Douglas Lundholm

Local Exclusion and Lieb–Thirring Inequalities for Intermediate and Fractional Statistics

In one and two spatial dimensions there is a logical possibility for identical quantum particles different from bosons and fermions, obeying intermediate or fractional (anyon) statistics. We consider applications of a recent Lieb–Thirring inequality for anyons in two dimensions, and derive new Lieb–Thirring inequalities for intermediate statistics in one dimension with implications for models of Lieb–Liniger and Calogero–Sutherland type. These inequalities follow from a local form of the exclusion principle valid for such generalized exchange statistics.

Local exclusion principle for identical particles obeying intermediate and fractional statistics

A local exclusion principle is observed for identical particles obeying intermediate and fractional exchange statistics in one and two dimensions, leading to bounds for the kinetic energy in terms of the density. This has implications for models of Lieb-Liniger and Calogero-Sutherland type and implies a nontrivial lower bound for the energy of the anyon gas whenever the statistics parameter is an odd numerator fraction. We discuss whether this is actually a necessary requirement.

Emergence of Fractional Statistics for Tracer Particles in a Laughlin Liquid.

We consider a thought experiment where two distinct species of 2D particles in a perpendicular magnetic field interact via repulsive potentials. If the magnetic field and the interactions are strong enough, one type of particles forms a Laughlin state and the other type couples to Laughlin quasiholes. We show that, in this situation, the motion of the second type of particles is described by an effective Hamiltonian, corresponding to the magnetic gauge picture for noninteracting anyons. The argument is in accord with, but distinct from, the Berry phase calculation of Arovas, Schrieffer, and Wilczek. It suggests possibilities to observe the influence of effective anyon statistics in fractional quantum Hall systems.

Geometric extensions of many-particle Hardy inequalities

Certain many-particle Hardy inequalities are derived in a simple and systematic way using the so-called ground state representation for the Laplacian on a subdomain of . This includes geometric extensions of the standard Hardy inequalities to involve volumes of simplices spanned by a subset of points. Clifford/multilinear algebra is employed to simplify geometric computations. These results and the techniques involved are relevant for classes of exactly solvable quantum systems such as the Calogero–Sutherland models and their higher-dimensional generalizations, as well as for membrane matrix models, and models of more complicated particle interactions of geometric character.

Construction of the zero-energy state of SU(2)-matrix theory: Near the origin

We explicitly construct a (unique) Spin(9) x SU(2) singlet state, phi, involving only the fermionic degrees of freedom of the supersymmetric matrix-model corresponding to reduced 10-dimensional sup ...

Fractional Hardy–Lieb–Thirring and Related Inequalities for Interacting Systems

We prove analogues of the Lieb–Thirring and Hardy–Lieb–Thirring inequalities for many-body quantum systems with fractional kinetic operators and homogeneous interaction potentials, where no anti-symmetry on the wave functions is assumed. These many-body inequalities imply interesting one-body interpolation inequalities, and we show that the corresponding one- and many-body inequalities are actually equivalent in certain cases.

On the geometry of supersymmetric quantum mechanical systems

We consider some simple examples of supersymmetric quantum mechanical systems and explore their possible geometric interpretation with the help of geometric aspects of real Clifford algebras. This leads to natural extensions of the considered systems to higher dimensions and more complicated potentials.

Lieb-Thirring Bounds for Interacting Bose Gases

We study interacting Bose gases and prove lower bounds for the kinetic plus interaction energy of a many-body wave function in terms of its particle density. These general estimates are then applied to various types of interactions, including hard sphere (in 3D) and hard disk (in 2D) as well as a general class of homogeneous potentials.

Weighted Supermembrane Toy Model

A weighted Hilbert space approach to the study of zero-energy states of supersymmetric matrix models is introduced. Applied to a related but technically simpler model, it is shown that the spectrum of the corresponding weighted Hamiltonian simplifies to become purely discrete for sufficient weights. This follows from a bound for the number of negative eigenvalues of an associated matrix-valued Schrödinger operator.

Octonionic Twists for Supermembrane Matrix Models

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