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arXiv:2312.01630 · Journal of the Physical Society of Japan

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

Kazuo Hida

Published Mar 8, 2024• 2 citations
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Abstract

The ground-state phases of mixed diamond chains with ($S, τ^{(1)}, τ^{(2)})=(1/2,1/2,1)$, 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 coupled by an exchange coupling $λ$. The vertex spins are coupled with the top and bottom apical spins by exchange couplings $1+δ$ and $1-δ$, respectively. Although this model has an infinite number of local conservation laws for $δ=0$, they are lost for finite $δ$. The ground-state phase diagram is determined using the numerical exact diagonalization and DMRG method in addition to the analytical approximations in various limiting cases. The phase diagram consists of a nonmagnetic phase and several kinds of ferrimagnetic phases. We find two different ferrimagnetic phases without spontaneous translational symmetry breakdown. It is also found that the quantized ferrimagnetic phases with large spatial periodicities present for $δ=0$ are easily destroyed by small $δ$ and replaced by a partial ferrimagnetic phase. The nonmagnetic phase is considered to be a gapless Tomonaga-Luttinger liquid phase based on the recently extended Lieb-Schultz-Mattis theorem to the site-reflection invariant spin chains and numerical diagonalization results.

DOI
10.7566/jpsj.93.044703

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Ground-State Phase Diagram of (1/2,1/2,1)(1/2,1/2,1)(1/2,1/2,1) Mixed Diamond Chains
The ground-state phases of mixed diamond chains with (S,τ(1),τ(2))=(1/2,1/2,1)S, τ^{(1)}, τ^{(2)})=(1/2,1/2,1)S,τ(1),τ(2))=(1/2,1/2,1), where SSS is the magnitude of vertex spins, and τ(1)τ^{(1)}τ(1) and τ(2)τ^{(2)}τ(2) are those of apical spins, are investigated. The two apical spins in each unit cell are coupled by an exchange coupling λλλ. The vertex spins are coupled with the top and bottom apical spins by exchange couplings 1+δ1+δ1+δ and 1−δ1-δ1−δ, respectively. Although this model has an infinite number of local conservation laws for δ=0δ=0δ=0, they are lost for finite δδδ. The ground-state phase diagram is determined using the numerical exact diagonalization and DMRG method in addition to the analytical approximations in various limiting cases. The phase diagram consists of a nonmagnetic phase and several kinds of ferrimagnetic phases. We find two different ferrimagnetic phases without spontaneous translational symmetry breakdown. It is also found that the quantized ferrimagnetic phases with large spatial periodicities present for δ=0δ=0δ=0 are easily destroyed by small δδδ and replaced by a partial ferrimagnetic phase. The nonmagnetic phase is considered to be a gapless Tomonaga-Luttinger liquid phase based on the recently extended Lieb-Schultz-Mattis theorem to the site-reflection invariant spin chains and numerical diagonalization results.
Ground-State Phase Diagram of (1/2,1/2,1)(1/2,1/2,1)(1/2,1/2,1) Mixed Diamond Chains with Single-Site Anisotropy
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First Order Transitions Between the Gapped Spin-Liquid and Ferrimagnetic Phases in (1/2,1/2,1)(1/2,1/2,1)(1/2,1/2,1) Mixed Diamond Chains with Bond Alternation
The ground-state phases of mixed diamond chains with bond alternation δδδ, and (S,τ(1),τ(2))=(1/2,1/2,1)S, τ^{(1)}, τ^{(2)})=(1/2,1/2,1)S,τ(1),τ(2))=(1/2,1/2,1), where SSS is the magnitude of vertex spins, and τ(1)τ^{(1)}τ(1) and τ(2)τ^{(2)}τ(2) are those of apical spins, are investigated. The two apical spins in each unit cell are connected by an exchange coupling λλλ. The exchange couplings between the apical spins and the vertex spins take the values 1+δ1+δ1+δ and 1−δ1-δ1−δ alternatingly. This model has an infinite number of local conservation laws. For large λλλ and δ≠0δ\neq 0δ=0, the ground state is equivalent to that of the spin 1/21/21/2 chain with bond alternation. Hence, the ground state is a gapped spin liquid. This energy gap vanishes for δ=0δ=0δ=0. With the decrease of λλλ, the ground state undergoes a transition at λ=λc0(δ)λ=λ_{\rm c0}(δ)λ=λc0​(δ) to a series of ferrimagnetic phases with a spontaneous magnetization msp=1/pm_{\rm sp}=1/pmsp​=1/p per unit cell where ppp is a positive integer. It is found that this transition is a first order transition for δ≠0δ\neq 0δ=0 with a discontinuous change in mspm_{\rm sp}msp​, while no discontinuity is found for δ=0δ=0δ=0. The critical behaviors of mspm_{\rm sp}msp​ and λc0(δ)λ_{\rm c0}(δ)λc0​(δ) around the critical point (δ,λ)=(0,λc0(δ))(δ,λ) =(0, λ_{\rm c0}(δ))(δ,λ)=(0,λc0​(δ)) are also discussed analytically.
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