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:1912.01967Letters in Mathematical PhysicsMar 9, 20213 citations

Approximate Method for the Free Energy

Akihide Oguchi

Progress of Theoretical PhysicsNov 1, 197628 citations

Letters in Mathematical Physics

Upper bound on the free energy of the spin 1/2 Heisenberg ferromagnet

Joseph G. Conlon, Jan Philip Solovej

Published Nov 1, 1991• 20 citations

Publisher

DOI: 10.1007/bf01885500

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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.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:1912.01967Letters in Mathematical PhysicsMar 9, 20213 citations

Approximate Method for the Free Energy

Akihide Oguchi

We propose an upper bound and a lower bound of the Helmholtz free energy in the statistical physics. A new approximate expression of the free energy is obtained. This approximate value of the free energy is proved to be greater than the lower bound and less than the upper bound. A systematic method which can be extended to improve the approximation is studied. The upper bound and the lower bound of the approximate free energy converge to the true free energy as the successive approximation proceeds. The method is first explained with the Ising ferromagnet and then applied to the Heisenberg ferromagnet. In the simplest approximation the result agrees with the Bethe-Peierls approximation for the Ising model and the constant coupling approximation for the Heisenberg model. A more accurate approximation is studied for the Ising ferromagnet.We propose an upper bound and a lower bound of the Helmholtz free energy in the statistical physics. A new approximate expression of the free energy is obtained. This approximate value of the free energy is proved to be greater than the lower bound and less than the upper bound. A systematic method which can be extended to improve the approximation is studied. The upper bound and the lower bound of the approximate free energy converge to the true free energy as the successive approximation proceeds. The method is first explained with the Ising ferromagnet and then applied to the Heisenberg ferromagnet. In the simplest approximation the result agrees with the Bethe-Peierls approximation for the Ising model and the constant coupling approximation for the Heisenberg model. A more accurate approximation is studied for the Ising ferromagnet.

Progress of Theoretical PhysicsNov 1, 197628 citations

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.