Adiabatic construction of hierarchical quantum Hall states

Abstract

We propose an exact model of anyon ground states including higher Landau levels, and use it to obtain fractionally quantized Hall states at filling fractions ν = p/(p(m − 1) + 1) with m odd, from integer Hall states at ν = p through adiabatic localization of magnetic flux. For appropriately chosen two-body potential interactions, the energy gap remains intact during the process, as we explicitly show for the p = 2 series of states. The construction hence provides a major step towards establishing the existence of incompressible states at these fillings.

Fractional Statistics

“…A more subtle contribution comes from the second term in (11), the CS term. Substituting (12) and evaluating it along the same lines, we see immediately that its contribution is minus one half of the Aharonov-Bohm phase (16). So the total phase we obtain when we adiabatically interchange two anyons via counterclockwise winding is exp (iθ ), and hence exactly what we obtain when we just count the phase of one particle in the fictitious vector potential of the other in (10) with (9).…”

“…9 On a more mundane level, this counting is consistent with both the statistics we obtain for the quasiparticles in the fractional quantum Hall effect 10,11 (see (37) below), and an exact model of charge-flux tube composites which connects integer and fractional quantum Hall states through an adiabatic process of attaching flux tubes to the charges. 12 The connection between anyons and charge-flux tube composites becomes even more apparent on closed surfaces, such as spheres and tori. 13,12 There, the allowed values for the number of anyons N and the statistical parameter θ are restricted such that the total fluxthat is the sum of the fictitious and the electromagnetic flux-through the surface seen by each particle is a multiple of the Dirac flux quantum Φ 0 .…”

“…12 The connection between anyons and charge-flux tube composites becomes even more apparent on closed surfaces, such as spheres and tori. 13,12 There, the allowed values for the number of anyons N and the statistical parameter θ are restricted such that the total fluxthat is the sum of the fictitious and the electromagnetic flux-through the surface seen by each particle is a multiple of the Dirac flux quantum Φ 0 .…”

Composite anyons on a torus

“…Composite fermions [1], the topological bound states of electrons and an even number (2s) of quantized vortices, lead to an explanation of the fractional quantum Hall effect [2,3] at fractions ν = n/(2sn ± 1) as the integer quantum Hall effect of composite fermions, and allow a calculation of the topological and non-topological features of these fractional quantum Hall states [4][5][6][7]. Greiter and Wilczek [8][9][10] proposed an adiabatic approach, wherein the fractional quantum Hall effect is connected to the integer quantum Hall effect by continuously tuning the strength of the vortex attached to electrons from zero to 2s, while at the same time varying the external magnetic field in such a manner that the effective magnetic field remains unchanged. The two limiting cases are familiar and well studied.…”

Inner workings of fractional quantum Hall parent Hamiltonians: A matrix product state point of view

“…This seemingly bypasses the traditional Hamiltonian starting point of condensed matter physics. In truth, many of the most preeminent FQH trial wave functions admit 3,5,[7][8][9][10][11][12] parent Hamiltonians with very useful properties. Such Hamiltonians do not only cement the status of certain states as incompressibile representatives of viable phases in the FQH regime.…”