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.KkWe 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

Jan 1, 19931 citation

Some remarks on anyons on the two-sphere

Alain Comtet, John McCabe, Stéphane Ouvry

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.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.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsJan 15, 199210 citations

Ground state energy of two anyons in harmonic potential by path integration

Weerachart Kilenthong, Virulh Sa-yakanit

In this thesis, the case of two anyons in a harmonic potential in two dimensions is studied by using Feynman path integration. Following Wilczek's ideas, in the concept of the anyon the extraordinary spin and statistics are fractional. Significant applications of anyons are to the fractional quantum Hall effect and anyon superconductivity. Moreover, the configuration space of anyonic systems is multiply connected, and then the path integral is very useful to solve such problems. Nevertheless, there is the alternative approach that can be applied to the same problem, namely, the Chern-Simons construction. By adding the Chern-Simons term into the ordinary Lagrangian, normal particles, particles with the usual statistics, cna become anyons. In this thesis we show that the ground-state energy of two anyons in a harmonic potential can be evaluated both by applying the path integral technique and by the use of the Chern-Simons theory. We show that the complexity of the topology of the anyonic system can be handled readily. It is shown that the ground-state energy of two anyons in a harmonic potential depends on the statistical parameter, alpha, representing the statistics of the anyon.In this thesis, the case of two anyons in a harmonic potential in two dimensions is studied by using Feynman path integration. Following Wilczek's ideas, in the concept of the anyon the extraordinary spin and statistics are fractional. Significant applications of anyons are to the fractional quantum Hall effect and anyon superconductivity. Moreover, the configuration space of anyonic systems is multiply connected, and then the path integral is very useful to solve such problems. Nevertheless, there is the alternative approach that can be applied to the same problem, namely, the Chern-Simons construction. By adding the Chern-Simons term into the ordinary Lagrangian, normal particles, particles with the usual statistics, cna become anyons. In this thesis we show that the ground-state energy of two anyons in a harmonic potential can be evaluated both by applying the path integral technique and by the use of the Chern-Simons theory. We show that the complexity of the topology of the anyonic system can be handled readily. It is shown that the ground-state energy of two anyons in a harmonic potential depends on the statistical parameter, alpha, representing the statistics of the anyon.

Jan 1, 19990 citations

A semiclassical treatment of path integrals for the spin system

Hiroshi Kuratsuji, Yutaka Mizobuchi

Starting with path integrals in the SU(2) coherent state representation, the semiclassical approximation of the propagator for the spin system is investigated. By extending the idea of the semiclassical expansion method, which was developed in the usual phase-space path integrals, to the path integrals in the curved phase space, which is characteristic of the SU(2) coherent states, we obtain a closed form for the semiclassical propagator. As an application, we discuss the semiclassical quantization condition for the spin system.Starting with path integrals in the SU(2) coherent state representation, the semiclassical approximation of the propagator for the spin system is investigated. By extending the idea of the semiclassical expansion method, which was developed in the usual phase-space path integrals, to the path integrals in the curved phase space, which is characteristic of the SU(2) coherent states, we obtain a closed form for the semiclassical propagator. As an application, we discuss the semiclassical quantization condition for the spin system.

Journal of Mathematical PhysicsApr 1, 198140 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 $\mathrm{Sp}{(2N)}_{1}.$ 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/0008372Physical review. B, Condensed matterFeb 21, 200148 citations

Bridging conformal field theory and parton approaches to SU ( n ) k chiral spin liquids

Tong Liu, Ying-Hai Wu, Hong-Hao Tu +1 more

We employ the $\mathrm{SU}{(n)}_{k}$ Wess-Zumino-Witten (WZW) model in conformal field theory to construct lattice wave functions in both one and two dimensions. The spins on all lattice sites are chosen to transform under the $\mathrm{SU}(n)$ irreducible representation with a single row and $k$ boxes in the Young tableau. It is demonstrated that the wave functions can be reinterpreted as parton states, which enables efficient conversion to matrix product states, such that many physical properties can be evaluated directly. In one dimension, these wave functions describe critical spin chains whose universality classes are in one-to-one correspondence with the WZW models used in the construction. In two dimensions, our constructions yield model wave functions for chiral spin liquids, and we show how to find all topological sectors of them in a systematic way. Using the null vectors of Kac-Moody algebras, parent Hamiltonians of the $\mathrm{SU}{(3)}_{k}$ series are derived. The $\mathrm{SU}{(3)}_{k}$ chiral spin liquids are lattice analogs of non-Abelian spin-singlet fractional quantum Hall states, and the $k=2$ member hosts Fibonacci anyons.We employ the $\mathrm{SU}{(n)}_{k}$ Wess-Zumino-Witten (WZW) model in conformal field theory to construct lattice wave functions in both one and two dimensions. The spins on all lattice sites are chosen to transform under the $\mathrm{SU}(n)$ irreducible representation with a single row and $k$ boxes in the Young tableau. It is demonstrated that the wave functions can be reinterpreted as parton states, which enables efficient conversion to matrix product states, such that many physical properties can be evaluated directly. In one dimension, these wave functions describe critical spin chains whose universality classes are in one-to-one correspondence with the WZW models used in the construction. In two dimensions, our constructions yield model wave functions for chiral spin liquids, and we show how to find all topological sectors of them in a systematic way. Using the null vectors of Kac-Moody algebras, parent Hamiltonians of the $\mathrm{SU}{(3)}_{k}$ series are derived. The $\mathrm{SU}{(3)}_{k}$ chiral spin liquids are lattice analogs of non-Abelian spin-singlet fractional quantum Hall states, and the $k=2$ member hosts Fibonacci anyons.

arXiv:2501.09567Physical review. B./Physical review. BMay 27, 20253 citations

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.

\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}.

We employ the

\mathrm{SU}{(n)}_{k}

Wess-Zumino-Witten (WZW) model in conformal field theory to construct lattice wave functions in both one and two dimensions. The spins on all lattice sites are chosen to transform under the

\mathrm{SU}(n)

irreducible representation with a single row and

k

boxes in the Young tableau. It is demonstrated that the wave functions can be reinterpreted as parton states, which enables efficient conversion to matrix product states, such that many physical properties can be evaluated directly. In one dimension, these wave functions describe critical spin chains whose universality classes are in one-to-one correspondence with the WZW models used in the construction. In two dimensions, our constructions yield model wave functions for chiral spin liquids, and we show how to find all topological sectors of them in a systematic way. Using the null vectors of Kac-Moody algebras, parent Hamiltonians of the

\mathrm{SU}{(3)}_{k}

series are derived. The

\mathrm{SU}{(3)}_{k}

chiral spin liquids are lattice analogs of non-Abelian spin-singlet fractional quantum Hall states, and the

k=2

member hosts Fibonacci anyons.