Frustration-induced quantum phases in mixed spin chain with frustrated side chains

A mixed Heisenberg spin chain with frustrated side chains is investigated by numerical and perturbational calculations. A frustration-induced quantum partially polarized ferrimagnetic phase and a nonmagnetic spin quadrupolar phase are found adjacent to the conventional Lieb-Mattis-type ferrimagnetic phase or the nonmagnetic singlet cluster solid phases. The partially polarized ferrimagnetic phase has an incommensurate spin structure. Similar structures are commonly found in other frustration-induced partially polarized ferrimagnetic phases. Numerical results also suggest a series of almost critical nonmagnetic ground states in a highly frustrated regime if the side chain spins weakly couple to the main chain.

Figure 1

The quantum spin chain with side chains, which we study in this paper. ${\mathbf{S}}_1\left(p\right)$, ${\mathbf{S}}_2\left(p\right)$, and $\mathbf{T}\left(p\right)$ are spins in the $p\text{th}$ unit cell. These magnitudes are $S_1=1$, $S_2=\frac{1}{2}$, and $T=\frac{1}{2}$, respectively.

Figure 2

Phase diagram for small $j$ with $r=1.2$.

Figure 3

$\Delta$ chain realized in the limit $j⪢1$.

Figure 4

Phase diagram in large $j$ limit. The shaded region is the almost critical nonmagnetic phase.

Figure 5

Phase diagram on the $j/r-r\Delta k$ plane, where $\Delta k=k-k_c\left(j/r\right)$.

Figure 6

Phase diagram for $r=10$ using the approximation in Sec. 3C.

Figure 7

Ground-state phase diagram of (1) for $r=2.0$. In this and in the following figures, phase boundaries determined from the numerical diagonalization data for $N=18$ and 24 are shown. The solid lines are guide for eye. The broken lines are the results of the small-$j$ approximation in $A$.

Figure 8

Ground-state phase diagram of (1) for $r=1.5$.

Figure 9

Ground-state phase diagram of (1) for $r=1.3$.

Figure 10

Ground-state phase diagram of (1) for $r=1.2$.

Figure 11

(a) The $j$ dependence of the scaled singlet triplet energy gap $N\Delta E$ with periodic boundary condition and (b) the energy difference between the different parity ground states $E_{+}-E_{-}$ with twisted boundary condition. The parameters are $r=1.3$, $k=0.12$, and $N=12$, 18, and 24. The triangles indicate the values of $j$ where the ground-state parity changes under the twisted boundary condition for $N=24$.

Figure 12

The $j$ dependence of the spontaneous magnetization for $k=0.5$ and $r=1.3$. The results are calculated in the $N=72$ system by the DMRG method. The magnetization is normalized by the saturated magnetization $M_s\equiv 2N/3$.

Figure 13

Magnetization curve in the PPF phase with $k=0.4$, $j=1.7$, and $r=1.5$ for $N=192$ calculated by the DMRG method.

Figure 14

Local magnetization profile in PPF phase with $k=0.4$, $j=1.7$, and $r=1.5$ for $N=192$ calculated by the DMRG method.

Figure 15

Classical planar spin configuration in a triangle.

Figure 16

Classical phase diagram on the $k\text{−}r$ plane.

Figure 17

Examples of classical ground-state configurations (a) with finite and (b) vanishing magnetizations. The inner arrows indicate the direction of the rotation of spins along each triangle.

Figure 18

Classical magnetization curves for $k=0.2$, 1.0 and 2.0 with $j=1.7$ and $r=1.5$.