Virial coefficients of non-abelian anyons

We study a system of non-Abelian anyons in the lowest Landau level of a strong magnetic field. Using diagrammatic techniques, we prove that the virial coefficients do not depend on the statistics parameter. This is true for all representations of all non-Abelian groups for the statistics of the particles and relies solely on the fact that the effective statistical interaction is a traceless operator.     

On the thermodynamics of multispecies anyons

We address the problem of multispecies anyons, i.e. particles of different species whose wave functions are subject to anyonlike conditions. The cluster and virial coefficients are considered. Special attention is paid to the case of anyons in the lowest Landau level of a strong magnetic field, when it is possible (i) to prove microscopically the equation of state, in particular in terms of Aharonov-Bohm charge-flux composite systems, and (ii) to formulate the problem in terms of single-state statistical distributions.     

Anyonic quantum walks

The one dimensional quantum walk of anyonic systems is presented. The anyonic walker performs braiding operations with stationary anyons of the same type ordered canonically on the line of the walk. Abelian as well as non-Abelian anyons are studied and it is shown that they have very different properties. Abelian anyonic walks demonstrate the expected quadratic quantum speedup. Non-Abelian anyonic walks are much more subtle. The exponential increase of the system’s Hilbert space and the particular statistical evolution of non-Abelian anyons give a variety of new behaviors. The position distribution of the walker is related to Jones polynomials, topological invariants of the links created by the anyonic world-lines during the walk. Several examples such as the SU(2)_k and the quantum double models are considered that provide insight to the rich diffusion properties of anyons.     

Non-Abelian anyon superconductivity

Non-Abelian anyons exist in certain spin models and may exist in quantum Hall systems at certain filling fractions. In this work, we studied the ground state of dynamical SU(2) level-kappa Chern-Simons non-Abelian anyons at finite density and no external magnetic field. We find that, in the large-kappa limit, the topological interaction induces a pairing instability and the ground state is a superconductor with d+id gap symmetry. We also develop a picture of pairing for the special value kappa=2 and argue that the ground state is a superfluid of pairs for all values of kappa.     

Lattice gauge theories and motion in group space

We extend the SU(2) lattice gauge theory of Kogut and Susskind to a general non-Abelian gauge group. At the Lagrangian level, we find the theory to be related to the motion of a point in group space. We then quantise such a system using the natural geometric structure of group parameter space, and we apply our results to find the Hamiltonian for the general lattice gauge theory. We also discuss the large N behaviour of the theory.     

Critical magnetic field ratio of anisotropic magnetic superconductors

The upper critical field, the lower critical field and the critical magnetic field ratio of anisotropic magnetic superconductors are calculated by Ginzburg–Landau theory analytically. The effect of the Ginzburg–Landau parameter (κ₀), magnetic susceptibility (χ) and magnetic-to-anisotropic parameter ratio (θ) on the critical field ratio are considered. We find that the value of critical field ratio increases with increasing κ₀ and θ, and decreases with increasing χ. The highest and the lowest value of critical field ratio is found in the diamagnetic superconductors and the ferromagnetic superconductors, respectively.     

Anyons in an exactly solved model and beyond

A spin-1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z₂ gauge field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap and carries excitations that are Abelian anyons. The other phase is gapless, but acquires a gap in the presence of magnetic field. In the latter case excitations are non-Abelian anyons whose braiding rules coincide with those of conformal blocks for the Ising model. We also consider a general theory of free fermions with a gapped spectrum, which is characterized by a spectral Chern number ν. The Abelian and non-Abelian phases of the original model correspond to ν = 0 and ν = ±1, respectively. The anyonic properties of excitation depend on ν mod 16, whereas ν itself governs edge thermal transport. The paper also provides mathematical background on anyons as well as an elementary theory of Chern number for quasidiagonal matrices.     

Exact results for the spectra of bosons and fermions with contact interaction

An N-body bosonic model with delta-contact interactions projected on the lowest Landau level is considered. For a given number of particles in a given angular momentum sector, any energy level can be obtained exactly by means of diagonalizing a finite matrix: they are roots of algebraic equations. A complete solution of the three-body problem is presented, some general properties of the N-body spectrum are pointed out, and a number of novel exact analytic eigenstates are obtained. The FQHE N-fermion model with Laplacian-delta interactions is also considered along the same lines of analysis. New exact eigenstates are proposed, along with the Slater determinant, whose eigenvalues are shown to be related to Catalan numbers.     

Correlators and fractional statistics in the quantum Hall bulk

We derive one-particle and two-particle correlators of anyons in the lowest Landau level. We show that the two-particle correlator exhibits signatures of fractional statistics which can distinguish anyons from their fermionic and bosonic counterparts. These signatures include the zeros of the two-particle correlator and its exclusion behavior. We find that the one-particle correlator in finite geometries carries valuable information relevant to experiments in which quasiparticles on the edge of a quantum Hall system tunnel through its bulk.     

Two-parametric fractional statistics models for anyons

In the paper, two-parametric models of fractional statistics are proposed in order to determine the functional form of the distribution function of free anyons. From the expressions of the second and third virial coefficients, an approximate correspondence is shown to hold for three models, namely, the nonadditive Polychronakos statistics and both the incomplete and the nonadditive modifications of the Haldane-Wu statistics. The difference occurs only in the fourth virial coefficient leading to a small correction in the equation of state. For the two generalizations of the Haldane-Wu statistics, the solutions for the statistics parameters g, q exist in the whole domain of the anyonic parameter α ∈ [0; 1], unlike the nonadditive Polychronakos statistics. It is suggested that the search for the expression of the anyonic distribution function should be made within some modifications of the Haldane-Wu statistics.     

Distribution functions for a two-dimensional non-interacting quantum electron gas in an external magnetic field

The exact n-body distribution functions are calculated for a two-dimensional, non-interacting quantum electron gas in an external magnetic field for any temperature and density. At low tempertures and filled lowest Landau level (LLL), these functions are identical to the exact distribution functions obtained by Jancovici [1981, Phys. Rev. Lett., 46, 386] for the classical two-dimensional one-component plasma (2DOCP) at the special plasma parameter Γ = 2, thus establishing that the 2DOCP provides an exact classical Boltzmann factor which describes the ideal LLL quantum state associated with the integral quantum Hall effect.     

The Quantum N-Body Problem and the Auxiliary Field Method: New Developments

Approximate analytical energy formulas for N-body semirelativistic Hamiltonians with one- and two-body interactions are obtained within the framework of the auxiliary field method. We first review the method in the case of nonrelativistic two-body problems. A general procedure is then given for N-body systems and a connection is presented between the method and the generalized virial theorem. The procedure is applied to the case of baryons in the large-N _c limit.     

Supersymmetries of the spin-1/2 particle in the field of magnetic vortex, and anyons

The quantum non-relativistic spin-1/2 planar systems in the presence of a perpendicular magnetic field are known to possess the N = 2 supersymmetry. We consider such a system in the field of a magnetic vortex, and find that there are just two self-adjoint extensions of the Hamiltonian that are compatible with the standard N = 2 supersymmetry. We show that only in these two cases one of the subsystems coincides with the original spinless Aharonov-Bohm model and comes accompanied by the super-partner Hamiltonian which allows a singular behavior of the wave functions. We find a family of additional, nonlocal integrals of motion and treat them together with local supercharges in the unifying framework of the tri-supersymmetry. The inclusion of the dynamical conformal symmetries leads to an infinitely generated superalgebra, that contains several representations of the superconformal osp(2∣2) symmetry. We present the application of the results in the framework of the two-body model of identical anyons. The nontrivial contact interaction and the emerging N = 2 linear and nonlinear supersymmetries of the anyons are discussed.     

Interferometry of non-Abelian anyons

We develop the general quantum measurement theory of non-Abelian anyons through interference experiments. The paper starts with a terse introduction to the theory of anyon models, focusing on the basic formalism necessary to apply standard quantum measurement theory to such systems. This is then applied to give a detailed analysis of anyonic charge measurements using a Mach-Zehnder interferometer for arbitrary anyon models. We find that, as anyonic probes are sent through the legs of the interferometer, superpositions of the total anyonic charge located in the target region collapse when they are distinguishable via monodromy with the probe anyons, which also determines the rate of collapse. We give estimates on the number of probes needed to obtain a desired confidence level for the measurement outcome distinguishing between charges, and explicitly work out a number of examples for some significant anyon models. We apply the same techniques to describe interferometry measurements in a double point-contact interferometer realized in fractional quantum Hall systems. To lowest order in tunneling, these results essentially match those from the Mach-Zehnder interferometer, but we also provide the corrections due to processes involving multiple tunnelings. Finally, we give explicit predictions describing state measurements for experiments in the Abelian hierarchy states, the non-Abelian Moore-Read state at ν=5/2 and Read-Rezayi state at ν=12/5.     

Incompressible liquid state of rapidly rotating bosons at filling factor 3/2

Bosons in the lowest Landau level, such as rapidly rotating cold trapped atoms, are investigated numerically in the specially interesting case in which the filling factor (ratio of particle number to vortex number) is 3/2. When a moderate amount of a longer-range (e.g., dipolar) interaction is included, we find clear evidence that the ground state is in a phase constructed earlier by two of us, in which excitations possess non-Abelian statistics.     

One-Dimensional Models for Atoms in Strong Magnetic Fields,II: Anti-Symmetry in the Landau Levels

Electrons in strong magnetic fields can be described by one-dimensional models in which the Coulomb potential and interactions are replaced by regularizations associated with the lowest Landau band. For a large class of models of this type, we show that the maximum number of electrons that can be bound is less than aZ+Zf(Z). The function f(Z) represents a small non-linear growth which reduces to A _p Z(logZ)²when the magnetic field B=O(Z ^p ) grows polynomially with the nuclear charge Z. In contrast to earlier work, the models considered here include those arising from realistic cases in which the full trial wave function for N-electrons is the product of an N-electron trial function in one-dimension and an antisymmetric product of states in the lowest Landau level.     

On bound states in the continuum of N-body systems and the virial theorem

We prove generalized versions of the quantum mechanical virial theorem and apply them to the investigation of the spectrum of N body Hamiltonians. We show, in particular, that for N particles interacting through 2-body potentials which may have singularities but “don't wiggle too much,” no positive energy bound state can exist. We also prove results on the absence of bound states with energy bigger than some value E⁰ ? − ∞ and extend them to the case of N particles interacting through ν-body forces (ν = 1, 2,…, N) and with an external electromagnetic field. Also some remarks for the case of a Dirac electron in an external potential are given as well as for some problems with boundary conditions. A by-product of this investigation is the unitarity of the S matrix and the strong asymptotic completeness for systems of N particles interacting by 2-body forces which are not restricted to be purely repulsive.     

Effect of Landau level mixing on braiding statistics

We examine the effect of Landau level mixing on the braiding statistics of quasiparticles of Abelian and non-Abelian quantum Hall states. While path dependent geometric phases can perturb the Abelian part of the statistics, we find that the non-Abelian properties remain unchanged to an accuracy that is exponentially small in the distance between quasiparticles.     

Tunneling spectroscopy of the electron exchange-correlation interaction in a Schottky barrier in a quantizing magnetic field: n-GaAs/Me junctions

The magnetic-field-induced variation in the width of an anomalous resistance peak in the region of a zero-bias anomaly in n-GaAS/Me tunnel junctions was studied in the range of magnetic field strengths of up to 23 T at the temperatures T=4.2 and 1.5 K. The experimental curves depend neither on the method of a semiconductor substrate surface preparation (doped single crystal cut or epitaxially grown doped semiconductor film) nor on the type of dopant (Te, Si) and metal (Me=Au, Al). A comparison to a theoretical dependence of the exchange potential on the magnetic field strength for electrons on the lowest Landau level confirms that the anomalous resistance peak width can serve as a measure of the exchange-correlation potential jump on the surface of the degenerate electron gas. The results provide an explanation for the dependence of the zero-bias anomaly on the magnetic field and offer a possibility for directly measuring the electron exchange-correlation interaction in a Schottky barrier by means of tunneling spectroscopy.     

Hardy and Lieb-Thirring Inequalities for Anyons

We consider the many-particle quantum mechanics of anyons, i.e. identical particles in two space dimensions with a continuous statistics parameter ${\alpha \in [0, 1]}$ α ∈ [ 0 , 1 ] ranging from bosons (α = 0) to fermions (α = 1). We prove a (magnetic) Hardy inequality for anyons, which in the case that α is an odd numerator fraction implies a local exclusion principle for the kinetic energy of such anyons. From this result, and motivated by Dyson and Lenard’s original approach to the stability of fermionic matter in three dimensions, we prove a Lieb-Thirring inequality for these types of anyons.