Stability and absence of binding for multi-polaron systems
Frank, Rupert L.1 ; Lieb, Elliott H.2 ; Seiringer, Robert3 ; Thomas, Lawrence E.4
1 Department of Mathematics, Princeton University Washington Road, Princeton, NJ, 08544 USA
2 Departments of Mathematics and Physics, Princeton University P.O. Box 708, Princeton, NJ, 08544 USA
3 Department of Physics, Princeton University P.O. Box 708, Princeton, NJ, 08544 USA
4 Department of Mathematics, University of Virginia Charlottesville, VA, 22904 USA
Publications Mathématiques de l'IHÉS, Tome 113 (2011), pp. 39-67.

We resolve several longstanding problems concerning the stability and the absence of multi-particle binding for N≥2 polarons. Fröhlich’s 1937 polaron model describes non-relativistic particles interacting with a scalar quantized field with coupling $\sqrt{\alpha}$, and with each other by Coulomb repulsion of strength U. We prove the following: (i) While there is a known thermodynamic instability for U<2α, stability of matter does hold for U>2α, that is, the ground state energy per particle has a finite limit as N→∞. (ii) There is no binding of any kind if U exceeds a critical value that depends on α but not on N. The same results are shown to hold for the Pekar-Tomasevich model.

DOI : 10.1007/s10240-011-0031-5
Frank, Rupert L. 1 ; Lieb, Elliott H. 2 ; Seiringer, Robert 3 ; Thomas, Lawrence E. 4

1 Department of Mathematics, Princeton University Washington Road, Princeton, NJ, 08544 USA
2 Departments of Mathematics and Physics, Princeton University P.O. Box 708, Princeton, NJ, 08544 USA
3 Department of Physics, Princeton University P.O. Box 708, Princeton, NJ, 08544 USA
4 Department of Mathematics, University of Virginia Charlottesville, VA, 22904 USA 
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Frank, Rupert L.; Lieb, Elliott H.; Seiringer, Robert; Thomas, Lawrence E. Stability and absence of binding for multi-polaron systems. Publications Mathématiques de l'IHÉS, Tome 113 (2011), pp. 39-67. doi : 10.1007/s10240-011-0031-5. http://www.numdam.org/articles/10.1007/s10240-011-0031-5/

[1.] Alexandrov, A. S.; Devreese, J. T. Advances in Polaron Physics, Springer, Berlin, 2010 | DOI

[2.] Brosens, F.; Klimin, S. N.; Devreese, J. T. Variational path-integral treatment of a translation invariant many-polaron system, Phys. Rev. B, Volume 71 (2005), pp. 214301-214313 | DOI

[3.] Conlon, J. G.; Lieb, E. H.; Yau, H.-T. The N 7/5 law for charged bosons, Commun. Math. Phys., Volume 116 (1988), pp. 417-448 | DOI | MR

[4.] Cycon, H. L.; Froese, R. G.; Kirsch, W.; Simon, B. Schrödinger Operators. Springer Texts and Monographs in Physics, 1987 | Zbl

[5.] Dolbeault, J.; Laptev, A.; Loss, M. Lieb-Thirring inequalities with improved constants, J. Eur. Math. Soc. (JEMS), Volume 10 (2008), pp. 1121-1126 | DOI | MR | Zbl

[6.] Donsker, M.; Varadhan, S. R. S. Asymptotics for the polaron, Commun. Pure Appl. Math., Volume 36 (1983), pp. 505-528 | DOI | MR | Zbl

[7.] Feynman, R. P. Slow electrons in a polar crystal, Phys. Rev., Volume 97 (1955), pp. 660-665 | DOI | Zbl

[8.] Frank, R. L.; Lieb, E. H.; Seiringer, R.; Thomas, L. E. Bi-polaron and N-polaron binding energies, Phys. Rev. Lett., Volume 104 (2010) | DOI

[9.] Fröhlich, H. Theory of electrical breakdown in ionic crystals, Proc. R. Soc. Lond. A, Volume 160 (1937), pp. 230-241 | DOI

[10.] Fröhlich, J. Existence of dressed one-electron states in a class of persistent models, Fortschr. Phys., Volume 22 (1974), pp. 159-198 | DOI

[11.] Gallavotti, G.; Ginibre, J.; Velo, G. Statistical mechanics of the electron-phonon system, Lett. Nuovo Cimento, Volume 28B (1970), pp. 274-286

[12.] Gerlach, B.; Löwen, H. Absence of phonon-induced localization for the free optical polaron and the corresponding Wannier exciton-phonon system, Phys. Rev. B, Volume 37 (1988), pp. 8042-8047 | DOI

[13.] Gerlach, B.; Löwen, H. Analytical properties of polaron systems or: do polaronic phase transitions exist or not?, Rev. Mod. Phys., Volume 63 (1991), pp. 63-90 | DOI | MR

[14.] Griesemer, M.; Møller, J. S. Bounds on the minimal energy of translation invariant N-polaron systems, Commun. Math. Phys., Volume 297 (2010), pp. 283-297 | DOI | MR | Zbl

[15.] Gurari, M. Self-energy of slow electrons in polar materials, Philos. Mag. Ser. 7, Volume 44 (1953), pp. 329-336 | Zbl

[16.] M. Hirokawa, Stability of formation of large bipolaron: non-relativistic quantum field theory, preprint (2006), . | arXiv

[17.] Hoffmann-Ostenhof, M.; Hoffmann-Ostenhof, T. Schrödinger inequalities and asymptotics behavior of the electron density of atoms and molecules, Phys. Rev. A, Volume 16 (1977), pp. 1782-1785 | DOI | MR

[18.] Lee, T.-D.; Pines, D. The motion of slow electrons in polar crystals, Phys. Rev., Volume 88 (1952), pp. 960-961 | DOI

[19.] Lee, T.-D.; Low, F.; Pines, D. The motion of slow electrons in a polar crystal, Phys. Rev., Volume 90 (1953), pp. 297-302 | DOI | MR | Zbl

[20.] Lieb, E. H. Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Stud. Appl. Math., Volume 57 (1976/77), pp. 93-105 | MR | Zbl

[21.] Lieb, E. H.; Oxford, S. An improved lower bound on the indirect Coulomb energy, Int. J. Quant. Chem., Volume 19 (1981), pp. 427-439 | DOI

[22.] Lieb, E. H.; Seiringer, R. The Stability of Matter in Quantum Mechanics, Cambridge University Press, Cambridge, 2010 | MR | Zbl

[23.] Lieb, E. H.; Solovej, J. P. Ground state energy of the one-component charged Bose gas, Commun. Math. Phys., Volume 217 (2001), pp. 127-163 | DOI | MR | Zbl

[24.] Lieb, E. H.; Thirring, W. Bound for the kinetic energy of fermions which proves the stability of matter, Phys. Rev. Lett., Volume 35 (1975), pp. 687-689 Erratum: ibid., 35 (1975), 1116 | DOI

[25.] Lieb, E. H.; Thomas, L. E. Exact ground state energy of the strong-coupling polaron, Commun. Math. Phys., Volume 183 (1997), pp. 511-519 Erratum: ibid., 188 (1997), 499–500 | DOI | MR | Zbl

[26.] Lieb, E. H.; Yamazaki, K. Ground-state energy and effective mass of the polaron, Phys. Rev., Volume 111 (1958), p. 728-722 | DOI | Zbl

[27.] Miyake, S. J. Strong coupling limit of the polaron ground state, J. Phys. Soc. Jpn., Volume 38 (1975), pp. 181-182 | DOI

[28.] Miyao, T.; Spohn, H. The bipolaron in the strong coupling limit, Ann. Henri Poincaré, Volume 8 (2007), pp. 1333-1370 | DOI | MR | Zbl

[29.] Møller, J. S. The polaron revisited, Rev. Math. Phys., Volume 18 (2006), pp. 485-517 | DOI | MR | Zbl

[30.] Nelson, E. Interaction of non-relativistic particles with a quantized scalar field, J. Math. Phys., Volume 5 (1964), pp. 1190-1197 | DOI | MR

[31.] S. I. Pekar, Research in Electron Theory of Crystals, United States Atomic Energy Commission, Washington, DC, 1963.

[32.] Pekar, S. I.; Tomasevich, O. F. Theory of F centers, Zh. Eksp. Teor. Fys., Volume 21 (1951), pp. 1218-1222

[33.] Roepstorff, G. Path Integral Approach to Quantum Physics, Springer, Berlin-Heidelberg-New York, 1994 | MR | Zbl

[34.] Smondyrev, M. A.; Fomin, V. M. Pekar-Fröhlich bipolarons, Polarons and Applications, Proceedings in Nonlinear Science, Wiley, New York (1994)

[35.] Spohn, H. The polaron functional integral, Stochastic Processes and Their Applications, Kluwer, Dordrecht-Boston-London (1990) | MR | Zbl

[36.] Smondyrev, M. A.; Verbist, G.; Peeters, F. M.; Devreese, J. T. Stability of multi polaron matter, Phys. Rev. B, Volume 47 (1993), pp. 2596-2601 | DOI

[37.] Verbist, G.; Peeters, F. M.; Devreese, J. T. Large bipolarons in two and three dimensions, Phys. Rev. B, Volume 43 (1991), pp. 2712-2720 | DOI

[38.] Verbist, G.; Smondyrev, M. A.; Peeters, F. M.; Devreese, J. T. Strong-coupling analysis of large bipolarons in two and three dimensions, Phys. Rev. B, Volume 45 (1992), pp. 5262-5269 | DOI

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