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SEMICLASSICAL ANALYSIS OF TWO- AND THREE-SPIN ANTIFERROMAGNETS AND ANYONS ON A SPHERE

Diptiman Sen

Published Jan 1, 1993• 1 citation
OpenAlex

Abstract

We do a semiclassical analysis for two or three spins which are coupled antiferromagnetically to each other. The semiclassical wave functions transform correctly under permutations of the spins if one takes into account the Wess-Zumino term present in the path integral for spins. The Wess-Zumino term here is a total derivative which has no effect on the energy spectrum. The semiclassical problem is related to that of anyons moving on a sphere with the statistics parameter θ being 2πS for two spins and 3πS for three spins. Finally, we present a novel way of deriving the semiclassical wave functions from the spin wave functions. PACS numbers: 75.10.Jm, 03.65.Sq, 74.20.Kk

Related papers

SEMICLASSICAL ANALYSIS OF TWO- AND THREE-SPIN ANTIFERROMAGNETS AND ANYONS ON A SPHERE

DIPTIMAN SEN

We do a semiclassical analysis for two or three spins which are coupled antiferromagnetically to each other. The semiclassical wave functions transform correctly under permutations of the spins if one takes into account the Wess-Zumino term present in the path integral for spins. The Wess-Zumino term here is a total derivative which has no effect on the energy spectrum. The semiclassical problem is related to that of anyons moving on a sphere with the statistics parameter θ being 2πS for two spins and 3πS for three spins. Finally, we present a novel way of deriving the semiclassical wave functions from the spin wave functions.

arXiv:hep-th/9304151•Modern Physics Letters A•Jun 21, 1993•3 citations

Some remarks on anyons on the two-sphere

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We construct a Hamiltonian for $N$ anyons moving on the two-sphere. We show why the anyonic gauge coupling must satisfy a Dirac quantization condition, reproducing the braiding constraint on the statistical parameter. We then give a perturbative analysis of the $N$-body spectrum near Bose statistics.

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Semiclassical analysis of spin systems near critical energies

Pedro Ribeiro, Thierry Paul

The spectral properties of su(2) Hamiltonians are studied for energies near the critical classical energy ${\ensuremath{\epsilon}}_{c}$ for which the corresponding classical dynamics presents hyperbolic points. A general method leading to an algebraic relation for eigenvalues in the vicinity of ${\ensuremath{\epsilon}}_{c}$ is obtained in the thermodynamic limit, when the semiclassical parameter ${n}^{\ensuremath{-}1}={(2s)}^{\ensuremath{-}1}$ goes to zero (where $s$ is the total spin of the system). Two applications of this method are given and compared with numerics. Matrix elements of observables, computed between states with energy near ${\ensuremath{\epsilon}}_{c}$, are also computed and shown to be in agreement with the numerical results.

arXiv:0806.3192•Physical Review A•Mar 16, 2009•9 citations

Integrable sigma models with θ=π

Paul Fendley

A fundamental result relevant to spin chains and two-dimensional disordered systems is that the sphere sigma model with instanton coupling $\ensuremath{\theta}=\ensuremath{\pi}$ has a nontrivial low-energy fixed point and a gapless spectrum. This result is extended to two series of sigma models with $\ensuremath{\theta}=\ensuremath{\pi}:$ the $\mathrm{SU}(N)/\mathrm{SO}(N)$ sigma models flow to the $\mathrm{SU}{(N)}_{1}$ Wess-Zumino-Witten theory, while the $O(2N)/O(N)\ifmmode\times\else\texttimes\fi{}O(N)$ models flow to ${O(2N)}_{1}$ $(2N$-free Majorana fermions). These models are integrable, and the exact quasiparticle spectra and S matrices are found. One interesting feature is that charges fractionalize when $\ensuremath{\theta}=\ensuremath{\pi}.$ I compute the energy in a background field, and verify that the perturbative expansions for $\ensuremath{\theta}=0$ and \ensuremath{\pi} are the same as they must be. I discuss the flows between the two sequences of models, and also argue that the analogous sigma models with $\mathrm{Sp}(2N)$ symmetry, the $\mathrm{Sp}(2N)/U(N)$ models, flow to $\mathrm{Sp}{(2N)}_{1}.$

arXiv:cond-mat/0008372•Physical review. B, Condensed matter•Feb 21, 2001•48 citations

SOLUTIONS IN SPHERICAL COORDINATES OF GENERAL EQUATIONS OF PARTICLES WITH SPIN

André Jolivet

The wave functions in spherical coordinates were determined in the case of particles with any spin by utilizing a representntion allowing these functions to be expressed by simple formulas. (tr-auth)

•Oct 22, 1962•0 citations
We do a semiclassical analysis for two or three spins which are coupled antiferromagnetically to each other. The semiclassical wave functions transform correctly under permutations of the spins if one takes into account the Wess-Zumino term present in the path integral for spins. The Wess-Zumino term here is a total derivative which has no effect on the energy spectrum. The semiclassical problem is related to that of anyons moving on a sphere with the statistics parameter θ being 2πS for two spins and 3πS for three spins. Finally, we present a novel way of deriving the semiclassical wave functions from the spin wave functions. PACS numbers: 75.10.Jm, 03.65.Sq, 74.20.Kk
We do a semiclassical analysis for two or three spins which are coupled antiferromagnetically to each other. The semiclassical wave functions transform correctly under permutations of the spins if one takes into account the Wess-Zumino term present in the path integral for spins. The Wess-Zumino term here is a total derivative which has no effect on the energy spectrum. The semiclassical problem is related to that of anyons moving on a sphere with the statistics parameter θ being 2πS for two spins and 3πS for three spins. Finally, we present a novel way of deriving the semiclassical wave functions from the spin wave functions.
We construct a Hamiltonian for NNN anyons moving on the two-sphere. We show why the anyonic gauge coupling must satisfy a Dirac quantization condition, reproducing the braiding constraint on the statistical parameter. We then give a perturbative analysis of the NNN-body spectrum near Bose statistics.
The spectral properties of su(2) Hamiltonians are studied for energies near the critical classical energy ${\ensuremath{\epsilon}}_{c}$ for which the corresponding classical dynamics presents hyperbolic points. A general method leading to an algebraic relation for eigenvalues in the vicinity of ${\ensuremath{\epsilon}}_{c}$ is obtained in the thermodynamic limit, when the semiclassical parameter ${n}^{\ensuremath{-}1}={(2s)}^{\ensuremath{-}1}$ goes to zero (where sss is the total spin of the system). Two applications of this method are given and compared with numerics. Matrix elements of observables, computed between states with energy near ${\ensuremath{\epsilon}}_{c}$, are also computed and shown to be in agreement with the numerical results.
A fundamental result relevant to spin chains and two-dimensional disordered systems is that the sphere sigma model with instanton coupling $\ensuremath{\theta}=\ensuremath{\pi}$ has a nontrivial low-energy fixed point and a gapless spectrum. This result is extended to two series of sigma models with $\ensuremath{\theta}=\ensuremath{\pi}:$ the SU(N)/SO(N)\mathrm{SU}(N)/\mathrm{SO}(N)SU(N)/SO(N) sigma models flow to the SU(N)1\mathrm{SU}{(N)}_{1}SU(N)1​ Wess-Zumino-Witten theory, while the $O(2N)/O(N)\ifmmode\times\else\texttimes\fi{}O(N)$ models flow to O(2N)1{O(2N)}_{1}O(2N)1​ (2N(2N(2N-free Majorana fermions). These models are integrable, and the exact quasiparticle spectra and S matrices are found. One interesting feature is that charges fractionalize when $\ensuremath{\theta}=\ensuremath{\pi}.$ I compute the energy in a background field, and verify that the perturbative expansions for $\ensuremath{\theta}=0$ and \ensuremath{\pi} are the same as they must be. I discuss the flows between the two sequences of models, and also argue that the analogous sigma models with Sp(2N)\mathrm{Sp}(2N)Sp(2N) symmetry, the Sp(2N)/U(N)\mathrm{Sp}(2N)/U(N)Sp(2N)/U(N) models, flow to Sp(2N)1.\mathrm{Sp}{(2N)}_{1}.Sp(2N)1​.