Ground-State Phase Diagram of (1/2,1/2,1) Mixed Diamond Chains

We start with several limiting cases where we can examine the ground state analytically. Before going into the discussion of specific cases, we briefly introduce the Lieb-Mattis theorem which is useful to determine the spontaneous magnetization in the absence of frustration. Let us consider a Heisenberg model

defined on a lattice consisting of two sublattices A and B. Here, $\mib S_{i}$ is the spin operator on the $i$-th site with arbitrary magnitude. It is assumed that the magnetic interactions $J_{ij}$ satisfy the following condition:

1. 1.

   If two spins are on different sublattices, then $J_{ij}\geq 0$.

2. 2.

   If two spins are on the same sublattice, then $J_{ij}\leq 0$.

3. 3.

   All spins are connected by magnetic interaction.

These assumptions imply the absence of frustration. Then, the spontaneous magnetization $M_{\rm sp}$ of the ground state is given by $M_{\rm sp}=|S_{\rm A}-S_{\rm B}|$, where $S_{\rm A}$ and $S_{\rm B}$ are the sums of the magnitudes of the spins on the A-sublattice and B-sublattice, respectively.[8]

If $-1<\delta<1$, the system is unfrustrated and the ground state is the QF phase with $m_{\rm sp}=1$ according to the LM theorem.[8] In this case, the sublattice A consists of the sites occupied by $\mib S_{l}$, and the sublattice B, those occupied by $\mib\tau_{l}^{(1)}$ and $\mib\tau_{l}^{(2)}$. Hence, $S_{\rm A}=L/2$ and $S_{\rm B}=3L/2$ which gives $M_{\rm sp}=L$ and $m_{\rm sp}=M_{\rm sp}/L=1$. The numerical analysis in Sect. 5 shows that this phase survives even in the weakly frustrated regime of small $\lambda$. Hereafter, this phase is called the LM1 phase. Schematic spin configuration is presented in Fig. 2(a).

If $\lambda>0$, the ground state is the QF phase with $m_{\rm sp}=1$ according to the LM theorem. In this case, the sublattice A consists of the sites occupied by $\mib\tau_{l}^{(1)}$, and the sublattice B, those occupied by $\mib S_{l}$ and $\mib\tau_{l}^{(2)}$. Hence, $S_{\rm A}=L/2$ and $S_{\rm B}=3L/2$ which gives $M_{\rm sp}=L$ and $m_{\rm sp}=M_{\rm sp}/L=1$. The numerical analysis in Sect. 5 shows that this phase survives even in the weakly frustrated regime of small $1-\delta$. Hereafter, this phase is called the LM2 phase. The schematic spin configuration presented in Fig. 2(b) demonstrates that this phase is distinct from the LM1 phase. This is also numerically confirmed in Sect. 5.

In the limit $\lambda\rightarrow\infty$, all pairs of $\mib\tau^{(1)}_{l}$ and $\mib\tau^{(2)}_{l}$ form doublet states $\left|\left|{T_{l},T_{l}^{z}}\right.\right\rangle$ with $T_{l}=1/2$ and $T_{l}^{z}=\pm 1/2$ where $\mib T_{l}\equiv\mib\tau^{(1)}_{l}+\mib\tau^{(2)}_{l}$. They are expressed using the basis $\left|{{\tau}^{(1)z}_{l},{\tau}^{(2)z}_{l}}\right\rangle$ as

For large but finite $\lambda$, $\mib T_{l}$ and $\mib S_{l^{\prime}}$ ($l^{\prime}=l$ or $l+1$) are coupled by an effective Heisenberg interaction $J_{\rm eff}$. If $J_{\rm eff}$ is antiferromagnetic, the ground state is the nonmagnetic TLL state as schematically shown in Fig. 2(c). If it is ferromagnetic, the ground state is the QF state with $m_{\rm sp}=1$ as shown in Fig. 2(b’). This ground state configuration is continuously deformed to that of the LM2 phase depicted in Fig. 2(b) by reducing the coefficient of $\left|{0,1/2}\right\rangle$ in Eq. (3) continuously. Hence, this QF state also belongs to the LM2 phase. We estimate $J_{\rm eff}$ within the first order perturbation calculation with respect to $(1\pm\delta)/\lambda$ as

Hence, we find the phase boundary between the TLL phase and the LM2 phase is given by $\delta=0.6$ for large enough $\lambda$.

The system is unfrustrated and the ground state is the nonmagnetic phase with $m_{\rm sp}=0$ according to the LM theorem. In this case, the sublattice A consists of the sites occupied by $\mib\tau_{l}^{(2)}$, and the sublattice B, those occupied by $\mib S_{l}$ and $\mib\tau_{l}^{(1)}$. Hence, $S_{\rm A}=L$ and $S_{\rm B}=L$ which gives $M_{\rm sp}=0$ and $m_{\rm sp}=M_{\rm sp}/L=0$. According to the numerical calculation of Sect. 5, this phase continues to the TLL phase discussed in Sect. 3.4.

For $1+\delta=\lambda=0$, $\mib S_{l}$ and $\mib\tau_{l}^{(2)}$ form a ferrimagnetic chain with $m_{\rm sp}=1/2$, and $\mib\tau^{(1)}_{l}$ are free spins with magnitude $1/2$. We take the zeroth order Hamiltonian as

and the total spin of ${\mathcal{H}}_{0}$ is defined by

The perturbation part of the Hamiltonian is rewritten as,

Within the subspace of the ferrimagnetic ground states of ${\mathcal{H}}_{0}$ with ${S_{0}^{\rm tot}}=L/2$, each ground state is specified by ${{S_{0}^{\rm tot}}}^{z}=-L/2,...,L/2$. From rotational symmetry, the two-spin effective Hamiltonian ${\mathcal{H}}_{l}^{\rm eff}$ for $\mib S^{\rm tot}_{0}$ and $\mib\tau^{(1)}_{l}$ is written down by their inner product as

It should be noted that the terms of higher powers of $\mib S_{0}^{\rm tot}\mib\tau^{(1)}_{l}$ are unnecessary since $\mib S_{0}^{\rm tot}\mib\tau^{(1)}_{l}$ can take only two values ${S}_{0}^{\rm tot}/2$ and $-({S}_{0}^{\rm tot}+1)/2$. Then, the total effective Hamiltonian ${\mathcal{H}}^{\rm eff}$ is given by

To determine $J_{\rm eff}^{\prime}$, we estimate the expectation values of ${\mathcal{H}}_{1}$ and ${\mathcal{H}}^{\rm eff}$ in the state $\left|{F}\right\rangle=\left|{S^{\rm tot}_{0}=L/2;S^{{\rm tot}z}_{0}=L/2}\right\rangle$ to find

Comparing (11) and (12), we find

The expectation values $\left\langle{F}\right|{S}^{z}_{l}\left|{F}\right\rangle$ and $\left\langle{F}\right|{\tau}^{(2)z}_{l}\left|{F}\right\rangle$ are calculated by the numerical diagonalization for $L=2,4,6,12$ and 14. After two steps of Shanks transformation,[14] we find

Thus, we can determine the phase boundary between the LM1 phase and the nonmagnetic phase as $\lambda/(1+\delta)\simeq 0.73815$ where the sign of $J_{\rm eff}^{\prime}$ changes.

As in the classical case, the two QF phases (LM1, LM2) with $m_{\rm sp}=1$ do not form a single phase but are separated by the PF phase, the TLL phase, and the $p=2$ QF phase. The $\delta$-dependence of the spontaneous magnetization $m_{\rm sp}$ for $L=48$ calculated by the DMRG method is shown in Fig. 6 for $\lambda=0.6$. This behavior shows that a PF phase with $m_{\rm sp}<1$ intervenes between the two QF phases with $m_{\rm sp}=1$. The corresponding behavior in the classical limit is also shown in Fig. 7. The angles $\theta$ and $\varphi$ vary by $\pi$ across the PF phase. This behavior explicitly shows that the LM1 and LM2 phases are different phases.

Figure 8 shows the $\lambda$-dependence of the spontaneous magnetization for (a) $\delta=0.1$ and (b) 0.02. The corresponding figure for $\delta=0$ taken from Ref. \citenhida2021 that shows the presence of an infinite series of QF phases is also shown as Fig. 8(c) for comparison. For $\delta=0.02$, the QF phases with $m_{\rm sp}=1$ $(p=1)$ and $m_{\rm sp}=1/2$ $(p=2)$ remain finite and the structures survive around the magnetizations corresponding to $p=3$ and 4. For $\delta=0.1$, only the QF phases with $m_{\rm sp}=1$ $(p=1)$ and $m_{\rm sp}=1/2$ $(p=2)$ remain finite, while those with $p\geq 3$ are smeared out and replaced by a PF phase. These results suggest that the QF phases become more fragile with increasing $p$. The corresponding curves in the classical limit are shown in Fig. 9. The QF phases vanish in the classical case showing that these are essentially quantum effects.

The fragility of the QF phase with large $p$ can be understood in the following way: If the spontaneous breakdown of the translational invariance were absent, the QF ground states with $p\geq 2$ are absent and the ground state is a PF state that is regarded as a magnetized TLL.[15] The low energy effective Hamiltonian in this phase is given by the $U(1)$ compactified boson field theory with the TLL parameter $K$ as follows:

where $\phi$ is a boson field defined on a circle $\phi\in[0,\sqrt{2}\pi)$, and $\Pi$ is the momentum density field conjugate to $\phi$. The spin wave velocity is set equal to unity. Extending the bosonization procedure of Ref. \citenoya to the case of mixed spin chains, a translation by one unit cell results in the shift of the boson field as

where $S_{\rm uc}$ is the sum of the spin magnitudes in a unit cell. In the present case, $S_{\rm uc}=2$. We consider the case $m_{\rm sp}=1/p$ that corresponds to the value of $m_{\rm sp}$ in the QF phase with period $p$. Then, taking the compactification condition into account, the shift (32) is rewritten as

The leading perturbation invariant under the shift (33) is given by

where $c$ is a constant. Although this operator is translationally invariant, if it is relevant, the phase $\phi$ is pinned to one of the minima of $c\cos(\sqrt{2}p\phi)$ and the translational invariance is spontaneously broken. Since the scaling dimension of the operator (34) is $x_{p}=p^{2}K/2$, it is relevant if $x_{p}<2$. Although the $p$-dependence of $K$ is unknown, assuming that it is moderate, the main $p$ dependence of $x_{p}$ comes from the factor of $p^{2}$. This explains why the QF phases with large $p$ are more fragile than those with small $p$.

For larger $\lambda$, the nonmagnetic phase appears and it continues to the TLL phase for large $\lambda$ discussed in Sect. 3.4. Since the sum of the spin magnitudes in a unit cell is an integer, the conventional LSM theorem[9, 10] does not exclude the unique gapped phase. However, our model (1) is invariant under the site-reflection about the vertex spin $\mib S_{l}$ whose magnitude is 1/2. Hence, our model satisfies the condition to exclude the unique gapped phase in the recent extension of the LSM theorem to the site-reflection invariant spin chains.[11, 12, 13] Taking the continuity to the TLL phase in the limit $\lambda\rightarrow\infty$ into account, the whole nonmagnetic phase is considered to be the TLL phase. This is confirmed by the numerical diagonalization calculation of the singlet-triplet energy gap $\Delta E$. It is checked that $\Delta E$ approximately scales with the system size $L$ as $\Delta E\sim 1/L$ as shown in Fig.10(a) for $\delta=-0.3$ and (b) for 0.3. Similar analyses are also carried out for several other values of $\delta$. In the vicinity of the PF phase indicated by the shaded area of Fig. 5, however, the deviation from the scaling relation $\Delta E\sim 1/L$ is significant as shown in Fig. 10(c). Nevertheless, this area shrinks with the system size. Hence, it is likely that the whole nonmagnetic phase is a TLL phase. It should be also remarked that the nonmagnetic ground state for $\delta=0$ is also a TLL phase[7] since it is exactly mapped onto the ground state of the spin-1/2 antiferromagnetic Heisenberg chain. For $\lambda\gtrsim{\lambda_{\rm c}}$, where ${\lambda_{\rm c}}$ is the nonmagnetic-ferrimagnetic transition point, the ferrimagnetic state with total spin ${S^{\rm tot}}>1$ comes down resulting in the level-crossing with the nonmagnetic state at ${\lambda_{\rm c}}$. Their energies measured from the ground state are plotted by the filled symbols in Fig. 10(c). These ferrimagnetic states are macroscopically different from the nonmagnetic state in the thermodynamic limit and cannot be regarded as elementary excitations in the nonmagnetic state, even though they have the next-lowest energy for finite-size systems. Unfortunately, in the region where this type of state has the next-lowest energy, it is difficult to identify the singlet-triplet gap, since we can calculate only several lowest eigenvalues by the Lanczos method we employ in this work.