arXiv:cond-mat/9812250 · Physical review. B, Condensed matter

Residual entropy and spin gap in a one-dimensional frustrated antiferromagnet

Matthieu Mambrini, Julien Trébosc, Frédéric Mila

Published Jun 1, 1999• 30 citations

Abstract

We show that the low-energy sector of the $S=1/2$, antiferromagnetic Heisenberg model on a one-dimensional lattice of coupled tetrahedra consists of ${2}^{N}$ replica of the spectrum of the dimerized Heisenberg chain, where N is the number of tetrahedra. This provides a proof of the following properties: (i) there is a residual ground-state entropy per spin equal to $(1/4)\mathrm{ln}2$; (ii) there is a singlet-triplet gap as long as the coupling between the tetrahedra is smaller than the internal one. These properties are compared to available results on the pyrochlore lattice.

DOI: 10.1103/physrevb.59.13806

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arXiv:cond-mat/9812250 · Physical review. B, Condensed matter

Residual entropy and spin gap in a one-dimensional frustrated antiferromagnet

Matthieu Mambrini, Julien Trébosc, Frédéric Mila

Published Jun 1, 1999• 30 citations

arXiv HTML PDF

Abstract

We show that the low-energy sector of the

S=1/2

, antiferromagnetic Heisenberg model on a one-dimensional lattice of coupled tetrahedra consists of

{2}^{N}

replica of the spectrum of the dimerized Heisenberg chain, where N is the number of tetrahedra. This provides a proof of the following properties: (i) there is a residual ground-state entropy per spin equal to

(1/4)\mathrm{ln}2

; (ii) there is a singlet-triplet gap as long as the coupling between the tetrahedra is smaller than the internal one. These properties are compared to available results on the pyrochlore lattice.

DOI: 10.1103/physrevb.59.13806

Related papers

Frustrated spin models on two- and three-dimensional decorated lattices with high residual entropy

D. V. Dmitriev, V. Ya. Krivnov, O. A. Vasilyev

We study the ground-state properties of a family of frustrated spin-1/2 Heisenberg models on two- and three-dimensional decorated lattices composed of connected star-shaped units. Each star is built from edge-sharing triangles with an antiferromagnetic interaction on the shared side and ferromagnetic interactions on the others. At a critical coupling ratio, the ideal star model - defined by equal ferromagnetic interactions - exhibits a macroscopically degenerate ground state, which we map onto a site percolation problem on the Lieb lattice. This mapping enables the calculation of exponential ground-state degeneracy and the corresponding residual entropy for square, triangular, honeycomb, and cubic lattices. Remarkably, the residual entropy remains high for all studied lattices, exceeding 60\% of the maximal value ln(2). Despite a gapless quadratic one-magnon spectrum, the low-temperature thermodynamics is governed by exponentially numerous gapped excitations. For a distorted-star variant of the model, the ground-state manifold is equivalent to that of decoupled ferromagnetic clusters, leading to exponential degeneracy with a lower, yet still substantial, residual entropy. At low temperature the system mimics a paramagnetic crystal of non-interacting spins with high spin value ( $s=4$ for a square lattice). The obtained results establish a structural design principle for engineering quantum magnets with a high ground-state degeneracy, suggesting promising candidates for enhanced magnetocaloric cooling and quantum thermal machines.We study the ground-state properties of a family of frustrated spin-1/2 Heisenberg models on two- and three-dimensional decorated lattices composed of connected star-shaped units. Each star is built from edge-sharing triangles with an antiferromagnetic interaction on the shared side and ferromagnetic interactions on the others. At a critical coupling ratio, the ideal star model - defined by equal ferromagnetic interactions - exhibits a macroscopically degenerate ground state, which we map onto a site percolation problem on the Lieb lattice. This mapping enables the calculation of exponential ground-state degeneracy and the corresponding residual entropy for square, triangular, honeycomb, and cubic lattices. Remarkably, the residual entropy remains high for all studied lattices, exceeding 60\% of the maximal value ln(2). Despite a gapless quadratic one-magnon spectrum, the low-temperature thermodynamics is governed by exponentially numerous gapped excitations. For a distorted-star variant of the model, the ground-state manifold is equivalent to that of decoupled ferromagnetic clusters, leading to exponential degeneracy with a lower, yet still substantial, residual entropy. At low temperature the system mimics a paramagnetic crystal of non-interacting spins with high spin value ($s=4$ for a square lattice). The obtained results establish a structural design principle for engineering quantum magnets with a high ground-state degeneracy, suggesting promising candidates for enhanced magnetocaloric cooling and quantum thermal machines.

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We have studied the low-energy excitation spectrum of a dimerized and frustrated antiferromagnetic Heisenberg chain. We use an analytic approach, based on a description of the excitations as triplets above a strong-coupling singlet ground state. The quasiparticle spectrum is calculated by treating the excitations as a dilute Bose gas with infinite on-site repulsion. Additional singlet $(S=0)$ and triplet $(S=1)$ modes are found as two-particle bound states of the elementary triplets. We have also calculated the contributions of the elementary and collective excitations into the spin structure factor. Our results are in excellent agreement with exact diagonalizations and dimer series expansions data as long as the dimerization parameter \ensuremath{\delta} is not too small $(\ensuremath{\delta}>0.1),$ i.e., while the elementary triplets can be treated as localized objects.We have studied the low-energy excitation spectrum of a dimerized and frustrated antiferromagnetic Heisenberg chain. We use an analytic approach, based on a description of the excitations as triplets above a strong-coupling singlet ground state. The quasiparticle spectrum is calculated by treating the excitations as a dilute Bose gas with infinite on-site repulsion. Additional singlet $(S=0)$ and triplet $(S=1)$ modes are found as two-particle bound states of the elementary triplets. We have also calculated the contributions of the elementary and collective excitations into the spin structure factor. Our results are in excellent agreement with exact diagonalizations and dimer series expansions data as long as the dimerization parameter \ensuremath{\delta} is not too small $(\ensuremath{\delta}>0.1),$ i.e., while the elementary triplets can be treated as localized objects.

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Takaaki Hirano, Yasuhiro Hatsugai

We investigate the entanglement entropy (EE) of gapped S=1 and $S=1/2$ spin chains with dimerization. We find that the effective boundary degrees of freedom as edge states contribute significantly to the EE. For the $S=1/2$ dimerized Heisenberg chain, the EE of the sufficiently long chain is essentially explained by the localized $S=1/2$ effective spins on the boundaries. As for S=1, the effective spins are also $S=1/2$ causing a Kennedy triplet that yields a lower bound for the EE. In this case, the residual entanglement reduces substantially by a continuous deformation of the Heisenberg model to that of the AKLT Hamiltonian.We investigate the entanglement entropy (EE) of gapped S=1 and $S=1/2$ spin chains with dimerization. We find that the effective boundary degrees of freedom as edge states contribute significantly to the EE. For the $S=1/2$ dimerized Heisenberg chain, the EE of the sufficiently long chain is essentially explained by the localized $S=1/2$ effective spins on the boundaries. As for S=1, the effective spins are also $S=1/2$ causing a Kennedy triplet that yields a lower bound for the EE. In this case, the residual entanglement reduces substantially by a continuous deformation of the Heisenberg model to that of the AKLT Hamiltonian.

arXiv:cond-mat/0703642Journal of the Physical Society of JapanJul 10, 200721 citations

We show that the low-energy sector of the

S=1/2

, antiferromagnetic Heisenberg model on a one-dimensional lattice of coupled tetrahedra consists of

{2}^{N}

(1/4)\mathrm{ln}2

; (ii) there is a singlet-triplet gap as long as the coupling between the tetrahedra is smaller than the internal one. These properties are compared to available results on the pyrochlore lattice.

s=4

for a square lattice). The obtained results establish a structural design principle for engineering quantum magnets with a high ground-state degeneracy, suggesting promising candidates for enhanced magnetocaloric cooling and quantum thermal machines.

(S=0)

and triplet

(S=1)

modes are found as two-particle bound states of the elementary triplets. We have also calculated the contributions of the elementary and collective excitations into the spin structure factor. Our results are in excellent agreement with exact diagonalizations and dimer series expansions data as long as the dimerization parameter \ensuremath{\delta} is not too small $(\ensuremath{\delta}>0.1),$ i.e., while the elementary triplets can be treated as localized objects.

We investigate the entanglement entropy (EE) of gapped S=1 and

S=1/2

spin chains with dimerization. We find that the effective boundary degrees of freedom as edge states contribute significantly to the EE. For the

S=1/2

dimerized Heisenberg chain, the EE of the sufficiently long chain is essentially explained by the localized

S=1/2

effective spins on the boundaries. As for S=1, the effective spins are also

S=1/2

causing a Kennedy triplet that yields a lower bound for the EE. In this case, the residual entanglement reduces substantially by a continuous deformation of the Heisenberg model to that of the AKLT Hamiltonian.