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arXiv:cond-mat/9903044 · APS

Spin-gap physics, ground state degeneracy, and bound states on the depleted kagome lattice

C. Hooley, Alexei M. Tsvelik

Published Mar 2, 1999• 0 citations
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Abstract

We analyse the antiferromagnetic spin-1/2 Heisenberg model on a depleted kagome lattice, where some bonds have been reduced to exchange integral J_2 << J_1. The fully depleted system consists of 1D chains, each with a doubly degenerate singlet-pair ground state and a spectral gap (like the Majumdar-Ghosh model). There are localised and itinerant low-energy excitations. The modes from the lowest branch of excitations are incapable of lifting the 2D system's ground state degeneracy at finite J_2/J_1 << 1. Low-energy excitations of the 2D system are dominated by coherently propagating bound states of the 1D excitations.

DOI
10.48550/arxiv.cond-mat/9903044

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We analyse the antiferromagnetic spin-1/2 Heisenberg model on a depleted kagome lattice, where some bonds have been reduced to exchange integral J_2 << J_1. The fully depleted system consists of 1D chains, each with a doubly degenerate singlet-pair ground state and a spectral gap (like the Majumdar-Ghosh model). There are localised and itinerant low-energy excitations. The modes from the lowest branch of excitations are incapable of lifting the 2D system's ground state degeneracy at finite J_2/J_1 << 1. Low-energy excitations of the 2D system are dominated by coherently propagating bound states of the 1D excitations.
Using exact numerical diagonalizations as well as the semiclassical spin-wave approach, we study the quantum phase diagram of the spin-12 Heisenberg kagome strip, which is a particular cut-out from the kagome lattice. The unit cell contains five spins placed on the central cite and on the four end sites building two spin-12 along the strip. The model can be characterized by three different constants for the nearest-neighbor exchange bonds between (i) the central and chain spins (J1), (ii) the end spins in a cluster (J2 > 0), and (ii) the spins of neighboring clusters (J′2>0) In the present work we extend the previous studies of the model, which were restricted to antiferromagnetic J1 bonds (J1 > 0), by including the region with ferromagnetic J1 couplings (J1<0). It is demonstrated that for dominating ferromagnetic J1 couplings there is another gapless phase, which can be related to the critical ground state of the spin-52 Heisenberg chain. The phase boundaries separating the discussed critical phases, as well as the behavior of their low-lying excitations close to these boundaries, are examined as well.
We study ground-state degeneracy and spin-wave excitations in a 1D version of a Kagome antiferromagnet —a Heisenberg antiferromagnet on a Kagome stripe. We show that for nearest-neighbor interaction, the classical ground state is infinitely degenerate. For any spin configuration from the degenerate set, the classical spin-wave spectrum contains, in addition to Goldstone modes, a branch of zero-energy excitations, and a zero mode in another branch. We demonstrate that the interactions beyond nearest neighbors lift the degeneracy, eliminate a zero mode, and give a finite dispersion to formerly zero-energy branch, leaving only Goldstone modes as zero-energy excitations.
We study the low-energy properties of Heisenberg antiferromagnetic spin-1/2 and spin-1 systems on various clusters made up of triangles. Some of the clusters have a geometry similar to representative pieces of the Kagome lattice, while others have the geometry of a sawtooth chain. While the ground state always has the lowest possible spin (0 or 1/2), the nature of the low-energy excitations depends on the geometry and the site spin. For the Kagome clusters with spin-1 sites, the lowest excitations are gapped, with singlet and triplet excitations having similar gaps. This is in contrast to Kagome clusters with spin-1/2 sites where there are several low-energy singlet excitations (possibly gapless in the thermodynamic limit), while triplet excitations have a gap. For the sawtooth chain with spin-1 sites, the lowest excitations are triplets with a gap; the gap to singlet excitations is about twice the triplet gap.
We study a Heisenberg antiferromagnet (AF) on a one-dimensional strip composed of two rows of corner sharing triangular plaquettes. The geometry of this ladder-like lattice naturally allows for two different exchange couplings, one between pairs of spins on the outer legs (J) and a different one between the spins on the axis and their nearest neighbors on the legs ${(J}^{\ensuremath{'}}).$ For ${J/J}^{\ensuremath{'}}\ensuremath{\lesssim}0.5$ the model is a ferrimagnet. Our main interest is in the region ${J/J}^{\ensuremath{'}}\ensuremath{\gtrsim}0.5,$ where the classical ground state of this system shows the same macroscopic degeneracy as the classical ground state of the antiferromagnet on the kagom\'e lattice. To explore to which extent this similarity between the classical ground states of these two models carries over to their quantum states we have applied exact diagonalization techniques and density-matrix renormalization group (DMRG) methods to our model. Exact diagonalization is restricted to system sizes of up to N=30N=30N=30 sites. For ${J/J}^{\ensuremath{'}}=1,$ the results obtained by this technique, low-energy spectra, correlation functions and the specific heat, agree qualitatively with the results obtained for finite samples of the kagom\'e AF. As in the case of the kagom\'e AF, these finite size results suggest that our model is a spin liquid with a gap between the ground state and the lowest spin excitation. However, extrapolations from DMRG data for strips of up to N=120N=120N=120 sites point towards a vanishing spin gap in a wide range of couplings, $0.5\ensuremath{\lesssim}{J/J}^{\ensuremath{'}}\ensuremath{\lesssim}1.25,$ so that contrary to the above conjecture, our one-dimensional model may in fact be critical in this parameter range. Moreover, our DMRG data indicate that the model undergoes a transition to a gapped state as the ratio ${J/J}^{\ensuremath{'}}$ increases through the value ${J/J}^{\ensuremath{'}}\ensuremath{\simeq}1.25.$ In the vicinity of the transition point, a high density of low-lying singlets develops in the spin gap.
Motivated by dipolar-coupled artificial spin systems, we present a theoretical study of the classical ${J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{3}$ Ising antiferromagnet on the kagome lattice. We establish the ground-state phase diagram of this model for J1>∣J2∣,∣J3∣{J}_{1}>|{J}_{2}|,|{J}_{3}|J1​>∣J2​∣,∣J3​∣ based on exact results for the ground-state energies. When all the couplings are antiferromagnetic, the model has three macroscopically degenerate ground-state phases, and using tensor networks, we can calculate the entropies of these phases and of their boundaries very accurately. In two cases, the entropy appears to be a fraction of that of the triangular lattice Ising antiferromagnet, and we provide analytical arguments to support this observation. We also notice that, surprisingly enough, the dipolar ground state is not a ground state of the truncated model, but of the model with smaller J3{J}_{3}J3​ interactions, an indication of a very strong competition between low-energy states in this model.
This paper is concerned with physics of the low-energy singlet excitations found to exist below the spin gap in numerical studies of the Kagome lattice quantum Heisenberg antiferromagnet. Insight into the nature of these excitations is obtained by exploiting an approximate mapping to a fully frustrated transverse-field Ising model on the dual dice lattice. This Ising model is shown to possess at least two phases---an ordered phase that also breaks translational symmetry with a large unit cell, and a paramagnetic phase. The former is argued to be a likely candidate for the ground state of the original Kagome magnet which thereby exhibits a specific pattern of dimer ordering with a large unit cell. Comparisons with available numerical results are made.