We study the ground state of a large number N of 2D extended anyons in an external magnetic field. We consider a scaling limit where the statistics parameter α is proportional to N−1 when N, which allows the statistics to be seen as a “perturbation around the bosonic end.” Our model is that of bosons in a magnetic field interacting through long-range magnetic potential generated by magnetic charges carried by each particle, smeared over discs of radius R. Our method allows us to take R → 0 not too fast at the same time as N : R = N−1/4+ɛ. We use the information theoretic version of the de Finetti theorem of Brandão and Harrow to justify the so-called “average field approximation”: the particles behave like independent, identically distributed bosons interacting via a self-consistent magnetic field.

1.
D.
Lundholm
and
N.
Rougerie
, “
The average field approximation for almost bosonic extended anyons
,”
J. Stat. Phys.
161
,
1236
1267
(
2015
).
2.
F. G. S. L.
Brandão
and
A. W.
Harrow
, “
Quantum de Finetti theorems under local measurements with applications
,”
Commun. Math. Phys.
353
,
469
506
(
2017
).
3.
K.
Li
and
G.
Smith
, “
Quantum de Finetti theorems under fully-one-way adaptative measurements
,”
Phys. Rev. Lett.
114
,
160503
(
2015
).
4.
N.
Rougerie
, “
Nonlinear Schrödinger limit of bosonic ground states, again
,” arXiv:1901.09561 (
2019
).
5.
D.
Lundholm
and
J. P.
Solovej
, “
Local exclusion principle for identical particles obeying intermediate and fractional statistics
,”
Phys. Rev. A
88
,
062106
(
2013
).
6.
D.
Lundholm
and
J. P.
Solovej
, “
Local exclusion and Lieb–Thirring inequalities for intermediate and fractional statistics
,”
Ann. Henri Poincaré
15
,
1061
1107
(
2014
).
7.
D.
Lundholm
and
J. P.
Solovej
, “
Hardy and Lieb-Thirring inequalities for anyons
,”
Commun. Math. Phys.
322
,
883
908
(
2013
).
8.
D.
Arovas
,
J. R.
Schrieffer
, and
F.
Wilczek
, “
Fractional statistics and the quantum Hall effect
,”
Phys. Rev. Lett.
53
,
722
723
(
1984
).
9.
M. O.
Goerbig
, “
Quantum Hall effects
,” arXiv:0909.1998 (
2009
).
10.
R. B.
Laughlin
, “
Nobel lecture: Fractional quantization
,”
Rev. Mod. Phys.
71
,
863
874
(
1999
).
11.
D.
Lundholm
and
N.
Rougerie
, “
Emergence of fractional statistics for tracer particles in a Laughlin liquid
,”
Phys. Rev. Lett.
116
,
170401
(
2016
).
12.
M.
Reed
and
B.
Simon
,
Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness
(
Academic Press
,
New York
,
1975
).
13.
I.
Avron
,
J.
Herbst
, and
B.
Simon
, “
Schrödinger operators with magnetic fields. 1. General interactions
,”
Duke Math. J.
45
(
4
),
847
(
1978
).
14.
M.
Correggi
and
L.
Oddis
, “
Hamiltonians for two-anyon systems
,”
Rend. Mat. Appl.
39
,
277
292
(
2018
).
15.
P.
Dabrowski
and
L.
Stovicek
, “
Aharonov-Bohm effect with δ–type interaction
,”
J. Math. Phys.
39
,
47
62
(
1998
).
16.
A.
Adami
and
R.
Teta
, “
On the Aharonov-Bohm effect
,”
Lett. Math. Phys.
43
,
43
53
(
1998
).
17.
M.
Bourdeau
and
R. D.
Sorkin
, “
When can identical particles collide?
,”
Phys. Rev. D
45
,
687
696
(
1992
).
18.
S.
Larson
and
D.
Lundholm
, “
Exclusion bounds for extended anyons
,”
Arch. Ration. Mech. Anal.
227
,
309
365
(
2018
).
19.
M.
Correggi
,
D.
Lundholm
, and
N.
Rougerie
, “
Local density approximation for the almost-bosonic anyon gas
,”
Anal. PDE
10
,
1169
1200
(
2017
).
20.
M.
Correggi
,
R.
Duboscq
,
N.
Rougerie
, and
D.
Lundholm
, “
Vortex patterns in the almost-bosonic anyon gas
,”
Europhys. Lett.
126
,
20005
(
2019
).
21.
E. H.
Lieb
and
M.
Loss
,
Analysis
, Graduate Studies in Mathematics Vol. 14, 2nd ed. (
American Mathematical Society
,
Providence, RI
,
2001
).
22.
R.
Bhatia
,
Matrix Analysis
(
Springer
,
1997
), Vol. 169.
23.
M.
Hoffmann-Ostenhof
,
T.
Hoffmann-Ostenhof
,
A.
Laptev
, and
J.
Tidblom
, “
Many-particle Hardy inequalities
,”
J. London Math. Soc.
77
,
99
114
(
2008
).
24.
M.
Lewin
,
P. T.
Nam
, and
N.
Rougerie
, “
Derivation of Hartree’s theory for generic mean-field Bose systems
,”
Adv. Math.
254
,
570
621
(
2014
).
25.
J. M.
Combes
,
R.
Schrader
, and
R.
Seiler
, “
Classical bounds and limits for energy distributions of Hamilton operators in electromagnetic fields
,”
Ann. Phys.
111
,
1
18
(
1978
).
26.
B.
Simon
,
Functional Integration and Quantum Physics
, 2nd ed. (
AMS Chelsea Publishing
,
Providence, RI
,
2005
).
27.
N.
Rougerie
,
Théorèmes de De Finetti, Limites de Champ Moyen et Condensation de Bose-Einstein
, Les cours Peccot, Spartacus IDH (
Cours Peccot, Collège de France
,
Paris
,
2016
), Février-Mars, 2014.
28.
M.
Lewin
,
P. T.
Nam
, and
N.
Rougerie
, “
A note on 2D focusing many-boson systems
,”
Proc. Am. Math. Soc.
145
,
2441
2454
(
2017
).
29.
R. L.
Hudson
and
G. R.
Moody
, “
Locally normal symmetric states and an analogue of de Finetti’s theorem
,”
Z. Wahrscheinlichkeitstheor. Verw. Gebiete
33
,
343
351
(
1975-1976
).
30.
M.
Reed
and
B.
Simon
,
Methods of Modern Mathematical Physics. I. Functional Analysis
(
Academic Press
,
1972
).
You do not currently have access to this content.