Random Aharonov-Bohm vortices and some exactly solvable families of integrals. (English) Zbl 1456.82528
Summary: A review of the random magnetic impurity model, introduced in the context of the quantum Hall effect, is presented. It models an electron moving in a plane and coupled to random Aharonov-Bohm vortices carrying a fraction of the flux quantum. Recent results on its perturbative expansion are given. In particular, some peculiar families of integrals turn out to be related to the Riemann \(\zeta(3)\) and \(\zeta(2)\).
MSC:
| 82B44 | Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics |
| 11Z05 | Miscellaneous applications of number theory |
| 81V70 | Many-body theory; quantum Hall effect |
Keywords:
disordered systems (theory)References:
| [1] | Aharonov Y and Bohm D 1959 Phys. Rev.115 485 · Zbl 0099.43102 · doi:10.1103/PhysRev.115.485 |
| [2] | For an earlier description of the same effect see Ehrenberg W and Siday R W 1949 Proc. Phys. Soc. London B 62 8 · Zbl 0032.23301 |
| [3] | Stovicek P 1989 Phys. Lett. A 142 5 · doi:10.1016/0375-9601(89)90702-0 |
| [4] | Mashkevich S, Myrheim J and Ouvry S 2004 Phys. Lett. A 330 41 · Zbl 1209.81227 · doi:10.1016/j.physleta.2004.07.040 |
| [5] | Desbois J, Furtlehner C and Ouvry S 1995 Nucl. Phys. B 453 759 · doi:10.1016/0550-3213(95)00478-B |
| [6] | Comtet A, Desbois J and Ouvry S 1990 J. Phys. A: Math. Gen.23 3563 · Zbl 0714.60066 |
| [7] | Werner W 1993 Thèse Université Paris 7 |
| [8] | Werner W 1994 Probab. Theory Relat. Fields 111 |
| [9] | Comtet A, Georgelin Y and Ouvry S 1989 J. Phys. A: Math. Gen.22 3917 |
| [10] | See Gradshteyn I S and Ryzhik I M 1994 Table of Integrals Series and Products 5th edn (New York: Academic) p 8.432 1 and 8.432 6 · Zbl 0918.65002 |
| [11] | Samaj L and Travenec I 2000 J. Stat. Phys101 713 · Zbl 0966.82015 · doi:10.1023/A:1026489924895 |
| [12] | Furtlehner C and Ouvry S 2003 Integrals involving four Macdonald functions and their relation to 7ζ(3)/2 Preprint math-ph/0306004 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
