Quantum Field Lagrangian model for charge density waves in   one-dimensional systems at finite temperature : A Thermofield Dynamics   Approach
  We consider the Thermofield Dynamics bosonization to perform a field theory analysis of the effective Lagrangian model for incommensurate charge density waves (ICDW) in one-dimensional systems at finite temperature. The phonon degree of freedom is carryied by a dynamical phase field, contributing to the quantum dynamics and symmetry related features of the ICDW phenomenon. The electron chiral density and the phase of the phonon field condensate as a thermal soliton, carrying the symmetry under the linked electron-phonon $U_e^5(1) \otimes U_{ph}(1)$ global transformations. Using the Gell'Mann-Low formula for finite temperature, the perturbative series of the phonon thermal correlation function is obtained. Due to the electron-phonon charge selection rule we obtain for the thermal vacuum expectation value for the order parameter $<0 (\beta) |\Phi| 0 (\beta)> = 0$, in accordance with the cluster decomposition property of the corresponding correlation function. This reflects the fact that the quantum description of the ICDW corresponds to a local charge transport through the lattice which is accomplished by an electron-lattice energy redistribution, which accounts for a thermal dynamical mass gap generation. The electron-phonon coupling can be rewritten in terms of a mass operator for the "physical" fermion operator $\Psi$ such that $<0 (\beta)| \bar\Psi (x)\Psi (x) | 0 (\beta)> \neq 0$, without the breakdown of the linked electron-phonon symmetry. 
Luiz Victorio Belvedere, Rubens Gurgel Amaral, Armando Flavio   Rodrigues
cond-mat.stat-mech quant-ph