Variational Approach for Many-Body Systems at Finite Temperature

We introduce an equation for density matrices that ensures a monotonic decrease of the free energy and reaches a fixed point at the Gibbs thermal. We build a variational approach for many-body systems that can be applied to a broad class of states, including all bosonic and fermionic Gaussian, as well as their generalizations obtained by unitary transformations, such as polaron transformations in electron-phonon systems. We apply it to the Holstein model on $20\times 20$ and $50\times 50$ square lattices, and predict phase separation between the superconducting and charge-density wave phases in the strong interaction regime.

Figure 1

(a) Phase diagrams for the Holstein model in a $50\times 50$ lattice for ${\omega }_b/t=10$ and $\nu =0.6$. The inset displays the filling factor $\nu$ as a function of the chemical potential at the point $P$, where $g/t=5$, $T/t=0.2$. (b) Phase separations at $P$ for $\nu =0.56$ (left panel) and $\nu =0.6$ (right panel) in a $20\times 20$ lattice. The first and second rows display the electron density and the SC order parameter.

Figure 2

Lattice BCS model: $s$-wave order parameter (a) and free energy density (b) as a function of the temperature $T/t$, for $U/t=-2$ and a $50\times 50$ lattice at half filling. The red curve gives the result of the FEFVM, which is on top of the mean field result in the thermodynamic limit. The black dashed line and the green curve (see insert) correspond to the ITVM for a symmetry-breaking field with $\epsilon ={10}^{-8}$, ${10}^{-12}$, respectively.

Figure 3

(a)–(b) The SC order parameters and the compressibility along three vertical lines in Fig. 1 obtained using homogeneous Ansätze. Negative compressibility indicates thermodynamically unstable states and corresponds to phase separation. (c)–(d) The SC order parameters, the free energy, and the chemical potential versus the filling factor $\nu$ for ${\omega }_b/t=10$, $g/t=5$, $T/t=0.2$. In (d) the free energy is plotted by subtracting a term proportional to the electron density, which does not affect the compressibility and the Maxwell construction. All plots have been obtained with the homogeneous Ansatz.