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arXiv:cond-mat/9607052 · Journal of Physics Condensed Matter

Ground states with cluster structures in a frustrated Heisenberg chain

K Takano, K Kubo, H Sakamoto

Published Aug 26, 1996• 144 citations
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Abstract

We examine the ground state of a Heisenberg model with arbitrary spin S on a one-dimensional lattice composed of diamond-shaped units. A unit includes two types of antiferromagnetic exchange interaction which frustrate each other. The system undergoes phase changes when the ratio between the exchange parameters varies. In some phases, strong frustration leads to larger local structures or clusters of spins than a dimer. We prove for arbitrary S that there exists a phase with four-spin cluster states, which was previously found numerically for a special value of in the S = 1/2 case. For S = 1/2 we show that there are three ground-state phases, and determine their boundaries.

DOI
10.1088/0953-8984/8/35/009

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We examine the ground state of a Heisenberg model with arbitrary spin S on a one-dimensional lattice composed of diamond-shaped units. A unit includes two types of antiferromagnetic exchange interaction which frustrate each other. The system undergoes phase changes when the ratio between the exchange parameters varies. In some phases, strong frustration leads to larger local structures or clusters of spins than a dimer. We prove for arbitrary S that there exists a phase with four-spin cluster states, which was previously found numerically for a special value of in the S = 1/2 case. For S = 1/2 we show that there are three ground-state phases, and determine their boundaries.
We study the ground-state phase diagram of a Heisenberg model with spin S=12S=\frac{1}{2}S=21​ on a diamond-like decorated square lattice. A diamond unit has two types of antiferromagnetic exchange interactions, and the ratio λλλ between the length of the diagonal bond and that of the other four edges determines the strength of frustration. It has been pointed out [J. Phys. Soc. Jpn {\bf85}, 033705 (2016)] that the so-called tetramer-dimer states, which are expected to be stabilized in an intermediate region of $λ_{\rm c}
The mixed diamond chain is a frustrated Heisenberg chain composed of successive diamond-shaped units with two kinds of spins of magnitudes SSS and S/2S/2S/2 (SSS: integer). Ratio $\ensuremath{\lambda}$ of two exchange parameters controls the strength of frustration. With varying $\ensuremath{\lambda}$, the Haldane state and several spin-cluster states appear as the ground state. A spin-cluster state is a tensor product of exact local eigenstates of cluster spins. We prove that a spin-cluster state is the ground state in a finite interval of $\ensuremath{\lambda}$. For S=1S=1S=1, we numerically determine the total phase diagram consisting of five phases.
Abstract We study the ground state properties of the anisotropic spin-1/2 Heisenberg model on lattices built from ideal diamond units with competing ferro- and antiferromagnetic interactions. The study covers the one-dimensional diamond chain and its two- and three-dimensional generalizations. The ground-state phase diagram contains four distinct phases: ferromagnetic (F), critical (C), monomer-dimer (MD), and tetramer-dimer (TD), which converge at a quadruple point. We demonstrate the presence of macroscopic ground-state degeneracy and corresponding residual entropy, which is maximal at the quadruple point and also extends throughout the MD phase and its boundaries with TD and F phases. For the diamond chain, we derive exact degeneracies, while for higher-dimensional lattices, we map the problem onto a bond percolation model or used transfer-matrix approach, enabling the numerical computation of the ground state degeneracy.
We study the ground-state phase diagrams and properties of spin-1/2 Heisenberg models on the diamond-like-decorated square and triangular lattices. The diamond-like-decorated square (triangular) lattice is a lattice in which the bonds of a square (triangular) lattice are replaced with diamond units. The diamond unit has two types of antiferromagnetic exchange interactions, and the ratio λ of the strength of the diagonal bond to that of the other four edges determines the ground-state properties. In particular, the macroscopically degenerated tetramer-dimer states, which are equivalent to the dimer covering states of the original lattices, are stabilized for λc < λ < 2, where the value of λc depends on the lattices. To determine the phase diagrams and boundaries λc, we employ the modified spin wave (MSW) method and the quantum Monte Carlo (QMC) method to estimate the ground-state energies of the ferrimagnetic states for λ < λc, where we can consider the mixed spin-1 and spin-1/2 Lieb-lattice and triangular Lieb-lattice Heisenberg antiferromagnets instead, and obtain λc(square)=0.974 and λc(triangular)=0.988. We also calculate the long-range order (LRO) parameters using the MSW and QMC methods and find the scaling relations where the spin reductions of each sublattice are inversely proportional to the number of sublattice sites. We prove these scaling relations by applying an infinitesimal uniform magnetic field. Furthermore, by examining the calculation process in the MSW, we clarify the mathematical structure behind the scaling relations for the sublattice LROs.
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The spin-12\frac{1}{2}21​ Heisenberg diamond chain with ferro- and antiferromagnetic exchange interactions is studied. The phase boundary in the parametric space of these interactions is determined, where the transition between the ferromagnetic and other (singlet or ferrimagnetic) ground state phases occurs. On this phase boundary there is a dispersionless (flat) energy band in the one-magnon spectrum and these states can be represented as localized magnons in the trapping cells between neighboring diamonds. The ground state consists of the localized magnons and special magnon complexes and it is macroscopically degenerate. A remarkable feature of the model is the existence, on a certain part of the phase boundary, of two- and three-magnon localized states, forming flat bands in two- and three-magnon spectrum. All these localized states also belong to the ground state manifold, which turns out to be exactly the same as for the system of independent spins-32\frac{3}{2}23​. This implies a large macroscopic degeneracy of the ground state 4n{4}^{n}4n (nnn is number of diamonds in the chain) and a high residual entropy per spin s0=23ln2{s}_{0}=\frac{2}{3}ln2s0​=32​ln2.