09-106 Marcel Griesemer, Jacob Schach Moller

Bounds on the Minimal Energy of Translation Invariant $N$-Polaron Systems (301K, pdf) Jul 4, 09

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Abstract. For systems of $N$ charged fermions (e.g. electrons) interacting with longitudinal optical quantized lattice vibrations of a polar crystal we derive upper and lower bounds on the minimal energy within the model of H.~Fr\"ohlich. The only parameters of this model, after removing the ultraviolet cutoff, are the constants $U>0$ and $\alpha>0$ measuring the electron-electron and the electron-phonon coupling strengths. They are constrained by the condition $\sqrt{2}\alpha<U$, which follows from the dependence of $U$ and $\alpha$ on electrical properties of the crystal. We show that the large $N$ asymptotic behavior of the minimal energy $E_N$ changes at $\sqrt{2}\alpha=U$ and that $\sqrt{2}\alpha\leq U$ is necessary for thermodynamic stability: for $\sqrt{2}\alpha > U$ the phonon-mediated electron-electron attraction overcomes the Coulomb repulsion and $E_N$ behaves like $-N^{7/3}$.

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