We present analytic formulas for the ground state energy of two-dimensional (2D) anyon gas in strong magnetic field (Landau level filling factor νL ≤ 1). The formulas are obtained by applying harmonic potential regularization for the vanishing confinement to harmonically confined Coulomb anyon gas. In a case of absence of the Coulomb interaction our analytic result provides an exact solution. It contains a contribution of the anyon gauge field characterized by the anyon parameters ν and νL. In a case of presence of the Coulomb interaction we introduce a function depending on the parameters ν, νL and the density parameter rs. The function is determined by fitting the Fano-Ortolani interpolation equation in the fractional quantum Hall effect regime for spin-polarized electrons and by consistence requirement with known results for the ground state energy of the 2D Coulomb Bose gas in strong magnetic field. We show that our formulas are valid not only for fermions (ν=1) but quite generally for anyons (0 ≤ ν ≤ 1).
We present analytic formulas for the ground state energy of two-dimensional (2D) anyon gas in strong magnetic field (Landau level filling factor νL ≤ 1). The formulas are obtained by applying harmonic potential regularization for the vanishing confinement to harmonically confined Coulomb anyon gas. In a case of absence of the Coulomb interaction our analytic result provides an exact solution. It contains a contribution of the anyon gauge field characterized by the anyon parameters ν and νL. In a case of presence of the Coulomb interaction we introduce a function depending on the parameters ν, νL and the density parameter rs. The function is determined by fitting the Fano-Ortolani interpolation equation in the fractional quantum Hall effect regime for spin-polarized electrons and by consistence requirement with known results for the ground state energy of the 2D Coulomb Bose gas in strong magnetic field. We show that our formulas are valid not only for fermions (ν=1) but quite generally for anyons (0 ≤ ν ≤ 1).
We derive an approximate analytic formula for the ground-state energy of the charged anyon gas. Our approach is based on the harmonically confined two-dimensional (2D) Coulomb anyon gas and a regularization procedure for vanishing confinement. To take into account the fractional statistics and Coulomb interaction we introduce a function, which depends on both the statistics and density parameters ($\ensuremath{\nu}$ and rs, respectively). We determine this function by fitting to the ground-state energies of the classical electron crystal at very large rs (the 2D Wigner crystal), and to the Hartree-Fock (HF) energy of the spin-polarized 2D electron gas, and the dense 2D Coulomb Bose gas at very small rs. The latter is calculated by use of the Bogoliubov approximation. Applied to the boson system $(\ensuremath{\nu}=0)$ our results are very close to recent results from Monte Carlo (MC) calculations. For spin-polarized electron systems $(\ensuremath{\nu}=1)$ our comparison leads to a critical judgment concerning the density range, to which the HF approximation and MC simulations apply. In dependence on $\ensuremath{\nu}$, our analytic formula yields ground-state energies, which monotonously increase from the bosonic to the fermionic side if rs>1. For ${r}_{s}\ensuremath{\leqslant}1$ it shows a nonmonotonous behavior indicating a breakdown of the assumed continuous transformation of bosons into fermions by variation of the parameter $\ensuremath{\nu}$.
The statistical mechanics of an anyon gas in a magnetic field is addressed. A harmonic regulator is used to define a proper thermodynamic limit. When the magnetic field is sufficiently strong, only exact N-anyon ground states, where anyons occupy the lowest Landau level, contribute to the equation of state. Particular attention is paid to the interval of definition of the statistical parameter \ensuremath{\alpha}\ensuremath{\in}[-1,0] where a gap exists. Interestingly enough, one finds that at the critical filling \ensuremath{\nu}=-1/\ensuremath{\alpha} where the pressure diverges, the external magnetic field is entirely screened by the flux tubes carried by the anyons.
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.
The energy of 2D Bloch electrons in a magnetic field is studied as a function of the filling fraction ν and the magnetic flux Φ. Using a new semi-classical quantization method the total energy E(Φ, ν) is calculated and shown to have an absolute minimum which corresponds to one flux quantum per particle. This optimal flux phenomenon is shown to occur under large conditions and for different lattices. An explicit cusp-like behavior of E νs. Φ at fixed ν is found both for the absolute and for the relative minima of E. Furthermore, the ground state energy is shown to be a smooth function of ν. The implications of our results for the stabilization of Anyons and the flux states are discussed.
An anyon wavefunction (characterized by the statistical factor n ) projected onto the lowest Landau level is derived for the fractional quantum Hall effect states at filling factor ν = n /(2 pn +1) ( p and n are integers). We study the properties of the anyon wavefunction by using detailed Monte Carlo simulations in disc geometry and show that the anyon ground-state energy is a lower bound to the composite fermion one. Our results suggest that the composite fermions can be viewed as a combination of anyons and a fluid of charge–neutral dipoles.
This thesis addresses the dimensional reduction of two-dimensional (2D) Abelian anyons under strong anisotropic confinement, studied both at the microscopic many-body level and at the mean-field/almost-bosonic level.Anyons are quasi-particles with fractional statistics that interpolate between bosons and fermions, characterized by an exchange phase e^{i·π·α} for α in (0, 1). The theoretical concept of anyons dates back to the work of Leinaas and Myrheim in 1977, while convincing experimental evidence for their existence emerged only in 2020, through their realization in fractional quantum Hall systems. Investigating their behavior under confinement not only clarifies the connections between one- and two-dimensional anyons, but also offers new insights into the mathematical structure of anyonic systems.At the microscopic level, we consider N anyons in the magnetic gauge picture with a harmonic trapping potential that freezes the motion in the longitudinal (y) direction. We rigorously prove that, after subtracting the frozen longitudinal contribution, the associated 2D energy Γ-converges to that of the one-dimensional (1D) Tonks-Girardeau (TG) gas. This establishes a precise mathematical connection between 2D anyons and 1D impenetrable bosons, and, via the Bose-Fermi mapping, to a free Fermi gas. Importantly, this 1D limit model is independent of the exchange parameter α at leading order, indicating that the effect of the magnetic gauge field is gauged away up to o(1) corrections. Equivalently, the long-range Aharonov-Bohm interactions collapse into a hard-core exclusion constraint in this dimensional reduction limit.At the mean-field level, in the average-field regime (N → ∞ with α proportional to N^{-1}), the 2D anyon system is described by the Chern-Simons-Schrödinger (CSS) functional, a nonlinear magnetic Schrödinger model with a self-consistent gauge field. Performing the same anisotropic dimensional reduction limit, the resulting 1D effective model is described by a quintic nonlinear Schrödinger (NLS) equation, whose quintic nonlinearity stems from the magnetic self-consistent gauge in the CSS model. This dimensional reduction is established both at the level of ground-state energies and, under suitable well-posedness assumptions on the CSS dynamics, at the level of time evolution. Unlike the TG gas, the effective coupling constant g in the quintic NLS retains a dependence on the exchange parameter α through g = π·β² for β proportional to α·N.Taken together, these results yield two distinct effective 1D descriptions arising from different orders of limits. Crucially, the mean-field/almost-bosonic limit and the dimensional reduction (strong confinement) limit do not commute: the TG gas limit is α-independent, whereas the quintic NLS limit keeps an α-dependence through β proportional to α·N. This non-commutativity highlights the subtle interplay between scaling limits and confinement in anyonic systems. Future work might investigate whether a critical regime separates these two situations.