Phonons are elementary bosonic excitations of an elastic medium -- microscopically, they correspond to atomic vibrations as measured from their static equilibrium positions. From this perspective, a deformation of a system under strain from its equilibrium form can be expressed in terms of a particular population of the phonon modes of the undeformed system that deviates from the equilibrium phonon population at a given temperature. The number of phonons is not conserved and they are always there, even at zero temperature, representing the zero-point vibrations of the atoms in the system. In a theoretical description, one may or may not take account of the coupling between the electronic degrees of freedom of the system and the phonon modes that represent the dynamics of the atoms in the system. In the Hubbard-Holstein model this coupling is taken into account (in a very specific way), alongside the electron-electron interaction (also in a very specific way).

Lattice deformation can come about through changes in the external conditions to which the system under investigation is exposed. Such changes, such as the change in the temperature, can bring about structural phase transition, leading to deformation of the system. Straining the system also leads to its elastic and plastic deformation, without necessarily inducing structural phase transition in it. The Hubbard-Holstein model does not describe such deformations. It is a model that describes the low-energy physics of an electronic system whose behaviour is well-described in the limit of the vanishing electron-electron and electron-phonon interaction; the model accounts for the low-energy physics of such well-specified idealised system upon the switching-on of the electron-electron and electron-phonon interaction.  

You should probably provide one (or more?) references which introduce/use the terminology (such as w0) you are alluding to. Alternatively give the definition of w0 and its physical significance within the model(s) in question. this will help in finding an appropriate answer for you.

Phonons are elementary bosonic excitations of an elastic medium -- microscopically, they correspond to atomic vibrations as measured from their static equilibrium positions. From this perspective, a deformation of a system under strain from its equilibrium form can be expressed in terms of a particular population of the phonon modes of the undeformed system that deviates from the equilibrium phonon population at a given temperature. The number of phonons is not conserved and they are always there, even at zero temperature, representing the zero-point vibrations of the atoms in the system. In a theoretical description, one may or may not take account of the coupling between the electronic degrees of freedom of the system and the phonon modes that represent the dynamics of the atoms in the system. In the Hubbard-Holstein model this coupling is taken into account (in a very specific way), alongside the electron-electron interaction (also in a very specific way).

Lattice deformation can come about through changes in the external conditions to which the system under investigation is exposed. Such changes, such as the change in the temperature, can bring about structural phase transition, leading to deformation of the system. Straining the system also leads to its elastic and plastic deformation, without necessarily inducing structural phase transition in it. The Hubbard-Holstein model does not describe such deformations. It is a model that describes the low-energy physics of an electronic system whose behaviour is well-described in the limit of the vanishing electron-electron and electron-phonon interaction; the model accounts for the low-energy physics of such well-specified idealised system upon the switching-on of the electron-electron and electron-phonon interaction.  

Dear Kai, naturally Subhadip is to respond to your comment, however in the prevalent notation of the Hubbard-Holstein model, ω0 usually denotes the coupling constant of the free-phonon Hamiltonian in the total Hamiltonian, consisting of the electronic (i.e. the Hubbard) Hamiltonian and one (or two, or ...) other term (terms), with coupling constant(s) g (g1, g2, ...) representing the on-site (on-site, nearest-neighbour, ...) electron-phonon interaction. The term ω0 Σi bi+ bi in the full Hamiltonian describes a non-dispersive Einstein phonon mode at frequency ω0.

Dear Kai, I completely agree with Mr Farid. The terms explained by Dr. Farid is complete, However reference regarding the terminologies can be obtained from references  

http://journals.aps.org/prb/abstract/10.1103/PhysRevB.65.174306

http://journals.aps.org/prb/abstract/10.1103/PhysRevB.53.9666

w0 gives dispersion less phonon frequency.

My question is, Is it true that for zero EPI (g1, g2,,,=0) bt non zero w0 can give zero lattice deformation?

While calculation so, in Hubbard holstein model it is obtained that for w0=3.0t 0r 0.4t but g1, g2=0.0 gives non zero values of lattice deformation.

But, it has been argued that the lattice deformation is zero.What is that fact behind the result? Is it due to finite size effect? Or the deformation can be zero.