Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-29T17:14:08.187Z Has data issue: false hasContentIssue false
  • English
  • Français

On the binding of polarons in a mean-field quantum crystal∗

Published online by Cambridge University Press:  28 March 2013

Mathieu Lewin  and
Nicolas Rougerie
Mathieu Lewin
Affiliation:
UniversitéGrenoble 1 and CNRS, LPMMC (UMR 5493), B.P. 166, 38 042 Grenoble, France. [email protected]
Nicolas Rougerie
Affiliation:
CNRS and Department of Mathematics (UMR 8088), University of Cergy-Pontoise, 95 000 Cergy-Pontoise, France; [email protected]
Get access

Abstract

We consider a multi-polaron model obtained by coupling the many-body Schrödinger equationfor N interacting electrons with the energy functional of a mean-fieldcrystal with a localized defect, obtaining a highly non linear many-body problem. Thephysical picture is that the electrons constitute a charge defect in an otherwise perfectperiodic crystal. A remarkable feature of such a system is the possibility to form a boundstate of electrons via their interaction with the polarizable background. We prove firstthat a single polaron always binds, i.e. the energy functional has aminimizer for N = 1. Then we discuss the case of multi-polaronscontaining N ≥ 2 electrons. We show that their existence is guaranteedwhen certain quantized binding inequalities of HVZ type are satisfied.

Keywords

Polaronquantum crystalbinding inequalitiesHVZ theoremChoquard-Pekar equation
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 3 , July 2013 , pp. 629 - 656
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

A. Alexandrov and J. Devreese, Advances in Polaron Physics. Springer Series in Solid-State Sciences, Springer (2009).
Cancès, É., Deleurence, A. and Lewin, M., A new approach to the modelling of local defects in crystals: the reduced Hartree-Fock case. Commun. Math. Phys. 281 (2008) 129–177. Google Scholar
Cancès, É., Deleurence, A. and Lewin, M., Non-perturbative embedding of local defects in crystalline materials. J. Phys. Condens. Matter 20 (2008) 294213. Google Scholar
Cancès, É. and Lewin, M., The dielectric permittivity of crystals in the reduced Hartree-Fock approximation. Arch. Ration. Mech. Anal. 197 (2010) 139–177. Google Scholar
Catto, I., Le Bris, C. and Lions, P.-L., On the thermodynamic limit for Hartree-Fock type models. Ann. Inst. Henri Poincaré Anal. Non Linéaire 18 (2001) 687–760. Google Scholar
Frank, R.L., Lieb, E.H., Seiringer, R. and Thomas, L.E., Bi-polaron and N-polaron binding energies. Phys. Rev. Lett. 104 (2010) 210402. Google Scholar
Frank, R.L., Lieb, E.H., Seiringer, R. and Thomas, L.E., Stability and absence of binding for multi-polaron systems. Publ. Math. Inst. Hautes Études Sci. 113 (2011) 39–67. Google Scholar
Fröhlich, H., Theory of Electrical Breakdown in Ionic Crystals. Proc. of R. Soc. London A 160 (1937) 230–241. Google Scholar
Fröhlich, H., Interaction of electrons with lattice vibrations. Proc. of R. Soc. London A 215 (1952) 291–298. Google Scholar
Griesemer, M. and Møller, J.S., Bounds on the minimal energy of translation invariant n-polaron systems. Commun. Math. Phys. 297 (2010) 283–297. Google Scholar
Hainzl, C., Lewin, M. and Séré, É., Existence of atoms and molecules in the mean-field approximation of no-photon quantum electrodynamics. Arch. Ration. Mech. Anal. 192 (2009) 453–499. Google Scholar
Hunziker, W., On the spectra of Schrödinger multiparticle Hamiltonians. Helv. Phys. Acta 39 (1966) 451–462. Google Scholar
Lewin, M., Geometric methods for nonlinear many-body quantum systems. J. Funct. Anal. 260 (2011) 3535–3595. Google Scholar
M. Lewin and N. Rougerie, Derivation of Pekar’s Polarons from a Microscopic Model of Quantum Crystals (2011).
Lieb, E.H., Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57 (1977) 93–105. Google Scholar
E.H. Lieb and M. Loss, Analysis, in Graduate Studies in Mathematics, 2nd edition, Vol. 14. AMS, Providence, RI. (2001).
Lieb, E.H. and Thomas, L.E., Exact ground state energy of the strong-coupling polaron. Commun. Math. Phys. 183 (1997) 511–519. Google Scholar
Lions, P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case, Part I. Ann. Inst. Henri Poincaré Anal. Non Linéaire 1 (1984) 109–149. Google Scholar
Lions, P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case, Part II. Ann. Inst. Henri Poincaré Anal. Non Linéaire 1 (1984) 223–283. Google Scholar
Miyao, T. and Spohn, H., The bipolaron in the strong coupling limit. Ann. Henri Poincaré 8 (2007) 1333–1370. Google Scholar
S. Pekar, Untersuchungen fiber die Elektronen Theorie der Kristalle. Berlin, Akademie-Verlag (1954).
S. Pekar, Research in electron theory of crystals. Tech. Report AEC-tr-5575. United States Atomic Energy Commission, Washington, DC (1963).
Pekar, S. and Tomasevich, O., Theory of F centers. Zh. Eksp. Teor. Fys. 21 (1951) 1218–1222. Google Scholar
M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional analysis. Academic Press (1972).
B. Simon, Trace ideals and their applications, in Lect. Note Ser., Vol. 35. London Mathematical Society. Cambridge University Press, Cambridge (1979).
Van Winter, C., Theory of finite systems of particles. I. The Green function. Mat.-Fys. Skr. Danske Vid. Selsk. 2 (1964). Google Scholar
Zhislin, G.M., Discussion of the spectrum of Schrödinger operators for systems of many particles. In Russian. Trudy Moskovskogo matematiceskogo obscestva 9 (1960) 81–120. Google Scholar