We investigate the mixed diamond chain composed of spins 1 and 1/2 when the exchange interaction is alternatingly distorted. Depending on the strengths of frustration and distortion, this system has various ground states. Each ground state consists of an array of spin clusters separated by singlet dimers by virtue of an infinite number of local conservation laws. We determine the ground state phase diagram by numerically analyzing each spin cluster. In particular for strong distortion, we find an infinite series of quantum phase transitions by the cluster expansion method and conformal field theory. This leads to the infinite series of steps in the behavior of the Curie constant and residual entropy.
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=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-
D 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 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=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=λc0(δ) to a series of ferrimagnetic phases with a spontaneous magnetization
msp=1/p per unit cell where
p is a positive integer. It is found that this transition is a first order transition for
δ=0 with a discontinuous change in
msp, while no discontinuity is found for
δ=0. The critical behaviors of
msp and
λc0(δ) around the critical point
(δ,λ)=(0,λc0(δ)) are also discussed analytically.
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.
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 apical spins
τ(1) and
τ(2) are connected with each other by an exchange coupling
λ. Other exchange couplings are set equal to unity. This model has an infinite number of local conservation laws. For large
λ, the ground state is equivalent to that of the uniform spin
1/2 chain. Hence, the ground state is a gapless spin liquid. For
λ≤0, the ground state is a Lieb-Mattis ferrimagnetic phase with spontaneous magnetization
msp=1 per unit cell. For intermediate
λ, we find a series of ferrimagnetic phases with
msp=1/p where
p takes positive integer values. The phases with
p≥2 are accompanied by the spontaneous breakdown of the
p-fold translational symmetry. It is suggested that the phase with arbitrarily large
p, namely infinitesimal spontaneous magnetization, is allowed as
λ approaches the transition point to the gapless spin liquid phase.
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)-fold STSB (
n = 1, 2, and 3) belongs to the same universality class as the
(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.