To gain insight into the influence of temperature, I recommend you to consider the paper by M Cyrot (J. Phys. (France) 33, 125 (1972)) [link attached below]; despite the adopted approximations, the treatment by Cyrot provides a lucid picture of what happens in a Mott insulator as modelled by the Hubbard Hamiltonian. Naturally, the phase diagram of the Hubbard Hamiltonian (the T-U phase diagram in the work by Cyrot) is determined by energetic considerations, that is, by the thermodynamics of the underlying interacting system. For the reasons specified in the paper, Cyrot neglects charge fluctuation in his considerations and focuses on spin fluctuation. Applying subsequently the Hubbard-Stratonovich transformation, he takes account of the two-body interaction term in the Hamiltonian by means of an effective Hamiltonian in which electrons interact with a fluctuating magnetic moment. Energetic considerations establish whether at different temperatures and interaction strengths the moments order or not and how each possibility affects the electronic state of the system. 

For more recent and accurate calculations, based of the quantum Monte Carlo and dynamical mean-filed theory, consult the papers by A Georges, and W Krauth (Phys. Rev. B 48, 7167 (1993)), by MJ Rozenberg, G Kotliar, and XY Zhang (Phys. Rev. B 49, 10181 (1994)), and by MJ Rozenberg, R Chitra, and G Kotliar (Phys. Rev. Lett. 83, 3498 (1999)). Consult also the review article by A Georges, G Kotliar, W Krauth, and MJ Rozenberg (Rev. Mod. Phys. 68, 13 (1996), which of course does not cover the latter 1999 paper. See in particular Section VII.C, p. 62, and VII.D, p. 65 (Phase diagram and thermodynamics) of the latter review article. The physical system focussed on here is V2O3.

Naturally, physics of the Mott insulators is also partly determined by lattice dynamics (interaction of electrons and phonons). The temperature dependence of the lattice dynamics is of significant relevance to the electronic state of the system.

Lastly, I should emphasise that theoretically, an insulator is well-defined only for T = 0, where one may or may not have an insulating ground state (to be contrasted with a thermal ensemble of states - a phase, not a pure state),  however depending on the magnitude of the insulating gap, there is a window of temperatures where one can meaningfully talk about an insulator. □

http://jphys.journaldephysique.org/articles/jphys/abs/1972/01/jphys_1972__33_1_125_0/jphys_1972__33_1_125_0.html

To gain insight into the influence of temperature, I recommend you to consider the paper by M Cyrot (J. Phys. (France) 33, 125 (1972)) [link attached below]; despite the adopted approximations, the treatment by Cyrot provides a lucid picture of what happens in a Mott insulator as modelled by the Hubbard Hamiltonian. Naturally, the phase diagram of the Hubbard Hamiltonian (the T-U phase diagram in the work by Cyrot) is determined by energetic considerations, that is, by the thermodynamics of the underlying interacting system. For the reasons specified in the paper, Cyrot neglects charge fluctuation in his considerations and focuses on spin fluctuation. Applying subsequently the Hubbard-Stratonovich transformation, he takes account of the two-body interaction term in the Hamiltonian by means of an effective Hamiltonian in which electrons interact with a fluctuating magnetic moment. Energetic considerations establish whether at different temperatures and interaction strengths the moments order or not and how each possibility affects the electronic state of the system. 

For more recent and accurate calculations, based of the quantum Monte Carlo and dynamical mean-filed theory, consult the papers by A Georges, and W Krauth (Phys. Rev. B 48, 7167 (1993)), by MJ Rozenberg, G Kotliar, and XY Zhang (Phys. Rev. B 49, 10181 (1994)), and by MJ Rozenberg, R Chitra, and G Kotliar (Phys. Rev. Lett. 83, 3498 (1999)). Consult also the review article by A Georges, G Kotliar, W Krauth, and MJ Rozenberg (Rev. Mod. Phys. 68, 13 (1996), which of course does not cover the latter 1999 paper. See in particular Section VII.C, p. 62, and VII.D, p. 65 (Phase diagram and thermodynamics) of the latter review article. The physical system focussed on here is V2O3.

Naturally, physics of the Mott insulators is also partly determined by lattice dynamics (interaction of electrons and phonons). The temperature dependence of the lattice dynamics is of significant relevance to the electronic state of the system.

Lastly, I should emphasise that theoretically, an insulator is well-defined only for T = 0, where one may or may not have an insulating ground state (to be contrasted with a thermal ensemble of states - a phase, not a pure state),  however depending on the magnitude of the insulating gap, there is a window of temperatures where one can meaningfully talk about an insulator. □

http://jphys.journaldephysique.org/articles/jphys/abs/1972/01/jphys_1972__33_1_125_0/jphys_1972__33_1_125_0.html

I should like to complement my above writing on this page with two more comments. Firstly, in real systems, on changing the temperature the system under consideration may undergo a structural phase transition, in which case the parameters of the model describing the electronic degrees of freedom are to be changed (by hand). Secondly, at the level of the Hubbard model, the electron-phonon interaction is taken account of by means of the Hubbard-Holstein model, which minimally couples the electronic degrees of freedom with an Einstein phonon mode. □

 Dear Dr. Behnam Farid.

Sorry for replying you so latter, and thanks very much for your answers and the recommended papers. The transform from insulator to metal and from metal to insulator of Mott insulator is the result of the competition between band width "W" and Hubbard energy "U". For the interaction between electron and lattice fluctuation dependence of the change of temperature, is it the mechanism of this interaction, which make Mott insulator transform into metal, to change the value of U?

When materials was treat with heating or cooling, its size will increase or decrease. May be this change can result in the increase of decrease of its lattice constant, and then influence the value of band width "W". Just a guess, and what about your opinion?

Yue Wang  

You are welcome Yue.

The constant W, the electronic bandwidth, is a measure of the kinetic energy of the electrons, to be contrasted with the interaction energy, which in the case of the Hubbard Hamiltonian is the on-site Coulomb energy U. It follows that the dimensionless parameter U/W  is a measure of the importance of the interaction energy in relation to the kinetic energy of the particles. The ratio U/W can increase by either increasing U for a constant value of W, or decreasing W  for a constant value of U, or increasing U and decreasing W both at the same time. Putting a crystal under hydrostatic pressure for instance, leads to a larger overlap of the atomic orbitals on the neighbouring sites, and thus to an increase in W (one can explicitly verify this by considering a simple tight-binding Hamiltonian). Assuming that by doing so U remains constant, or very nearly constant, one clearly realises that application of a hydrostatic pressure to a crystal will lead to a decrease in the ratio U/W. Systems that are Mott insulating indeed undergo an insulating-to-metal transition under the application of a sufficiently large hydrostatic pressure.

To appreciate the role of phonons, one has to realise that physical properties of systems are characterised by dynamic correlation functions. Under the adiabatic assumption (the Born-Oppenheimer approximation), which is reasonable, but not generally valid, for systems consisting of relatively heavy atoms, to leading order some correlation functions are additive, consisting of a contribution arising from the dynamics of the interacting electrons, and a contribution arising from the dynamics of the vibrating ions. In the cases where this additivity does not apply, the electronic and ionic degrees of freedom get entangled, making theoretical computations extremely difficult to perform. There is a Migdal theorem which provides a quantitative measure for deciding whether the adiabatic approximation is reasonable / valid or not. This measure involves W  in combination with an energy scale related to the dynamics of the phonons, so that the ratio U/W  is no longer the only dimensionless parameter to take account of. Thus, whereas in the case of a system consisting of electrons and rigid ions the magnitude of W  is irrelevant for so long as the ration U/W  is equal to a given constant, in an electron-phonon coupled system the magnitude of W  is relevant, as it is to be measured also with respect to a relevant energy scale distinct from U.

The number of phonons not being a constant, but its average being an increasing function of the temperature, it is not difficult to appreciate the added significance of the temperature to electron-phonon coupled systems. Importantly, as T  increases, phonons increasingly interact with each other, rendering the harmonic approximation invalid -- taking account of the anharmonic effects is nothing but taking account of the phonon-phonon interaction effects. As crystals are heated, the closer one gets to the melting temperature, the more important become the anharmonic effects. This illustrates the fundamental limitation of the model Hamiltonians (such as the Hubbard-Holstein Hamiltonian) that consider phonons as independent harmonic oscillators minimally coupled to the electronic degrees of freedom. □

Thank you again, Dr. Behnam Farid, for your answer.