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/0701715 · Physical review. B, Condensed matter

Statistical interparticle potential between two anyons

Wung-Hong Huang

Published Dec 1, 1995• 7 citations
arXivHTMLPDF

Abstract

The density matrix of a two-anyon system is evaluated and used to investigate the "statistical interparticle potential" following the theory of Uhlenbeck. The main purpose is to see how the statistical potential will depend on the fractional statistical parameter $\ensuremath{\alpha}$. The result shows that the statistical potential for a two-anyon system with $\ensuremath{\alpha}>~\frac{1}{2}$ is always repulsive. For the system with $0<\ensuremath{\alpha}<\frac{1}{2}$, the potential is repulsive at short distances and becomes attractive at long distances. It remains only in the boson system ($\ensuremath{\alpha}=0$) that the repulsive potential arising from the exclusion principle can disappear and lead to an attractive potential at all distances.

DOI
10.1103/physrevb.52.15090

Related papers

Statistical potential in an ideal anyon gas

B. Kahng, K. Park

We calculate the e/Istatistical potentials between particles in an ideal anyon gas arising from the symmetry property of the wave functions. The statistical potential is known to be purely repulsive for fermions, and to be purely attractive for bosons. In contrast, the statistical potentials for anyons are found to behave differently; the statistical potential for quasibosons is not purely attractive, displaying a minimum at finite interparticle distance. That implies there exists a stable equilibrium interparticle distance between two quasibosons. The statistical potential for quasifermions behaves similarly to that for fermions. Our result supports that the probability amplitude of finding two anyons at the same space point is zero.

Physical review. B, Condensed matter•Apr 1, 1992•3 citations

Statistical interparticle potential of an ideal gas of non-Abelian anyons

Francesco Mancarella, Andrea Trombettoni, Giuseppe Mussardo

We determine and study the statistical interparticle potential of an ideal system of non-Abelian Chern-Simons (NACS) particles, comparing our results with the corresponding results of an ideal gas of Abelian anyons. In the Abelian case, the statistical potential depends on the statistical parameter and it has a "quasi-bosonic" behaviour for statistical parameter in the range (0,1/2) (non-monotonic with a minimum) and a "quasi-fermionic" behaviour for statistical parameter in the range (1/2,1) (monotonically decreasing without a minimum). In the non-Abelian case the behavior of the statistical potential depends on the Chern- Simons coupling and the isospin quantum number: as a function of these two parameters, a phase diagram with quasi-bosonic, quasi-fermionic and bosonic-like regions is obtained and investigated. Finally, using the obtained expression for the statistical potential, we compute the second virial coefficient of the NACS gas, which correctly reproduces the results available in literature.

arXiv:1207.1724•Journal of Physics A Mathematical and Theoretical•Jun 18, 2013•5 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.

•Jan 1, 1999•0 citations

Anyons and Lowest Landau Level Anyons

Stéphane Ouvry

Intermediate statistics interpolating from Bose statistics to Fermi statistics are allowed in two dimensions. This is due to the particular topology of the two dimensional configuration space of identical particles, leading to non trivial braiding of particles around each other. One arrives at quantum many-body states with a multivalued phase factor, which encodes the any- onic nature of particle windings. Bosons and fermions appear as two limiting cases. Gauging away the phase leads to the so-called anyon model, where the charge of each particle interacts “à la Aharonov-Bohm” with the fluxes carried by the other particles. The multivaluedness of the wave function has been traded off for topological interactions between ordinary particles. An alternative Lagrangian formulation uses a topological Chern-Simons term in 2+1 dimensions. Taking into account the short distance repulsion between particles leads to an Hamiltonian well defined in perturbation theory, where all perturbative divergences have disappeared. Together with numerical and semi-classical studies, perturbation theory is a basic analytical tool at disposal to study the model, since finding the exact N-body spectrum seems out of reach (except in the 2-body case which is solvable, or for particular classes of N-body eigenstates which generalize some 2-body eigenstates). However, a simplification arises when the anyons are coupled to an external homogeneous magnetic field. In the case of a strong field, by projecting the system on its lowest Landau level (LLL, thus the LLL-anyon model), the anyon model becomes solvable, i.e., the classes of exact eigenstates alluded to above provide for a complete interpolation from the LLL-Bose spectrum to the LLL-Fermi spectrum. Being a solvable model allows for an explicit knowledge of the equation of state and of the mean occupation number in the LLL, which do interpolate from the Bose to the Fermi cases. It also provides for a combinatorial interpretation of LLL-anyon braiding statistics in terms of occupation of single particle states. The LLL-anyon model might also be relevant experimentally: a gas of electrons in a strong magnetic field is known to exhibit a quantized Hall conductance, leading to the integer and fractional quantum Hall effects. Haldane/exclusion statistics, introduced to describe FQHE edge excitations, is a priori different from anyon statistics, since it is not defined by braiding considerations, but rather by counting arguments in the space of available states. However, it has been shown to lead to the same kind of thermodynamics as the LLL-anyon thermodynamics (or, in other words, the LLL-anyon model is a microscopic quantum mechanical realization of Hal- dane’s statistics). The one dimensional Calogero model is also shown to have the same kind of thermodynamics as the LLL-anyons thermodynamics. This is not a coincidence: the LLL-anyon model and the Calogero model are intimately related, the latter being a particular limit of the former. Finally, on the purely combinatorial side, the minimal difference partition problem — partition of integers with minimal difference constraints on their parts — can also be mapped on an abstract exclusion statistics model with a constant one-body density of states, which is neither the LLL-anyon model nor the Calogero model. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

arXiv:0712.2174•Birkhäuser Basel eBooks•Jan 1, 2009•32 citations

Comment on “Quantum Statistical Mechanics of an Ideal Gas with Fractional Exclusion Statistics in Arbitrary Dimension”

Wung-Hong Huang

It is mentioned that anyon thermodynamic potential $Q(\alpha, N)$ could not be factorized in terms characteristic of the ideal boson $\alpha =0$ and fermion $\alpha =1$ gases by the relation $Q(\alpha, N) = (1-\alpha) Q(0, N_b)+ \alpha Q(1, N_f)$ in which $N=N_f +N_b$, that claimed in Phys. Rev. Lett. 78, 3233 (1997). Our analyses indicate that the thermodynamic quantities of anyon gas may be factorized as $Q(\alpha) = \alpha Q(1) + (1-\alpha) Q(0)$ only in the two-dimension system.

arXiv:cond-mat/0702075•Physical Review Letters•Sep 14, 1998•12 citations

Kohn-Sham density functional theory of Abelian anyons

Yayun Hu, Ganpathy N. Murthy, Sumathi Rao +1 more

We develop a density functional treatment of noninteracting Abelian anyons, which is capable, in principle, of dealing with a system of a large number of anyons in an external potential. Comparison with exact results for a few particles shows that the model captures the behavior qualitatively and semiquantitatively, especially in the vicinity of the fermionic statistics. We then study anyons with statistics parameter $1+1/n$, which are thought to condense into a superconducting state. An indication of the superconducting behavior is the mean-field result that, for uniform density systems, the ground-state energy increases under the application of an external magnetic field independent of its direction. Our density functional theory based analysis does not find that to be the case for finite systems of anyons, which can accommodate a weak external magnetic field through density transfer between the bulk and the boundary rather than through transitions across effective Landau levels, but the ``Meissner repulsion'' of the external magnetic field is recovered in the thermodynamic limit as the effect of the boundary becomes negligible. We also consider the quantum Hall effect of anyons, and show that its topological properties, such as the charge and statistics of the excitations and the quantized Hall conductance, arise in a self-consistent fashion.

arXiv:2010.09872•Physical review. B./Physical review. B•Jan 19, 2021•11 citations
The density matrix of a two-anyon system is evaluated and used to investigate the "statistical interparticle potential" following the theory of Uhlenbeck. The main purpose is to see how the statistical potential will depend on the fractional statistical parameter \ensuremathα\ensuremath{\alpha}\ensuremathα. The result shows that the statistical potential for a two-anyon system with \ensuremathα> 12\ensuremath{\alpha}>~\frac{1}{2}\ensuremathα> 21​ is always repulsive. For the system with 0<\ensuremathα<120<\ensuremath{\alpha}<\frac{1}{2}0<\ensuremathα<21​, the potential is repulsive at short distances and becomes attractive at long distances. It remains only in the boson system (\ensuremathα=0\ensuremath{\alpha}=0\ensuremathα=0) that the repulsive potential arising from the exclusion principle can disappear and lead to an attractive potential at all distances.
We calculate the e/Istatistical potentials between particles in an ideal anyon gas arising from the symmetry property of the wave functions. The statistical potential is known to be purely repulsive for fermions, and to be purely attractive for bosons. In contrast, the statistical potentials for anyons are found to behave differently; the statistical potential for quasibosons is not purely attractive, displaying a minimum at finite interparticle distance. That implies there exists a stable equilibrium interparticle distance between two quasibosons. The statistical potential for quasifermions behaves similarly to that for fermions. Our result supports that the probability amplitude of finding two anyons at the same space point is zero.
We determine and study the statistical interparticle potential of an ideal system of non-Abelian Chern-Simons (NACS) particles, comparing our results with the corresponding results of an ideal gas of Abelian anyons. In the Abelian case, the statistical potential depends on the statistical parameter and it has a "quasi-bosonic" behaviour for statistical parameter in the range (0,1/2) (non-monotonic with a minimum) and a "quasi-fermionic" behaviour for statistical parameter in the range (1/2,1) (monotonically decreasing without a minimum). In the non-Abelian case the behavior of the statistical potential depends on the Chern- Simons coupling and the isospin quantum number: as a function of these two parameters, a phase diagram with quasi-bosonic, quasi-fermionic and bosonic-like regions is obtained and investigated. Finally, using the obtained expression for the statistical potential, we compute the second virial coefficient of the NACS gas, which correctly reproduces the results available in literature.
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.
Intermediate statistics interpolating from Bose statistics to Fermi statistics are allowed in two dimensions. This is due to the particular topology of the two dimensional configuration space of identical particles, leading to non trivial braiding of particles around each other. One arrives at quantum many-body states with a multivalued phase factor, which encodes the any- onic nature of particle windings. Bosons and fermions appear as two limiting cases. Gauging away the phase leads to the so-called anyon model, where the charge of each particle interacts “à la Aharonov-Bohm” with the fluxes carried by the other particles. The multivaluedness of the wave function has been traded off for topological interactions between ordinary particles. An alternative Lagrangian formulation uses a topological Chern-Simons term in 2+1 dimensions. Taking into account the short distance repulsion between particles leads to an Hamiltonian well defined in perturbation theory, where all perturbative divergences have disappeared. Together with numerical and semi-classical studies, perturbation theory is a basic analytical tool at disposal to study the model, since finding the exact N-body spectrum seems out of reach (except in the 2-body case which is solvable, or for particular classes of N-body eigenstates which generalize some 2-body eigenstates). However, a simplification arises when the anyons are coupled to an external homogeneous magnetic field. In the case of a strong field, by projecting the system on its lowest Landau level (LLL, thus the LLL-anyon model), the anyon model becomes solvable, i.e., the classes of exact eigenstates alluded to above provide for a complete interpolation from the LLL-Bose spectrum to the LLL-Fermi spectrum. Being a solvable model allows for an explicit knowledge of the equation of state and of the mean occupation number in the LLL, which do interpolate from the Bose to the Fermi cases. It also provides for a combinatorial interpretation of LLL-anyon braiding statistics in terms of occupation of single particle states. The LLL-anyon model might also be relevant experimentally: a gas of electrons in a strong magnetic field is known to exhibit a quantized Hall conductance, leading to the integer and fractional quantum Hall effects. Haldane/exclusion statistics, introduced to describe FQHE edge excitations, is a priori different from anyon statistics, since it is not defined by braiding considerations, but rather by counting arguments in the space of available states. However, it has been shown to lead to the same kind of thermodynamics as the LLL-anyon thermodynamics (or, in other words, the LLL-anyon model is a microscopic quantum mechanical realization of Hal- dane’s statistics). The one dimensional Calogero model is also shown to have the same kind of thermodynamics as the LLL-anyons thermodynamics. This is not a coincidence: the LLL-anyon model and the Calogero model are intimately related, the latter being a particular limit of the former. Finally, on the purely combinatorial side, the minimal difference partition problem — partition of integers with minimal difference constraints on their parts — can also be mapped on an abstract exclusion statistics model with a constant one-body density of states, which is neither the LLL-anyon model nor the Calogero model. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
It is mentioned that anyon thermodynamic potential Q(α,N)Q(\alpha, N)Q(α,N) could not be factorized in terms characteristic of the ideal boson α=0\alpha =0α=0 and fermion α=1\alpha =1α=1 gases by the relation Q(α,N)=(1−α)Q(0,Nb)+αQ(1,Nf)Q(\alpha, N) = (1-\alpha) Q(0, N_b)+ \alpha Q(1, N_f)Q(α,N)=(1−α)Q(0,Nb​)+αQ(1,Nf​) in which N=Nf+NbN=N_f +N_bN=Nf​+Nb​, that claimed in Phys. Rev. Lett. 78, 3233 (1997). Our analyses indicate that the thermodynamic quantities of anyon gas may be factorized as Q(α)=αQ(1)+(1−α)Q(0)Q(\alpha) = \alpha Q(1) + (1-\alpha) Q(0)Q(α)=αQ(1)+(1−α)Q(0) only in the two-dimension system.
We develop a density functional treatment of noninteracting Abelian anyons, which is capable, in principle, of dealing with a system of a large number of anyons in an external potential. Comparison with exact results for a few particles shows that the model captures the behavior qualitatively and semiquantitatively, especially in the vicinity of the fermionic statistics. We then study anyons with statistics parameter 1+1/n1+1/n1+1/n, which are thought to condense into a superconducting state. An indication of the superconducting behavior is the mean-field result that, for uniform density systems, the ground-state energy increases under the application of an external magnetic field independent of its direction. Our density functional theory based analysis does not find that to be the case for finite systems of anyons, which can accommodate a weak external magnetic field through density transfer between the bulk and the boundary rather than through transitions across effective Landau levels, but the ``Meissner repulsion'' of the external magnetic field is recovered in the thermodynamic limit as the effect of the boundary becomes negligible. We also consider the quantum Hall effect of anyons, and show that its topological properties, such as the charge and statistics of the excitations and the quantized Hall conductance, arise in a self-consistent fashion.