We consider a multi-polaron model obtained by coupling the many-body Schrödinger equation for N interacting electrons with the energy functional of a mean-field crystal with a localized defect, obtaining a highly non linear many-body problem. The physical picture is that the electrons constitute a charge defect in an otherwise perfect periodic crystal. A remarkable feature of such a system is the possibility to form a bound state of electrons via their interaction with the polarizable background. We prove first that a single polaron always binds, i.e. the energy functional has a minimizer for N = 1. Then we discuss the case of multi-polarons containing N ≥ 2 electrons. We show that their existence is guaranteed when certain quantized binding inequalities of HVZ type are satisfied.
Mots-clés : polaron, quantum crystal, binding inequalities, hvz theorem, choquard-pekar equation
@article{COCV_2013__19_3_629_0, author = {Lewin, Mathieu and Rougerie, Nicolas}, title = {On the binding of polarons in a mean-field quantum crystal}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {629--656}, publisher = {EDP-Sciences}, volume = {19}, number = {3}, year = {2013}, doi = {10.1051/cocv/2012025}, mrnumber = {3092354}, zbl = {1291.35248}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2012025/} }
TY - JOUR AU - Lewin, Mathieu AU - Rougerie, Nicolas TI - On the binding of polarons in a mean-field quantum crystal JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2013 SP - 629 EP - 656 VL - 19 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2012025/ DO - 10.1051/cocv/2012025 LA - en ID - COCV_2013__19_3_629_0 ER -
%0 Journal Article %A Lewin, Mathieu %A Rougerie, Nicolas %T On the binding of polarons in a mean-field quantum crystal %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 629-656 %V 19 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2012025/ %R 10.1051/cocv/2012025 %G en %F COCV_2013__19_3_629_0
Lewin, Mathieu; Rougerie, Nicolas. On the binding of polarons in a mean-field quantum crystal. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 3, pp. 629-656. doi : 10.1051/cocv/2012025. http://www.numdam.org/articles/10.1051/cocv/2012025/
[1] Advances in Polaron Physics. Springer Series in Solid-State Sciences, Springer (2009).
and ,[2] A new approach to the modelling of local defects in crystals: the reduced Hartree-Fock case. Commun. Math. Phys. 281 (2008) 129-177. | MR | Zbl
, and ,[3] Non-perturbative embedding of local defects in crystalline materials. J. Phys. Condens. Matter 20 (2008) 294213.
, and ,[4] The dielectric permittivity of crystals in the reduced Hartree-Fock approximation. Arch. Ration. Mech. Anal. 197 (2010) 139-177. | MR | Zbl
and ,[5] On the thermodynamic limit for Hartree-Fock type models. Ann. Inst. Henri Poincaré Anal. Non Linéaire 18 (2001) 687-760. | Numdam | MR | Zbl
, and ,[6] Bi-polaron and N-polaron binding energies. Phys. Rev. Lett. 104 (2010) 210402.
, , and ,[7] Stability and absence of binding for multi-polaron systems. Publ. Math. Inst. Hautes Études Sci. 113 (2011) 39-67. | Numdam | MR | Zbl
, , and ,[8] Theory of Electrical Breakdown in Ionic Crystals. Proc. of R. Soc. London A 160 (1937) 230-241. | Zbl
,[9] Interaction of electrons with lattice vibrations. Proc. of R. Soc. London A 215 (1952) 291-298. | Zbl
,[10] Bounds on the minimal energy of translation invariant n-polaron systems. Commun. Math. Phys. 297 (2010) 283-297. | MR | Zbl
and ,[11] Existence of atoms and molecules in the mean-field approximation of no-photon quantum electrodynamics. Arch. Ration. Mech. Anal. 192 (2009) 453-499. | MR | Zbl
, and ,[12] On the spectra of Schrödinger multiparticle Hamiltonians. Helv. Phys. Acta 39 (1966) 451-462. | MR | Zbl
,[13] Geometric methods for nonlinear many-body quantum systems. J. Funct. Anal. 260 (2011) 3535-3595. | MR | Zbl
,[14] Derivation of Pekar's Polarons from a Microscopic Model of Quantum Crystals (2011). | Zbl
and ,[15] Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation. Stud. Appl. Math. 57 (1977) 93-105. | MR | Zbl
,[16] Analysis, in Graduate Studies in Mathematics, 2nd edition, Vol. 14. AMS, Providence, RI. (2001). | MR | Zbl
and ,[17] Exact ground state energy of the strong-coupling polaron. Commun. Math. Phys. 183 (1997) 511-519. | MR | Zbl
and ,[18] 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. | Numdam | MR | Zbl
,[19] 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. | Numdam | MR | Zbl
,[20] The bipolaron in the strong coupling limit. Ann. Henri Poincaré 8 (2007) 1333-1370. | MR | Zbl
and ,[21] Untersuchungen fiber die Elektronen Theorie der Kristalle. Berlin, Akademie-Verlag (1954). | Zbl
,[22] Research in electron theory of crystals. Tech. Report AEC-tr-5575. United States Atomic Energy Commission, Washington, DC (1963).
,[23] Theory of F centers. Zh. Eksp. Teor. Fys. 21 (1951) 1218-1222.
and ,[24] Methods of Modern Mathematical Physics. I. Functional analysis. Academic Press (1972). | MR | Zbl
and ,[25] Trace ideals and their applications, in Lect. Note Ser., Vol. 35. London Mathematical Society. Cambridge University Press, Cambridge (1979). | MR | Zbl
,[26] Theory of finite systems of particles. I. The Green function. Mat.-Fys. Skr. Danske Vid. Selsk. 2 (1964). | MR | Zbl
,[27] Discussion of the spectrum of Schrödinger operators for systems of many particles. In Russian. Trudy Moskovskogo matematiceskogo obscestva 9 (1960) 81-120. | Zbl
,Cité par Sources :