Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review

Free Energy of the Antiferromagnetic Linear Chain

Robert B. Griffiths

Published Nov 2, 1964• 34 citations
Publisher

Abstract

Rigorous upper and lower bounds for the free energy of the antiferromagnetic Heisenberg linear chain are obtained from general convexity properties of the free energy. The lower bound is here derived; the upper bound has been obtained previously by Bulaevskii. Both are compared with the calculations of Bonner and Fisher for finite chains.

DOI
10.1103/physrev.136.a751

Related papers

On the ground state energy of the linear antiferromagnetic Heisenberg chain with nearest- and next-nearest-neighbour interactions

P.M. van den Broek

The author calculates lower bounds for the ground state energy of the linear antiferromagnetic Heisenberg chain with nearest- and next-nearest-neighbour interactions. From these lower bounds new upper bounds are derived. Several approximations of the ground state energy given in the literature are compared with these upper and lower bounds. It is found that the best approximation is given by extrapolation from finite cyclic chains. This extrapolation, however, is restricted to certain values of the relative strength of the interactions. Here he extrapolates the ground state energy from finite open chains where no such restriction occurs.

Journal of Physics C Solid State Physics•Oct 20, 1980•7 citations

Influence of third-nearest-neighbor interactions on the ground state of Heisenberg linear chains

Jorge M. Mesías, Eugenio Emilio Vogel

Antiferromagnetic ground-state energies for infinite Heisenberg linear chains with up to third-nearest-neighbor interactions have been determined by means of numerical extrapolations. On the other hand, new conditions to produce a ferromagnetic ground state have been established for finite chains. These conditions are expressed as lower bounds for the coupling constants. The particular case in which the coupling constant for third-nearest-neighbor interactions is zero is compared in two different ways with previous calculations by other authors, showing good correspondence.

Physical review. B, Condensed matter•Dec 1, 1986•1 citation

A CLASS OF ENERGY LEVELS FOR THE HEISENBERG LINEAR CHAIN. 2. LEVELS NEAR THE ANTIFERRO MAGNETIC GROUND STATE

Robert B. Griffiths

The distribution of 'unbound' states in the Bethe formalism for the linear chain of spin 1/2 atoms with a Heisenbert exchange interaction between nearest neighbors is investigated in the vicinity of the antiferromagnetic ground state, utilizing the spin wave states discovered by des Cloizeaux and Pearson. An upper bound is obtained for the free energy of the anti ferromagnetic chain at very low temperatures. Plausible arguments are presented to show that the 'unbound' states actually make a negligible contribution to the partition function of an infinite chain, at least in the absence of a magnetic field.

•Jan 1, 1963•0 citations

Free energy asymptotics of the quantum Heisenberg spin chain

Marcin Napiórkowski, Robert Seiringer

Abstract We consider the ferromagnetic quantum Heisenberg model in one dimension, for any spin $$S\ge 1/2$$ S ≥ 1 / 2 . We give upper and lower bounds on the free energy, proving that at low temperature it is asymptotically equal to the one of an ideal Bose gas of magnons, as predicted by the spin-wave approximation. The trial state used in the upper bound yields an analogous estimate also in the case of two spatial dimensions, which is believed to be sharp at low temperature.

arXiv:1912.01967•Letters in Mathematical Physics•Mar 9, 2021•3 citations

Note on the Singular Terms in the Magnetic Susceptibility of the Antiferromagnetic Heisenberg Linear Chains

知司 山田

At low temperatures it is shown theoretically that there exist the singular terms in the magnetic susceptibility of the antiferromagnetic Heisenberg linear chains with spin S =1/2. Furthermore, a discussion is made in analogy with the critical phenomena. The free energy which describes the properties of the present system is composed of two parts: The regular and the singular parts. The singular terms in the magnetic susceptibility come from the latter part.

Journal of the Physical Society of Japan•May 15, 1974•0 citations

Rigorous Bounds on the Free Energy of Electron–Phonon Models

Hans A. De Raedt, Kristel F. L. Michielsen, Luuk Van Dijk

We present a collection of rigorous upper and lower bounds to the free energy of electron–phonon models with linear electron–phonon interaction. These bounds are used to compare different variational approaches. It is shown rigorously that the ground states corresponding to the sharpest bounds do not exhibit Off-Diagonal Long-Range Order in the two-particle density matrix.

International Journal of Modern Physics B•May 30, 1997•1 citation
Rigorous upper and lower bounds for the free energy of the antiferromagnetic Heisenberg linear chain are obtained from general convexity properties of the free energy. The lower bound is here derived; the upper bound has been obtained previously by Bulaevskii. Both are compared with the calculations of Bonner and Fisher for finite chains.
The author calculates lower bounds for the ground state energy of the linear antiferromagnetic Heisenberg chain with nearest- and next-nearest-neighbour interactions. From these lower bounds new upper bounds are derived. Several approximations of the ground state energy given in the literature are compared with these upper and lower bounds. It is found that the best approximation is given by extrapolation from finite cyclic chains. This extrapolation, however, is restricted to certain values of the relative strength of the interactions. Here he extrapolates the ground state energy from finite open chains where no such restriction occurs.
Antiferromagnetic ground-state energies for infinite Heisenberg linear chains with up to third-nearest-neighbor interactions have been determined by means of numerical extrapolations. On the other hand, new conditions to produce a ferromagnetic ground state have been established for finite chains. These conditions are expressed as lower bounds for the coupling constants. The particular case in which the coupling constant for third-nearest-neighbor interactions is zero is compared in two different ways with previous calculations by other authors, showing good correspondence.
The distribution of 'unbound' states in the Bethe formalism for the linear chain of spin 1/2 atoms with a Heisenbert exchange interaction between nearest neighbors is investigated in the vicinity of the antiferromagnetic ground state, utilizing the spin wave states discovered by des Cloizeaux and Pearson. An upper bound is obtained for the free energy of the anti ferromagnetic chain at very low temperatures. Plausible arguments are presented to show that the 'unbound' states actually make a negligible contribution to the partition function of an infinite chain, at least in the absence of a magnetic field.
Abstract We consider the ferromagnetic quantum Heisenberg model in one dimension, for any spin S≥1/2S\ge 1/2S≥1/2 S ≥ 1 / 2 . We give upper and lower bounds on the free energy, proving that at low temperature it is asymptotically equal to the one of an ideal Bose gas of magnons, as predicted by the spin-wave approximation. The trial state used in the upper bound yields an analogous estimate also in the case of two spatial dimensions, which is believed to be sharp at low temperature.