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First Order Transitions Between the Gapped Spin-Liquid and Ferrimagnetic Phases in (1/2,1/2,1) Mixed Diamond Chains with Bond Alternation

The ground-state phases of mixed diamond chains with bond alternation $δ$, and ($S, τ^{(1)}, τ^{(2)})=(1/2,1/2,1)$, where $S$ is the magnitude of vertex spins, and $τ^{(1)}$ and $τ^{(2)}$ are those of apical spins, are investigated. The two apical spins in each unit cell are connected by an exchange coupling $λ$. The exchange couplings between the apical spins and the vertex spins take the values $1+δ$ and $1-δ$ alternatingly. This model has an infinite number of local conservation laws. For large $λ$ and $δ\neq 0$, the ground state is equivalent to that of the spin $1/2$ chain with bond alternation. Hence, the ground state is a gapped spin liquid. This energy gap vanishes for $δ=0$. With the decrease of $λ$, the ground state undergoes a transition at $λ=λ_{\rm c0}(δ)$ to a series of ferrimagnetic phases with a spontaneous magnetization $m_{\rm sp}=1/p$ per unit cell where $p$ is a positive integer. It is found that this transition is a first order transition for $δ\neq 0$ with a discontinuous change in $m_{\rm sp}$, while no discontinuity is found for $δ=0$. The critical behaviors of $m_{\rm sp}$ and $λ_{\rm c0}(δ)$ around the critical point $(δ,λ) =(0, λ_{\rm c0}(δ))$ are also discussed analytically.