 (phase III). The 1/5 plateaus are clearly visible in both conditions. These magnetization curves show little variation with size N and boundary conditions. However, in Fig. 3 (a) (phase II), a plateau deviating from 1/5 just one step, that is, M/M sat = 1/5 + 2/N , appears under the OBC. We confirmed that this deviation appears in other sets of J 1 and J 2 on the phase II, implying the presence of edge excitations that appear in the Haldane chain. The phases I, IV, V, and VI have been examined in the previous study. Therefore, we investigate the phases II and III in the following. denoting the position of the 5-site unit. This plot comes from the following relation that holds between the central charge c and the position-dependent EE, EE(j), where a c is a nonuniversal constant, and b c = 6 (3) for the OBC (PBC) [3][4][5][6]. The value of c becomes finite when the spin-spin correlation exhibits power-law decays, and gives c = 1 in the TLL. On the other hand, when spinspin correlation decays exponentially, for example, in the Haldane phase, c = 0. The value of c in Fig. 4 is obtained by fitting the EE data using straight lines. Since the c for the phase II is nearly unity as in the TLL, the phase II is expected to be the gapless spin liquid. Since plateau phases should have an energy gap, there should be a gapless excitation only in the subspace where total S z does not change. This feature has also been observed in the 1/3 magnetization plateau in frustrated three-leg spin tubes [40][41][42]. We also confirm that the spin-spin correlations of the S z and S x components exhibit power decay corresponding to the gapless excitation and exponential decay corresponding to the energy gap, respectively (see Supplemental Material [39]).
In the phase III, c is nearly zero. This indicates that there is an energy gap in this phase. Moreover, we confirm that the ground state has no degeneracy and there is no symmetry breaking as shown in Fig. 2(b).
In order to investigate the phase II in more detail, we π π calculate the DSSF, S αβ (q, ω) defined by where q is the momentum for the lattice geometry shown in Fig. 1, |0 is the ground state with energy E 0 , and η is a broadening factor.S with x i being the position of spin i and α(β) = +, −, z. Figure 5 shows S zz (q, ω) corresponding to the S z component and [S +− (q, ω)+S −+ (q, ω)]/2 corresponding to the S x and S y components for L = 12 (N = 5 × 12) with η = 0.05 under the PBC. Since the position x i is defined in Fig. 1, the range of 0 ≤ q/π ≤ 4 corresponds to a half of the extended Brillouin zone. As shown in Fig. 5(a), a gapless excitation in the S z component emerges around q/π = 2. This excitation is consistent with the result expected from the fact that c = 1. We confirm that the value of q showing the lowest-energy excitation is size dependent. As the size is increased, the q value tends to move toward 2π (see Supplemental Material [39]). We thus expect that in the thermodynamic limit, a gapless excitation emerges at q/π = 2. In addition, low energy excitations at ω 0.2 show less dispersive feature. This feature will be discussed later.
As shown in Fig. 5(b), [S +− (q, ω)+S −+ (q, ω)]/2 corresponding to the excitations of the S + and S − components, has a gap. Here, the external magnetic field h is set to be at the center of the 1/5 magnetization plateau. The minimum excitation gap is found to exist around q/π = 2 and q/π = 3. Continuous excitations exist up to the high energy region more than ω = 1.5 at q/π = 2.
Finally, we calculate the ES to investigate of each phase in more detail. The ground state can be Schmidt decomposed as follows; where Φ L α and Φ R α are orthonormal basis vectors of the left and right part of the chain, respectively [7]. The λ 2 α are the eigenvalues of the reduced density matrix. The ES is defined as −2 ln(λ α ). Figure 6 shows the results of the ES as a function of J 2 at J 1 = 0.9. The phases I, III and V are trivial phases due to the mixture of singly and doubly degenerate states of the ES [7]. Whereas, in the phase II, all the ES are doubly degenerate. This is identical to the feature of the spin-1 Haldane chain. Therefore we conclude that the phase II is a non-trivial topological phase [7]. We also confirm that the double degeneracy is independent of the L, J 1 , and J 2 . The degeneracy is obtained even for L = 2 (the shortest chain). The phases II and III are newly identified magnetization plateau phases in the present study. The phase III is concluded to be a trivial phase based on i) c = 0, ii) no degeneracy in the ground state, and iii) trivial distribution of the ES. On the other hand, the phase II is a novel phase that has the characteristics of both the gapless spin liquid and the spin-1 Haldane state, because of c = 1 and the double degeneracy of the ES. Why does the phase II have these characteristics? We anticipate that a mixed state of singlet dimers and up-spin monomers, which reveals a liquid behavior like the RVB, is the ground state of the phase II. We refer to this state as RDML. The states in Fig. 7 correspond to snapshots of the RDML state. They have one monomer in all units except at both ends, and there is no nearest-neighboring monomer. The ES in these states shows double degeneracy. The RDML state whose typical components are shown in Fig. 7 can generate infinitesimal energy excitations by swapping any dimers with monomers. This corresponds to local excitations, which form a dispersionless structure in the S zz (q, ω). The DMRG result shown in Fig. 5 confirms this property. As the size is increased, the dispersionless excitations are more pronounced around q/π = 2 (see Supplemental Material [39]). Assuming the RDML state, we can explain the gapless exsitetion, dou- ble degeneracy of the ES, and dispersionless low-energy excitations in the S zz (q, ω). Therefore, we believe that the phase II is the RDML state.
In summary, we obtained the phase diagram of the 1/5 plateaus of the KSC using the DMRG method. We found two new plateau phases (II and III). The most surprising result is that the phase II exhibits the gapless spin liquid with c = 1 as in the half-integer spin chain and the double degeneracy of the ES as in the spin-1 chain. We also calculated the DSSF in the phase II using the DDMRG. We found that the DSSF of the S zz (q, ω) exhibits the gapless excitation corresponding to c = 1 and dispersionless lowenergy excitations corresponding to the local excitations. Finally, we proposed the RDML state that can explain the gapless exsitetion, double degeneracy of the ES, and dispersionless low-energy excitations in the S zz (q, ω) in the phase II.
The KSC compounds with five exchange interactions have already been reported [43]. Therefore, there is a possibility that compounds with our model will be synthesized in the future. Since M/M sat = 1/5 is a relatively low magnetization, the 1/5 plateau may be observed experimentally. Accordingly the experimental study of the KSC will be vigorously pursued in the future.
We acknowledge discussions with Gonzalo Alvarez from the Center For Nanophase Materials Sciences at ORNL. This work was supported in part by MEXT as a social and scientific priority issue "Creation of new functional devices and high-performance materials to support next-generation industries" (CDMSI) to be tackled by using post-K computer and by MEXT HPCI Strategic Programs for Innovative Research (SPIRE) (hp190198). The numerical calculation was partly carried out at the K Computer and HOKUSAI, RIKEN Advanced Institute for Computational Science, the facilities of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo, and the Information Technology Center, The University of Tokyo. This work was also supported by the Japan Society for the Promotion of Science, KAK-ENHI (Grants No. 19H01829 and No. JP19H05825) Here, we show the results of the calculations of the local magnetization (LM) (Fig. S1), spin-spin correlations (SSCs) (Fig. S2), and dynamical spin structure factor (DSSF) (Fig. S3) in the phase II at J 1 = 1.0 and J 2 = 0.7 (see Fig. 1 in the main text). In our density-matrix renormalization group (DMRG) calculation for the LM and SSCs, we use the kagome-strip chain (KSC) under the open boundary condition (OBC) for L = 200 (total number of sites is N = 5 × L; see Fig. 1 in the main text). In order to obtain sufficiently accurate results, we set the number of states m to be 2500 and resulting truncation error is less than 10 −9 .
Figures S1(a) and S1(b) show the LM S z i at sites along the upper edge and at central sites in the KSC, respectively. Both ends of upper edge (i = 1 and i = 400) have almost full moment S z i=1(400) = 0.42, corresponding to the edge excitation. We note that the LM at sites along the lower edge is the same as those along the upper edge plotted in Fig. S1(a). This means that there is no spontaneous breaking of the mirror symmetry with respect to the central axis along the chain. Plateau phases usually have some long-range orders such as an up-updown structure in the zigzag spin chain, while the phase II is completely different. From the results in Figs. S1(a) and S1(b), we can see that the phase II is not a simple ordered state but a spin-density-wave-like state.
To investigate the phase II that has both gapless excitation and magnetization plateau in detail, we calculate the SSCs of the S z and S x components, which are defined by S z i S z j − S z i S z j and S x i S x j , respectively. The SSCs of the S z and S x components exhibit power decay corresponding to the gapless excitation with c = 1 as shown in Figs. S2(a) and S2(b) and exponential decay corresponding to the magnetization plateau as shown in Fig. S2(c) and S2(d), respectively. These results are consistent with the fact that the phase II has c = 1 and the magnetization plateau.
Finally, to confirm the finite size effect for the DSSF S zz (q, ω), we calculate S zz (q, ω) of the KSC under the periodic boundary condition (PBC) for L = 12 and L = 16 using dynamical DMRG (DDMRG). Figure S3 shows these results around q/π = 2 in the low-energy region. The value of q showing lowest-energy excitation for L = 16 (2.25π) is closer to 2π than that for L = 12 (2.33π). Furthermore, we find that low-energy excitations for L = 16 are lower in energy and flatter than those for L = 12. We thus expect that in the thermodynamic limit, gapless excitation occurs at q/π = 2 accompanied by flat-band excitations around q/π = 2.