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

arXiv:cond-mat/0111581 · Europhysics Letters (EPL)

Reply to the Comment by O. Waldmann on “Bounding and approximating parabolas for the spectrum of Heisenberg spin systems”

H-J Schmidt, Jürgen Schnack, Marshall Luban

Published Feb 1, 2002• 4 citations
arXivHTMLPDF

Abstract

O. Waldmann has shown that some spin systems, which fulfill the condition of a weakly homogeneous coupling matrix, have a spectrum whose minimal or maximal energies are rather poorly approximated by a quadratic dependence on the total spin quantum number. We comment on this observation and provide the new argument that, under certain conditions, the approximating parabolas appear as natural bounds of the spectrum generated by spin coherent states.

DOI
10.1209/epl/i2002-00507-2

Related papers

Comment on “Bounding and approximating parabolas for the spectrum of Heisenberg spin systems” by H.-J. Schmidt, J. Schnack and M. Luban

O. Waldmann

Recently, Schmidt et al. proved that the energy spectrum of a Heisenberg spin system (HSS) is bounded by two parabolas, i.e. lines which depend on the total spin quantum number S as S(S+1). The prove holds for homonuclear HSSs which fulfill a weak homogenity condition. Moreover, the extremal values of the exact spectrum of various HSS which were studied numerically were found to lie on approximate parabolas, named rotational bands, which could be obtained by a shift of the boundary parabolas. In view of this, it has been claimed that the rotational band structure (RBS) of the energy spectrum is a general behavior of HSSs. Furthermore, since the approximate parabolas are very close to the true boundaries of the spectrum for the examples discussed, it has been claimed that the methods allow to predict the detailed shape of the spectrum and related properties for a general HSS. In this comment I will show by means of examples that the RBS hypothesis is not valid for general HSSs. In particular, weak homogenity is neither a necessary nor a sufficient condition for a HSS to exhibit a spectrum with RBS.

arXiv:cond-mat/0201096•Europhysics Letters (EPL)•Feb 1, 2002•8 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

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

Reply to “Comment on ‘Gapless spin liquid ground state of the spin- 12J1−J2 Heisenberg model on square lattices' ”

Wen-Yuan Liu, Shaojun Dong, Chao Wang +4 more

In a recent comments [arXiv:1909.12788 (2019)], Zhao et al. argue that the definition of dimer orders used in our paper [Phys. Rev. B 98, 241109 (2018)] may not rule out valence bond solid (VBS) orders of $J_1$-$J_2$ model on the open boundary conditions (OBC). In this reply, we show that their argument does not apply to our case.

arXiv:1910.09130•Physical review. B./Physical review. B•Apr 9, 2020•1 citation

Reply to Comment on Semiclassical approximations in phase space with coherent states

Michel Baranger, Marcus A. M. de Aguiar, F Keck +2 more

The Herman–Kluk (HK) formula was shown in (Baranger et al J. Phys. A: Math. Gen. 34 7227 ) not to be a correct semiclassical limit of an exact quantum mechanical formula. Two previous attempts to derive it using semiclassical arguments contain serious errors. These statements are left totally untouched by Herman and Grossmann's comment. They argue that the formula which we found to be at fault is not the one that should be called the HK formula. However, the formula we criticized is definitely one of the steps, in fact the main step, in these two published derivations of the HK formula. Very recently, a new derivation was published by Miller. It is interesting, but it is not semiclassical.

Journal of Physics A Mathematical and General•Oct 21, 2002•29 citations

Reply to a Comment by J. Bolte, R. Glaser and S. Keppeler on: Semiclassical theory of spin-orbit interactions using spin coherent states

Mikhail Pletyukhov, Ch. Amann, Mehta, M. +1 more

We reply to a Comment on our recently proposed semiclassical theory for systems with spin-orbit interactions.

arXiv:nlin/0209028•Sep 13, 2002•0 citations
O. Waldmann has shown that some spin systems, which fulfill the condition of a weakly homogeneous coupling matrix, have a spectrum whose minimal or maximal energies are rather poorly approximated by a quadratic dependence on the total spin quantum number. We comment on this observation and provide the new argument that, under certain conditions, the approximating parabolas appear as natural bounds of the spectrum generated by spin coherent states.
Recently, Schmidt et al. proved that the energy spectrum of a Heisenberg spin system (HSS) is bounded by two parabolas, i.e. lines which depend on the total spin quantum number S as S(S+1). The prove holds for homonuclear HSSs which fulfill a weak homogenity condition. Moreover, the extremal values of the exact spectrum of various HSS which were studied numerically were found to lie on approximate parabolas, named rotational bands, which could be obtained by a shift of the boundary parabolas. In view of this, it has been claimed that the rotational band structure (RBS) of the energy spectrum is a general behavior of HSSs. Furthermore, since the approximate parabolas are very close to the true boundaries of the spectrum for the examples discussed, it has been claimed that the methods allow to predict the detailed shape of the spectrum and related properties for a general HSS. In this comment I will show by means of examples that the RBS hypothesis is not valid for general HSSs. In particular, weak homogenity is neither a necessary nor a sufficient condition for a HSS to exhibit a spectrum with RBS.
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.
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
In a recent comments [arXiv:1909.12788 (2019)], Zhao et al. argue that the definition of dimer orders used in our paper [Phys. Rev. B 98, 241109 (2018)] may not rule out valence bond solid (VBS) orders of J1J_1J1​-J2J_2J2​ model on the open boundary conditions (OBC). In this reply, we show that their argument does not apply to our case.
The Herman–Kluk (HK) formula was shown in (Baranger et al J. Phys. A: Math. Gen. 34 7227 ) not to be a correct semiclassical limit of an exact quantum mechanical formula. Two previous attempts to derive it using semiclassical arguments contain serious errors. These statements are left totally untouched by Herman and Grossmann's comment. They argue that the formula which we found to be at fault is not the one that should be called the HK formula. However, the formula we criticized is definitely one of the steps, in fact the main step, in these two published derivations of the HK formula. Very recently, a new derivation was published by Miller. It is interesting, but it is not semiclassical.