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arXiv:1905.02117 · Journal of the Physical Society of Japan

Ground State Phases of Distorted S = 1 Diamond Chains

Kazuo Hida

Published Jun 21, 2019• 5 citations
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Abstract

The ground states of distorted $S=1$ diamond chains are investigated for two types of distortion called type A and B [J. Phys. Soc. Jpn. 79 (2010) 114703]. For the type A distortion, Haldane phases with and without spontaneous translational symmetry breakdown are present for large values of parameter $λ$ that parametrize the strength of frustration. For small $λ$, the Haldane phase and two quantized ferrimagnetic phases in the undistorted chain remain stable even for strong distortion. In contrast, for the type B distortion, the quantized ferrimagnetic phases with and without spontaneous translational symmetry breakdown are present for large $λ$. The partial ferrimagnetic phases emerge between them. For small $λ$, two quantized ferrimagnetic phases remain and the partial ferrimagnetic phases also emerge between them. The Haldane phase between the two kinds of ferrimagnetic phases turns into a topologically trivial double Haldane phase for strong distortion.

DOI
10.7566/jpsj.88.074705

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The ground states of distorted S=1S=1S=1 diamond chains are investigated for two types of distortion called type A and B [J. Phys. Soc. Jpn. 79 (2010) 114703]. For the type A distortion, Haldane phases with and without spontaneous translational symmetry breakdown are present for large values of parameter λλλ that parametrize the strength of frustration. For small λλλ, the Haldane phase and two quantized ferrimagnetic phases in the undistorted chain remain stable even for strong distortion. In contrast, for the type B distortion, the quantized ferrimagnetic phases with and without spontaneous translational symmetry breakdown are present for large λλλ. The partial ferrimagnetic phases emerge between them. For small λλλ, two quantized ferrimagnetic phases remain and the partial ferrimagnetic phases also emerge between them. The Haldane phase between the two kinds of ferrimagnetic phases turns into a topologically trivial double Haldane phase for strong distortion.
The ground states of two types of distorted mixed diamond chains with spins 1 and 1/2 are investigated using exact diagonalization, DMRG, and mapping onto low-energy effective models. In the undistorted case, the ground state consists of an array of independent spin-1 clusters separated by singlet dimers. The lattice distortion induces an effective interaction between cluster spins. When this effective interaction is antiferromagnetic, several Haldane phases appear with or without spontaneous translational symmetry breakdown (STSB). The transition between the Haldane phase without STSB and that with (n+1)(n+1)(n+1)-fold STSB (nnn = 1, 2, and 3) belongs to the same universality class as the (n+1)(n+1)(n+1)-clock model. In contrast, when the effective interaction is ferromagnetic, the quantized and partial ferrimagnetic phases appear with or without STSB. An effective low-energy theory for the partial ferrimagnetic phase is presented.
The ground-state phases of alternating-bond spin-1 diamond chains are investigated. Each ground state consists of an array of spin clusters separated by singlet dimers owing to an infinite number of local conservation laws. If no singlet dimers are present, the ground state is equivalent to that of a spin chain with infinite length.For strong frustration, we find a series of quantum phase transitions as in the case of alternating-bond mixed diamond chains with spins 1 and 1/2. For intermediate frustration, we find the nonmagnetic Haldane or dimer phases according to whether the bond alternation is weak or strong. For weak frustration and weak bond alternation, we find the ferrimagnetic states with spontaneous magnetizations m=1/6m=1/6m=1/6 and 1/3 per site. The ferrimagnetic state with m=1/6m=1/6m=1/6 is accompanied by a spontaneous translational symmetry breakdown. This phase vanishes for strong bond alternation.
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.
We investigate the ground state properties of an S=1/2 distorted diamond chain described by the Hamiltonian ${\cal H}=J_1\sum_\ell\bigl\{\bigl(\vecS_{3\ell-1}\cdot\vecS_{3\ell} +\vecS_{3\ell}\cdot\vecS_{3\ell+1}\bigr)+J_2\vecS_{3\ell-2}\cdot\vecS_{3\ell-1} +J_3\bigl(\vecS_{3\ell-2}\cdot\vecS_{3\ell}+\vecS_{3\ell}\cdot\vecS_{3\ ell+2}\bigr)-H S_{\ell}^z\bigr\}$ (J1, J2, J3≥0J_1,~J_2,~J_3\geq0J1​, J2​, J3​≥0), which models well a trimerized S=1/2 spin chain system Cu3_33​Cl6_66​(H2_22​O)2_22​$\cdot2H2H2H_8CCC_4SOSOSO_2$. Using an exact diagonalization method by means of the Lanczös technique, we determine the ground state phase diagram in the H=0 case, composed of the dimerized, spin fluid, and ferrimagnetic phases. Performing a degenerate perturbation calculation, we analyze the phase boundary line between the latter two phases in the $J_2,~J_3\llJ_1$ limit, the result of which is in good agreement with the numerical result. We calculate, by the use of the density matrix renormalization group method, the ground state magnetization curve for the case (a) where J1=1.0J_1=1.0J1​=1.0, J2=0.8J_2=0.8J2​=0.8, and J3=0.5J_3=0.5J3​=0.5, and the case (b) where J1=1.0J_1=1.0J1​=1.0, J2=0.8J_2=0.8J2​=0.8, and J3=0.3J_3=0.3J3​=0.3. We find that in the case (b) the 2/3-plateau appears in addition to the 1/3-plateau which also appears in the case (a). The translational symmetry of the Hamiltonian H\cal HH is spontaneously broken in the 2/3-plateau state as well as in the dimerized state.
The ground-state phases of anisotropic mixed diamond chains with spins 1 and 1/2 are investigated. Both single-site and exchange anisotropies are considered. We find the phases consisting of an array of uncorrelated spin-1 clusters separated by singlet dimers. Except in the simplest case where the cluster consists of a single S=1S=1S=1 spin, this type of ground state breaks the translational symmetry spontaneously. Although the mechanism leading to this type of ground state is the same as that in the isotropic case, it is nonmagnetic or paramagnetic depending on the competition between two types of anisotropy. We also find the Néel, period-doubled Néel, Haldane, and large-DDD phases, where the ground state is a single spin cluster of infinite size equivalent to the spin-1 Heisenberg chain with alternating anisotropies. The ground-state phase diagrams are determined for typical sets of parameters by numerical analysis. In various limiting cases, the ground-state phase diagrams are determined analytically. The low-temperature behaviors of magnetic susceptibility and entropy are investigated to distinguish each phase by observable quantities. The relationship of the present model with the anisotropic rung-alternating ladder with spin-1/2 is also discussed.
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 apical spins τ(1)τ^{(1)}τ(1) and τ(2)τ^{(2)}τ(2) are connected with each other by an exchange coupling λλλ. Other exchange couplings are set equal to unity. This model has an infinite number of local conservation laws. For large λλλ, the ground state is equivalent to that of the uniform spin 1/21/21/2 chain. Hence, the ground state is a gapless spin liquid. For λ≤0λ\leq 0λ≤0, the ground state is a Lieb-Mattis ferrimagnetic phase with spontaneous magnetization msp=1m_{\rm sp}=1msp​=1 per unit cell. For intermediate λλλ, we find a series of ferrimagnetic phases with msp=1/pm_{\rm sp}=1/pmsp​=1/p where ppp takes positive integer values. The phases with p≥2p \geq 2p≥2 are accompanied by the spontaneous breakdown of the ppp-fold translational symmetry. It is suggested that the phase with arbitrarily large ppp, namely infinitesimal spontaneous magnetization, is allowed as λλλ approaches the transition point to the gapless spin liquid phase.