Phonons and the Isotopically Induced Mott transition - Physics

not strictly meet this criterion as we will see in **the** following chapters, we will start

from this assumption **and** explore **the** limits of **the** parameter range over which this is

valid.

5.2 Analytic techniques for **the** two-site Holstein model

5.2.1 The adiabatic limit t >> h ω.

In **the** strong coupling limit all electrons in **the** conduction b**and** are ‘dressed’ by

phonons **and** **the** hopping integral is renormalized. In his original work on small

polaron dynamics using a two-site model Holstein defined two regimes, **the** adiabatic

**and** non-adiabatic, to which different analytical approaches are applied to study this

renormalization. 28

In **the** adiabatic limit it is assumed that **the** lattice ions move much more slowly than

**the** valence electrons. Physically, this corresponds to an electron spending much

less time on a given site (**the** localization time) than required for **the** lattice to deform

into **the** polaronic configuration (**the** relaxation time). Because **the** localization time is

inversely proportional to **the** electronic kinetic energy (represented by **the** hopping

integral t within **the** Holstein framework) **and** **the** relaxation time is of **the** same order

of magnitude as **the** period of **the** lattice vibration,

in terms of **the** Holstein parameters: 30 hω