Abstract
Ideal Fermion systems represent an issue of great recent interest. They consist of N-Fermion gases without interaction but subject to an exterior field or potential. In this paper we review recent studies of such systems in the following situations:
-
the low temprerature behavior of confined fermion gases in contact with an exterior reservoir
-
the action of a magnetic field on two-dimensional electronic devices
-
the transport and dissipation for time-dependent scatterers connected to several leads, known as “quantum pumps”.
The response of these different systems under the influence of the exterior potential or field is considered. In the first situation the response is calculated in terms of the “dynamical susceptibility” in the linear response framework, which yields a generalized Kubo formula. In the second situation the response is expressed by the magnetic susceptibility and the magnetization which are estimated semiclassically for various regimes of the temperature as compared with powers of the Planck constant. In the last situation the response is a measurable current estimated in an adiabatic framework.
All these results are contained by a series of papers by D. Robert and myself, and in recent works by Avron, Elgart, Graf and Sadun.
Chapter PDF
Similar content being viewed by others
Key words
Mathematics Subject Classification (2000)
References
Agam O., J Phys. I (France) 4, 697–730, (1994).
Avron J., Elgart A., Graf G.M., Sadun L. Schnee K., Adiabatic charge pumping in open quantum systems, Comm. Pre Appl. Math., 57, 528, (2004).
Avron J., Elgart A., Graf G.M., Sadun L. Transport and Dissipation in “Quantum Pumps”, J. Stat. Phys. 116, 425–473, (2004).
Avron J., Gutkin B., Oaknin D., Adiabatic Swimming in an Ideal Quantum Gas, Phys. Rev. Lett., 96, 130602, (2006).
Brack M., Murthy M.V., Harmonically trapped fermion gases: exact and asymptotic results in arbitrary dimensions, J. Phys. A: Math. Gen. 36, 1111–1133, (2003).
Bruun G.M., Clark C.W., Ideal gases in time-dependent traps, Phys. Rev. A, 61, 061601(R), (2000).
Büttiker M., Prêtre A., Thomas H., Phys. Rev. Lett. 70, 4114, (1993).
Butts D.A., Rokhsar D.S., Trapped Fermi Gases, Phys. Rev. A, 55, 4346–4350, (1997).
Combescure M., Robert D., Rigorous semiclassical results for the magnetic response of an electron gas, Rev. Math. Phys., 13, 1055–1073, (2001).
Combescure M., Robert D., Semiclassical results in the linear response theory, Ann. Phys., 305, 45–59, (2003).
Eckhardt B., Eigenvalue Statistics in Quantum Ideal Gases, arXiv: chao-dyn/ 9809005, (1998).
Fock V., Z. Physik, 47, 446–450, (1928).
Gat O., Avron J., Semiclassical Analysis and the Magnetization of the Hofstadter Model, Phys. Rev. Lett. 91, 186801, (2003).
Gat O., Avron J., Magnetic fingerprints of fractal spectra and the duality of Hofstadter models, New J. Phys., 5, 44.1–44.8, (2003).
Gleisberg F., Wonneberger W., Noninteracting fermions in a one-dimensional harmonic atom trap: Exact one-particle properties at zero temperature, Phys. Rev. A 62, 063602, (2000).
Helffer B., Sjöstrand J., On diamagnetism and the de Haas-Van Alphen effect, Ann. I. H.P., Phys. Théor., 52, 303–352, (1990).
Kubasiak A., Korbicz J., Zakrzewski J., Lewenstein M., Fermi-Dirac statistics and the number theory, Europhysics Letters, (2005).
Landau L., Z. Physik, 64, 629, (1930).
Leboeuf P., Monastra A., Quantum thermodynamic fluctuations of a chaotic Fermigas model, Nuclear Physics A, 724, 69–84, (2003).
Lert P.W., Weare J.H., Static semiclassical response of a bounded electron gas. 1, J. Chem. Phys., 68, 2221–2227, (1978).
Levy L.P., Reich D.H., Pfeiffer L., West K., Physica B 189, 204, (1993).
Li M., Yan Z., Chen J., Chen L., Chen C., Thermodynamic properties of an ideal Fermi gas in an external potential with U=br t in any dimensional space, Phys. Rev. A 58, 1445–1449, (1998).
Molinari V., Sumini M., Rocchi F., Fermion gases in magnetic fields: a semiclassical treatment, Eur. Phys. J. D, 12, 211–217, (2000).
Molinari V., Rocchi F., Sumini M., Kinetic Description of Rotating Gases in External Magnetic Fields in the Framework of the Thomas-Fermi-Dirac Approach, Transport Theory and Stat. Phys., 32, 607–621, (2003).
Peierls R.E., Z. Physik, 80, 763–791, (1933).
Pezzè L., Pitaevskii L., Smerzi A., Stringari S., Insulating Behaviour of a Trapped Ideal fermi Gas, Phys. Rev. Lett. 93, 120401, (2004).
Quang D.N., Tung N.H., Semiclassical approach to the density of states of the disordered electron gas in a quantum wire, Phys. Rev. B 60, 13648–136358, (1999).
Richter K., Ullmo D., Jalabert R., Orbital magnetism in the ballistic regime: geometrical effects, Phys. Rep. 276, 1–83, (1996).
Salasnich L., Ideal quantum gases in D-dimensional space and power-law potentials, Journ. Math. Phys., 41, 8016–8024, (2000).
Switkes M., Marcus C.M., Campman K., Gossard A.G., Science, 283, 1907, (1999).
Tran M.N.,Exact ground-state number fluctuations of trapped ideal and interacting fermions, J. Phys. A: Math. Gen. 36, 961–973, (2003).
Tran M.N., Murthy M.V., Bhaduri R.K., Ground-state fluctuations in finite Fermi systems, Phys. Rev. E, 63 031105, (2001).
van Faassen E., Dielectric response of a nondegenerate electron gas in semiconductor nancorystallites, Phys. Rev. B, 58, 15729–15735, (1998).
van Zyl B.P., Analytical expression for the first-order density matrix of a d-dimensional harmonically confined Fermi gas at finite temperature, Phys. Rev. A, 68, 033601, (2003).
Yajima K., Existence of solutions for Schrödinger evolution equations, Commun. Math. Phys., 110, 415–426, (1987).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Combescure, M. (2008). Semiclassical Results for Ideal Fermion Systems. A Review. In: Janas, J., Kurasov, P., Naboko, S., Laptev, A., Stolz, G. (eds) Methods of Spectral Analysis in Mathematical Physics. Operator Theory: Advances and Applications, vol 186. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8755-6_4
Download citation
DOI: https://doi.org/10.1007/978-3-7643-8755-6_4
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-8754-9
Online ISBN: 978-3-7643-8755-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)