Local density approximation for the almost-bosonic anyon gas

Abstract

We study the minimizers of an energy functional with a self-consistent magnetic field, which describes a quantum gas of almost-bosonic anyons in the average-field approximation. For the homogeneous gas we prove the existence of the thermodynamic limit of the energy at fixed effective statistics parameter, and the independence of such a limit from the shape of the domain. This result is then used in a local density approximation to derive an effective Thomas-Fermi-like model for the trapped anyon gas in the limit of a large effective statistics parameter (i.e., "less-bosonic" anyons).

Many-anyon trial states

“…In a recent series of works [42][43][44][45][46][47], the question concerning the manyanyon ground state has been investigated in the light of modern mathematical methods. Interestingly, it was found that the ground-state energy for the free ideal anyon gas can be non-trivially bounded from below, but only under the assumption that α is an odd-numerator rational number (in contrast to electron FQHE which typically involves odd-denominator filling factors).…”

Local density approximation for almost-bosonic anyons

“…In [CLR17] Theorem 2.1 (and consequently Theorem 2.2) is proven by a direct inspection of the large L limit of the r.h.s. of (3.1).…”

“…We present here a synthetic exposition of the main arguments used in the proofs of the results stated in the previous Section. We refer to [CLR17] for further details. The starting point is the discussion of the homogeneous gas (see Section 2.1), which will be used as a key tool to take into account the inhomogeneity introduced by the trapping potential V .…”

Vortex patterns in the almost-bosonic anyon gas

“…The corresponding magnetic field curl βA[|u| 2 ] = 2πβ|u| 2 has a total flux 2πβ. Hence, regarding the statistics parameter α in (1) as the number of attached flux units per particle, the total magnetic flux β (referred to in [9,24] as the scaled statistics parameter) should be a large number, proportional to the number of particles. We return to this in the next paragraph.…”

“…This phase circulation cancels the long-range component of A[|u| 2 ] and allows for the TF profile to emerge on the macroscopic length scale β 1/(s+2) . We refer to [9] for details, in particular for the discussion of a trial state giving the correct energy by developing phase circulations on length scales O(β −s/(2(s+2)) ). It is intuitively clear that the phase circulation responsible for the validity of the LDA must come from quantized vortices in the gas.…”