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

Linear energy bounds for Heisenberg spin systems

Published Jan 1, 2002• 0 citations
OpenAlex

Abstract

Abstract. Recently obtained results on linear energy bounds are generalized to arbitrary spin quantum numbers and coupling schemes. Thereby the class of so– called independent magnon states, for which the relative ground–state property can be rigorously established, is considerably enlarged. We still require that the matrix of exchange parameters has constant row sums, but this can be achieved by means of a suitable gauge and need not be considered as a physical restriction. PACS numbers: 75.10.Jm, 75.50.XxLinear energy bounds 2 1.

Related papers

Linear energy bounds for Heisenberg spin systems

Heinz–Jürgen Schmidt

Recently obtained results on linear energy bounds are generalized to arbitrary spin quantum numbers and coupling schemes. Thereby, the class of so-called independent magnon states, for which the relative ground-state property can be rigorously established, is considerably enlarged. We still require that the matrix of exchange parameters has constant row sums, but this can be achieved by means of a suitable gauge and need not be considered as a physical restriction.

arXiv:cond-mat/0203270•Journal of Physics A Mathematical and General•Jul 25, 2002•33 citations

Bounding and approximating parabolas for the spectrum of Heisenberg spin systems

Heinz-jürgen Schmidt, Jürgen Schnack, Marshall Luban

PACS. 75.10.Jm – Quantized spin models. PACS. 75.10.Hk – Classical spin models. PACS. 75.50.Xx – Molecular magnets. Abstract. – We prove that for a wide class of quantum spin systems with isotropic Heisenberg coupling the energy eigenvalues which belong to a total spin quantum number S have upper and lower bounds depending at most quadratically on S. The only assumption adopted is that the mean coupling strength of any spin w. r. t. its neighbours is constant for all N spins. The coefficients of the bounding parabolas are given in terms of special eigenvalues of the N × N coupling matrix which are usually easily evaluated. In addition we show that the bounding parabolas, if properly shifted, provide very good approximations of the true boundaries of the spectrum. We present numerical examples of frustrated rings, a cube, and an icosahedron. Introduction. – The study of finite spin systems is not only interesting in its own right but has also important applications for understanding so-called “molecular magnets”. The synthesis of these systems has undergone rapid progress in recent years building on successes in coordination and polyoxometalate chemistry [1–4]. Each of the identical molecular units can

•Jan 1, 2001•9 citations

6 Linear independence of localized magnon states

Schmidt, Heinz-Jürgen, Johannes Richter, Roderich Moessner

Abstract. At the magnetic saturation field, certain frustrated lattices have a class of states known as “localized multi-magnon states ” as exact ground states. The number of these states scales exponentially with the number N of spins and hence they have a finite entropy also in the thermodynamic limit N → ∞ provided they are sufficiently linearly independent. In this article we present rigorous results concerning the linear dependence or independence of localized magnon states and investigate special examples. For large classes of spin lattices including what we called the orthogonal type and the isolated type as well as the kagomé, the checkerboard and the star lattice we have proven linear independence of all localized multi-magnon states. On the other hand the pyrochlore lattice provides an example of a spin lattice having localized multi-magnon states with considerable linear dependence. ‡ To whom correspondence should be addressed (hschmidt@uos.de) Localized magnon states 2 1. Introduction and

•Jan 1, 2006•27 citations

2 The Free Energy of the Quantum Heisenberg Ferromagnet at Large Spin

Michele Correggi, Alessandro Giuliani

We consider the spin-S ferromagnetic Heisenberg model in three dimensions, in the absence of an external field. Spin wave theory suggests that in a suitable temperature regime the system behaves effectively as a system of non-interacting bosons (magnons). We prove this fact at the level of the specific free energy: if S → ∞ and the inverse temperature β → 0 in such a way that βS stays constant, we rigorously show that the free energy per unit volume converges to the one suggested by spin wave theory. The proof is based on the localization of the system in small boxes and on upper and lower bounds on the local free energy, and it also provides explicit error bounds on the remainder. 1

•Jan 1, 2012•6 citations

Linear independence of localized magnon states

Heinz-Jürgen Schmidt, Johannes Richter, Roderich Moessner

At the magnetic saturation field, certain frustrated lattices have a class of states known as "localized multi-magnon states" as exact ground states. The number of these states scales exponentially with the number $N$ of spins and hence they have a finite entropy also in the thermodynamic limit $N\to \infty$ provided they are sufficiently linearly independent. In this article we present rigorous results concerning the linear dependence or independence of localized magnon states and investigate special examples. For large classes of spin lattices including what we called the orthogonal type and the isolated type as well as the kagomé, the checkerboard and the star lattice we have proven linear independence of all localized multi-magnon states. On the other hand the pyrochlore lattice provides an example of a spin lattice having localized multi-magnon states with considerable linear dependence.

arXiv:cond-mat/0604649•Journal of Physics A Mathematical and General•Aug 9, 2006•50 citations

Bounding and approximating parabolas for the spectrum of Heisenberg spin systems

H.-J Schmidt, J Schnack, M Luban

We prove that for a wide class of quantum spin systems with isotropic Heisenberg coupling the energy eigenvalues which belong to a total spin quantum number S have upper and lower bounds depending at most quadratically on S . The only assumption adopted is that the mean coupling strength of any spin with respect to its neighbours is constant for all N spins. The coefficients of the bounding parabolas are given in terms of special eigenvalues of the N × N coupling matrix which are usually easily evaluated. In addition, we show that the bounding parabolas, if properly shifted, provide very good approximations of the true boundaries of the spectrum. We present numerical examples of frustrated rings, a cube, and an icosahedron.

arXiv:cond-mat/0101228•Europhysics Letters (EPL)•Jul 1, 2001•12 citations
Abstract. Recently obtained results on linear energy bounds are generalized to arbitrary spin quantum numbers and coupling schemes. Thereby the class of so– called independent magnon states, for which the relative ground–state property can be rigorously established, is considerably enlarged. We still require that the matrix of exchange parameters has constant row sums, but this can be achieved by means of a suitable gauge and need not be considered as a physical restriction. PACS numbers: 75.10.Jm, 75.50.XxLinear energy bounds 2 1.
PACS. 75.10.Jm – Quantized spin models. PACS. 75.10.Hk – Classical spin models. PACS. 75.50.Xx – Molecular magnets. Abstract. – We prove that for a wide class of quantum spin systems with isotropic Heisenberg coupling the energy eigenvalues which belong to a total spin quantum number S have upper and lower bounds depending at most quadratically on S. The only assumption adopted is that the mean coupling strength of any spin w. r. t. its neighbours is constant for all N spins. The coefficients of the bounding parabolas are given in terms of special eigenvalues of the N × N coupling matrix which are usually easily evaluated. In addition we show that the bounding parabolas, if properly shifted, provide very good approximations of the true boundaries of the spectrum. We present numerical examples of frustrated rings, a cube, and an icosahedron. Introduction. – The study of finite spin systems is not only interesting in its own right but has also important applications for understanding so-called “molecular magnets”. The synthesis of these systems has undergone rapid progress in recent years building on successes in coordination and polyoxometalate chemistry [1–4]. Each of the identical molecular units can
Abstract. At the magnetic saturation field, certain frustrated lattices have a class of states known as “localized multi-magnon states ” as exact ground states. The number of these states scales exponentially with the number N of spins and hence they have a finite entropy also in the thermodynamic limit N → ∞ provided they are sufficiently linearly independent. In this article we present rigorous results concerning the linear dependence or independence of localized magnon states and investigate special examples. For large classes of spin lattices including what we called the orthogonal type and the isolated type as well as the kagomé, the checkerboard and the star lattice we have proven linear independence of all localized multi-magnon states. On the other hand the pyrochlore lattice provides an example of a spin lattice having localized multi-magnon states with considerable linear dependence. ‡ To whom correspondence should be addressed (hschmidt@uos.de) Localized magnon states 2 1. Introduction and
We consider the spin-S ferromagnetic Heisenberg model in three dimensions, in the absence of an external field. Spin wave theory suggests that in a suitable temperature regime the system behaves effectively as a system of non-interacting bosons (magnons). We prove this fact at the level of the specific free energy: if S → ∞ and the inverse temperature β → 0 in such a way that βS stays constant, we rigorously show that the free energy per unit volume converges to the one suggested by spin wave theory. The proof is based on the localization of the system in small boxes and on upper and lower bounds on the local free energy, and it also provides explicit error bounds on the remainder. 1
At the magnetic saturation field, certain frustrated lattices have a class of states known as "localized multi-magnon states" as exact ground states. The number of these states scales exponentially with the number NNN of spins and hence they have a finite entropy also in the thermodynamic limit N→∞N\to \inftyN→∞ provided they are sufficiently linearly independent. In this article we present rigorous results concerning the linear dependence or independence of localized magnon states and investigate special examples. For large classes of spin lattices including what we called the orthogonal type and the isolated type as well as the kagomé, the checkerboard and the star lattice we have proven linear independence of all localized multi-magnon states. On the other hand the pyrochlore lattice provides an example of a spin lattice having localized multi-magnon states with considerable linear dependence.
We prove that for a wide class of quantum spin systems with isotropic Heisenberg coupling the energy eigenvalues which belong to a total spin quantum number S have upper and lower bounds depending at most quadratically on S . The only assumption adopted is that the mean coupling strength of any spin with respect to its neighbours is constant for all N spins. The coefficients of the bounding parabolas are given in terms of special eigenvalues of the N × N coupling matrix which are usually easily evaluated. In addition, we show that the bounding parabolas, if properly shifted, provide very good approximations of the true boundaries of the spectrum. We present numerical examples of frustrated rings, a cube, and an icosahedron.