Disordered ground states in a quantum frustrated spin chain with side chains

We study a frustrated mixed spin chain with side chains, where the spin species and the exchange interactions are spatially varied. A nonlinear $\sigma$ model method is formulated for this model, and a phase diagram with two disordered spin-gap phases is obtained for typical cases. Among them, we examine the case with a main chain, which consists of an alternating array of spin-1 and spin-$\frac{1}{2}$ sites, and side chains, each of which consists of a single spin-$\frac{1}{2}$ site, in great detail. Based on numerical, perturbational, and variational approaches, we propose a singlet cluster solid picture for each phase, where the ground state is expressed as a tensor product of local singlet states.

Figure 1

A quantum spin chain with side chains; two unit cells are presented. ${\mathbf{S}}_1$, ${\mathbf{S}}_2$, and $\mathbf{T}$ are spins whose magnitudes satisfy Eq. (3). The case where $S_1=1$, $S_2=\frac{1}{2}$, and $T=\frac{1}{2}$ is particularly studied in detail.

Figure 2

Phase boundaries for several values of $r$ in the $k\text{−}j$ plane by the NLSM method for $S_1=1$, $S_2=\frac{1}{2}$, and $T=\frac{1}{2}$. For each value of $r$ $\left(=K_2/2K_1\right)$, the region of $0<k<\frac{1}{r}-1$ is meaningful in the NLSM method.

Figure 3

Phase boundaries in the $k\text{−}j$ plane by the NLSM method in the case of (a) $S_1=\frac{3}{2}$, $S_2=1$, and $T=\frac{1}{2}$, and (b) in the case of $S_1=\frac{3}{2}$, $S_2=\frac{1}{2}$, and $T=1$. For each value of $r$, the region of $0<k<\frac{1}{r}-1$ is meaningful.

Figure 4

Phase boundaries by numerical diagonalization for $S_1=1$, $S_2=\frac{1}{2}$, and $T=\frac{1}{2}$. Each point of a boundary is determined by extrapolation for the total spin number $3N=12$, 18, and 24. Inset: Phase boundaries for small $k$.

Figure 5

(a) Three-spin units for $j\to 0$ and (b) those for $j\to \infty$ and $r\to 0$ in the case of $S_1=1$, $S_2=\frac{1}{2}$, and $T=\frac{1}{2}$. Three spins (circles) in a shadowed region form a three-spin unit.

Figure 6

Four regions in a typical phase diagram for $S_1=1$, $S_2=\frac{1}{2}$, and $T=\frac{1}{2}$; the phase diagram for $r=0.2$ by the NLSM method is shown. The solid line is the phase boundary between the Gap I phase and the Gap II phase. In each phase, the dashed line of $j=1$ means a crossover between regions with different features. The Gap I phase consists of regions I-A and I-B, while th Gap II phase consists of regions II-A and II-B. They are explained by SCS pictures.

Figure 7

VBS pictures in the limiting cases for $S_1=1$, $S_2=\frac{1}{2}$, and $T=\frac{1}{2}$: (a) VBS I represents the ground state for $j\to 0$ and $k<k_c$, and (b) VBS II does so for $j\to \infty$. Small circles are $\frac{1}{2}$ spins and bold gray lines are valence bonds. Spin ${\mathbf{S}}_1$ is expressed by two $\frac{1}{2}$ spins as ${\mathbf{S}}_1={\mathbf{S}}_1^{\left(1\right)}+{\mathbf{S}}_1^{\left(2\right)}$. Loops including two valence bonds correspond to SCS pictures (see text).

Figure 8

(a) SCS I, the SCS picture for the Gap I phase ($S_1=1$, $S_2=\frac{1}{2}$, and $T=\frac{1}{2}$). SCS I is a tensor product form of singlet clusters. (b) A singlet cluster $|{\psi }_{\text{I}}\left(c\right)⟩$ (the left hand side) in SCS I. It is represented as a linear combination of two valence bond states, $|{\alpha }_{\text{I}}⟩$ and $|{\beta }_{\text{I}}⟩$ (the right hand side). $c$ is the coefficient of the linear combination.

Figure 9

Short range correlation functions for $r=0.3$ and $k=0.2$ numerically calculated for $N=12$, 18, and 24 ($N$ is the number of spins).

Figure 10

(a) SCS II, the SCS picture for the Gap II phase ($S_1=1$, $S_2=\frac{1}{2}$, and $T=\frac{1}{2}$). SCS II is a tensor product form of singlet clusters. (b) A singlet cluster $|{\psi }_{\text{II}}\left(c\right)⟩$ (the left hand side) in SCS II. It is represented as a linear combination of two valence bond states, $|{\alpha }_{\text{II}}⟩$ and $|{\beta }_{\text{II}}⟩$ (the right hand side). $c$ is the coefficient of the linear combination. (c) Identity among local valence bond states. $|{\beta }_{\text{II}}⟩$ (the left hand side) is exactly the sum of $|{\alpha }_{\text{II}}⟩$ and $|{\gamma }_{\text{II}}⟩$ (the right hand side).

Figure 11

Coefficient $c_{\text{I}}$ of the linear combination in the variational wave function for SCS I ($S_1=1$, $S_2=\frac{1}{2}$, and $T=\frac{1}{2}$). The optimal coefficient $c$ does not depend on $r$ in the wave function. The lines are in the order of ascending $k$ from top to bottom.

Figure 12

Coefficient $c_{\text{II}}$ of the linear combination in the variational wave function for SCS I in the case of $r=0.2$ ($S_1=1$, $S_2=\frac{1}{2}$, and $T=\frac{1}{2}$). The lines are in the order of ascending $k$ from top to bottom.

Figure 13

Phase boundaries by the variational calculation for the SCS wave functions ($S_1=1$, $S_2=\frac{1}{2}$, and $T=\frac{1}{2}$).